{ "claims": [ { "claim": "The Glossary of Mathematical Notation defines, with worked examples, the common notation that the main text uses without defining it.", "claim_id": "P001-back-H1", "related_corrections": [ "P001-BACK-E001" ], "tied_to": "Glossary of Mathematical Notation, table on pp. 282-285 (PDF pp. 300-303); opening sentence on p. 282; Introduction p. xviii" }, { "claim": "The Glossary of Common Abbreviations gives the unabbreviated form of the abbreviations used in the book and points to where each is defined; abbreviated terms are defined at first use in the text.", "claim_id": "P001-back-H2", "related_corrections": [ "P001-BACK-N002", "P001-FRONT-N001" ], "tied_to": "Glossary of Common Abbreviations, table on pp. 286-287 (PDF pp. 304-305); sentence above the table on p. 286; Introduction p. xviii (“gives the meanings of all common abbreviations used in this book, along with references to where they are defined in the text”)" }, { "claim": "The References list every work cited in the book and give its bibliographic details accurately (implicit claim of a reference list).", "claim_id": "P001-back-H3", "related_corrections": [ "P001-BACK-E002", "P001-BACK-N003", "P001-BACK-N005", "P001-BACK-N006", "P001-BACK-N007", "P001-BACK-N008", "P001-BACK-N009", "P001-BACK-N010", "P001-BACK-N011", "P001-BACK-N012", "P001-BACK-N013", "P001-BACK-N014", "P001-BACK-N015", "P001-BACK-N016", "P001-BACK-N017" ], "tied_to": "References, pp. 288-292 (PDF pp. 306-310); 101 entries (87 author headings and 14 further same-author works)" }, { "claim": "The Index sends the reader to the pages on which each term is defined or discussed (implicit claim of an index).", "claim_id": "P001-back-H4", "related_corrections": [ "P001-BACK-N019", "P001-BACK-N022", "P001-BACK-N023", "P001-BACK-N024", "P001-BACK-N026" ], "tied_to": "Index, pp. 293-298 (PDF pp. 311-316); 512 entries and 747 page or footnote locators" }, { "claim": "Random generative processes are formalized as probability spaces, and the Kolmogorov axioms imply monotonicity, the subtraction rule, $P(\\emptyset)=0$, the bounds $0\\le P(A)\\le1$, the complement rule and the addition rule.", "claim_id": "P001-ch01-H1", "related_corrections": [ "P001-C01-E009", "P001-C01-E007" ], "tied_to": "Definitions 1.1.1-1.1.2; Theorems 1.1.4 and 1.1.7; Examples 1.1.3 and 1.1.6" }, { "claim": "Conditional probability satisfies the Multiplicative Law, Bayes' Rule and the Law of Total Probability. Events $A$ and $B$ are independent when $P(A\\cap B)=P(A)P(B)$, equivalently $P(A\\mid B)=P(A)$ when $P(B)>0$. In the coin-and-die example, $P(B)=5/24$ and $P(A\\mid B)=3/5$, and heads is independent of neither a roll of 3 nor a roll of 6.", "claim_id": "P001-ch01-H2", "related_corrections": [ "P001-C01-E008" ], "tied_to": "Definitions 1.1.8, 1.1.12, 1.1.15; Theorems 1.1.9, 1.1.10, 1.1.13, 1.1.14, 1.1.16; Example 1.1.11" }, { "claim": "Random variables, functions of them and operators on them are mathematical objects; two random variables are equal when they agree at every outcome, and $g(X)=h(X)$ if and only if $g$ and $h$ agree on the range $X(\\Omega)$.", "claim_id": "P001-ch01-H3", "related_corrections": [ "P001-C01-E010" ], "tied_to": "Definitions 1.2.1-1.2.3 and 1.3.1; Theorem 1.3.2; footnote 28" }, { "claim": "The PMF characterizes a discrete distribution and the CDF a general distribution. For a continuous variable, the book states that the PDF is unique and equals $dF/dx$ at every $x$, its integral gives interval probabilities, and $\\Pr[X=x]=0$. The support is defined as the set where the PMF or PDF is positive.", "claim_id": "P001-ch01-H4", "related_corrections": [ "P001-C01-E006", "P001-C01-E005", "P001-C01-E002", "P001-C01-E004", "P001-C01-E003", "P001-C01-N005" ], "tied_to": "Definitions 1.2.4, 1.2.5, 1.2.11, 1.2.14, 1.2.15, 1.2.20; Theorems 1.2.9, 1.2.10, 1.2.12, 1.2.16, 1.2.17; Examples 1.2.6-1.2.8, 1.2.13, 1.2.18, 1.2.19; Figures 1.2.1-1.2.5" }, { "claim": "Joint PMFs, PDFs and CDFs describe bivariate distributions and yield marginal and conditional distributions and event probabilities. The book characterizes independence of $X$ and $Y$ by factorization of the joint PMF or PDF into the marginals, by the corresponding conditional distributions, by independence of all events $\\{X\\in D\\}$ and $\\{Y\\in E\\}$, and by independence of $g(X)$ and $h(Y)$ for all functions $g$ and $h$.", "claim_id": "P001-ch01-H5", "related_corrections": [ "P001-C01-E001", "P001-C01-E002", "P001-C01-N005", "P001-C01-N007", "P001-C01-N008" ], "tied_to": "Definitions 1.3.3, 1.3.4, 1.3.7, 1.3.10, 1.3.11, 1.3.14, 1.3.16; Theorems 1.3.6, 1.3.9, 1.3.12, 1.3.13, 1.3.15, 1.3.17; Examples 1.3.5, 1.3.8" }, { "claim": "The bivariate framework (joint CDF, joint PMF/PDF, joint continuity, marginal and conditional PMFs/PDFs) extends to random vectors of any length.", "claim_id": "P001-ch01-H6", "related_corrections": [ "P001-C01-N009", "P001-C01-E002", "P001-C01-N007" ], "tied_to": "Definitions 1.4.1-1.4.7" }, { "claim": "Expected values, moments, variances and standard deviations summarize a distribution without a parametric model. Expectations are linear: $\\mathrm{E}[c]=c$, $\\mathrm{E}[aX]=a\\mathrm{E}[X]$ and $\\mathrm{E}[aX+bY+c]=a\\mathrm{E}[X]+b\\mathrm{E}[Y]+c$. Expectations of functions can be computed from the PMF or PDF. The variance and standard-deviation rules include $\\mathrm{V}[X]=\\mathrm{E}[X^2]-\\mathrm{E}[X]^2$, $\\mathrm{V}[X+c]=\\mathrm{V}[X]$, $\\mathrm{V}[aX]=a^2\\mathrm{V}[X]$ and $\\sigma[aX]=|a|\\sigma[X]$. Chebyshev's inequality bounds the probability of a deviation of at least $\\epsilon$ standard deviations by $1/\\epsilon^2$; a normal distribution is determined by its mean and standard deviation and is closed under affine maps and independent sums.", "claim_id": "P001-ch02-H1", "related_corrections": [ "P001-C02-E005" ], "tied_to": "Definitions 2.1.1, 2.1.10-2.1.12, 2.1.15, 2.1.19; Theorems 2.1.5, 2.1.6, 2.1.8, 2.1.9, 2.1.13-2.1.18, 2.1.20, 2.1.21; Examples 2.1.2-2.1.4, 2.1.17 (printed pp. 44-56, PDF pp. 62-74)" }, { "claim": "The mean squared error of $X$ about $c$ is $\\mathrm{V}[X]+(\\mathrm{E}[X]-c)^2$, so the expected value is the constant prediction that minimizes mean squared error. For a fair coin flip, the minimizer is $c=1/2$.", "claim_id": "P001-ch02-H2", "related_corrections": [ "P001-C02-E002" ], "tied_to": "Theorems 2.1.23, 2.1.24; Example 2.1.25; Figure 2.1.1 (printed pp. 56-59, PDF pp. 74-77)" }, { "claim": "Covariance satisfies $\\mathrm{V}[X+Y]=\\mathrm{V}[X]+2\\operatorname{Cov}[X,Y]+\\mathrm{V}[Y]$, $\\operatorname{Cov}[X,X]=\\mathrm{V}[X]$ and bilinearity. Correlation summarizes linear relationships, lies in $[-1,1]$ and, according to the book, equals $\\pm1$ exactly under linear dependence. Independence implies zero covariance, zero correlation when defined, and additive variances; zero correlation does not imply independence.", "claim_id": "P001-ch02-H3", "related_corrections": [ "P001-C02-A002", "P001-C02-E004", "P001-C02-E005", "P001-C02-N002", "P001-C02-N003" ], "tied_to": "Theorems 2.2.2-2.2.8; Definition 2.2.5; Example 2.2.9; footnotes 12 and 13 (printed pp. 59-66, PDF pp. 77-84)" }, { "claim": "Conditional expectations are linear; the Law of Iterated Expectations gives $\\mathrm{E}[Y]=\\mathrm{E}[\\mathrm{E}[Y\\mid X]]$, and the Law of Total Variance gives $\\mathrm{V}[Y]=\\mathrm{E}[\\mathrm{V}[Y\\mid X]]+\\mathrm{V}[\\mathrm{E}[Y\\mid X]]$. Deviations from the CEF have conditional mean zero, are uncorrelated with every function of $X$ and have variance $\\mathrm{E}[\\mathrm{V}[Y\\mid X]]$. The CEF minimizes prediction mean squared error given $X$. If $X$ and $Y$ are independent, then $\\mathrm{E}[Y\\mid X]=\\mathrm{E}[Y]$, $\\mathrm{V}[Y\\mid X]=\\mathrm{V}[Y]$ and the BLP of $Y$ given $X$ is $\\mathrm{E}[Y]$; the book also extends these statements to functions of the two variables.", "claim_id": "P001-ch02-H4", "related_corrections": [ "P001-C02-E006", "P001-C02-N002", "P001-C02-N004", "P001-C02-N005" ], "tied_to": "Definitions 2.2.10, 2.2.12, 2.2.15; Theorems 2.2.11, 2.2.13, 2.2.14, 2.2.17-2.2.20, 2.2.25; Example 2.2.16 (printed pp. 67-76 and 82-84, PDF pp. 85-94 and 100-102)" }, { "claim": "If $\\mathrm{V}[X]>0$, the best linear predictor of $Y$ given $X$ is $\\alpha+\\beta X$, where $\\beta=\\operatorname{Cov}[X,Y]/\\mathrm{V}[X]$ and $\\alpha=\\mathrm{E}[Y]-\\beta\\mathrm{E}[X]$. It is also the best linear approximation to the CEF and equals the CEF when that is linear. Its deviations satisfy $\\mathrm{E}[\\varepsilon]=\\mathrm{E}[X\\varepsilon]=\\operatorname{Cov}[X,\\varepsilon]=0$. For $Y=10X^2+W$, with $X\\sim U(0,1)$ and independent standard normal $W$, the CEF is $10X^2$ and the BLP is $-5/3+10X$. For a nonlinear CEF, the BLP depends on the distribution of $X$ and may approximate it well only where $X$ has mass.", "claim_id": "P001-ch02-H5", "related_corrections": [ "P001-C02-E001", "P001-C02-E003", "P001-C02-E007", "P001-C02-N002", "P001-C02-N006", "P001-C02-N007", "P001-C02-N008" ], "tied_to": "Theorem 2.2.21 and its two corollaries; Theorem 2.2.22; footnotes 17-21; Examples 2.2.23, 2.2.24; Figures 2.2.1, 2.2.2 (printed pp. 76-83, PDF pp. 94-101)" }, { "claim": "The concepts extend to the multivariate case: the covariance matrix, the multivariate variance rule, the multivariate CEF as best predictor, and the multivariate BLP as the linear function minimizing mean squared error. Its slope coefficients are its partial derivatives, and deviations from it are uncorrelated with every $X_{[k]}$.", "claim_id": "P001-ch02-H6", "related_corrections": [ "P001-C02-E008", "P001-C02-N009", "P001-C02-N010" ], "tied_to": "Definitions 2.3.1, 2.3.3, 2.3.4, 2.3.6; Theorems 2.3.2, 2.3.5, 2.3.7, 2.3.8 (printed pp. 84-87, PDF pp. 102-105)" }, { "claim": "Under equal-probability sampling of one unit, the PMF of $X$ is the finite population mass function, so $\\mathrm{E}[X]$ is the population mean and $\\mathrm{V}[X]$ the population variance. Draws with replacement are i.i.d.; draws without replacement from a large population are approximately so.", "claim_id": "P001-ch03-H1", "related_corrections": [ "P001-C03-E003", "P001-C03-N001" ], "tied_to": "Definition 3.1.1; Definition 3.1.2; Example 3.1.3 (Section 3.1.1, printed pp. 92-94)" }, { "claim": "For i.i.d. data, the sample mean is unbiased for $\\mathrm{E}[X]$, has sampling variance $\\mathrm{V}[X]/n$, satisfies a Chebyshev bound and is consistent by the Weak Law of Large Numbers. The empirical CDF at any fixed point is consequently consistent for $F(x)$.", "claim_id": "P001-ch03-H2", "related_corrections": [ "P001-C03-E005", "P001-C03-N004" ], "tied_to": "Theorems 3.2.3, 3.2.4, 3.2.5, 3.2.8, 3.2.9; Figure 3.2.1 (Section 3.2.1, printed pp. 96-102)" }, { "claim": "The plug-in sample variance has expectation $((n-1)/n)\\mathrm{V}[X]$ and is consistent; the unbiased sample variance is unbiased and consistent. The estimator $\\widehat{\\mathrm{V}}[X]/n$ is unbiased and consistent for the sampling variance of the sample mean, and the estimated standard error is consistent though biased downward.", "claim_id": "P001-ch03-H3", "related_corrections": [ "P001-C03-E006", "P001-C03-N007", "P001-C03-N008" ], "tied_to": "Theorems 3.2.19, 3.2.21, 3.2.33, 3.2.35; Definition 3.2.32; footnote 25 (Sections 3.2.3 and 3.2.6, printed pp. 105-108 and 114-116)" }, { "claim": "The standardized sample mean converges in distribution to $N(0,1)$ by the Central Limit Theorem; Slutsky's theorem makes the two forms of the CLT equivalent. Asymptotic unbiasedness, normality, standard error, mean squared error and consistency of variance estimators are defined through the limit distribution of $\\sqrt{n}(\\hat\\theta-\\theta)$.", "claim_id": "P001-ch03-H4", "related_corrections": [ "P001-C03-N005", "P001-C03-N006", "P001-C03-N007" ], "tied_to": "Theorems 3.2.24, 3.2.25; Definitions 3.2.22, 3.2.23, 3.2.26-3.2.32; Example 3.2.27 (Sections 3.2.4-3.2.5, printed pp. 108-114)" }, { "claim": "Estimands can be written as statistical functionals $T(F)$, with plug-in estimator $T(\\hat F)$. The sample mean and plug-in sample variance are examples. The book states that the plug-in estimator is asymptotically normal when $T$ satisfies the usual plug-in regularity conditions at $F$, and presents the mean of a finite-variance distribution as satisfying those conditions.", "claim_id": "P001-ch03-H5", "related_corrections": [ "P001-C03-E001", "P001-C03-E002", "P001-C03-E003", "P001-C03-N009" ], "tied_to": "Definitions 3.3.1-3.3.3; Examples 3.3.4, 3.3.5; Theorem 3.3.6 and footnote 31 (Sections 3.3 and 3.3.1, printed pp. 116-120)" }, { "claim": "For a continuous PDF, the rectangular-kernel density estimator is consistent for the PDF at every $x$ when $h\\to0$ and $nh\\to\\infty$. Kernel plug-in estimators follow by integrating the estimated density.", "claim_id": "P001-ch03-H6", "related_corrections": [ "P001-C03-A001", "P001-C03-N011", "P001-C03-N012" ], "tied_to": "Definition 3.3.7; Theorem 3.3.8; Definition 3.3.9; footnote 33; Figure 3.3.2 (Section 3.3.2, printed pp. 121-124)" }, { "claim": "For an asymptotically normal estimator with a consistent sampling variance estimator, $\\hat\\theta\\pm z_{1-\\alpha/2}\\sqrt{\\widehat{\\mathrm{V}}}$ is an asymptotically valid confidence interval; $\\Phi(t)$, $1-\\Phi(t)$ and $2(1-\\Phi(|t|))$ give asymptotically valid p-values for the corresponding tests.", "claim_id": "P001-ch03-H7", "related_corrections": [ "P001-C03-N014", "P001-C03-N015" ], "tied_to": "Definition 3.4.1; Theorem 3.4.2; Example 3.4.3; Definitions 3.4.4, 3.4.5; Theorem 3.4.6 (Sections 3.4.1-3.4.2, printed pp. 124-130)" }, { "claim": "The bootstrap resamples $n$ observations with replacement and takes the standard deviation of the re-estimates. The book states that this consistently estimates the standard error under the usual plug-in regularity conditions and presents percentile intervals as an alternative.", "claim_id": "P001-ch03-H8", "related_corrections": [ "P001-C03-N016", "P001-C03-N017" ], "tied_to": "The four-step procedure on printed p. 131; footnotes 41-42 (Section 3.4.3, printed pp. 130-132)" }, { "claim": "The uniform and Bernoulli simulations illustrate undercoverage of normal-approximation confidence intervals for the mean at small $n$, a nearly unbiased but variable standard error estimator, and coverage approaching its nominal level as $n$ grows, with some nonmonotonicity.", "claim_id": "P001-ch03-H9", "related_corrections": [ "P001-C03-E007", "P001-C03-N018" ], "tied_to": "Example 3.4.7; Table 3.4.1; Table 3.4.2 (Section 3.4.4, printed pp. 132-135)" }, { "claim": "With $l$ i.i.d. clusters of equal size $m$, the cluster sample mean equals the mean of cluster averages, is unbiased and consistent for $\\mathrm{E}[W]$, and is asymptotically normal as $l$ grows. Its sampling variance is $(lm^2)^{-1}\\sum_{j,j\\prime}\\operatorname{Cov}[X_{ij},X_{ij\\prime}]$, which under equal variances and common correlation is $\\sigma^2(1+\\rho(m-1))/(lm)$. The variance estimator $\\widehat{\\mathrm{V}}[W]/l$ and the block bootstrap provide inference; positive intracluster correlation can make a large unit count effectively a small sample.", "claim_id": "P001-ch03-H10", "related_corrections": [ "P001-C03-E004", "P001-C03-N019" ], "tied_to": "Definitions 3.5.1, 3.5.2; Theorems 3.5.3-3.5.7; Example 3.5.8; block bootstrap procedure (Section 3.5, printed pp. 135-141)" }, { "claim": "Regression is the plug-in estimator of the best linear predictor: replacing expectations by sample means in the definition of the BLP gives the OLS estimator, which is consistent for the BLP coefficients under i.i.d. sampling and the usual regularity conditions, whatever the form of the CEF.", "claim_id": "P001-ch04-H1", "related_corrections": [ "P001-C04-N002", "P001-C04-E005", "P001-C04-N007" ], "tied_to": "Chapter introduction and Section 4.1 (pp. 143-147); Section 4.1.1 formulas; Definition 4.1.2; Example 4.1.1; Table 4.1.1; Figure 4.1.2" }, { "claim": "When $\\mathbb{X}^T\\mathbb{X}$ is invertible, the OLS coefficient estimator is $(\\mathbb{X}^T\\mathbb{X})^{-1}\\mathbb{X}^T\\mathbf{Y}$. OLS residuals have mean zero and zero sample covariance with every regressor, paralleling the properties of deviations from the BLP.", "claim_id": "P001-ch04-H2", "related_corrections": [ "P001-C04-E005", "P001-C04-N002" ], "tied_to": "Section 4.1.3 (pp. 147-151); Theorem 4.1.4; Theorem 4.1.5" }, { "claim": "Under i.i.d. sampling and the usual plug-in regularity conditions, $\\sqrt{n}(\\hat{\\boldsymbol\\beta}-\\boldsymbol\\beta)$ is asymptotically normal with sandwich covariance. The robust variance estimator is its plug-in estimator, and robust or bootstrap standard errors support normal-approximation confidence intervals and p-values without a linearity assumption on the CEF.", "claim_id": "P001-ch04-H3", "related_corrections": [ "P001-C04-N003" ], "tied_to": "Sections 4.2.1-4.2.2 (pp. 151-154); Definition 4.2.1; Example 4.2.2" }, { "claim": "Classical standard errors are derived under a linear CEF and conditional homoskedasticity. The closing discussion states that classical and robust variance estimates agree in large samples under homoskedasticity and recommends robust or bootstrap standard errors for large $n$.", "claim_id": "P001-ch04-H4", "related_corrections": [ "P001-C04-N003", "P001-C04-N005", "P001-C04-N006" ], "tied_to": "Section 4.2.3 (pp. 155-156); Definition 4.2.3; the display preceding it; footnote 8" }, { "claim": "If the CEF is continuous and the support of $X$ is a closed interval, the BLP of $Y$ on $X,X^2,\\ldots,X^K$ approximates the CEF to any desired mean-squared precision for sufficiently large $K$. Polynomial and interaction terms therefore allow OLS to approximate nonlinear CEFs.", "claim_id": "P001-ch04-H5", "related_corrections": [ "P001-C04-N007" ], "tied_to": "Sections 4.3 and 4.3.1 (pp. 156-161); Theorems 4.3.2-4.3.3; Example 4.3.1; Figures 4.3.1-4.3.3; footnote 11" }, { "claim": "The examples illustrate overfitting at small $n$: with ten observations, a fifth-order polynomial tracks noise whether or not the CEF is linear, while linear regression estimates the BLP. The book presents the overfitting problem as less pronounced at $n=100$.", "claim_id": "P001-ch04-H6", "related_corrections": [ "P001-C04-E006" ], "tied_to": "Section 4.3.2 (pp. 161-164); Examples 4.3.4 and 4.3.5; Figures 4.3.4 and 4.3.5" }, { "claim": "Partial derivatives of an estimated CEF can be summarized by the average partial derivative (APD) or the partial derivative at the average (PDA); the two differ in general and the APD is usually the better summary of an overall relationship.", "claim_id": "P001-ch04-H7", "related_corrections": [ "P001-C04-E002", "P001-C04-N009", "P001-C04-N010" ], "tied_to": "Sections 4.3.3-4.3.4 (pp. 164-168); Definitions 4.3.6 and 4.3.7; Example 4.3.8; Table 4.4.5" }, { "claim": "A polynomial least-squares sieve whose degree satisfies $J_n\\to\\infty$ and $J_n/n\\to0$ is presented as a consistent estimator of a smooth CEF.", "claim_id": "P001-ch04-H8", "related_corrections": [ "P001-C04-E001", "P001-C04-E003" ], "tied_to": "Section 4.3.5 (pp. 168-169); Definition 4.3.9 and the paragraph after it" }, { "claim": "The lasso penalizes the mean squared residual by $\\lambda$ times the sum of absolute slope coefficients. It equals OLS at $\\lambda=0$, is presented as consistent for a linear CEF when $\\lambda\\to0$ with fixed $K$, and permits selection of $\\lambda$ by leave-one-out cross-validation.", "claim_id": "P001-ch04-H9", "related_corrections": [ "P001-C04-E004" ], "tied_to": "Section 4.3.6 (pp. 169-170); Definition 4.3.10; Table 4.4.5 column (8); Figure 4.4.4 panel (8)" }, { "claim": "In the application to 184 countries, an additional percentage point of clean-water access is associated with about 0.6 fewer infant deaths per 1,000 births. Conditioning on electricity access nearly halves the average partial derivative, to about $-0.35$. The book reports significant quadratic curvature in water but no significant simple water-electricity interaction, describes the APD as stable across specifications, including the lasso, while the standard errors of the APD and PDA grow with the complexity of the specification, and describes the PDA as more sensitive to specification.", "claim_id": "P001-ch04-H10", "related_corrections": [ "P001-C04-E006", "P001-C04-N012", "P001-C04-N015" ], "tied_to": "Section 4.4 (pp. 170-176); Table 4.4.5; Figures 4.4.1-4.4.4" }, { "claim": "A parametric model is a family of conditional distributions indexed by a finite-dimensional parameter. The classical linear model implies CEF $\\mathbf{x}\\boldsymbol\\beta$; a binary choice model with mean function $h$ implies CEF $h(\\mathbf{x}\\boldsymbol\\beta)$, with predictions in $[0,1]$. Logit and probit are standard examples.", "claim_id": "P001-ch05-H1", "related_corrections": [ "P001-C05-E004", "P001-C05-E009", "P001-C05-E010", "P001-C05-N002" ], "tied_to": "Definition 5.1.1; Definition 5.1.4 and Theorem 5.1.5; Definition 5.1.7 and Theorem 5.1.8; Definitions 5.1.9 and 5.1.10; Figure 5.1.1" }, { "claim": "For $n$ i.i.d. flips of a biased coin, the ML estimate of $p$ is the sample mean; for $\\mathbf{Y}=(1,0,1)$ it is $2/3$.", "claim_id": "P001-ch05-H2", "related_corrections": [ "P001-C05-N003" ], "tied_to": "Example 5.2.3; footnote 9; Figure 5.2.1" }, { "claim": "Under the classical linear model, the ML estimator of $\\boldsymbol\\beta$ is the OLS estimator, which minimizes the sum of squared residuals.", "claim_id": "P001-ch05-H3", "related_corrections": [ "P001-C05-E010", "P001-C05-E005" ], "tied_to": "Example 5.2.4; footnote 10" }, { "claim": "If the parametric model is true and regularity conditions hold, the ML estimator is consistent for $\\boldsymbol\\theta$; the book also asserts asymptotic normality and efficiency without proof. ML is presented as a plug-in estimator: the maximizer of the sample mean log-likelihood estimates the maximizer of the expected log-likelihood.", "claim_id": "P001-ch05-H4", "related_corrections": [ "P001-C05-E008", "P001-C05-E010", "P001-C05-E005" ], "tied_to": "Theorem 5.2.6 and its proof sketch; Examples 5.2.3-5.2.5" }, { "claim": "Theorem 5.2.8 states that ML consistently estimates the unique parameter minimizing the Kullback-Leibler divergence between the true conditional distribution and the model at a fixed covariate value $\\mathbf{x}$. The surrounding discussion presents this as a best parametric approximation, paralleling OLS, and says that the target is the true parameter when the model is true.", "claim_id": "P001-ch05-H5", "related_corrections": [ "P001-C05-E001", "P001-C05-E003" ], "tied_to": "Definition 5.2.7; Theorem 5.2.8; Example 5.2.9 and Figure 5.2.2" }, { "claim": "The ML plug-in estimator applies a functional to $\\hat F_{ML}$, combining the empirical distribution of $\\mathbf{X}$ with the fitted conditional model. For the CEF it reduces to $\\mathbf{x}\\hat{\\boldsymbol\\beta}_{ML}$ or $h(\\mathbf{x}\\hat{\\boldsymbol\\beta}_{ML})$. The book gives consistency under regularity conditions when the model is true, with no general guarantee otherwise.", "claim_id": "P001-ch05-H6", "related_corrections": [ "P001-C05-E007", "P001-C05-E009", "P001-C05-E004", "P001-C05-E010" ], "tied_to": "Definition 5.2.10; Theorems 5.1.5 and 5.1.8" }, { "claim": "Mixtures of parametric models can be fitted by ML; the estimator is said to be consistent and asymptotically efficient if any subset of the component models jointly characterizes the conditional distribution, and is otherwise an approximation. A normal/log-normal/exponential mixture approximates the “unusual distribution” markedly better than any one component.", "claim_id": "P001-ch05-H7", "related_corrections": [ "P001-C05-E002", "P001-C05-N005", "P001-C05-N004" ], "tied_to": "Section 5.2.5; Example 5.2.11; Figures 5.2.3, 5.2.4, 5.2.5" }, { "claim": "ML can be penalized with an $\\ell^1$ penalty on a subvector of $\\boldsymbol\\theta$. The book states that the penalized estimator is usually consistent for the minimum KL divergence predictor when the penalty parameter does not grow too quickly, and presents cross-validation as a reasonable way to choose $\\lambda$.", "claim_id": "P001-ch05-H8", "related_corrections": [ "P001-C05-E001", "P001-C05-N006", "P001-C05-N003", "P001-C05-E010" ], "tied_to": "the $\\ell^1$-penalized ML display on p. 202; Section 4.3.6 (Definition 4.3.10)" }, { "claim": "Model-based standard errors are valid only if the model is true; robust standard errors and the bootstrap remain valid in large samples under misspecification (under regularity), so they are typically preferable. Models are approximations that are useful when interpreted as such.", "claim_id": "P001-ch05-H9", "related_corrections": [], "tied_to": "Section 5.2.7; Section 5.3; Section 4.2.3 (Definition 4.2.3) for the regression analogue; Example 5.2.4 (ML/OLS equivalence); no numbered variance result in the chapter" }, { "claim": "With only bounded support, $\\mathrm{Supp}[Y_i]\\subseteq[a,b]$, and stable outcomes, $\\mathrm{E}[Y_i]$ is partially identified: it lies in $[\\mathrm{E}[Y_i^*|R_i=1]\\Pr[R_i=1]+a\\Pr[R_i=0],\\ \\mathrm{E}[Y_i^*|R_i=1]\\Pr[R_i=1]+b\\Pr[R_i=0]]$; these bounds are sharp, and the plug-in estimators of the two endpoints (impute $a$ or $b$ for every missing outcome and take the sample mean) are unbiased and consistent.", "claim_id": "P001-ch06-H1", "related_corrections": [ "P001-C06-E001", "P001-C06-N001" ], "tied_to": "Theorem 6.1.3; Theorem 6.1.4; Example 6.1.2 (Section 6.1.1, pp. 209-213, PDF pp. 227-231)" }, { "claim": "Under MCAR ($Y_i$ independent of $R_i$, $\\Pr[R_i=1]>0$), $\\mathrm{E}[Y_i]=\\mathrm{E}[Y_i^*|R_i=1]$, so the mean of the observed outcomes is the plug-in estimator.", "claim_id": "P001-ch06-H2", "related_corrections": [ "P001-C06-N001" ], "tied_to": "Definition 6.1.5; Theorem 6.1.6; Example 6.1.7 (Section 6.1.2, pp. 213-215, PDF pp. 231-233)" }, { "claim": "Under MAR ($Y_i$ independent of $R_i$ given $\\mathbf{X}_i$, with the response probability bounded away from zero), $\\mathrm{E}[Y_i]$ is point identified as $\\sum_x\\mathrm{E}[Y_i^*|R_i=1,\\mathbf{X}_i=x]\\Pr[\\mathbf{X}_i=x]$, and the CEF satisfies $\\mathrm{E}[Y_i|\\mathbf{X}_i]=\\mathrm{E}[Y_i^*|R_i=1,\\mathbf{X}_i]$; the post-stratification plug-in estimator follows.", "claim_id": "P001-ch06-H3", "related_corrections": [ "P001-C06-N001", "P001-C06-N003", "P001-C06-N005" ], "tied_to": "Definition 6.1.8; Theorem 6.1.9; Theorem 6.1.10; Examples 6.2.1 and 6.2.2 (Sections 6.1.3 and 6.2.1, pp. 215-222, PDF pp. 233-240)" }, { "claim": "Under MAR given $\\mathbf{X}_i$, the propensity score $p_R(\\mathbf{X}_i)$ balances the covariates ($R_i$ independent of $\\mathbf{X}_i$ given $p_R(\\mathbf{X}_i)$), $Y_i$ is independent of $R_i$ given $p_R(\\mathbf{X}_i)$, and positivity carries over; so it is enough to condition on the propensity score.", "claim_id": "P001-ch06-H4", "related_corrections": [ "P001-C06-N002" ], "tied_to": "Definition 6.1.11; Theorem 6.1.12 (Section 6.1.4, pp. 217-219, PDF pp. 235-237)" }, { "claim": "Because the CEF of the observed outcomes equals the CEF of all outcomes under MAR, missing outcomes can be imputed by regression (imputing only the missing values or all values gives the same OLS-based estimate) or by hot deck matching on the propensity score; propensity scores can be estimated by maximum likelihood from a binary choice model.", "claim_id": "P001-ch06-H5", "related_corrections": [ "P001-C06-E002", "P001-C06-E006", "P001-C06-N002", "P001-C06-N003", "P001-C06-N004" ], "tied_to": "Examples 6.2.3 and 6.2.5; Definition 6.2.4 (Sections 6.2.2-6.2.4, pp. 222-226, PDF pp. 240-244)" }, { "claim": "Under MAR, $\\mathrm{E}[Y_i]=\\mathrm{E}[Y_i^*R_i/p_R(\\mathbf{X}_i)]$; this gives the IPW estimator and its stabilized (ratio) version, whose denominator has population expectation one; and with a discrete, finite $\\mathbf{X}_i$ the IPW estimator is equivalent to the post-stratification plug-in estimator.", "claim_id": "P001-ch06-H6", "related_corrections": [ "P001-C06-E003", "P001-C06-E004", "P001-C06-E005", "P001-C06-N002", "P001-C06-N005" ], "tied_to": "Theorem 6.2.6; Definitions 6.2.7 and 6.2.8 (Section 6.2.5, pp. 226-228, PDF pp. 244-246)" }, { "claim": "Double robustness: under MAR, $\\mathrm{E}[Y_i]=\\mathrm{E}\\left[\\tilde{\\mathrm{E}}[Y_i^*|R_i=1,\\mathbf{X}_i]+R_i(Y_i^*-\\tilde{\\mathrm{E}}[Y_i^*|R_i=1,\\mathbf{X}_i])/\\tilde p_R(\\mathbf{X}_i)\\right]$ if either the CEF approximation is exactly right (with $\\tilde p_R$ bounded away from 0 and 1) or the propensity approximation is exactly right; the DR estimator is the plug-in version and is consistent under the stated plug-in conditions.", "claim_id": "P001-ch06-H7", "related_corrections": [ "P001-C06-N005" ], "tied_to": "Theorem 6.2.9; Definition 6.2.10 (Section 6.2.6, pp. 228-231, PDF pp. 246-249)" }, { "claim": "For the cross-national average of clean energy use (184 countries, 50 with missing outcomes): estimated bounds 7.110 to 34.283, with a 95% lower confidence limit of 5.109; MCAR estimate 9.762 (95% CI 7.157 to 12.367); seven MAR-based estimates between 8 and 9, all inside the estimated bounds and below the MCAR estimate; the IPW and DR estimates have the smallest bootstrap standard errors.", "claim_id": "P001-ch06-H8", "related_corrections": [ "P001-C06-E006", "P001-C06-E007", "P001-C06-N006", "P001-C06-N007" ], "tied_to": "Table 6.3.1 (Section 6.3, pp. 231-234, PDF pp. 249-252)" }, { "claim": "If MAR is believed, the estimates imply that countries with missing outcome data use, on average, less clean energy than countries with data; this depends on MAR, which cannot be verified.", "claim_id": "P001-ch06-H9", "related_corrections": [], "tied_to": "Table 6.3.1; Section 6.3, closing paragraph, p. 234 (PDF p. 252)" }, { "claim": "The chapter is built to parallel Chapter 7: causal inference can be viewed as a missing data problem, and the bracketed section numbers point to the analogous sections of Chapter 7.", "claim_id": "P001-ch06-H10", "related_corrections": [ "P001-C06-E003", "P001-C06-N002", "P001-C06-N003" ], "tied_to": "Chapter introduction, p. 208 (PDF p. 226); section headings 6.1 to 6.2.6" }, { "claim": "Causal inference compares potential outcomes that cannot jointly be observed, so causal identification requires assumptions beyond the distribution of the observables; the identification question can be separated from estimation and uncertainty.", "claim_id": "P001-ch07-H1", "related_corrections": [], "tied_to": "Chapter introduction (pp. 235-236); Definitions 7.1.1-7.1.3; Theorem 7.1.4; Example 7.1.5; closing paragraph of Section 7.4 (p. 276)" }, { "claim": "With outcomes bounded in $[a,b]$, the ATE $\\mathrm{E}[\\tau_i]$ lies in a sharp interval computable from the distribution of $(Y_i,D_i)$ whose width is $b-a$. The plug-in estimators of the two endpoints are unbiased and consistent.", "claim_id": "P001-ch07-H2", "related_corrections": [ "P001-C07-E001" ], "tied_to": "Section 7.1.3 (pp. 240-244); Theorems 7.1.7 and 7.1.8; Example 7.1.6" }, { "claim": "Under random assignment, the ATE equals $\\mathrm{E}[Y_i\\mid D_i=1]-\\mathrm{E}[Y_i\\mid D_i=0]$ and is estimated by the difference in means. The book identifies the mean of $\\tau_i$, rather than its median, variance or CDF. Without random assignment, the difference in means decomposes into the ATT and selection bias.", "claim_id": "P001-ch07-H3", "related_corrections": [], "tied_to": "Section 7.1.4 (pp. 244-247); Definition 7.1.9; Theorem 7.1.10; Example 7.1.11; decomposition display on p. 247" }, { "claim": "Under strong ignorability given $\\mathbf{X}_i$, the ATE and conditional ATE are identified by covariate-specific contrasts. The propensity score $p_D(\\mathbf{X}_i)$ is a sufficient conditioning variable: balance, conditional independence and positivity hold given it. Covariate balance alone does not establish ignorability.", "claim_id": "P001-ch07-H4", "related_corrections": [ "P001-C07-N002" ], "tied_to": "Sections 7.1.5-7.1.6 (pp. 247-252); Definition 7.1.12; Theorems 7.1.13, 7.1.14 and 7.1.16" }, { "claim": "Random assignment need not imply strong ignorability given a post-treatment variable: the example gives an adjusted effect of $-1/4$ after conditioning on a variable affected by treatment, although the true effect is zero.", "claim_id": "P001-ch07-H5", "related_corrections": [ "P001-C07-E002", "P001-C07-N003" ], "tied_to": "Section 7.1.7 (pp. 252-254); Example 7.1.17" }, { "claim": "The book extends the potential-outcome framework to multivalued and continuous treatments and states that, under ignorability, unconditional and conditional mean effects are identified for any $d,d\\prime\\in\\operatorname{Supp}[D_i]$. Under strong ignorability and differentiability, it identifies the average marginal causal effect with the average partial derivative of the CEF with respect to $D_i$.", "claim_id": "P001-ch07-H6", "related_corrections": [ "P001-C07-A001", "P001-C07-N004" ], "tied_to": "Section 7.1.8 (pp. 254-256); Definition 7.1.18; generalizations of Theorems 7.1.13-7.1.14; Definition 7.1.19; display $\\mathrm{AMCE}=\\mathrm{APD}_{D_i}$" }, { "claim": "Under strong ignorability, the ATE can be estimated by post-stratification for discrete $\\mathbf{X}$ or by regression imputation using OLS, unpooled OLS or another CEF estimator, including the ML plug-in. These approaches use CEF approximation without requiring a particular error distribution; in the pooled linear case, the estimate is the coefficient on $D_i$.", "claim_id": "P001-ch07-H7", "related_corrections": [ "P001-C07-E003", "P001-C07-E004", "P001-C07-N005" ], "tied_to": "Sections 7.2.1-7.2.3 (pp. 256-262); Examples 7.2.1 and 7.2.2; Definition 7.2.3; ML plug-in displays on pp. 261-262" }, { "claim": "One-to-one nearest-neighbor propensity-score matching imputes each missing potential outcome from the closest unit in the other arm and is not a plug-in estimator. The book describes matching with estimated scores as not generally root-$n$ consistent, attributes difficulties with confidence intervals and the bootstrap to slowly shrinking bias, and states that bootstrap standard errors are nevertheless likely to characterize sampling variability adequately.", "claim_id": "P001-ch07-H8", "related_corrections": [ "P001-C07-E011", "P001-C07-N006" ], "tied_to": "Section 7.2.4 (pp. 262-264); Example 7.2.4; footnote 13 (Abadie and Imbens 2008); footnote 20 (p. 279)" }, { "claim": "Under strong ignorability, $\\mathrm{E}[\\tau_i]=\\mathrm{E}[Y_iD_i/p_D(\\mathbf{X}_i)-Y_i(1-D_i)/(1-p_D(\\mathbf{X}_i))]$, motivating IPW and stabilized IPW estimators. The doubly robust representation holds if either the outcome regression or the propensity score is correct.", "claim_id": "P001-ch07-H9", "related_corrections": [ "P001-C07-E005", "P001-C07-E006" ], "tied_to": "Sections 7.2.6-7.2.7 (pp. 264-270); Theorem 7.2.5; Definitions 7.2.6, 7.2.7 and 7.2.9; Theorem 7.2.8" }, { "claim": "Ignorability is not directly testable from observables, but placebo variables can provide indirect evidence about a design. The chapter proposes an added-variable $F$ test of the placebo null.", "claim_id": "P001-ch07-H10", "related_corrections": [ "P001-C07-E007", "P001-C07-N007" ], "tied_to": "Section 7.2.8 (pp. 270-271): placebo null and $F$-test procedure; footnote 14" }, { "claim": "The book states that positivity implies complete population overlap and suggests a near converse. With limited overlap, it proposes changing the target population, for example to the ATT by propensity-score weighting or to a $p_D(1-p_D)$-weighted effect by regressing $Y$ on $D$ and $p_D(\\mathbf{X})$. It describes empirical overlap failures as a micronumerosity problem and discusses trimming or Winsorizing estimated propensity scores.", "claim_id": "P001-ch07-H11", "related_corrections": [ "P001-C07-E008", "P001-C07-E009", "P001-C07-N008", "P001-C07-N009" ], "tied_to": "Section 7.3 (pp. 271-276); Definition 7.3.1; Theorem 7.3.2; Examples 7.3.3 and 7.3.4; Figures 7.3.1 and 7.3.2; Section 7.3.1 ATT formula; Definition 7.3.5" }, { "claim": "For Swiss naturalization referenda with $n=2{,}419$, the application reports bounds $[-54.156,45.844]$ on the effect of male identification on the share of no votes and an unadjusted difference of $1.335$ with $p=0.051$. It describes ignorability as quite plausible and most covariate-adjusted estimates as smaller, positive and farther from statistical significance, while assessing matching and IPW unfavorably.", "claim_id": "P001-ch07-H12", "related_corrections": [ "P001-C07-E010", "P001-C07-N012", "P001-C07-E011", "P001-C07-N013" ], "tied_to": "Section 7.5 (pp. 276-280); Table 7.5.1" }, { "claim": "The book develops “agnostic statistics”: what can be learned about the world without assuming that there is a simple generative model that can be known to be true, with assumptions and targets of inquiry kept explicit, as foundations for researchers unwilling to assume more than they or their audience find credible.", "claim_id": "P001-front-H1", "related_corrections": [], "tied_to": "Half-title p. i (PDF p. 1); Introduction p. xv (PDF p. 15); the book as a whole; ch. 4 opening (p. 143); Theorem 3.3.6; Definition 4.1.2; Theorem 5.2.8" }, { "claim": "The preceding three decades brought a credibility revolution in social and health science research, which motivates a statistical presentation less dependent on unsustainable modeling assumptions.", "claim_id": "P001-front-H2", "related_corrections": [], "tied_to": "Half-title p. i (PDF p. 1); opening paragraph of the Introduction, p. xv (PDF p. 15); no numbered result, table or figure is offered as evidence" }, { "claim": "Part I (Probability, Chapters 1 and 2): the set-theoretic formulation of probability and random variables gives a language for describing features of random generative processes whose full structure is not known.", "claim_id": "P001-front-H3", "related_corrections": [], "tied_to": "Introduction p. xvi (PDF p. 16), Part I paragraph; Chapters 1-2 (Definitions 1.1.2, 1.2.1, 1.2.11; Definition 2.2.15; Theorem 2.2.21)" }, { "claim": "Part II (Statistics, Chapters 3-5): from repeated independent draws from a common process one can learn about some features of that process “without any further assumptions”, estimate them, and quantify uncertainty with confidence statements.", "claim_id": "P001-front-H4", "related_corrections": [], "tied_to": "Introduction p. xvi (PDF p. 16), Part II paragraph; Theorem 3.2.8 (WLLN); Theorem 3.2.24 (CLT); Section 3.3 and Theorem 3.3.6; Definition 3.4.1; Theorem 3.4.2; Example 3.4.3" }, { "claim": "Part II, continued: plug-in methods, including OLS regression and maximum likelihood, estimate useful approximation targets without assuming a restrictive linear or parametric model to be true.", "claim_id": "P001-front-H5", "related_corrections": [], "tied_to": "Introduction p. xvi (PDF p. 16), final sentences of the Part II paragraph; Section 4.1 and Definition 4.1.2; Theorem 5.2.8; Figure 5.2.2" }, { "claim": "Part III (Identification, Chapters 6 and 7): missing-data and causal questions are tied to features of a probability distribution through identification assumptions that are substantive and separable from the assumptions of statistical inference.", "claim_id": "P001-front-H6", "related_corrections": [], "tied_to": "Introduction pp. xv-xvi (PDF pp. 15-16), Part III paragraph; Chapters 6-7 (Sections 6.1-6.2 and 7.1-7.2; Definitions 6.1.5, 6.1.8, 7.1.9, 7.1.12; Theorems 6.1.9, 7.1.13)" }, { "claim": "Prerequisites and presentation: mild calculus and elementary set theory suffice; linear algebra is not used gratuitously and needs no prior training; measure theory is confined to technical footnotes; proofs are given for as many results as possible.", "claim_id": "P001-front-H7", "related_corrections": [], "tied_to": "Introduction pp. xvi-xvii (PDF pp. 16-17); the book as a whole; Section 4.1.3; Glossary of Mathematical Notation" }, { "claim": "Reader aids: references at the end of each chapter; a Glossary of Mathematical Notation; and a Glossary of Common Abbreviations that gives the meanings of all common abbreviations used in the book, with references to where they are defined.", "claim_id": "P001-front-H8", "related_corrections": [ "P001-FRONT-N001" ], "tied_to": "Introduction p. xvii (PDF p. 17) and p. xviii (PDF p. 18), first paragraph; Sections 1.5, 2.4, 3.6, 4.5, 5.4, 6.4, 7.6 (Further Readings); glossaries pp. 282-287" }, { "claim": "The Contents and the list of tables and figures map the book: 7 chapters and 120 numbered sections and subsections in three parts, two glossaries, references and index; 6 tables and 30 figures.", "claim_id": "P001-front-H9", "related_corrections": [], "tied_to": "Contents pp. vii-xi (PDF pp. 7-11); Tables and Figures pp. xii-xiii (PDF pp. 12-13); every heading and every table and figure caption in the body" }, { "claim": "The book's title is an homage to Lin (2013), which itself references the “agnostic regression” of Angrist and Imbens (2002).", "claim_id": "P001-front-H10", "related_corrections": [], "tied_to": "Introduction p. xviii (PDF p. 18), acknowledgments; References, pp. 288 and 290" } ], "date_published": "2026-10-25", "findings": [ { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of one row of the glossary table. The left cell holds the set-subtraction sign. The right cell reads: Set subtraction. A set-minus B denotes the set that contains all the elements in A except any that are also in B. Formally, A set-minus B is the set of s in A such that s is not in B. A set-minus B is called the relative complement of B in A, since A set-minus B equals A union B superscript C. A red outline surrounds the union sign in the last formula.", "caption": "Glossary of Mathematical Notation, row for set subtraction, p. 283 (PDF p. 301). The outline marks the union sign in $A\\backslash B = A \\cup B^C$, which should be an intersection sign.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-E001.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T101", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T101" } ], "headline": "The set difference is the intersection with the complement, $A\\backslash B = A \\cap B^C$; the glossary prints a union.", "id": "P001-BACK-E001", "location": "Glossary of Mathematical Notation, row for set subtraction, last sentence, p. 283 (PDF p. 301)", "pdf_page": 301, "printed_page": "283", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T101" ], "text": { "as_printed": "\\emph{Set subtraction.} $A\\backslash B$ (read ``$A$ set-minus $B$'') denotes the set that contains all the elements in $A$ \\emph{except} any that are also in $B$. Formally, $A\\backslash B = \\{s \\in A : s \\notin B\\}$. $A\\backslash B$ is called the \\emph{relative complement} of $B$ in $A$, since $A\\backslash B = A \\hl{\\cup} B^C$.", "corrected": "\\emph{Set subtraction.} $A\\backslash B$ (read ``$A$ set-minus $B$'') denotes the set that contains all the elements in $A$ \\emph{except} any that are also in $B$. Formally, $A\\backslash B = \\{s \\in A : s \\notin B\\}$. $A\\backslash B$ is called the \\emph{relative complement} of $B$ in $A$, since $A\\backslash B = A \\hl{\\cap} B^C$.\n\nHere $A$ and $B$ are subsets of the universal set $U$, and $B^C = \\{s \\in U : s \\notin B\\}$, as in the glossary's entry for the complement.", "evidence": "I derive the corrected identity and give a counterexample to the printed one. Let $A$ and $B$ be subsets of the universal set $U$, and let $s \\in U$. By the definitions of set subtraction and of the complement in the glossary, I have\n\\[\ns \\in A\\backslash B \\iff (s \\in A \\text{ and } s \\notin B) \\iff (s \\in A \\text{ and } s \\in B^C) \\iff s \\in A \\cap B^C ,\n\\]\nwhich implies that $A\\backslash B = A \\cap B^C$. For the printed form, take $U = \\{1,2,3\\}$, $A = \\{1,2\\}$ and $B = \\{2,3\\}$. Then $A\\backslash B = \\{1\\}$ and $B^C = \\{1\\}$, so $A \\cup B^C = \\{1,2\\} \\neq \\{1\\}$, while $A \\cap B^C = \\{1\\}$. Among the $64$ pairs of subsets of $\\{1,2,3\\}$, the printed identity fails for $56$ and the corrected identity holds for all $64$.", "problem": "The last identity is false. The union $A \\cup B^C$ contains every element of $A$, including those that are in $B$, and every element outside $B$, including those that are outside $A$. It contradicts the set-builder definition given one sentence earlier in the same row, which is correct.", "what_survives": "The verbal and set-builder definitions in the row, the glossary entries for the complement, the union and the intersection, and the proof of the Addition Rule on p.~10, which uses $A \\cap B^C$, are correct. Nothing in the book depends on the misprinted identity." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-E001", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of two reference entries from page 288. The second reads: Cameron, A. Colin and Pravin K. Trivedi. 2005. Microeconomics: Methods and applications. New York, NY: Cambridge University Press. A red outline surrounds the word Microeconomics.", "caption": "References, p. 288 (PDF p. 306). The outline marks “Microeconomics” in the entry for Cameron and Trivedi (2005); the book is Microeconometrics.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-E002.png" }, { "alt": "Crop of five lines from the references on page 291: the last line of one entry, the entry for Newey, and the entry for Pearl (2000). The entry for Newey reads: Newey, Whitney K. 1990. Semiparametric efficiency bounds. Journal of Applied Economics. 5(2): 99 to 135. A red outline surrounds the word Economics.", "caption": "References, p. 291 (PDF p. 309). The outline marks “Economics” in the journal name of the entry for Newey (1990); the journal is the Journal of Applied Econometrics.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-E002-b.png" }, { "alt": "Crop of five lines from the references on page 292: the entry for Van der Laan and Rose (2011), the entry for VanderWeele, and the first line of the entry for Varian (2014). The entry for VanderWeele reads: VanderWeele, Tyler J. 2015. Causal inference: Methods for mediation and interaction. New York, NY: Oxford University Press. A red outline surrounds the words Causal inference.", "caption": "References, p. 292 (PDF p. 310). The outline marks “Causal inference” at the start of the title in the entry for VanderWeele (2015); the title begins Explanation in Causal Inference.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-E002-c.png" } ], "formal_release": null, "formal_statements": [], "headline": "Three reference entries misprint a book title or a journal name: the correct forms are Microeconometrics, Journal of Applied Econometrics and Explanation in Causal Inference.", "id": "P001-BACK-E002", "location": "References, entries for Cameron and Trivedi (2005), p. 288 (PDF p. 306), Newey (1990), p. 291 (PDF p. 309), and VanderWeele (2015), p. 292 (PDF p. 310)", "pdf_page": 306, "printed_page": "288", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Cameron, A. Colin and Pravin K. Trivedi. 2005. \\emph{\\hl{Microeconomics}: Methods and applications}. New York, NY: Cambridge University Press.\n\nNewey, Whitney K. 1990. Semiparametric efficiency bounds. \\emph{Journal of Applied \\hl{Economics}}. 5(2): 99--135.\n\nVanderWeele, Tyler J. 2015. \\emph{\\hl{Causal inference}: Methods for mediation and interaction}. New York, NY: Oxford University Press.", "corrected": "Cameron, A. Colin and Pravin K. Trivedi. 2005. \\emph{\\hl{Microeconometrics}: Methods and applications}. New York, NY: Cambridge University Press.\n\nNewey, Whitney K. 1990. Semiparametric efficiency bounds. \\emph{Journal of Applied \\hl{Econometrics}}. 5(2): 99--135.\n\nVanderWeele, Tyler J. 2015. \\emph{\\hl{Explanation in causal inference}: Methods for mediation and interaction}. New York, NY: Oxford University Press.", "evidence": "I checked the three entries on the page against the publishers' own listings: the Cambridge University Press page for the book by Cameron and Trivedi, the journal's page for the article by Newey, and the title page of the book by VanderWeele.", "problem": "Two book titles and one journal name are misprinted. The book by Cameron and Trivedi is \\emph{Microeconometrics: Methods and Applications}, the article by Newey appeared in the \\emph{Journal of Applied Econometrics}, and the book by VanderWeele is \\emph{Explanation in Causal Inference: Methods for Mediation and Interaction}. Anyone who follows the entries as printed searches for the wrong title or in the wrong journal; the authors and years still identify each work.", "what_survives": "The passages of the book that cite these works are unaffected, and no mathematical or empirical claim depends on the three entries." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-E002", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the top of the Glossary of Common Abbreviations. A red outline surrounds the sentence Commonly abbreviated terms are defined when first used in the text. The next sentence reads We provide the unabbreviated terms here for reference. Below is a table with the headings Abbreviation, and Term and Reference in Text. Its first five rows are ATE, Average Treatment Effect, Section 7.1.1; BLP, Best Linear Predictor, Theorem 2.2.21; CDF, Cumulative Distribution Function, Definition 1.2.11; CEF, Conditional Expectation Function, Definition 2.2.15; and CMT, Continuous Mapping Theorem, Theorem 3.2.7.", "caption": "Glossary of Common Abbreviations, p. 286 (PDF p. 304). The outline marks the sentence saying that abbreviations are defined at first use. In the first rows of the table there is no row for APD before ATE, for ATC or ATT after it, or for CIA after CEF.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N002.png" } ], "formal_release": null, "formal_statements": [], "headline": "The Glossary of Common Abbreviations omits CIA, APD, PDA, ATT and ATC, and CIA is not expanded anywhere in the text.", "id": "P001-BACK-N002", "location": "Glossary of Common Abbreviations, sentence above the table, and the table, pp. 286–287 (PDF pp. 304–305)", "pdf_page": 304, "printed_page": "286–287", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\emph{Glossary of Common Abbreviations (p.~286).} Commonly abbreviated terms are \\hl{defined when first used in the text}. We provide the unabbreviated terms here for reference.\n\n\\emph{Abbreviations in the table (pp.~286--287).} ATE, BLP, CDF, CEF, CMT, DR, i.i.d., IPW, KL, MAR, MCAR, ML, MSE, NRCM, OLS, PDF, PMF, WLLN.\n\n\\emph{Introduction (p.~xviii).} In addition, we provide a Glossary of Common Abbreviations, which gives the meanings of \\hl{all common abbreviations used in this book}, along with references to where they are defined in the text.\n\n\\emph{Section 7.3.1 (p.~273), first use.} The ATT is nonparametrically identified under the \\hl{CIA} and a weaker positivity assumption, $p_D(\\mathbf{X}_i) < \\epsilon < 1$.\n\n\\emph{Section 7.3.1 (p.~273), second use.} Thus, if the usual plug-in regularity conditions hold, the treatment propensity function is known (or can be estimated consistently), and the \\hl{CIA} holds, then regression will consistently estimate the average treatment effect for a reweighted population.", "corrected": "At its first use on p.~273, write ``\\hl{conditional independence assumption (CIA)}''. Add five rows to the table:\n\\begin{audittable}{ll}{Rows to add to the Glossary of Common Abbreviations.}\nAbbreviation & Term and Reference in Text \\\\\nAPD & Average Partial Derivative; Definition 4.3.6 \\\\\nATC & Average Treatment Effect among the Control Subjects; Section 7.1.1 \\\\\nATT & Average Treatment Effect among the Treated; Section 7.1.1 \\\\\nCIA & Conditional Independence Assumption; Definition 7.1.12 \\\\\nPDA & Partial Derivative at the Average; Definition 4.3.7 \\\\\n\\end{audittable}", "evidence": "I checked the whole table on pp.~286--287, the definitions on pp.~166--167 and 238, and the two uses of CIA on p.~273. I also searched the full text, finding CIA on p.~273 only.", "problem": "Five abbreviations that the text uses are missing from the table: APD and PDA (Definitions 4.3.6 and 4.3.7, pp.~166--167), ATT and ATC (expanded on p.~238), and CIA. Since CIA is used twice on p.~273 and is expanded nowhere, the statement that abbreviations are defined at first use is untrue for it. Its meaning, the conditional independence assumption of Definition 7.1.12, can be inferred from the discussion around it (see \\fid{P001-C07-N009}).", "what_survives": "The eighteen entries that the table lists and the statistical exposition are unaffected, and no result depends on the glossary." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N002", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Seven lines of the References. A red box marks 'In press.' in the entry 'Aronow, Peter M. and Cyrus Samii. In press. Estimating average causal effects under general interference'. The next entry begins 'Athey, Susan and Guido W. Imbens. Working paper.'", "caption": "References, p. 288 (PDF p. 306): the Aronow and Samii entry, with “In press.” marked. The Athey and Imbens entry follows it.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N003.png" }, { "alt": "Six lines of the References. A red box marks 'In press.' in the entry 'Imbens, Guido W. and Michel Kolesar. In press. Robust standard errors in small samples'. The next entry reads 'Imbens, Guido W. and Donald B. Rubin. 2011. Causal inference in statistics and social sciences. Unpublished manuscript. Harvard University.'", "caption": "References, p. 290 (PDF p. 308): the Imbens and Kolesar entry, with “In press.” marked. The Imbens and Rubin entry follows it.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N003-p290.png" } ], "formal_release": null, "formal_statements": [], "headline": "Two entries marked “In press” are for articles that appeared in 2017 and 2016, before the book; two further entries cite a working paper and a 2011 manuscript of works that were published in 2017 and 2015.", "id": "P001-BACK-N003", "location": "References, p. 288 (PDF p. 306), entries for Aronow and Samii and for Athey and Imbens; p. 290 (PDF p. 308), entries for Imbens and Kolesar and for Imbens and Rubin", "pdf_page": 306, "printed_page": "288", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\emph{p.~288.} Aronow, Peter M. and Cyrus Samii. \\hl{In press.} Estimating average causal effects under general interference, with application to a social network experiment. \\emph{Annals of Applied Statistics}.\n\nAthey, Susan and Guido W. Imbens. Working paper. The state of applied econometrics: Causality and policy evaluation.\n\n\\emph{p.~290.} Imbens, Guido W. and Michel Kolesar. \\hl{In press.} Robust standard errors in small samples: Some practical advice. \\emph{The Review of Economics and Statistics}.\n\nImbens, Guido W. and Donald B. Rubin. 2011. \\emph{Causal inference in statistics and social sciences}. Unpublished manuscript. Harvard University.", "corrected": "\\emph{p.~288.} P. M. Aronow and Cyrus Samii. \\hl{2017.} Estimating average causal effects under general interference, with application to a social network experiment. \\emph{Annals of Applied Statistics}. \\hl{11(4): 1912--1947.}\n\n\\emph{p.~290.} Imbens, Guido W. and Michel Kolesar. \\hl{2016.} Robust standard errors in small samples: Some practical advice. \\emph{The Review of Economics and Statistics}. \\hl{98(4): 701--712.}\n\nThe citations on pp.~281 and~177 become Aronow and Samii (2017) and Imbens and Kolesar (2016). For the other two entries, the published versions are Athey and Imbens (2017), \\emph{Journal of Economic Perspectives} 31(2): 3--32, and Imbens and Rubin (2015), \\emph{Causal Inference for Statistics, Social, and Biomedical Sciences: An Introduction}, Cambridge University Press. If the earlier versions are the ones meant, the entries should say so explicitly.", "evidence": "I checked the pages against the published versions. The first page of the article in the \\emph{Annals of Applied Statistics} is headed ``2017, Vol.~11, No.~4, 1912--1947'' (DOI 10.1214/16-AOAS1005), and the first page of the article in \\emph{The Review of Economics and Statistics} carries ``October 2016, 98(4): 701--712''. The publisher's listing for the article by Athey and Imbens places it in vol.~31, no.~2 (Spring 2017) of the \\emph{Journal of Economic Perspectives}, and the listing of Cambridge University Press for the book by Imbens and Rubin gives 6 April 2015 as the publication date of the hardback.", "problem": "Both articles marked ``In press'' had appeared when the book was published in 2019, so the two labels were out of date, and the two entries lack the year, volume and pages. The citations of these articles on pp.~281 and~177 carry the same label. The other two entries cite earlier versions of works that had also been published by then: the article in 2017, and the book in 2015 under a different title. A citation may deliberately identify an earlier version. The later publication does not by itself imply that these two entries are wrong.", "what_survives": "Each of the four works can be identified from its authors and title. What the book says about these works does not depend on the publication details, and no result is affected." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N003", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Five lines of the References. A red box marks '(28)2, 2014:' in the last line of the entry, 'Economic Perspectives. (28)2, 2014: 29-50.'", "caption": "References, p. 288 (PDF p. 306): the entry for Belloni, Chernozhukov and Hansen, with “(28)2, 2014:” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N005.png" } ], "formal_release": null, "formal_statements": [], "headline": "In the entry for Belloni, Chernozhukov and Hansen (2014), “(28)2, 2014:” should read “28(2):”.", "id": "P001-BACK-N005", "location": "References, p. 288 (PDF p. 306), entry for Belloni, Chernozhukov and Hansen (2014)", "pdf_page": 306, "printed_page": "288", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Belloni, Alexandre, Victor Chernozhukov, and Christian Hansen. 2014. High-dimensional methods and inference on structural and treatment effects. \\emph{Journal of Economic Perspectives}. \\hl{(28)2, 2014:} 29--50.", "corrected": "Belloni, Alexandre, Victor Chernozhukov, and Christian Hansen. 2014. High-dimensional methods and inference on structural and treatment effects. \\emph{Journal of Economic Perspectives}. \\hl{28(2):} 29--50.", "evidence": "I checked pp.~288 and 292.", "problem": "The parentheses are set around the volume and not around the issue, and the year is printed a second time. Elsewhere, where an issue is given, the References set an article as volume(issue): pages. In particular, the Varian (2014) entry on p.~292, which is for an article in the same issue of the same journal, reads ``\\emph{Journal of Economic Perspectives}. 28(2): 3--28.''", "what_survives": "The volume, issue, year and pages are not in doubt, and the article can be identified." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N005", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Five lines of the References. A red box marks 'in estimation' in the title 'Moving the goalposts: Addressing limited overlap in estimation of average treatment effects by changing the estimand'.", "caption": "References, p. 289 (PDF p. 307): the entry for Crump, Hotz, Imbens and Mitnik, with “in estimation” marked, where the title has “in the estimation”.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N006.png" }, { "alt": "Five lines of the References. A red box marks '972' in 'Biometrics. 61: 962-972.' at the end of the entry for Bang and Robins.", "caption": "References, p. 288 (PDF p. 306): the entry for Bang and Robins, with the last page, “972”, marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N006-p288.png" } ], "formal_release": null, "formal_statements": [], "headline": "The title of Crump et al. (2006) reads “in the estimation of”, and the article by Bang and Robins (2005) ends on p. 973, not p. 972; the year 2001 printed for Hayashi’s Econometrics may be that of a later printing of a book first published in 2000.", "id": "P001-BACK-N006", "location": "References, p. 289 (PDF p. 307), entry for Crump, Hotz, Imbens and Mitnik (2006); p. 288 (PDF p. 306), entry for Bang and Robins (2005); p. 290 (PDF p. 308), entry for Hayashi (2001)", "pdf_page": 307, "printed_page": "289", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\emph{p.~289.} Crump, Richard K., V. Joseph Hotz, Guido W. Imbens, and Oscar Mitnik. 2006. Moving the goalposts: Addressing limited overlap \\hl{in estimation} of average treatment effects by changing the estimand. NBER Technical Working Paper No. 330.\n\n\\emph{p.~288.} Bang, Heejung and James M. Robins. 2005. Doubly robust estimation in missing data and causal inference models. \\emph{Biometrics}. 61: 962--\\hl{972}.\n\n\\emph{p.~290.} Hayashi, Fumio. 2001. \\emph{Econometrics}. Princeton, NJ: Princeton University Press.", "corrected": "\\emph{p.~289.} Crump, Richard K., V. Joseph Hotz, Guido W. Imbens, and Oscar Mitnik. 2006. Moving the goalposts: Addressing limited overlap in \\hl{the} estimation of average treatment effects by changing the estimand. NBER Technical Working Paper No. 330.\n\n\\emph{p.~288.} Bang, Heejung and James M. Robins. 2005. Doubly robust estimation in missing data and causal inference models. \\emph{Biometrics}. 61: 962--\\hl{973}.\n\nIf the first edition of Hayashi's book is the one meant, then the year is 2000 both in the entry on p.~290 and in the citation on p.~204. If a printing of 2001 is meant, the entry should identify it.", "evidence": "I checked the pages against the publishers' listings. The NBER listing for Technical Working Paper 330 (\\url{https://www.nber.org/papers/t0330}) is headed ``Moving the Goalposts: Addressing Limited Overlap in the Estimation of Average Treatment Effects by Changing the Estimand'', and the publisher's listing for the article in \\emph{Biometrics} reads ``Pages 962-973''. The economics catalogue of Princeton University Press for 2017 lists Hayashi's book as ``2000. 712 pages'', and the copyright page of the book carries the year 2000.", "problem": "The title of the working paper has ``in the estimation of'', and the entry omits ``the''. The article in \\emph{Biometrics} occupies pp.~962--973, and the entry stops one page short. The first edition of Hayashi's book is dated 2000. The entry and the citation ``Hayashi (2001)'' on p.~204 give 2001. Since a printing dated 2001 is not excluded, the evidence does not show that the year is an error.", "what_survives": "The three works can be identified from the entries as printed. No statement in the text and no result depends on these details." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N006", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Five lines of the References. A red box marks the surname 'Dasgupta' at the start of the entry 'Dasgupta, Anirban. 2008. Asymptotic theory of statistics and probability.'", "caption": "References, p. 289 (PDF p. 307): the entry for the book by DasGupta, with the surname “Dasgupta” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N007.png" } ], "formal_release": null, "formal_statements": [], "headline": "The surname in the entry should be “DasGupta”, as it is in the two citations in the text.", "id": "P001-BACK-N007", "location": "References, p. 289 (PDF p. 307), entry “Dasgupta, Anirban. 2008.”", "pdf_page": 307, "printed_page": "289", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\hl{Dasgupta}, Anirban. 2008. \\emph{Asymptotic theory of statistics and probability}. New York, NY: Springer-Verlag New York.", "corrected": "\\hl{DasGupta}, Anirban. 2008. \\emph{Asymptotic theory of statistics and probability}. New York, NY: Springer-Verlag New York.", "evidence": "I checked the pages and the author field of the publisher's listing for the book (\\url{https://link.springer.com/book/10.1007/978-0-387-75971-5}).", "problem": "The text cites the book as DasGupta (2008), with a capital G, in footnote~9 on p.~56 and on p.~141. The entry has a lower-case g. The publisher gives the author's name as Anirban DasGupta, and I conclude that the entry is the one that is misspelled.", "what_survives": "The book that is cited is not in doubt, and nothing else depends on the spelling." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N007", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Four lines of the References. A red box marks 'Vol. 5' at the start of the second line of the entry 'Davidson, Russell and James G. MacKinnon. 2004. Econometric theory and methods, Vol. 5. New York, NY: Oxford University Press.'", "caption": "References, p. 289 (PDF p. 307): the entry for Davidson and MacKinnon, with “Vol. 5” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N008.png" } ], "formal_release": null, "formal_statements": [], "headline": "“Vol. 5” should be deleted from the entry for Davidson and MacKinnon (2004): the book is a single volume.", "id": "P001-BACK-N008", "location": "References, p. 289 (PDF p. 307), entry for Davidson and MacKinnon (2004)", "pdf_page": 307, "printed_page": "289", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Davidson, Russell and James G. MacKinnon. 2004. \\emph{Econometric theory and methods}\\hl{, Vol. 5}. New York, NY: Oxford University Press.", "corrected": "Davidson, Russell and James G. MacKinnon. 2004. \\emph{Econometric theory and methods}\\del{, Vol. 5}. New York, NY: Oxford University Press.", "evidence": "I checked the page against the description of the book in a library catalogue (Oxford University Press, 2004, xviii + 750 pages, ISBN 0195123727), and against the indexed front matter of the copy that the authors have posted, which carries the same ISBN and no volume number.", "problem": "The book is a single volume of xviii + 750 pages and is not the fifth volume of a larger work, so no volume number belongs in the entry. Where the ``5'' came from is not known.", "what_survives": "The citation on p.~177 gives authors and year only and is unaffected. The book can be identified from the rest of the entry." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N008", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Four lines of the References. A red box marks 'Sarah' in the entry 'Diaconis, Persi, Sarah Holmes, and Richard Montgomery. 2007. Dynamical bias in the coin toss.'", "caption": "References, p. 289 (PDF p. 307): the entry for Diaconis, Holmes and Montgomery, with the given name “Sarah” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N009.png" } ], "formal_release": null, "formal_statements": [], "headline": "The second author of “Dynamical bias in the coin toss” is Susan Holmes, not Sarah Holmes.", "id": "P001-BACK-N009", "location": "References, p. 289 (PDF p. 307), entry for Diaconis, Holmes and Montgomery (2007)", "pdf_page": 307, "printed_page": "289", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Diaconis, Persi, \\hl{Sarah} Holmes, and Richard Montgomery. 2007. Dynamical bias in the coin toss. \\emph{Society for Industrial and Applied Mathematics Review}. 49(2): 211--235.", "corrected": "Diaconis, Persi, \\hl{Susan} Holmes, and Richard Montgomery. 2007. Dynamical bias in the coin toss. \\emph{Society for Industrial and Applied Mathematics Review}. 49(2): 211--235.", "evidence": "I checked the page against the journal's listing for the article (DOI 10.1137/S0036144504446436), whose list of authors reads ``Persi Diaconis, Susan Holmes, and Richard Montgomery''.", "problem": "The given name of the second author is wrong in the entry. The citation in footnote~3 on p.~5 gives surnames only and is unaffected.", "what_survives": "The article can be identified from its title, journal, volume and pages, and nothing else is affected." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N009", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Four lines of the References. A red box marks the second '2016.' in the entry 'Greenland, Sander et al. 2016. Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations. 2016. European Journal of Epidemiology.'", "caption": "References, p. 289 (PDF p. 307): the entry for Greenland et al., with the second “2016.” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N010.png" }, { "alt": "Four lines of the References. A red box marks 'applicationsin' in the title 'An introduction to statistical learning with applicationsin R.'", "caption": "References, p. 290 (PDF p. 308): the entry for James et al., with “applicationsin” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N010-p290.png" }, { "alt": "Five lines of the References. A red box marks 'Model- based', printed with a space after the hyphen, in the entry 'Rosenbaum, Paul R. 1987. Model- based direct adjustment.'", "caption": "References, p. 291 (PDF p. 309): the entry for Rosenbaum (1987), with “Model- based” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N010-p291.png" } ], "formal_release": null, "formal_statements": [], "headline": "Three slips of typesetting: “2016.” is printed twice in the entry for Greenland et al., “applicationsin” should be “applications in” in the entry for James et al., and “Model- based” should be “Model-based” in the entry for Rosenbaum (1987).", "id": "P001-BACK-N010", "location": "References, p. 289 (PDF p. 307), entry for Greenland et al. (2016); p. 290 (PDF p. 308), entry for James et al. (2014); p. 291 (PDF p. 309), entry for Rosenbaum (1987)", "pdf_page": 307, "printed_page": "289", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\emph{p.~289.} Greenland, Sander et al. 2016. Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations. \\hl{2016.} \\emph{European Journal of Epidemiology}. 31(4): 337--350.\n\n\\emph{p.~290.} James, Gareth et al. 2014. \\emph{An introduction to statistical learning with \\hl{applicationsin} R}. New York, NY: Springer Science+Business Media New York.\n\n\\emph{p.~291.} Rosenbaum, Paul R. 1987. \\hl{Model-~based} direct adjustment. \\emph{Journal of the American Statistical Association}. 82(398): 387--394.", "corrected": "\\emph{p.~289.} Greenland, Sander et al. 2016. Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations. \\del{2016.} \\emph{European Journal of Epidemiology}. 31(4): 337--350.\n\n\\emph{p.~290.} James, Gareth et al. 2014. \\emph{An introduction to statistical learning with \\hl{applications in} R}. New York, NY: Springer Science+Business Media New York.\n\n\\emph{p.~291.} Rosenbaum, Paul R. 1987. \\hl{Model-based} direct adjustment. \\emph{Journal of the American Statistical Association}. 82(398): 387--394.", "evidence": "I checked the three pages.", "problem": "The year 2016 is printed after the authors and again after the title. A space is missing between ``applications'' and ``in''. A space follows the hyphen of ``Model-based'', in the middle of a line. The References also print Hernan and Kolesar without accents while they keep the accents of Hájek and Pötscher; I have not checked the accented spellings against a source, and I do not correct them here.", "what_survives": "The three works can be identified, and in each case the intended text is plain." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N010", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Five lines of the References. A red box marks '968' in 'Statistical Association. 81(396): 945-968.' at the end of the entry for Holland.", "caption": "References, p. 290 (PDF p. 308): the entry for Holland (1986), with the last page of the range, “968”, marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N011.png" } ], "formal_release": null, "formal_statements": [], "headline": "The article by Holland (1986) occupies pp. 945–960; the printed range, 945–968, also takes in the comments and the first page of Holland’s rejoinder.", "id": "P001-BACK-N011", "location": "References, p. 290 (PDF p. 308), entry for Holland (1986)", "pdf_page": 308, "printed_page": "290", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Holland, Paul W. 1986. Statistics and causal inference. \\emph{Journal of the American Statistical Association}. 81(396): 945--\\hl{968}.", "corrected": "Holland, Paul W. 1986. Statistics and causal inference. \\emph{Journal of the American Statistical Association}. 81(396): 945--\\hl{960}.\n\nIf the comments or the rejoinder are also meant, they should be cited separately, with their own pages.", "evidence": "I checked the page against an indexed version of the publisher's table of contents for the issue (\\url{https://www.tandfonline.com/toc/uasa20/81/396}), which has ``Pages: 945-960'' for the article and ``Pages: 967-968'' for the comment by Granger, and against the cover page of a scan of the rejoinder, which gives 968--970.", "problem": "The article named in the entry occupies pp.~945--960. The published comments follow on pp.~961--968, and Holland's rejoinder is on pp.~968--970, so the printed range reaches its first page. The printed range therefore fits neither the article alone nor the article together with the comments and the rejoinder. Whether the comments were meant to be included is not known.", "what_survives": "The citation on p.~236 gives author and year only. The article can be identified from its title, volume and first page, and nothing in the text depends on the range." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N011", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Five lines of the References. A red box marks '684' in 'Association. 47: 663-684.' at the end of the entry for Horvitz and Thompson.", "caption": "References, p. 290 (PDF p. 308): the entry for Horvitz and Thompson, with the last page, “684”, marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N012.png" } ], "formal_release": null, "formal_statements": [], "headline": "The article by Horvitz and Thompson (1952) ends on p. 685, not p. 684.", "id": "P001-BACK-N012", "location": "References, p. 290 (PDF p. 308), entry for Horvitz and Thompson (1952)", "pdf_page": 308, "printed_page": "290", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Horvitz, Daniel G. and Donovan J. Thompson. 1952. A generalization of sampling without replacement from a finite universe. \\emph{Journal of the American Statistical Association}. 47: 663--\\hl{684}.", "corrected": "Horvitz, Daniel G. and Donovan J. Thompson. 1952. A generalization of sampling without replacement from a finite universe. \\emph{Journal of the American Statistical Association}. 47: 663--\\hl{685}.", "evidence": "I checked the page against the cover page of a scan of the published article, which gives the source as Vol.~47, No.~260 (Dec., 1952), pp.~663--685.", "problem": "The last page of the article is 685, and the entry stops one page short.", "what_survives": "The citation on p.~227 gives authors and year only. The article can be identified, and what the book says about the estimator is unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N012", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Five lines of the References. A red box marks 'population means' in the title 'Demystifying double robustness: A comparison of alternative strategies for estimating population means from incomplete data'.", "caption": "References, p. 290 (PDF p. 308): the entry for Kang and Schafer, with “population means” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N013.png" } ], "formal_release": null, "formal_statements": [], "headline": "The title of Kang and Schafer (2007) has “estimating a population mean”, not “estimating population means”.", "id": "P001-BACK-N013", "location": "References, p. 290 (PDF p. 308), entry for Kang and Schafer (2007)", "pdf_page": 308, "printed_page": "290", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Kang, Joseph D. Y. and Joseph L. Schafer. 2007. Demystifying double robustness: A comparison of alternative strategies for estimating \\hl{population means} from incomplete data. \\emph{Statistical Science}. 22(4): 523--539.", "corrected": "Kang, Joseph D. Y. and Joseph L. Schafer. 2007. Demystifying double robustness: A comparison of alternative strategies for estimating \\hl{a population mean} from incomplete data. \\emph{Statistical Science}. 22(4): 523--539.", "evidence": "I checked the entry against the first page of the electronic reprint of the published article (\\url{https://arxiv.org/pdf/0804.2958}), whose title reads ``Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data''.", "problem": "The title of the article has the singular with the indefinite article, ``a population mean'', where the entry has the plural.", "what_survives": "The article can be identified. The citation on p.~234 does not quote the title, and no result depends on it." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N013", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Five lines of the References. A red box marks '4: 117' in 'Psychological Methods. 4: 117.' at the end of the entry for Reichardt and Gollob.", "caption": "References, p. 291 (PDF p. 309): the entry for Reichardt and Gollob, with “4: 117” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N014.png" } ], "formal_release": null, "formal_statements": [], "headline": "The entry for Reichardt and Gollob (1999) gives the first page only and no issue; the article is in 4(1): 117–128.", "id": "P001-BACK-N014", "location": "References, p. 291 (PDF p. 309), entry for Reichardt and Gollob (1999)", "pdf_page": 309, "printed_page": "291", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Reichardt, Charles S. and Harry F. Gollob. 1999. Justifying the use and increasing the power of a \\emph{t} test for a randomized experiment with a convenience sample. \\emph{Psychological Methods}. \\hl{4: 117}.", "corrected": "Reichardt, Charles S. and Harry F. Gollob. 1999. Justifying the use and increasing the power of a \\emph{t} test for a randomized experiment with a convenience sample. \\emph{Psychological Methods}. \\hl{4(1): 117--128}.", "evidence": "I checked the entry against the bibliographic details in the publisher's preview of the article, which give 4(1): 117--128 (DOI 10.1037/1082-989X.4.1.117).", "problem": "The entry is incomplete and not false: 117 is the first page of the article, which runs to p.~128 in issue~1 of volume~4. Elsewhere the References give a page range for an article in a journal.", "what_survives": "Nothing that the entry prints is wrong, and the citation on p.~281 is unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N014", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Eight lines from the reference list. In the entry for Richardson and Robins, a red box marks 'Working paper.' where other entries give a year, and a second red box marks the word 'the' in 'Center for the Statistics and the Social Sciences'.", "caption": "References, p. 291 (PDF p. 309): the entry for Richardson and Robins, with “Working paper.” in the position of the year and the extra “the” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N015.png" } ], "formal_release": null, "formal_statements": [], "headline": "In the entry for Richardson and Robins, the issuing body is the Center for Statistics and the Social Sciences, with no “the” before “Statistics”, and the working paper is dated 2013.", "id": "P001-BACK-N015", "location": "References, p. 291 (PDF p. 309), entry for Richardson and Robins", "pdf_page": 309, "printed_page": "291", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Richardson, Thomas S. and James M. Robins. \\hl{Working paper.} Single-world intervention graphs (SWIGs): A unification of the counterfactual and graphical approaches to causality. Working Paper Number 128. Center for \\hl{the} Statistics and the Social Sciences, University of Washington, Seattle.", "corrected": "Richardson, Thomas S. and James M. Robins. \\hl{2013.} Single-world intervention graphs (SWIGs): A unification of the counterfactual and graphical approaches to causality. Working Paper Number 128. Center \\hl{for Statistics} and the Social Sciences, University of Washington, Seattle.", "evidence": "I checked the entry against the center's list of working papers, which is headed ``Center for Statistics and the Social Sciences'' and has the line ``WP 128 $\\cdot$ April 2013'' (\\url{https://csss.uw.edu/research/working-papers?items_per_page=25&page=1}).", "problem": "The name of the center has no ``the'' before ``Statistics''. The entry also gives ``Working paper'' in the position of the year, although the working paper is dated: the center lists it as Working Paper 128, April 2013. The entry omits the year; it does not give a false one.", "what_survives": "No result or argument depends on the name of the center or on the date, and the working paper remains identifiable by its authors, title and number. The two citations of it in the text, on pp.~247 and 281, give ``mimeo.'' in place of a year and still lead to the entry." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N015", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Six lines from the reference list: the last two lines of the preceding entry, the entry 'Richardson, Thomas S. and Andrea Rotnitzky. 2014. Causal etiology of the research of James M. Robins. Statistical Science. 29(4): 259-284.', and the entry that follows. A red box marks the page range '259-284'.", "caption": "References, p. 291 (PDF p. 309): the entry for Richardson and Rotnitzky (2014), with the page range “259–284” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N016.png" } ], "formal_release": null, "formal_statements": [], "headline": "The article by Richardson and Rotnitzky (2014) is on pp. 459–484 of Statistical Science 29(4), not on pp. 259–284.", "id": "P001-BACK-N016", "location": "References, p. 291 (PDF p. 309), entry for Richardson and Rotnitzky (2014)", "pdf_page": 309, "printed_page": "291", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Richardson, Thomas S. and Andrea Rotnitzky. 2014. Causal etiology of the research of James M. Robins. \\emph{Statistical Science}. 29(4): \\hl{259--284}.", "corrected": "Richardson, Thomas S. and Andrea Rotnitzky. 2014. Causal etiology of the research of James M. Robins. \\emph{Statistical Science}. 29(4): \\hl{459--484}.", "evidence": "I checked the entry against the first page of the article (DOI 10.1214/14-STS505), read in the IMS electronic reprint at \\url{https://arxiv.org/pdf/1503.02894}, which carries the journal header ``2014, Vol. 29, No. 4, 459--484''.", "problem": "The first digit of both page numbers is wrong; the year, the volume and the issue are right.", "what_survives": "The entry still identifies the article by its authors, title, journal, volume and issue. The text cites the article by author and year only (p.~281), and nothing depends on the page numbers." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N016", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Eight lines from the reference list. In the entry 'Robins, James M. 1986. A new approach to causal inference in mortality studies with a sustained exposure period? Application to control of the healthy worker survivor effect', a small red box marks the question mark after 'period'.", "caption": "References, p. 291 (PDF p. 309): the entry for Robins (1986), with the question mark after “period” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N017.png" } ], "formal_release": null, "formal_statements": [], "headline": "In the title of Robins (1986), the question mark after “period” should be a dash.", "id": "P001-BACK-N017", "location": "References, p. 291 (PDF p. 309), entry for Robins (1986)", "pdf_page": 309, "printed_page": "291", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Robins, James M. 1986. A new approach to causal inference in mortality studies with a sustained exposure period\\hl{? A}pplication to control of the healthy worker survivor effect. Mathematical models in medicine: Diseases and epidemics. Part 2. \\emph{Mathematical Modelling}. 7: 1393--1512.", "corrected": "Robins, James M. 1986. A new approach to causal inference in mortality studies with a sustained exposure period\\hl{---a}pplication to control of the healthy worker survivor effect. Mathematical models in medicine: Diseases and epidemics. Part 2. \\emph{Mathematical Modelling}. 7: 1393--1512.", "evidence": "I checked the entry against the publisher's listing of the article (\\url{https://www.sciencedirect.com/science/article/pii/0270025586900886}), which gives \\emph{Mathematical Modelling} 7(9--12): 1393--1512 and the DOI 10.1016/0270-0255(86)90088-6. The title there ends ``with a sustained exposure period---application to control of the healthy worker survivor effect''.", "problem": "The title of the article is not a question. In the publisher's listing of the article, a dash joins ``period'' to ``application'', which is in lower case. How the question mark entered the entry is not known.", "what_survives": "Only the punctuation and following capitalization of this title change, and no result or argument in the book depends on them." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N017", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Four index lines: 'geometric distribution, 19'; 'Glivenko-Cantelli Theorem, 102n14, 117n27,' continued on the next line by '120n31'; and, after a gap, 'heteroskedasticity, 156'. A red box marks the locator '117n27'.", "caption": "Index, p. 295 (PDF p. 313), left column: the entry “Glivenko–Cantelli Theorem”, with the locator “117n27” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N019.png" } ], "formal_release": null, "formal_statements": [], "headline": "The index locator for the Glivenko–Cantelli Theorem on p. 117 should give footnote 29, not footnote 27.", "id": "P001-BACK-N019", "location": "Index, p. 295 (PDF p. 313), left column, entry “Glivenko–Cantelli Theorem”", "pdf_page": 313, "printed_page": "295", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Glivenko--Cantelli Theorem, 102n14, 117n\\hl{27}, 120n31", "corrected": "Glivenko--Cantelli Theorem, 102n14, 117n\\hl{29}, 120n31", "evidence": "I checked the pages. Footnote 29 begins ``In fact, an even stronger claim holds. The Glivenko--Cantelli Theorem:''. Footnote 31 on p.~120 refers to ``the Glivenko--Cantelli Theorem in Footnote 29'', and the index entries for almost sure convergence on pp.~293 and 294 give 117n29.", "problem": "Footnote 29 on p.~117 names and states the theorem, while footnote 27 is on p.~116 and concerns the multivariate empirical CDF.", "what_survives": "The page number is right, and the other two locators lead to footnotes that name the theorem. The theorem is unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N019", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Ten index lines: the heading 'independence' and its eight subentries, from 'CEFs and BLPs under, 82-84' to 'of random variables, 38-39'. A red box marks '59-60' in the subentry 'covariance and, 59-60', which stands directly below 'correlation and, 64-66'.", "caption": "Index, p. 295 (PDF p. 313), right column: the heading “independence” and its subentries, with the locator “59–60” of “covariance and” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N022.png" } ], "formal_release": null, "formal_statements": [], "headline": "Under “independence”, the subentry “covariance and” should point to p. 64, not to pp. 59–60.", "id": "P001-BACK-N022", "location": "Index, p. 295 (PDF p. 313), right column, heading “independence”, subentry “covariance and”", "pdf_page": 313, "printed_page": "295", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\emph{Under ``independence''.} covariance and, \\hl{59--60}", "corrected": "\\emph{Under ``independence''.} covariance and, \\hl{64}", "evidence": "I checked the pages. Section 2.2.2 is titled ``Covariance, Correlation, and Independence'', and Theorem 2.2.8 states that $\\operatorname{Cov}[X,Y]=0$ for independent random variables $X$ and $Y$. The corresponding subentry under ``covariance'' on p.~294, ``and independence, 64'', gives the right page.", "problem": "Pages 59--60 define covariance, derive its alternative formula and state the variance rule; they do not discuss independence. The relation between covariance and independence is Theorem 2.2.8, in Section 2.2.2 on p.~64. The printed locator repeats that of the heading ``covariance, 59--60'' on p.~294.", "what_survives": "The subject of the subentry is right, and only its locator changes. Theorem 2.2.8 is unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N022", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Eight index lines under the heading 'maximum likelihood (ML) estimation, 185-203, 233'. Red boxes mark the space and the comma in 'inference with , 202-203' and in 'of the classical linear model , 189-192'. The subentry below, 'of the probit model, 192-193', has no such space.", "caption": "Index, p. 296 (PDF p. 314), left column: the heading “maximum likelihood (ML) estimation”, with the space before the comma marked in two subentries.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N023.png" }, { "alt": "Five index lines: 'probit model, 183'; 'maximum likelihood estimation of ,' continued on the next line by '192-193'; 'propensity score'; and 'for causal inference, 250-252'. A red box marks the space and the comma after 'estimation of'.", "caption": "Index, p. 297 (PDF p. 315), right column: the heading “probit model”, with the space before the comma marked in its subentry.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N023-p297.png" }, { "alt": "Six index lines, all subentries of 'estimator': 'kernel plug-in, 123'; 'lasso, 169-170, 174, 202'; 'MSE of, 104-105'; 'plug in, see plug-in estimator'; 'post-stratification, 219, 256'; and 'regression, see regression'. A red box marks 'plug in', which is set without a hyphen.", "caption": "Index, p. 295 (PDF p. 313), left column: six subentries of the heading “estimator”, which begins on p. 294, with “plug in” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N023-p295.png" } ], "formal_release": null, "formal_statements": [], "headline": "Three index subentries have a stray space before the comma, and the subentry “plug in” under “estimator” should be hyphenated.", "id": "P001-BACK-N023", "location": "Index, p. 296 (PDF p. 314), left column, heading “maximum likelihood (ML) estimation”; p. 297 (PDF p. 315), right column, heading “probit model”; p. 295 (PDF p. 313), left column, subentry “plug in” of the heading “estimator”", "pdf_page": 314, "printed_page": "296", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\emph{p.~296, under ``maximum likelihood (ML) estimation''.} inference with\\hl{ ,} 202--203 \\ldots\\ of the classical linear model\\hl{ ,} 189--192\n\n\\emph{p.~297, under ``probit model''.} maximum likelihood estimation of\\hl{ ,} 192--193\n\n\\emph{p.~295, under ``estimator''.} \\hl{plug in}, \\emph{see} plug-in estimator", "corrected": "\\emph{p.~296, under ``maximum likelihood (ML) estimation''.} inference with\\hl{,} 202--203 \\ldots\\ of the classical linear model\\hl{,} 189--192\n\n\\emph{p.~297, under ``probit model''.} maximum likelihood estimation of\\hl{,} 192--193\n\n\\emph{p.~295, under ``estimator''.} \\hl{plug-in}, \\emph{see} plug-in estimator", "evidence": "I checked the pages. The neighboring subentries ``of the probit model, 192--193'' on p.~296 and ``kernel plug-in, 123'' on p.~295 show the usual forms.", "problem": "In three subentries a space stands between the last word and the comma. Under ``estimator'', the subentry ``plug in'' lacks the hyphen of the entry ``plug-in estimator'' to which it refers.", "what_survives": "Only the typography changes, and no locator is affected. The abbreviation ``CLT'' occurs only in the index, where the heading ``Central Limit Theorem (CLT)'' on p.~293 expands it, and it needs no change." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N023", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Eight index lines: the heading 'missing data, 208-219' and its six subentries, the last of which runs over two lines. A red box marks '230, 267' in the first subentry, 'Double Robustness Theorem for, 230, 267'.", "caption": "Index, p. 296 (PDF p. 314), right column: the heading “missing data” and its subentries, with the locators “230, 267” of “Double Robustness Theorem for” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N024.png" } ], "formal_release": null, "formal_statements": [], "headline": "Under “missing data”, the subentry “Double Robustness Theorem for” should point to pp. 229–230, where the theorem for missing data appears, not to pp. 230 and 267.", "id": "P001-BACK-N024", "location": "Index, p. 296 (PDF p. 314), right column, heading “missing data”, subentry “Double Robustness Theorem for”", "pdf_page": 314, "printed_page": "296", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\emph{Under ``missing data, 208--219''.} Double Robustness Theorem for, \\hl{230, 267}", "corrected": "\\emph{Under ``missing data, 208--219''.} Double Robustness Theorem for, \\hl{229--230}", "evidence": "I checked the pages. Under the heading ``Double Robustness Theorem'' on p.~294, the index gives ``for causal inference, 267--269'' and ``for missing data, 229--230''.", "problem": "Since the subentry stands under ``missing data'', it refers to the Double Robustness Theorem for Missing Data. That theorem is Theorem 6.2.9, on pp.~229--230, and the printed locators omit p.~229, where it is stated. Page 267 states Theorem 7.2.8, the Double Robustness Theorem for Causal Inference. The passage that introduces Theorem 7.2.8 compares it with the case of missing data, but the theorem on that page is the causal one.", "what_survives": "Only this locator changes, and both theorems are unaffected. I examined five further cases in which an index entry differs from the entry for the same topic elsewhere in the index, or omits the page of a definition. They are ``regression estimator for, 224'' under ``missing data''; ``joint, 36, 40'' under ``probability density function (PDF)''; ``conditional, 42'' under ``probability mass function (PMF)''; ``kernel density, 122'' under ``estimator''; and the two entries for the conditional independence assumption, which omit p.~247. In each case the printed locator points to a page that treats its subject, and none is changed." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N024", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Eleven index lines: the heading 'plug-in estimator, 120', its eight subentries, and the next heading, 'plug-in principle, 116-124'. A red box marks '118' in the last subentry, 'sample mean as, 118, 210, 240'. Three lines above it is the subentry 'plug-in sample variance as, 119'.", "caption": "Index, p. 297 (PDF p. 315), left column: the heading “plug-in estimator” and its subentries, with the locator “118” of “sample mean as” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N026.png" }, { "alt": "Four index lines: the heading 'sample mean' and the subentries 'as plug-in estimator, 118, 210, 240', 'Chebyshev's Inequality for, 98-99' and 'consistency of, 105'. A red box marks '118'.", "caption": "Index, p. 298 (PDF p. 316), left column: the heading “sample mean”, with the locator “118” of “as plug-in estimator” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-BACK-N026-p298.png" } ], "formal_release": null, "formal_statements": [], "headline": "In both index subentries for the sample mean as a plug-in estimator, the first locator should be p. 119, not p. 118.", "id": "P001-BACK-N026", "location": "Index, p. 297 (PDF p. 315), left column, heading “plug-in estimator”, subentry “sample mean as”; repeated on p. 298 (PDF p. 316), left column, heading “sample mean”, subentry “as plug-in estimator”", "pdf_page": 315, "printed_page": "297", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\emph{p.~297, under ``plug-in estimator, 120''.} sample mean as, \\hl{118}, 210, 240\n\n\\emph{p.~298, under ``sample mean''.} as plug-in estimator, \\hl{118}, 210, 240", "corrected": "\\emph{p.~297, under ``plug-in estimator, 120''.} sample mean as, \\hl{119}, 210, 240\n\n\\emph{p.~298, under ``sample mean''.} as plug-in estimator, \\hl{119}, 210, 240", "evidence": "I checked the pages. The subentry for the companion example on the same page, ``plug-in sample variance as, 119'', gives the right page. Pages 210 and 240 each use the sample mean as a plug-in estimator.", "problem": "Example 3.3.4, ``Sample Mean as a Plug-In Estimator'', is on p.~119. Page 118 defines a plug-in estimator in general and announces two examples, but it does not identify the sample mean as one.", "what_survives": "The locators 210 and 240 are right in both subentries. Example 3.3.4 and its later uses are unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-BACK-N026", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Footnote 6 in full, nine lines of small type. A red box marks the words 'is impossible to define the probability of every subset of Omega' on the fifth line.", "caption": "Footnote 6, p. 6 (PDF p. 24): the clause on defining the probability of every subset of $\\Omega$ is marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C01-E009.png" } ], "formal_release": null, "formal_statements": [], "headline": "Footnote 6 should say that a specified probability law need not extend to every subset of the sample space, not that no probability can be defined on every subset.", "id": "P001-C01-E009", "location": "Footnote 6, p. 6 (PDF p. 24)", "pdf_page": 24, "printed_page": "6", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "Some readers well versed in set theory may wonder why we do not simply let $S=\\mathcal{P}(\\Omega)$, the power set (that is, the set of all subsets) of $\\Omega$. For reasons that we will not discuss in this book, this does not always work; for some sample spaces $\\Omega$, it is \\hl{impossible to define the probability of every subset} of $\\Omega$ in a manner consistent with the axioms of probability (see Definition 1.1.2).", "corrected": "For reasons that we will not discuss in this book, this does not always work; \\hl{a specified probability law need not extend to all subsets of the sample space while preserving its required properties}.", "intended_meaning": "The footnote explains why the event space is not always taken to be the power set. Read literally, the marked clause is too strong. For every nonempty $\\Omega$ and any fixed $\\omega_0\\in\\Omega$, the point mass $P(A)=\\mathbf{1}\\{\\omega_0\\in A\\}$ is defined for every $A\\subseteq\\Omega$ and satisfies the three axioms of Definition 1.1.2, since at most one set in a pairwise disjoint sequence contains $\\omega_0$. The intended statement is that a law with further required properties, such as the uniform law on $[0,1]$, need not extend to all subsets while keeping those properties. That statement is correct, and nothing in the book depends on the footnote." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C01-E009", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Five lines of text. A red box, spanning the end of the second line and the start of the third, marks the words 'the former always occurs whenever the latter does'.", "caption": "First paragraph of p. 9 (PDF p. 27): the parenthesis that explains monotonicity, with the reversed clause marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C01-E007.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T23", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T23" } ], "headline": "In the explanation of monotonicity, “former” and “latter” are interchanged: if one event is a subset of another, the latter occurs whenever the former does.", "id": "P001-C01-E007", "location": "First paragraph of p. 9 (PDF p. 27), the explanation of monotonicity that follows Theorem 1.1.4", "pdf_page": 27, "printed_page": "9", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T23" ], "text": { "as_printed": "Monotonicity implies that, if one event is a subset of another (so that \\hl{the former always occurs whenever the latter does}), then the probability of the former occurring is no greater than that of the latter.", "corrected": "Monotonicity implies that, if one event is a subset of another (so that \\hl{the latter always occurs whenever the former does}), then the probability of the former occurring is no greater than that of the latter.", "evidence": "An exact counterexample to the parenthesis: for a fair die with $A=\\{6\\}$ and $B=\\{4,5,6\\}$, the outcome $4$ lies in $B$ but not in $A$, although $A\\subseteq B$. Here $P(A)=1/6\\le 1/2=P(B)$, as the theorem requires.", "problem": "If $A\\subseteq B$, then $B$ occurs whenever $A$ does, and $B$ can occur without $A$. The parenthesis states the converse, which describes $B\\subseteq A$.", "what_survives": "Theorem 1.1.4 and its proof state the inclusion and the inequality correctly; only the parenthesis changes." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C01-E007", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "The last three lines of a proof, a displayed sum, and five lines of the next paragraph. A small red box marks the greater-than-or-equal sign in 'if P(A_i) >= 0 for i = 1, 2, 3, ...'. A second red box marks the words '{A, A^C} is a partition of Omega'.", "caption": "End of the proof of Theorem 1.1.13 and the next paragraph, p. 13 (PDF p. 31): the sign $\\ge$ in the condition on $P(A_i)$ and the remark on $\\{A,A^C\\}$ are marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C01-E008.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T24", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T24" } ], "headline": "The proof of Theorem 1.1.13 should require $P(A_i)>0$, not $P(A_i)\\ge 0$, before conditioning on $A_i$, and $\\{A,A^C\\}$ is a partition of $\\Omega$ only when $A$ is neither empty nor all of $\\Omega$.", "id": "P001-C01-E008", "location": "Proof of Theorem 1.1.13 and the paragraph that follows it, p. 13 (PDF p. 31)", "pdf_page": 31, "printed_page": "13", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T24" ], "text": { "as_printed": "Additionally, if $P(A_i)\\hl{\\ge}0$ for $i=1,2,3,\\ldots$, then we can apply the Multiplicative Law of Probability to each term in the summation to obtain\n\\[\nP(B)=\\sum_i P(B\\mid A_i)P(A_i).\n\\]\n\\ldots\\ Notice that \\hl{for any event} $A$, $\\{A,A^C\\}$ is a partition of $\\Omega$, so the method we used to calculate $P(B)$ above was indeed a special case of the Law of Total Probability.", "corrected": "Additionally, if $P(A_i)\\hl{>}0$ for $i=1,2,3,\\ldots$, then we can apply the Multiplicative Law of Probability to each term in the summation to obtain\n\\[\nP(B)=\\sum_i P(B\\mid A_i)P(A_i).\n\\]\n\\ldots\\ Notice that for any event $A$, \\hl{the nonempty members of} $\\{A,A^C\\}$ \\hl{form} a partition of $\\Omega$, so the method we used to calculate $P(B)$ above was indeed a special case of the Law of Total Probability.\n\nThe step in full: if $\\{A_1,A_2,A_3,\\ldots\\}$ is a partition of $\\Omega$, $B\\in S$, and $P(A_i)>0$ for $i=1,2,3,\\ldots$, then $P(B\\cap A_i)=P(B\\mid A_i)P(A_i)$ for each $i$, and\n\\[\nP(B)=\\sum_i P(B\\cap A_i)=\\sum_i P(B\\mid A_i)P(A_i).\n\\]", "evidence": "An exact counterexample to the printed condition. Let $\\Omega=\\{a,b\\}$ with $P(\\{a\\})=1$ and $P(\\{b\\})=0$, and set $A_1=\\{a\\}$, $A_2=\\{b\\}$ and $B=\\Omega$. Then $\\{A_1,A_2\\}$ is a partition of $\\Omega$ and both members have nonnegative probability. The first form of the theorem holds:\n\\[\nP(B)=P(B\\cap A_1)+P(B\\cap A_2)=1+0=1.\n\\]\nThe second form cannot be written, because\n\\[\nP(B\\mid A_2)=\\frac{P(B\\cap A_2)}{P(A_2)}=\\frac{0}{0}\n\\]\nis undefined. For the remark, $A=\\varnothing$ and $A=\\Omega$ both give $\\{A,A^C\\}=\\{\\varnothing,\\Omega\\}$, which contains the empty set.", "problem": "By Definition 1.1.8, the conditional probability $P(B\\mid A_i)$ is defined only when $P(A_i)>0$, and the statement of Theorem 1.1.13 on p.~12 carries that condition. The proof restates the condition as $P(A_i)\\ge 0$. This holds for every event, and it would permit conditioning on an event of probability zero. Separately, Definition 1.1.12 requires the members of a partition to be nonempty, and $\\{A,A^C\\}$ has an empty member when $A=\\varnothing$ or $A=\\Omega$.", "what_survives": "Theorem 1.1.13 as stated on p.~12, the proof of its first form, and the calculation for the coin flip and die roll are unaffected. With $P(A_i)>0$, the proof of the second form holds as printed. Dropping an empty member repairs the remark, but it does not permit conditioning on a nonempty member of probability zero." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C01-E008", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "A paragraph of five lines followed by three bulleted formulas. A red box marks the lower limit 'x=1' under the summation sign in the third bullet, 'Pr[X < 4] = sum from x=1 to 3 of f(x)'.", "caption": "First bulleted list on p. 20 (PDF p. 38): the lower limit $x=1$ of the sum in the third bullet is marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C01-E006.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T25", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T25" }, { "id": "P001-T26", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T26" } ], "headline": "For an integer-valued random variable $X$, the sum for $\\Pr[X<4]$ must start at $-\\infty$, not at $1$.", "id": "P001-C01-E006", "location": "First bulleted list on p. 20 (PDF p. 38), third bullet", "pdf_page": 38, "printed_page": "20", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T25", "P001-T26" ], "text": { "as_printed": "To illustrate how the PMF can fully describe a discrete random variable, consider a random variable $X$ such that $f(x)=0$ for all $x\\notin\\mathbb{Z}$ (that is, $X$ takes on only integer values). Then\n\\begin{itemize}\n\\item $\\Pr[X\\ge 3]=\\sum_{x=3}^{\\infty}f(x)$.\n\\item $\\Pr[X\\ge 3\\text{ or }X=1]=f(1)+\\sum_{x=3}^{\\infty}f(x)$.\n\\item $\\Pr[X<4]=\\sum_{x=\\hl{1}}^{3}f(x)$.\n\\end{itemize}", "corrected": "Let $X$ be a discrete random variable with PMF $f$ such that $f(x)=0$ for all $x\\notin\\mathbb{Z}$. Then\n\\[\n\\Pr[X<4]=\\sum_{x=\\hl{-\\infty}}^{3}f(x).\n\\]\nThe sum from $1$ is valid exactly when $\\Pr[X\\le 0]=0$.", "evidence": "Exact counterexamples and a derivation. If $X=0$ with probability one, then $f(0)=1$ and $\\Pr[X<4]=1$, while the printed sum is $f(1)+f(2)+f(3)=0$. For the Bernoulli random variable of Example 1.2.7 with $p=1/3$, I have $f(0)=2/3$ and $f(1)=1/3$, so that $\\Pr[X<4]=1$ and the printed sum is $1/3$.\n\n\\begin{audittable}{lccc}{The third bullet as printed and as corrected, in four exact examples.}\nDistribution of $X$ & $\\Pr[X<4]$ & $\\sum_{x=1}^{3}f(x)$ (printed) & $\\sum_{x=-\\infty}^{3}f(x)$ (corrected) \\\\\n$X=0$ with probability one & $1$ & $0$ & $1$ \\\\\n$X=-2$ with probability one & $1$ & $0$ & $1$ \\\\\nBernoulli, $p=1/3$ (Example 1.2.7) & $1$ & $1/3$ & $1$ \\\\\nFair die (Example 1.2.6) & $1/2$ & $1/2$ & $1/2$ \\\\\n\\end{audittable}\n\nFor the corrected identity, write $A_n=\\{X=n\\}$ for each integer $n$. These events are pairwise disjoint, and since $X$ is integer-valued,\n\\[\n\\{X<4\\}=\\bigcup_{n\\le 3}A_n .\n\\]\nCountable additivity gives $\\Pr[X<4]=\\sum_{n=-\\infty}^{3}f(n)$. The sum is the limit of $\\sum_{n=-m}^{3}f(n)$ as $m\\to\\infty$, which exists because the terms are nonnegative and the partial sums are at most $1$.", "problem": "The passage restricts $X$ only to integer values, so $X$ may put mass at $0$ and at the negative integers, as the Bernoulli random variable of Example 1.2.7 does at $0$. The printed sum omits that mass. By countable additivity, I have\n\\[\n\\Pr[X<4]-\\sum_{x=1}^{3}f(x)=\\Pr[X\\le 0],\n\\]\nand the third bullet holds if and only if $\\Pr[X\\le 0]=0$.", "what_survives": "The first two bullets, the three probabilities for the die roll in the list that follows ($2/3$, $5/6$ and $1/2$), Definitions 1.2.4 and 1.2.5, and Theorems 1.2.9 and 1.2.10 are unaffected. For the die roll $\\Pr[X\\le 0]=0$, and the printed sum from $1$ is correct there. I found no later use of the third bullet in the book." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C01-E006", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "The heading 'Example 1.2.13. CDF of a Fair Die Roll' and five lines of text. Red boxes mark the word 'necessarily' in 'the CDF is necessarily a step function' and, on the next line, the words 'flat everywhere except where it jumps'.", "caption": "Example 1.2.13, p. 23 (PDF p. 41): the first sentence, with “necessarily” and the description of a step function marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C01-E005.png" }, { "alt": "Four lines of text. A red box, spanning the end of the third line and the start of the fourth, marks the words 'like the CDF of a discrete random variable'.", "caption": "Paragraph after Definition 3.3.1, p. 116 (PDF p. 134): the comparison of the empirical CDF with the CDF of a discrete random variable is marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C01-E005-p116.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T27", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T27" }, { "id": "P001-T28", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T28" } ], "headline": "The CDF of a discrete random variable is the sum of its jumps; it is flat between jumps when the range is finite, as for a die roll, but not for every discrete random variable.", "id": "P001-C01-E005", "location": "Example 1.2.13, first sentence, p. 23 (PDF p. 41); the comparison is repeated after Definition 3.3.1, p. 116 (PDF p. 134)", "pdf_page": 41, "printed_page": "23", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T27", "P001-T28" ], "text": { "as_printed": "\\emph{p.~23.} For discrete random variables such as a die roll, the CDF is \\hl{necessarily} a step function, that is, a function that is \\hl{flat everywhere except where it} ``jumps'' from one value to another.\n\n\\emph{p.~116.} Note that the empirical CDF is always a nondecreasing step function, \\hl{like the CDF of a discrete random variable}.", "corrected": "\\emph{p.~23.} For \\hl{a discrete random variable, the CDF is the sum of the jumps contributed by its point masses. For finite range, as in a die roll, it is constant between the finitely many possible values.}\n\n\\emph{p.~116.} Note that the empirical CDF is always a nondecreasing step function, like the CDF of a discrete random variable \\hl{with finite range}.\n\nIn symbols: let $X$ be a discrete random variable with distinct possible values $q_i$ and masses $p_i=\\Pr[X=q_i]$, so that $\\sum_i p_i=1$. Then its CDF is\n\\[\nF(x)=\\sum_i p_i\\,\\mathbf{1}\\{q_i\\le x\\}\\quad\\text{for all }x\\in\\mathbb{R},\n\\]\nand the jump of $F$ at $q_i$ is $p_i$. If the range of $X$ is finite, then $F$ is constant between consecutive possible values.", "evidence": "An exact counterexample. Let $\\Omega$ be the set of pairs of integers $(m,k)$ such that $m\\ge 0$ and $0\\le k<2^m$, and define\n\\[\nP(\\{(m,k)\\})=2^{-(2m+1)},\\qquad X(m,k)=\\frac{2k+1}{2^{m+1}} .\n\\]\nThe masses are positive, and their sum is $\\sum_{m\\ge 0}2^m\\,2^{-(2m+1)}=\\sum_{m\\ge 0}2^{-(m+1)}=1$. The range of $X$ is the set of dyadic rationals in $(0,1)$, which is countable, so $X$ is discrete under Definition 1.2.4. Let $F$ denote the CDF of $X$, and fix real numbers $0\\le a0 .\n\\]\nThis implies that $F$ is strictly increasing on $(0,1)$ and constant on no subinterval. The point $1/3$ is not a value of $X$, and $F$ is continuous there: given $M$, every $y$ close enough to $1/3$ lies on the same side of each value $(2k+1)/2^{m+1}$ with $m\\le M$ as $1/3$ does, so $F(y)$ and $F(1/3)$ differ by at most the mass $2^{-(M+1)}$ of the remaining values. Evaluating the infinite sums exactly, I find $F(1/4)=3/16$, $F(1/3)=1/5$, $F(1/2)=3/4$, $F(2/3)=4/5$ and $F(3/4)=15/16$, with a jump of $1/2$ at $1/2$.", "problem": "Definition 1.2.4 calls $X$ discrete when its range is countable, and a countable range can be dense in an interval. The CDF of such a random variable still increases only by jumps, but it need not be constant on any interval. The description ``flat everywhere except where it jumps'', and the flat segments drawn in Figure 1.2.1, are correct when the range is finite, as for the die roll. They do not describe every discrete CDF.", "what_survives": "Figure 1.2.1 and the probabilities for the die roll, Definition 1.2.4, Theorems 1.2.10 and 1.2.12, and the empirical CDF of Definition 3.3.1 are unaffected; the empirical CDF has at most $n$ jumps and is a step function. I found no later proof or calculation that relies on the CDF of every discrete random variable being constant between its jumps. The correction weakens the general description: the CDF is said to be constant between jumps only when the range is finite." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C01-E005", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop showing Definition 1.2.14, which calls X continuous if there exists a non-negative function f such that F(x) equals the integral of f from minus infinity to x for all real x, followed by the sentence 'The function f is called the probability density function (PDF). The Fundamental Theorem of Calculus implies that, for any continuous random variable, the PDF is unique and can be defined as follows:'. A red box marks the words 'the PDF is unique and can be defined as follows:'.", "caption": "Definition 1.2.14 and the sentence after it, p. 24 (PDF p. 42). The marked words say that the PDF is unique and can be defined by the formula of Definition 1.2.15.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C01-E002.png" }, { "alt": "Crop of Definition 1.2.15: for a continuous random variable X with CDF F, the probability density function of X is f(x) = dF(u)/du evaluated at u = x, for all x in R. A red box marks the quantifier 'for all x in R'.", "caption": "Definition 1.2.15, p. 25 (PDF p. 43). The marked quantifier requires the derivative of the CDF to exist, and to equal the PDF, at every real $x$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C01-E002-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T29", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T29" }, { "id": "P001-T30", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T30" } ], "headline": "The PDF of a continuous random variable is unique only up to equality almost everywhere, and it equals the derivative of the CDF at almost every point, not at every point.", "id": "P001-C01-E002", "location": "Section 1.2.4, pp. 24–25 (PDF pp. 42–43): the sentence after Definition 1.2.14, and Definition 1.2.15. The same qualification applies to Definition 1.3.11 (p. 36), to Definition 1.4.5 (p. 41), and to Definition 1.3.16 and Theorem 1.3.17 (pp. 38–39).", "pdf_page": 42, "printed_page": "24–25", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T29", "P001-T30" ], "text": { "as_printed": "The function $f$ is called the \\emph{probability density function (PDF)}. The Fundamental Theorem of Calculus implies that, for any continuous random variable, the PDF is \\hl{unique} and can be defined as follows:\n\n\\textbf{Definition 1.2.15.} \\emph{Probability Density Function (PDF)}\n\nFor a continuous random variable $X$ with CDF $F$, the \\emph{probability density function} of $X$ is\n\\[\nf(x)=\\left.\\frac{dF(u)}{du}\\right|_{u=x},\\ \\hl{\\forall x\\in\\mathbb{R}}.\n\\]", "corrected": "The function $f$ is called the \\emph{probability density function (PDF)}. The Fundamental Theorem of Calculus\\hl{, in Lebesgue's form,} implies that, for any continuous random variable, the PDF is unique \\hl{up to equality almost everywhere} and \\hl{is related to the CDF} as follows:\n\n\\textbf{Definition 1.2.15.} \\emph{Probability Density Function (PDF)}\n\nFor a continuous random variable $X$ with CDF $F$, \\hl{a} \\emph{probability density function} of $X$ is \\hl{a non-negative function $f$ that satisfies Definition 1.2.14. Any two such functions are equal at Lebesgue-almost every point, and}\n\\[\nf(x)=\\left.\\frac{dF(u)}{du}\\right|_{u=x}\\quad\\hl{\\text{for Lebesgue-almost every }x\\in\\mathbb{R}}.\n\\]\n\\hl{At the remaining points the derivative may fail to exist, and $f$ may take any non-negative values there.}\n\nThe dependent statements change in the same way.\n\\begin{itemize}\n\\item Definition 1.3.11 (p.~36) and Definition 1.4.5 (p.~41): the joint PDF is the non-negative function $f$ of Definition 1.3.10 (respectively Definition 1.4.4), unique up to equality almost everywhere. It is the mixed derivative of the joint CDF $F$ in the weak sense, that is, $\\int F\\,\\partial_1\\cdots\\partial_K\\phi\\,dx=(-1)^K\\int f\\,\\phi\\,dx$ for every smooth function $\\phi$ with compact support. The formula stated at every point requires additional regularity of $f$.\n\\item Definition 1.3.16 and Theorem 1.3.17 (pp.~38--39), continuous case: $X$ and $Y$ are independent if $f(x,y)=f_X(x)f_Y(y)$ for Lebesgue-almost every $(x,y)$, or equivalently if some version of the joint PDF is the product of the marginal PDFs. The equality $f_{X|Y}(x|y)=f_X(x)$ holds for Lebesgue-almost every $x$, for almost every $y$ with respect to the distribution of $Y$. The statements for PMFs hold at every point, as printed.\n\\item Definition 1.2.20 (p.~31): the set $\\operatorname{Supp}[X]=\\{x\\in\\mathbb{R}:f(x)>0\\}$ refers to the specified version of the PDF.\n\\end{itemize}", "evidence": "The evidence is an exact counterexample taken from the book, followed by the derivation of the corrected statement. Let $X\\sim U(0,1)$, so that $F(x)=0$ for $x<0$, $F(x)=x$ for $0\\le x\\le 1$ and $F(x)=1$ for $x>1$. For $0 0}', with a red box around the formula. The paragraph below says that for continuous X, Supp[X] is the set of values over which X has nonzero probability density, with footnote marker 26, and gives the example Supp[X] = [0,1] for X uniform on (0,1).", "caption": "Definition 1.2.20 and the paragraph after it, p. 31 (PDF p. 49). The marked formula is the definition of support used in the book, the set where the PMF/PDF is positive; the footnote marker 26 follows the words “nonzero probability density”. The definition is correct.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C01-E003-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "Footnote 26 describes support by neighbourhoods, which can include points where the PDF is zero; the support used in the book is the set where the PMF/PDF is positive, and the footnote should say that the two notions can differ.", "id": "P001-C01-E003", "location": "Definition 1.2.20 and footnote 26, p. 31 (PDF p. 49)", "pdf_page": 49, "printed_page": "31", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "\\textbf{Definition 1.2.20.} \\emph{Support}\n\nFor a random variable $X$ with PMF/PDF $f$, the \\emph{support} of $X$ is\n\\[\n\\operatorname{Supp}[X]=\\bigl\\{x\\in\\mathbb{R}:f(x)>0\\bigr\\}.\n\\]\nFootnote 26: \\hl{As Manski (1999, p.~15) intuitively describes it,} ``A value $x_0$ is said to be on the \\emph{support} of the probability distribution of $x$ if there is a positive probability of observing $x$ arbitrarily close to $x_0$.''", "corrected": "Footnote 26: \\hl{A related but distinct notion of support is described by Manski (1999, p.~15):} ``A value $x_0$ is said to be on the \\emph{support} of the probability distribution of $x$ if there is a positive probability of observing $x$ arbitrarily close to $x_0$.'' \\hl{This notion can include zero-density boundary or limit points that are excluded by the positivity-set convention used here.}", "intended_meaning": "Definition 1.2.20 governs: the support is the set where the specified PMF or PDF is positive, and Definitions 1.3.14 and 1.4.7 define conditional PDFs for conditioning values in this set, which keeps their denominators nonzero. The quotation in footnote 26 describes a different notion, the set of points whose every neighbourhood has positive probability, and the two sets can differ at points where the PDF is zero. For the continuous PDF $f(x)=6x(1-x)$ on $[0,1]$, zero elsewhere, the support of Definition 1.2.20 is $(0,1)$. Since $\\Pr[|X|<\\epsilon]=3\\epsilon^2-2\\epsilon^3>0$ for $0<\\epsilon<1$ and, by symmetry, $\\Pr[|X-1|<\\epsilon]>0$, the set described in the footnote is $[0,1]$. No formula in the book uses the footnote, and I did not check the quotation against Manski (1999)." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C01-E003", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the sentence 'From this definition, it follows that two functions of a random variable, g(X) and h(X), are equal as random variables if and only if they are equal as functions on X(Omega)', followed by Theorem 1.3.2: 'Let X be a random variable, and let f and g be functions of X. Then g(X) = h(X) if and only if, for all x in X(Omega), g(x) = h(x).' A red box marks the words 'f and g'.", "caption": "Theorem 1.3.2 and the sentence before it, p. 32 (PDF p. 50). The marked words introduce the functions as $f$ and $g$; the sentence above the theorem and the display use $g$ and $h$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C01-E010.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T33", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T33" } ], "headline": "Theorem 1.3.2 introduces its functions as $f$ and $g$ but states the equivalence for $g$ and $h$; the functions should be introduced as $g$ and $h$.", "id": "P001-C01-E010", "location": "Theorem 1.3.2, p. 32 (PDF p. 50), first line of the statement", "pdf_page": 50, "printed_page": "32", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T33" ], "text": { "as_printed": "\\textbf{Theorem 1.3.2.} \\emph{Equality of Functions of a Random Variable}\n\nLet $X$ be a random variable, and let $\\hl{f}$ and $\\hl{g}$ be functions of $X$. Then\n\\[\ng(X)=h(X)\\iff\\forall x\\in X(\\Omega),\\ g(x)=h(x).\n\\]", "corrected": "Let $X$ be a random variable, and let $\\hl{g}$ and $\\hl{h}$ be functions of $X$. Then\n\\[\ng(X)=h(X)\\iff\\forall x\\in X(\\Omega),\\ g(x)=h(x).\n\\]", "evidence": "The evidence is a check of the page: the preceding paragraph, the theorem and its whole proof.", "problem": "The statement never introduces $h$ and never uses $f$. The sentence before the theorem, the displayed equivalence and both directions of the proof are written for $g$ and $h$.", "what_survives": "The theorem and its proof are correct once the names agree. Nothing else changes." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C01-E010", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of two sentences and a display. The second sentence reads 'So, for example, the probability that (X,Y) will fall within the triangular region {0 <= X <= 1, Y <= X} is', and the display reads 'Pr[0 <= X <= 1, Y <= X] = integral from 0 to 1 of the integral from 0 to x of f(x,y) dy dx'. Red boxes mark 'Y <= X' in the sentence and in the display.", "caption": "The illustration after Theorem 1.3.12, p. 37 (PDF p. 55). The two marked conditions $Y\\le X$ put no lower bound on $Y$, while the inner integral starts at $0$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C01-E001.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T34", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T34" }, { "id": "P001-T35", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T35" } ], "headline": "The event in the illustration after Theorem 1.3.12 omits the bound $0\\le Y$; the triangular region is $\\{0\\le X\\le 1,\\ 0\\le Y\\le X\\}$.", "id": "P001-C01-E001", "location": "p. 37 (PDF p. 55), the illustration after Theorem 1.3.12: the sentence beginning “So, for example,” and the display that follows it", "pdf_page": 55, "printed_page": "37", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T34", "P001-T35" ], "text": { "as_printed": "So, for example, the probability that $(X,Y)$ will fall within the triangular region $\\{0\\le X\\le 1,\\hl{Y\\le X}\\}$ is\n\\[\n\\Pr[0\\le X\\le 1,\\hl{Y\\le X}]=\\int_0^1\\int_0^x f(x,y)\\,dy\\,dx.\n\\]", "corrected": "For jointly continuous random variables $X$ and $Y$ with joint PDF $f$, as in Theorem 1.3.12:\n\nSo, for example, the probability that $(X,Y)$ will fall within the triangular region $\\{0\\le X\\le 1,\\hl{0\\le Y}\\le X\\}$ is\n\\[\n\\Pr[0\\le X\\le 1,\\hl{0\\le Y}\\le X]=\\int_0^1\\int_0^x f(x,y)\\,dy\\,dx.\n\\]\nAlternatively, the event can be kept as printed. Then ``triangular region'' becomes ``region'', since the set is unbounded, and the display becomes $\\Pr[0\\le X\\le 1,\\ Y\\le X]=\\int_0^1\\int_{-\\infty}^{x}f(x,y)\\,dy\\,dx$.", "evidence": "The evidence is a derivation and an exact counterexample. Let $A=\\{0\\le X\\le 1,\\ Y\\le X\\}$, $T=\\{0\\le X\\le 1,\\ 0\\le Y\\le X\\}$ and $N=\\{0\\le X\\le 1,\\ Y<0\\}$. The events $T$ and $N$ are disjoint, and their union is $A$. By Theorem 1.3.12, I obtain\n\\[\n\\Pr[A]=\\int_0^1\\int_{-\\infty}^{x}f(x,y)\\,dy\\,dx=\\int_0^1\\int_0^x f(x,y)\\,dy\\,dx+\\Pr[0\\le X\\le 1,\\ Y<0].\n\\]\nThe printed equality therefore holds if and only if $\\Pr[0\\le X\\le 1,\\ Y<0]=0$. For the counterexample, let $X$ and $Y$ be independent, with PDFs $6x(1-x)$ on $(0,1)$ and $-6y(1+y)$ on $(-1,0)$, each zero elsewhere. Then $Y<0n\\}\\le 1$ for every $n\\ge 1$, where $\\lambda$ denotes Lebesgue measure, and this implies that $\\lambda\\{y:g(y)=\\infty\\}=0$. Let $\\tilde g$ equal $g$ where $g$ is finite and take any finite non-negative values elsewhere. Then $\\Pr[Y\\in B]=\\int_B\\tilde g(y)\\,dy$ for every Borel set $B$, and $\\tilde g$ is a PDF of $Y$.", "problem": "Definition 1.2.14 makes a PDF a real-valued function, and Theorem 1.3.13 asserts that the integral of the joint PDF over $x$ is the value of the marginal PDF at every $y$. The hypotheses do not make this integral finite at every $y$. It is finite for almost every $y$, but it can be infinite at exceptional points even when the joint PDF is smooth, bounded and strictly positive. At such a point no real value of $f_Y(y)$ satisfies the printed equality. This is a qualification of the same kind as in \\fid{P001-C01-E002}.\n\nTwo later statements carry the same quantifier. The remark after Definition 1.3.14 (p.~38) says that conditional PDFs are univariate PDFs ``given any fixed value for the conditioning variable'', but the ratio is a PDF only where the integral in its denominator is positive and finite. The continuous case of Definition 1.4.6 (p.~42) states the formula with several variables integrated out, again for all $y$.", "what_survives": "Event probabilities and the marginal distributions are unchanged. Expectations and the Law of Iterated Expectations (Theorem 2.2.17) use these identities inside integrals, where equality almost everywhere suffices. What changes is the wording at fixed points. The conditional PDF of Definition 1.3.14 and the remark after it, Theorem 1.3.15, the continuous cases of Definitions 1.4.6 and 1.4.7, and the conditional expectations defined from them at a fixed conditioning value (Definitions 2.2.10, 2.2.15, 2.3.3 and 2.3.4) hold outside an exceptional set of conditioning values that has probability zero. No numerical result is shown to change." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C01-N007", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of Theorem 1.3.17, Implications of Independence (Part I), with its five bulleted statements. A red box marks the last two: 'for all D, E subsets of R, the events {X in D} and {Y in E} are independent' and 'For all functions g of X and h of Y, g(X) is independent of h(Y).'", "caption": "Theorem 1.3.17, p. 39 (PDF p. 57). The two marked statements quantify over all subsets $D,E\\subseteq\\mathbb{R}$ and over all functions $g$ of $X$ and $h$ of $Y$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C01-N008.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T47", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T47" } ], "headline": "The last two statements of Theorem 1.3.17 are meant for Borel sets $D$ and $E$ and for Borel measurable functions $g$ and $h$, with the independence of $g(X)$ and $h(Y)$ understood in the general sense defined by events.", "id": "P001-C01-N008", "location": "Theorem 1.3.17, p. 39 (PDF p. 57), fourth and fifth statements, read with Definition 1.3.16, p. 38 (PDF p. 56)", "pdf_page": 57, "printed_page": "39", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [ "P001-T47" ], "text": { "as_printed": "\\textbf{Theorem 1.3.17.} \\emph{Implications of Independence (Part I)}\n\nLet $X$ and $Y$ be either two discrete random variables with joint PMF $f$ or two jointly continuous random variables with joint PDF $f$. Then the following statements are equivalent (that is, each one implies all the others):\n\\begin{itemize}\n\\item $X\\perp\\!\\!\\!\\perp Y$.\n\\item $\\forall x,y\\in\\mathbb{R}$, $f(x,y)=f_X(x)f_Y(y)$.\n\\item $\\forall x\\in\\mathbb{R}$ and $\\forall y\\in\\operatorname{Supp}[Y]$, $f_{X|Y}(x|y)=f_X(x)$.\n\\item $\\hl{\\forall D,E\\subseteq\\mathbb{R}}$, the events $\\{X\\in D\\}$ and $\\{Y\\in E\\}$ are independent.\n\\item \\hl{For all functions} $g$ of $X$ and $h$ of $Y$, $g(X)\\perp\\!\\!\\!\\perp h(Y)$.\n\\end{itemize}", "corrected": "\\begin{itemize}\n\\item $\\forall D,E\\subseteq\\mathbb{R}$, the events $\\{X\\in D\\}$ and $\\{Y\\in E\\}$ are independent.\n\\item For all functions $g$ of $X$ and $h$ of $Y$, $g(X)\\perp\\!\\!\\!\\perp h(Y)$.\n\\end{itemize}\n\\hl{Footnote to these two statements: Sets and functions in these statements are understood to be Borel measurable. For arbitrary real-valued random variables $U$ and $V$, $U$ is independent of $V$ if $\\Pr[U\\in A,V\\in B]=\\Pr[U\\in A]\\Pr[V\\in B]$ for every pair of Borel sets $A,B\\subseteq\\mathbb{R}$.}", "intended_meaning": "The two marked statements are meant for Borel sets $D$ and $E$ and for Borel measurable functions $g$ and $h$, with $g(X)\\perp\\!\\!\\!\\perp h(Y)$ understood in the general sense that $\\Pr[g(X)\\in A,h(Y)\\in B]=\\Pr[g(X)\\in A]\\Pr[h(Y)\\in B]$ for all Borel sets $A$ and $B$. The qualifiers matter for two reasons: $\\{X\\in D\\}$ need not be an event when $D$ is not measurable, and $g(X)$ and $h(Y)$ can form a pair that is neither two discrete nor two jointly continuous random variables, for which Definition 1.3.16 does not define independence (for independent uniform $X$ and $Y$, let $g(x)=\\mathbf{1}\\{x>1/2\\}$ and $h(y)=y$). The book points to this reading: footnote 15 (p.~16) restricts a ``function of $X$'' to one for which $g(X)$ is a random variable, footnote 21 (p.~24) says that its definitions and theorems can almost always be generalized to all probability distributions, and the proofs of Theorem 3.2.9 (p.~102) and Theorem 3.2.19 (p.~106) cite ``a generalization of Theorem 1.3.17''. Under this reading the statements are correct, since for Borel sets $A$ and $B$ the event $\\{g(X)\\in A,h(Y)\\in B\\}$ equals $\\{X\\in g^{-1}(A),Y\\in h^{-1}(B)\\}$, the sets $g^{-1}(A)$ and $h^{-1}(B)$ are Borel, and by the fourth statement the probability of this event is the product of the two probabilities. The indicator and the square used in the two later proofs are Borel measurable, and the discrete case holds as printed.\n\nThe general measurable-transformation claim is supported by the preimage derivation above. The related formal statements in P001-T47 additionally assume finite second moments of the original and transformed variables." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C01-N008", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of Definition 1.4.7, Conditional PMF/PDF (Multivariate Case), with the discrete formula and the continuous formula for the conditional PMF/PDF of Y given the vector X = x. Red boxes mark 'Supp[X]' at the end of each formula, where X is printed in italic and not in bold.", "caption": "Definition 1.4.7, p. 42 (PDF p. 60). In both marked domain conditions the $X$ inside $\\operatorname{Supp}[\\,\\cdot\\,]$ is italic, while the conditioning vector $\\mathbf{X}$ is bold everywhere else in the definition.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C01-N009.png" } ], "formal_release": null, "formal_statements": [], "headline": "In Definition 1.4.7 the support is printed as $\\operatorname{Supp}[X]$ with an italic $X$; it should be $\\operatorname{Supp}[\\mathbf{X}]$, the support of the random vector.", "id": "P001-C01-N009", "location": "Definition 1.4.7, p. 42 (PDF p. 60), the two domain conditions", "pdf_page": 60, "printed_page": "42", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "For discrete random variables $X_{[1]}$, $X_{[2]}$, \\ldots, $X_{[K]}$, and $Y$ with joint PMF $f$, the \\emph{conditional PMF} of $Y$ given $\\mathbf{X}=\\mathbf{x}$ is\n\\[\nf_{Y|\\mathbf{X}}(y|\\mathbf{x})=\\Pr[Y=y|\\mathbf{X}=\\mathbf{x}]=\\frac{\\Pr[Y=y,\\mathbf{X}=\\mathbf{x}]}{\\Pr[\\mathbf{X}=\\mathbf{x}]}=\\frac{f(\\mathbf{x},y)}{f_{\\mathbf{X}}(\\mathbf{x})},\n\\]\n$\\forall y\\in\\mathbb{R}$ and $\\forall\\mathbf{x}\\in\\operatorname{Supp}[\\hl{X}]$.\n\nFor jointly continuous random variables $X_{[1]}$, $X_{[2]}$, \\ldots, $X_{[K]}$, and $Y$ with joint PDF $f$, the \\emph{conditional PDF} of $Y$ given $\\mathbf{X}=\\mathbf{x}$ is\n\\[\nf_{Y|\\mathbf{X}}(y|\\mathbf{x})=\\frac{f(\\mathbf{x},y)}{f_{\\mathbf{X}}(\\mathbf{x})},\\ \\forall y\\in\\mathbb{R}\\text{ and }\\forall\\mathbf{x}\\in\\operatorname{Supp}[\\hl{X}].\n\\]", "corrected": "In both displays, replace $\\operatorname{Supp}[X]$ with $\\operatorname{Supp}[\\hl{\\mathbf{X}}]$. An optional gloss is $\\operatorname{Supp}[\\mathbf{X}]=\\{\\mathbf{x}\\in\\mathbb{R}^K:f_{\\mathbf{X}}(\\mathbf{x})>0\\}$. \\hl{For the continuous formula, use the slice integral as the denominator where it is positive and finite, and choose one fixed PDF at the remaining conditioning values, which have probability zero.}", "evidence": "The evidence is a check of the page: in both displays the $\\mathbf{x}$ is bold and the $X$ inside the brackets is italic, while p.~40 reserves boldface letters for vectors.", "problem": "The conditioning object is the random vector $\\mathbf{X}=(X_{[1]},X_{[2]},\\ldots,X_{[K]})$, which is bold everywhere else in the definition, and no scalar $X$ appears in it. The set that is meant is the support of $\\mathbf{X}$, the set where its marginal PMF/PDF is positive.", "what_survives": "The typeface change identifies the intended vector support: Section 1.4 extends the constructions for one variable to vectors. The continuous formula also needs the qualification about exceptional conditioning values in \\fid{P001-C01-N007}." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C01-N009", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the box of Theorem 2.1.21, Properties of the Normal Distribution, whose first bullet reads: for all a, b in R with a not equal to 0, if W = aX + b, then W is normal. Below the box, the sentence 'We omit the proof. This theorem has an interesting and immediate implication: any linear combination of any number of mutually independent normal random variables must itself be normal.' A red box marks the words 'any linear combination'.", "caption": "Theorem 2.1.21 and the sentence after it, p. 56 (PDF p. 74). The marked words admit the combination whose coefficients are all zero, which is a constant; the theorem above them requires $a\\neq0$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-E005.png" }, { "alt": "Crop of footnote 13, which gives the joint PDF of the bivariate normal distribution. Three red boxes mark: the expression rho(X, Y) in the first line, written with parentheses; the symbol mu with a lower-case subscript y in the last term of the exponent; and the range 'minus 1 less than or equal to rho less than or equal to 1' in the line after the formula. The last line of the footnote writes the correlation as rho[X, Y], with square brackets.", "caption": "Footnote 13, p. 66 (PDF p. 84). The marked range includes $\\rho=\\pm1$, where the displayed density is undefined; the other two marks are the correlation written with parentheses and the lower-case subscript in $\\mu_y$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-E005-fn13.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T38", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T38" }, { "id": "P001-T39", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T39" } ], "headline": "The linear combination in the sentence after Theorem 2.1.21 must have a coefficient that is not zero, and the bivariate normal density of footnote 13 requires $-1<\\rho<1$; the footnote also has two slips of notation.", "id": "P001-C02-E005", "location": "Sentence after Theorem 2.1.21, p. 56 (PDF p. 74); footnote 13, p. 66 (PDF p. 84)", "pdf_page": 74, "printed_page": "56", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T38", "P001-T39" ], "text": { "as_printed": "Sentence after Theorem 2.1.21, p.~56:\n\nThis theorem has an interesting and immediate implication: \\hl{any linear combination} of any number of mutually independent normal random variables must itself be normal.\n\nFootnote 13, p.~66:\n\nWhile $\\hl{\\rho(X,Y)}=0$ does not generally imply $X\\perp\\!\\!\\!\\perp Y$ even when $X$ and $Y$ are both normal, it does in the special case where the joint distribution of $X$ and $Y$ is \\emph{bivariate normal}, that is, $X$ and $Y$ have the joint PDF\n\\[\nf(x,y)=\\frac{1}{2\\pi\\sigma_X\\sigma_Y\\sqrt{1-\\rho^2}}\\,e^{-\\frac{1}{2(1-\\rho^2)}\\left[\\frac{(x-\\mu_X)^2}{\\sigma_X^2}+\\frac{(y-\\mu_Y)^2}{\\sigma_Y^2}-\\frac{2\\rho(x-\\mu_X)(y-\\hl{\\mu_y})}{\\sigma_X\\sigma_Y}\\right]}\n\\]\nfor some $\\mu_X,\\mu_Y,\\sigma_X,\\sigma_Y,\\rho\\in\\mathbb{R}$ with $\\sigma_X>0$, $\\sigma_Y>0$, and $\\hl{-1\\le\\rho\\le1}$.", "corrected": "Sentence after Theorem 2.1.21, p.~56:\n\nThis theorem has an interesting and immediate implication: any \\hl{nontrivial} linear combination of any number of mutually independent normal random variables must itself be normal.\n\nHere a linear combination $\\sum_i c_iX_i$ is nontrivial if at least one coefficient $c_i$ is not zero.\n\nFootnote 13, p.~66:\n\nWhile $\\hl{\\rho[X,Y]}=0$ does not generally imply $X\\perp\\!\\!\\!\\perp Y$ even when $X$ and $Y$ are both normal, it does in the special case where the joint distribution of $X$ and $Y$ is \\emph{bivariate normal}, that is, $X$ and $Y$ have the joint PDF\n\\[\nf(x,y)=\\frac{1}{2\\pi\\sigma_X\\sigma_Y\\sqrt{1-\\rho^2}}\\,e^{-\\frac{1}{2(1-\\rho^2)}\\left[\\frac{(x-\\mu_X)^2}{\\sigma_X^2}+\\frac{(y-\\mu_Y)^2}{\\sigma_Y^2}-\\frac{2\\rho(x-\\mu_X)(y-\\hl{\\mu_Y})}{\\sigma_X\\sigma_Y}\\right]}\n\\]\nfor some $\\mu_X,\\mu_Y,\\sigma_X,\\sigma_Y,\\rho\\in\\mathbb{R}$ with $\\sigma_X>0$, $\\sigma_Y>0$, and $\\hl{-1<\\rho<1}$.\n\nAt $\\rho=\\pm1$ the joint distribution is singular and has no two-dimensional PDF.", "evidence": "The evidence is an exact counterexample for each passage, with a short derivation. Let $X_1,\\dots,X_n$ be mutually independent normal random variables with means $\\mu_1,\\dots,\\mu_n$ and variances $\\sigma_1^2,\\dots,\\sigma_n^2>0$, and let $c_1,\\dots,c_n\\in\\mathbb{R}$. Then\n\\[\n\\mathrm{V}\\Big[\\sum_{i=1}^n c_iX_i\\Big]=\\sum_{i=1}^n c_i^2\\sigma_i^2 .\n\\]\nThis variance is zero when every $c_i$ is zero, and the combination is then the constant $0$. Otherwise it is positive, and the combination is normal, since its characteristic function is $\\exp\\big(it\\sum_i c_i\\mu_i-t^2\\sum_i c_i^2\\sigma_i^2/2\\big)$.\n\nFor the footnote, let $X$ and $Y$ have standard deviations $\\sigma_X,\\sigma_Y>0$ and correlation $\\rho$, and set $A=(X-\\mu_X)/\\sigma_X$ and $B=(Y-\\mu_Y)/\\sigma_Y$. Then\n\\[\n\\mathrm{E}\\big[(A-B)^2\\big]=2-2\\rho,\\qquad \\mathrm{E}\\big[(A+B)^2\\big]=2+2\\rho .\n\\]\nAt $\\rho=1$ the first expectation is zero, and I have $A=B$ with probability one. At $\\rho=-1$ the second is zero, and I have $A=-B$ with probability one. In both cases the joint distribution is concentrated on a line and has no two-dimensional density.", "problem": "Since Definition 2.1.19 (p.~55) requires $\\sigma>0$, a constant is not a normal random variable. The linear combination with every coefficient equal to zero is the constant $0$, and the sentence on p.~56 includes it, although Theorem 2.1.21 itself requires $a\\neq0$. In footnote 13 the range $-1\\le\\rho\\le1$ includes $\\rho=\\pm1$, where $\\sqrt{1-\\rho^2}=0$ and the factor $1/(2(1-\\rho^2))$ is undefined, so that the displayed formula defines no density. The same footnote writes $\\rho(X,Y)$ where the book's notation is $\\rho[X,Y]$, and $\\mu_y$ where $\\mu_Y$ is meant.", "what_survives": "Theorems 2.1.20 and 2.1.21, the density of footnote 13 for $-1<\\rho<1$, and the footnote's statement that $\\rho[X,Y]=0$ implies independence for a bivariate normal pair are unaffected. Only the exclusions of degenerate distributions and the two symbols change." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-E005", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of Figure 2.1.1, 'Minimum MSE Solution for a Fair Coin Flip'. A solid U-shaped curve is plotted against 'Choice of c' from -2 to 2. It starts at a height of about 13 at c = -2, reaches its lowest point, at a height of about 0.5, where a dashed vertical line marks c = 0.5, and ends at a height of about 5 at c = 2. A red box marks the vertical axis: the label MSE and the tick labels 0, 2, 4, 6, 8, 10, 12.", "caption": "Figure 2.1.1, p. 59 (PDF p. 77). The marked axis is labelled MSE, but its scale is twice the mean squared error of Example 2.1.25: the curve bottoms out at about $0.5$, where the mean squared error is $0.25$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-E002.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T40", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T40" } ], "headline": "The curve in Figure 2.1.1 approximates twice the mean squared error named on its vertical axis; the tick labels of that axis should be halved, which puts the minimum at $1/4$ for $c=1/2$.", "id": "P001-C02-E002", "location": "Figure 2.1.1, p. 59 (PDF p. 77), vertical axis and solid curve", "pdf_page": 77, "printed_page": "59", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T40" ], "text": { "as_printed": "The figure, described: the vertical axis is labelled \\hl{MSE} and has the tick labels \\hl{0, 2, 4, 6, 8, 10, 12}. The horizontal axis is labelled ``Choice of c'' and has the tick labels $-2$, $-1$, $0$, $1$, $2$. The solid curve starts near $(-2,13)$, has its minimum near $(1/2,1/2)$ and ends near $(2,5)$, and a dashed vertical line marks $c=1/2$. These coordinates describe the drawn curve and are not printed on the figure. The caption reads:\n\nFIGURE 2.1.1. \\emph{Minimum MSE Solution for a Fair Coin Flip}\n\nExample 2.1.25, p.~58, for the fair coin flip $X\\in\\{0,1\\}$:\n\\[\n\\begin{aligned}\n\\mathrm{E}\\big[(X-c)^2\\big]&=\\frac12(0-c)^2+\\frac12(1-c)^2\\\\\n&=\\frac12(c^2+1+c^2-2c)\\\\\n&=\\frac12(1+2c^2-2c)\\\\\n&=\\frac12+c^2-c.\n\\end{aligned}\n\\]", "corrected": "In Figure 2.1.1 the vertical scale is twice the stated mean squared error. Halve the tick labels of the vertical axis, so that 0, 2, 4, 6, 8, 10, 12 become \\hl{0, 1, 2, 3, 4, 5, 6}, or redraw the curve with its heights divided by two. For the fair coin flip $X\\in\\{0,1\\}$ of Example 2.1.25,\n\\[\n\\mathrm{E}\\big[(X-c)^2\\big]=c^2-c+\\frac12=\\Big(c-\\frac12\\Big)^2+\\frac14 ,\n\\]\nwhose minimum is $\\hl{1/4}$ at $c=1/2$. The calculation in the example and the minimizing choice are unchanged. A repair of the label alone is also possible: keep the drawing and label the vertical axis $2\\times\\mathrm{MSE}$.", "evidence": "The evidence is an exact counterexample to the printed scale: the mean squared error of Example 2.1.25, computed exactly, set against the coordinates of the published curve. I took the coordinates of the curve and of the tick marks from the drawing commands of the published PDF and calibrated both axes on the printed tick labels.\n\\begin{audittable}{rrr}{Height of the curve in Figure 2.1.1 on the printed scale, and the mean squared error of Example 2.1.25.}\n$c$ & Height of the printed curve & $\\mathrm{E}[(X-c)^2]=c^2-c+1/2$ \\\\\n\\hline\n$-2$ & $13.00002$ & $13/2$ \\\\\n$1/2$ & $0.50010$ & $1/4$ \\\\\n$2$ & $5.00002$ & $5/2$ \\\\\n\\end{audittable}\nEvery point of the centre line of the drawn curve is within $0.011$ of $2(c^2-c+1/2)$ on the printed scale. Half the width of the stroke is about $0.023$ on that scale, and neither the thickness of the line nor rounding accounts for the factor of two.", "problem": "The figure illustrates Example 2.1.25, in which $X$ is a fair coin flip and the mean squared error about $c$ is $\\mathrm{E}[(X-c)^2]=c^2-c+1/2$ (Definition 2.1.22). This function equals $13/2$ at $c=-2$ and $5/2$ at $c=2$, and its minimum is $1/4$ at $c=1/2$. The printed curve has heights of about $13$, $5$ and $1/2$ at these points. On the printed scale it approximates twice the mean squared error across the plotted range. The unweighted sum $(0-c)^2+(1-c)^2=2\\,\\mathrm{E}[(X-c)^2]$ equals $13$, $5$ and $1/2$ at these points. The code that drew the figure is not available, and the cause of the factor is not established.", "what_survives": "Example 2.1.25, its solution $c=1/2$, Definition 2.1.22 and Theorems 2.1.23 and 2.1.24 do not depend on the figure. The figure still shows the minimizing choice, since doubling a function does not move its minimizer, and the dashed line stays at $c=1/2$. Only the heights read from the figure change. A search of the text finds the figure cited in the List of Figures, in the sentence on p.~58 that introduces it and in its caption, with no later reference." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-E002", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of the display that begins 'Likewise, if X and Y are jointly continuous, then E[Y] = ...'. Four lines are shown. A red box marks the fourth, the double integral from minus infinity to infinity of y times f of y given x times f_X(x), dy dx.", "caption": "Proof of Theorem 2.2.17, jointly continuous case, p. 72 (PDF p. 90); the display continues on p. 73. The marked line is the first in which the conditional PDF appears under an integral over all real $x$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-N002.png" } ], "formal_release": null, "formal_statements": [], "headline": "In the proof of the Law of Iterated Expectations, the integrals over $x$ whose integrands contain $f_{Y|X}(y|x)$ or $\\mathrm{E}[Y|X=x]$ are taken over $\\operatorname{Supp}[X]$, where these are defined.", "id": "P001-C02-N002", "location": "Proof of Theorem 2.2.17, jointly continuous case, pp. 72–73 (PDF pp. 90–91)", "pdf_page": 90, "printed_page": "72–73", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "Likewise, if $X$ and $Y$ are jointly continuous, then\n\\[\n\\begin{aligned}\n\\mathrm{E}[Y]&=\\int_{-\\infty}^{\\infty}yf_Y(y)dy\\\\\n&=\\int_{-\\infty}^{\\infty}y\\left(\\int_{-\\infty}^{\\infty}f(x,y)dx\\right)dy\\\\\n&=\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}yf(x,y)dydx\\\\\n&=\\hl{\\int_{-\\infty}^{\\infty}}\\int_{-\\infty}^{\\infty}yf_{Y|X}(y|x)f_X(x)dydx\\\\\n&=\\hl{\\int_{-\\infty}^{\\infty}}\\left(\\int_{-\\infty}^{\\infty}yf_{Y|X}(y|x)dy\\right)f_X(x)dx\\\\\n&=\\hl{\\int_{-\\infty}^{\\infty}}\\mathrm{E}[Y|X=x]f_X(x)dx\\\\\n&=\\mathrm{E}\\big[\\mathrm{E}[Y|X]\\big].\n\\end{aligned}\n\\]", "corrected": "\\[\n\\begin{aligned}\n\\mathrm{E}[Y]&=\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}yf(x,y)dydx\\\\\n&=\\hl{\\int_{\\operatorname{Supp}[X]}}\\int_{-\\infty}^{\\infty}yf_{Y|X}(y|x)f_X(x)dydx\\\\\n&=\\hl{\\int_{\\operatorname{Supp}[X]}}\\left(\\int_{-\\infty}^{\\infty}yf_{Y|X}(y|x)dy\\right)f_X(x)dx\\\\\n&=\\hl{\\int_{\\operatorname{Supp}[X]}}\\mathrm{E}[Y|X=x]f_X(x)dx\\\\\n&=\\mathrm{E}\\big[\\mathrm{E}[Y|X]\\big].\n\\end{aligned}\n\\]\nThe sums over $x$ in the discrete case are likewise over $\\operatorname{Supp}[X]$.", "intended_meaning": "Definition 1.3.14 (p.~38) defines $f_{Y|X}(y|x)$ only for $x\\in\\operatorname{Supp}[X]$, and $\\mathrm{E}[Y|X=x]$ is defined on the same set, while the marked integrals run over all real $x$. These integrals are meant over $\\operatorname{Supp}[X]$. That reading is correct: for $x$ outside $\\operatorname{Supp}[X]$, I have $\\int_{-\\infty}^{\\infty}f(x,y)dy=f_X(x)=0$ with $f\\ge0$, so these $x$ contribute nothing to $\\int\\int yf(x,y)dydx$. The theorem and the rest of the proof are unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-N002", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the box of Theorem 2.2.8, Implications of Independence (Part II), with its four bullets. A red box marks the third bullet, 'Correlation is zero: rho[X, Y] = 0.'", "caption": "Theorem 2.2.8, p. 64 (PDF p. 82). The marked bullet is stated for all independent $X$ and $Y$; the correlation in it is defined only when both variances are positive.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-A002.png" }, { "alt": "Crop of the box of Definition 2.2.5, Correlation, which defines rho[X, Y] as Cov[X, Y] divided by sigma[X] sigma[Y]. Red boxes mark the condition 'with sigma[X] > 0 and sigma[Y] > 0'. The paragraph below the box ends: '(Note that, unlike covariance, correlation is undefined when the variance of either variable is zero.)'", "caption": "Definition 2.2.5 and the paragraph after it, p. 62 (PDF p. 80). The marked condition is the domain on which the third bullet of Theorem 2.2.8 is read; the paragraph ends by saying that correlation is undefined when either variance is zero.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-A002-def.png" } ], "formal_release": null, "formal_statements": [], "headline": "The third bullet of Theorem 2.2.8 holds when the correlation is defined, that is, when both variables have positive variance.", "id": "P001-C02-A002", "location": "Theorem 2.2.8, third bullet, p. 64 (PDF p. 82), with the last line of its proof on p. 65 (PDF p. 83); Definition 2.2.5, p. 62 (PDF p. 80)", "pdf_page": 82, "printed_page": "64", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "\\textbf{Theorem 2.2.8.} \\emph{Implications of Independence (Part II)}\n\nIf $X$ and $Y$ are independent random variables, then\n\\begin{itemize}\n\\item $\\mathrm{E}[XY]=\\mathrm{E}[X]\\mathrm{E}[Y]$.\n\\item Covariance is zero: $\\operatorname{Cov}[X,Y]=0$.\n\\item \\hl{Correlation is zero}: $\\rho[X,Y]=0$.\n\\item Variances are additive: $\\mathrm{V}[X+Y]=\\mathrm{V}[X]+\\mathrm{V}[Y]$.\n\\end{itemize}", "corrected": "\\begin{itemize}\n\\item Correlation, \\hl{when defined,} is zero: $\\rho[X,Y]=0$.\n\\end{itemize}\nThe correlation line of the proof on p.~65 applies when $\\sigma[X]>0$ and $\\sigma[Y]>0$, and Theorems 2.2.6 and 2.2.7 likewise use correlation on the domain of Definition 2.2.5.", "intended_meaning": "Definition 2.2.5 (p.~62) defines $\\rho[X,Y]$ only when $\\sigma[X]>0$ and $\\sigma[Y]>0$. The third bullet is read on that domain, where it is correct, since $\\operatorname{Cov}[X,Y]=0$ gives $\\rho[X,Y]=0$. Independence alone does not place $X$ and $Y$ in that domain. If $X$ is the constant $3$ and $Y$ is a fair die roll, then $X$ and $Y$ are independent and the other three bullets hold, but $\\mathrm{V}[X]=0$ and $\\rho[X,Y]$ is undefined." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-A002", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the box of Theorem 2.2.6, Correlation and Linear Dependence, with its three bullets. Red boxes mark the equality Y = a + bX at the end of the second bullet and the equality Y = a - bX at the end of the third bullet. Below the box, the sentence 'We omit the proof of this theorem.'", "caption": "Theorem 2.2.6, p. 63 (PDF p. 81). The two marked equalities are stated without “almost surely”; by Definition 1.3.1 of the book they then hold at every outcome.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-E004.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T41", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T41" }, { "id": "P001-T42", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T42" } ], "headline": "Perfect correlation implies the affine relation only with probability one: in the second and third bullets of Theorem 2.2.6, $Y=a+bX$ and $Y=a-bX$ should hold almost surely.", "id": "P001-C02-E004", "location": "Theorem 2.2.6, p. 63 (PDF p. 81), second and third bullets", "pdf_page": 81, "printed_page": "63", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T41", "P001-T42" ], "text": { "as_printed": "\\textbf{Theorem 2.2.6.} \\emph{Correlation and Linear Dependence}\n\nFor random variable $X$ and $Y$,\n\\begin{itemize}\n\\item $\\rho[X,Y]\\in[-1,1]$.\n\\item $\\rho[X,Y]=1\\Longleftrightarrow\\exists a,b\\in\\mathbb{R}$ with $b>0$ such that $\\hl{Y=a+bX}$.\n\\item $\\rho[X,Y]=-1\\Longleftrightarrow\\exists a,b\\in\\mathbb{R}$ with $b>0$ such that $\\hl{Y=a-bX}$.\n\\end{itemize}", "corrected": "\\textbf{Theorem 2.2.6.} \\emph{Correlation and Linear Dependence}\n\nFor random \\hl{variables} $X$ and $Y$,\n\\begin{itemize}\n\\item $\\rho[X,Y]\\in[-1,1]$.\n\\item $\\rho[X,Y]=1\\Longleftrightarrow\\exists a,b\\in\\mathbb{R}$ with $b>0$ such that $Y=a+bX$ \\hl{almost surely}.\n\\item $\\rho[X,Y]=-1\\Longleftrightarrow\\exists a,b\\in\\mathbb{R}$ with $b>0$ such that $Y=a-bX$ \\hl{almost surely}.\n\\end{itemize}\nHere $X$ and $Y$ have finite, strictly positive variances, as Definition 2.2.5 requires for $\\rho[X,Y]$ to be defined, and ``$Y=a+bX$ almost surely'' means that $\\Pr[Y=a+bX]=1$. The plural in the opening line is the correction of \\fid{P001-C02-N003}. The same two words are needed where the statement is repeated: in the description of perfect positive and negative linear dependence on p.~62, and in the last sentence of footnote 21 on p.~78.", "evidence": "The evidence is an exact counterexample for each sign, with a derivation of the corrected statement. Let $\\Omega=\\{u,v,n\\}$ with the probabilities and random variables below.\n\\begin{audittable}{lrrrr}{The counterexample: three outcomes, one of probability zero.}\nOutcome & Probability & $X$ & $Y_+$ & $Y_-$ \\\\\n\\hline\n$u$ & $1/2$ & $0$ & $0$ & $1$ \\\\\n$v$ & $1/2$ & $1$ & $1$ & $0$ \\\\\n$n$ & $0$ & $0$ & $1$ & $0$ \\\\\n\\end{audittable}\nEach of $X$, $Y_+$ and $Y_-$ takes the values $0$ and $1$ with probability $1/2$; each has mean $1/2$ and variance $1/4$, and its range equals its support $\\{0,1\\}$. A direct computation gives $\\mathrm{E}[XY_+]=1/2$, $\\operatorname{Cov}[X,Y_+]=1/4$ and $\\rho[X,Y_+]=1$. If $Y_+=a+bX$ at every outcome, then from the outcomes $u$ and $v$ I obtain $a=0$ and $b=1$, and the outcome $n$ requires $1=0$. Similarly, I have $\\mathrm{E}[XY_-]=0$, $\\operatorname{Cov}[X,Y_-]=-1/4$ and $\\rho[X,Y_-]=-1$. If $Y_-=a-bX$ at every outcome, then $u$ and $v$ give $a=1$ and $b=1$, and $n$ requires $0=1$. Both relations hold with probability one.\n\nFor the corrected statement, let $X$ and $Y$ have finite second moments with $\\sigma[X]>0$ and $\\sigma[Y]>0$, and set $A=(X-\\mathrm{E}[X])/\\sigma[X]$ and $B=(Y-\\mathrm{E}[Y])/\\sigma[Y]$. Then\n\\[\n\\mathrm{E}\\big[(A-B)^2\\big]=2-2\\rho[X,Y],\\qquad \\mathrm{E}\\big[(A+B)^2\\big]=2+2\\rho[X,Y].\n\\]\nBoth are nonnegative, and the first bullet follows. If $\\rho[X,Y]=1$, then $(A-B)^2$ is a nonnegative random variable with expectation zero. I conclude that $A=B$ with probability one, and\n\\[\nY=a+bX\\ \\text{almost surely},\\qquad b=\\frac{\\sigma[Y]}{\\sigma[X]}>0,\\quad a=\\mathrm{E}[Y]-b\\,\\mathrm{E}[X].\n\\]\nIf $\\rho[X,Y]=-1$, I apply the same argument to $(A+B)^2$ and find that $Y=a-bX$ almost surely with $b=\\sigma[Y]/\\sigma[X]$ and $a=\\mathrm{E}[Y]+b\\,\\mathrm{E}[X]$. Conversely, if $Y=a\\pm bX$ almost surely with $b>0$, then $\\mathrm{V}[Y]=b^2\\,\\mathrm{V}[X]$ and $\\operatorname{Cov}[X,Y]=\\pm b\\,\\mathrm{V}[X]$, which implies that $\\rho[X,Y]=\\pm1$.", "problem": "The book defines equality of random variables at every outcome: by Definition 1.3.1 (p.~31), $X=Y$ means $X(\\omega)=Y(\\omega)$ for all $\\omega\\in\\Omega$, and footnote 28 on that page distinguishes this from $\\Pr[X=Y]=1$, which it calls equality almost surely. With this definition, the second and third bullets say that $\\rho[X,Y]=\\pm1$ forces $Y=a\\pm bX$ at every outcome. Under perfect correlation the relation holds only with probability one, and $Y$ may differ from $a\\pm bX$ on an event of probability zero. The first bullet and the implications from the affine relation to $\\rho[X,Y]=\\pm1$ are correct.", "what_survives": "The bound $\\rho[X,Y]\\in[-1,1]$ and the implications from the affine relation to $\\rho[X,Y]=\\pm1$ are unaffected, and both equivalences hold in the corrected form. The conclusion drawn from $\\rho[X,Y]=\\pm1$ weakens from equality at every outcome to equality with probability one, and this changes no distribution, no moment and no expected prediction loss. Theorem 2.2.7, the derivation of the best linear predictor on pp.~76--78 and the variance calculation of Example 3.5.8 on pp.~140--141 rest on moment formulas and do not use equality at every outcome." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-E004", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the boxes of Theorem 2.2.6 and Theorem 2.2.7. In the opening line of Theorem 2.2.6, 'For random variable X and Y,', a red box marks the word 'variable'. In the opening line of Theorem 2.2.7, 'For random variables X, Y, and Z,', a red box marks the letter Z. The four bullets of Theorem 2.2.7 involve only X, Y and the constants a, b, c, d.", "caption": "Theorems 2.2.6 and 2.2.7, p. 63 (PDF p. 81). The marked word should be plural, and the marked $Z$ appears in none of the four bullets below it.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-N003.png" } ], "formal_release": null, "formal_statements": [], "headline": "Theorem 2.2.6 should begin “For random variables $X$ and $Y$”, and Theorem 2.2.7 should not introduce $Z$, which it does not use.", "id": "P001-C02-N003", "location": "Opening lines of Theorem 2.2.6 and of Theorem 2.2.7, p. 63 (PDF p. 81)", "pdf_page": 81, "printed_page": "63", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Theorem 2.2.6: For random \\hl{variable} $X$ and $Y$,\n\nTheorem 2.2.7: For random variables $X$, $Y$, and $\\hl{Z}$,", "corrected": "Theorem 2.2.6: For random \\hl{variables} $X$ and $Y$,\n\nTheorem 2.2.7: For random variables $X$ \\hl{and} $Y$,", "evidence": "The evidence is a check of the page: I read both theorems and the whole proof of Theorem 2.2.7 on pp.~63--64, and $Z$ does not occur after the opening line.", "problem": "Theorem 2.2.6 concerns the two random variables $X$ and $Y$ and has ``variable'' in the singular. Theorem 2.2.7 introduces a third random variable $Z$, while its four bullets and its proof involve only $X$, $Y$ and the constants $a,b,c,d$.", "what_survives": "Nothing else in either theorem changes. The second and third bullets of Theorem 2.2.6 have their own correction, \\fid{P001-C02-E004}." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-N003", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the start of Example 2.2.16, Flipping a Coin and Rolling a Die: 'Consider, again, the generative process from Example 1.1.11. Recall from Example 1.3.5 that the conditional PMF of Y given X = x is', followed by the conditional PMF, equal to 1/4 for x = 0 and y in 1 to 4, to 1/6 for x = 1 and y in 1 to 6, and to 0 otherwise. A red box marks the words 'Example 1.3.5'.", "caption": "Example 2.2.16, p. 70 (PDF p. 88). The marked reference should be to Example 1.3.8, where this conditional PMF is derived.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-N004.png" } ], "formal_release": null, "formal_statements": [], "headline": "The conditional PMF recalled at the start of Example 2.2.16 comes from Example 1.3.8, not from Example 1.3.5.", "id": "P001-C02-N004", "location": "Example 2.2.16, second sentence, p. 70 (PDF p. 88)", "pdf_page": 88, "printed_page": "70", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Consider, again, the generative process from Example 1.1.11. Recall from \\hl{Example 1.3.5} that the conditional PMF of $Y$ given $X=x$ is\n\\[\nf_{Y|X}(y|x)=\\begin{cases}\\frac14 &:\\ x=0,\\,y\\in\\{1,2,3,4\\}\\\\[0.5ex] \\frac16 &:\\ x=1,\\,y\\in\\{1,2,3,4,5,6\\}\\\\[0.5ex] 0 &:\\ \\text{otherwise.}\\end{cases}\n\\]", "corrected": "Consider, again, the generative process from Example 1.1.11. Recall from \\hl{Example 1.3.8} that the conditional PMF of $Y$ given $X=x$ is", "evidence": "The evidence is a check of the pages: I compared the sentence on p.~70 with Examples 1.3.5 and 1.3.8 on pp.~33--35.", "problem": "Example 1.3.5 (p.~33) gives the joint PMF of $X$ and $Y$ and one value of the joint CDF. The conditional PMF is defined only afterwards, in Definition 1.3.7, and the function quoted here is displayed in Example 1.3.8 (pp.~34--35).", "what_survives": "The displayed conditional PMF is correct, and the rest of Example 2.2.16 is unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-N004", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of four rows of a display, each beginning with an equals sign: E[epsilon squared given X] + 2W E[epsilon given X] + W squared; E[epsilon squared given X] + 0 + W squared; E[epsilon squared given X] minus E[epsilon given X] squared + W squared; V[Y given X] + W squared. Below, the sentence 'where the third and fifth lines follow from Theorem 2.2.19.' A red box marks the words 'third and fifth lines'.", "caption": "Proof of Theorem 2.2.20, p. 76 (PDF p. 94): the last four rows of the display and the sentence after it. The marked words are right if the opening row of the display, on p. 75, is not counted.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-N005.png" }, { "alt": "Crop of the end of a paragraph and the first two rows of a display. A red box marks the first row, E[U squared given X] = E[(epsilon + W) squared given X]. The second row reads = E[epsilon squared + 2 epsilon W + W squared given X].", "caption": "Proof of Theorem 2.2.20, foot of p. 75 (PDF p. 93): the first two rows of the same display. The marked row is the opening row, which the count leaves out.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-N005-p75.png" } ], "formal_release": null, "formal_statements": [], "headline": "“The third and fifth lines” counts the lines after the opening row of the display; the two steps that use Theorem 2.2.19 are $\\mathrm{E}[\\varepsilon|X]=0$ and $\\mathrm{V}[\\varepsilon|X]=\\mathrm{V}[Y|X]$.", "id": "P001-C02-N005", "location": "Proof of Theorem 2.2.20, pp. 75–76 (PDF pp. 93–94), the sentence after the display that ends “$=\\mathrm{V}[Y|X]+W^2$”", "pdf_page": 93, "printed_page": "75–76", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "\\[\n\\begin{aligned}\n\\mathrm{E}\\big[U^2\\,\\big|\\,X\\big]&=\\mathrm{E}\\big[(\\varepsilon+W)^2\\,\\big|\\,X\\big]\\\\\n&=\\mathrm{E}\\big[\\varepsilon^2+2\\varepsilon W+W^2\\,\\big|\\,X\\big]\\\\\n&=\\mathrm{E}\\big[\\varepsilon^2\\,\\big|\\,X\\big]+2W\\mathrm{E}[\\varepsilon|X]+W^2\\\\\n&=\\mathrm{E}\\big[\\varepsilon^2\\,\\big|\\,X\\big]+0+W^2\\\\\n&=\\mathrm{E}\\big[\\varepsilon^2\\,\\big|\\,X\\big]-\\mathrm{E}[\\varepsilon|X]^2+W^2\\\\\n&=\\mathrm{V}[Y|X]+W^2,\n\\end{aligned}\n\\]\nwhere the \\hl{third and fifth lines} follow from Theorem 2.2.19.", "corrected": "where $\\hl{\\mathrm{E}[\\varepsilon|X]=0}$ and $\\hl{\\mathrm{V}[\\varepsilon|X]=\\mathrm{V}[Y|X]}$ follow from Theorem 2.2.19.", "intended_meaning": "The display has six rows, two on p.~75 and four on p.~76, and Theorem 2.2.19 (p.~74) states that $\\mathrm{E}[\\varepsilon|X]=0$ and $\\mathrm{V}[\\varepsilon|X]=\\mathrm{V}[Y|X]$. Counting from the row after the opening one, the third line is the row with $+0$ and the fifth is the last row, and both use the theorem. If the opening row is counted, the same rows are the fourth and the sixth. The proof is correct under either count." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-N005", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of the sentence 'By linearity of expectations and the chain rule, as well as the fact that derivatives pass through expectations (because they pass through sums and integrals), this becomes:' followed by two displayed lines in which the derivative of E[epsilon squared] is moved inside the expectation, ending in minus 2 E[epsilon] and minus 2 E[epsilon X]. Red boxes mark the words from 'derivatives' to 'integrals)'.", "caption": "Proof of Theorem 2.2.21, p. 77 (PDF p. 95). The marked words justify moving each derivative inside the expectation in the display below them.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-E007.png" } ], "formal_release": null, "formal_statements": [], "headline": "The remark that derivatives pass through expectations is not a general rule; the step holds here because $\\mathrm{E}[(Y-a-bX)^2]$ is a quadratic polynomial in $(a,b)$ with finite coefficients.", "id": "P001-C02-E007", "location": "Proof of Theorem 2.2.21, p. 77 (PDF p. 95), the sentence before the displayed normal equations", "pdf_page": 95, "printed_page": "77", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "By linearity of expectations and the chain rule, as well as the fact that \\hl{derivatives pass through expectations (because they pass through sums and integrals)}, this becomes:\n\\[\n\\begin{aligned}\n0&=\\frac{\\partial\\mathrm{E}[\\varepsilon^2]}{\\partial\\alpha}=\\mathrm{E}\\left[\\frac{\\partial\\varepsilon^2}{\\partial\\alpha}\\right]=\\mathrm{E}\\left[2\\varepsilon\\frac{\\partial\\varepsilon}{\\partial\\alpha}\\right]=-2\\mathrm{E}[\\varepsilon],\\\\\n0&=\\frac{\\partial\\mathrm{E}[\\varepsilon^2]}{\\partial\\beta}=\\mathrm{E}\\left[\\frac{\\partial\\varepsilon^2}{\\partial\\beta}\\right]=\\mathrm{E}\\left[2\\varepsilon\\frac{\\partial\\varepsilon}{\\partial\\beta}\\right]=-2\\mathrm{E}[\\varepsilon X].\n\\end{aligned}\n\\]", "corrected": "\\hl{Expanding the finite quadratic $\\mathrm{E}[(Y-a-bX)^2]$, differentiating in $a$ and $b$, and evaluating at $(a,b)=(\\alpha,\\beta)$ gives:}", "intended_meaning": "The marked words describe the step taken in this proof, and that step is valid. Under the chapter's convention that the expectations it uses converge absolutely (footnote 2, p.~45), the objective is the quadratic polynomial\n\\[\n\\mathrm{E}\\big[(Y-a-bX)^2\\big]=\\mathrm{E}[Y^2]-2a\\,\\mathrm{E}[Y]-2b\\,\\mathrm{E}[XY]+a^2+2ab\\,\\mathrm{E}[X]+b^2\\,\\mathrm{E}[X^2],\n\\]\nand differentiating it term by term gives the displayed equations. As a general statement the words are too broad, since a derivative does not pass through every sum or integral without further conditions. The normal equations and Theorem 2.2.21 are unaffected; the letters $\\alpha$ and $\\beta$ in the derivatives are the subject of \\fid{P001-C02-N006}." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-E007", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of the start of the proof: 'Let epsilon = Y - (a + bX). If g(X) = alpha + beta X is the best linear predictor of Y given X, then (alpha, beta) = arg min over (a, b) in R squared of E[epsilon squared]. The first-order conditions yield the system of equations', followed by two equations setting to zero the partial derivatives of E[epsilon squared]. Red boxes mark the definition epsilon = Y - (a + bX) and the two denominators, partial alpha and partial beta.", "caption": "Proof of Theorem 2.2.21, p. 77 (PDF p. 95). The first mark is the definition of $\\varepsilon$ in terms of $a$ and $b$; the other two are the derivatives, which are written in $\\alpha$ and $\\beta$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-N006.png" }, { "alt": "Crop of footnote 19, 'The second-order partial derivatives are', with three displayed lines. In the second line, the second derivative of E[epsilon squared] in beta, a red box marks a bold upright X inside the expression minus 2 E[epsilon X]. The other occurrences of X in that line are italic.", "caption": "Footnote 19, p. 77 (PDF p. 95). The marked $\\mathbf{X}$ is set in bold, the book’s notation for a random vector; every other $X$ in the line is the italic scalar.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-N006-fn19.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T43", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T43" } ], "headline": "In the proof of Theorem 2.2.21 the first-order conditions are derivatives in the free arguments $a$ and $b$, evaluated at $(\\alpha,\\beta)$, and not derivatives in $\\alpha$ and $\\beta$; in footnote 19 one bold $\\mathbf{X}$ should be the scalar $X$.", "id": "P001-C02-N006", "location": "Proof of Theorem 2.2.21, p. 77 (PDF p. 95), from its first sentence to the first-order conditions; footnote 19, second displayed line", "pdf_page": 95, "printed_page": "77", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T43" ], "text": { "as_printed": "\\textbf{Proof:} Let $\\varepsilon=Y-(a+bX)$. If $g(X)=\\alpha+\\beta X$ is the best linear predictor of $Y$ given $X$, then\n\\[\n(\\alpha,\\beta)=\\operatorname*{arg\\,min}_{(a,b)\\in\\mathbb{R}^2}\\mathrm{E}\\big[\\varepsilon^2\\big].\n\\]\nThe first-order conditions yield the system of equations\n\\[\n\\begin{aligned}\n0&=\\frac{\\partial\\mathrm{E}[\\varepsilon^2]}{\\hl{\\partial\\alpha}},\\\\\n0&=\\frac{\\partial\\mathrm{E}[\\varepsilon^2]}{\\hl{\\partial\\beta}}.\n\\end{aligned}\n\\]\nFootnote 19, second displayed line:\n\\[\n\\frac{\\partial^2\\mathrm{E}[\\varepsilon^2]}{\\partial\\beta^2}=\\frac{\\partial}{\\partial\\beta}\\big(-2\\mathrm{E}[\\varepsilon\\hl{\\mathbf{X}}]\\big)=-2\\mathrm{E}\\left[\\frac{\\partial\\varepsilon}{\\partial\\beta}X\\right]=-2\\mathrm{E}\\big[-X^2\\big]=2\\mathrm{E}\\big[X^2\\big],\n\\]", "corrected": "\\textbf{Proof:} Let $\\hl{Q(a,b)=\\mathrm{E}\\big[(Y-(a+bX))^2\\big]}$ for $(a,b)\\in\\mathbb{R}^2$. If $g(X)=\\alpha+\\beta X$ is the best linear predictor of $Y$ given $X$, then\n\\[\n(\\alpha,\\beta)=\\operatorname*{arg\\,min}_{(a,b)\\in\\mathbb{R}^2}\\hl{Q(a,b)}.\n\\]\nThe first-order conditions yield the system of equations\n\\[\n\\begin{aligned}\n0&=\\hl{\\frac{\\partial Q}{\\partial a}(\\alpha,\\beta)},\\\\\n0&=\\hl{\\frac{\\partial Q}{\\partial b}(\\alpha,\\beta)}.\n\\end{aligned}\n\\]\nThe later displays of the proof and footnote 19 are read in the same way: each derivative is taken in $a$ or $b$ and evaluated at $(\\alpha,\\beta)$, where $\\varepsilon=Y-(\\alpha+\\beta X)$. In the second line of footnote 19, $-2\\mathrm{E}[\\varepsilon\\mathbf{X}]$ becomes $-2\\mathrm{E}[\\varepsilon\\hl{X}]$, with the scalar $X$.", "evidence": "The evidence is a derivation of the first-order conditions in the corrected notation, and a check of the page for the bold letter. I write\n\\[\nQ(a,b)=\\mathrm{E}\\big[(Y-a-bX)^2\\big]=\\mathrm{E}[Y^2]-2a\\,\\mathrm{E}[Y]-2b\\,\\mathrm{E}[XY]+a^2+2ab\\,\\mathrm{E}[X]+b^2\\,\\mathrm{E}[X^2].\n\\]\nDifferentiating, I obtain\n\\[\n\\begin{aligned}\n\\frac{\\partial Q}{\\partial a}(a,b)&=2a+2b\\,\\mathrm{E}[X]-2\\,\\mathrm{E}[Y]=-2\\,\\mathrm{E}[Y-a-bX],\\\\\n\\frac{\\partial Q}{\\partial b}(a,b)&=2a\\,\\mathrm{E}[X]+2b\\,\\mathrm{E}[X^2]-2\\,\\mathrm{E}[XY]=-2\\,\\mathrm{E}\\big[X(Y-a-bX)\\big].\n\\end{aligned}\n\\]\nSetting both derivatives to zero at $(a,b)=(\\alpha,\\beta)$, I find $\\beta=\\operatorname{Cov}[X,Y]/\\mathrm{V}[X]$ and $\\alpha=\\mathrm{E}[Y]-\\beta\\,\\mathrm{E}[X]$, as in the theorem. The second derivatives of $Q$ are $2$ in $a$, $2\\,\\mathrm{E}[X^2]$ in $b$ and $2\\,\\mathrm{E}[X]$ mixed. The determinant of the Hessian is $4\\,\\mathrm{E}[X^2]-4\\,\\mathrm{E}[X]^2=4\\,\\mathrm{V}[X]>0$, which implies that $(\\alpha,\\beta)$ is the unique minimizer. In the published PDF the $X$ of $-2\\mathrm{E}[\\varepsilon X]$ in footnote 19 is set in the bold font, and the $X$ that follow it in the same line are in italic.", "problem": "The proof defines $\\varepsilon=Y-(a+bX)$, a function of the free arguments $a$ and $b$, and $(\\alpha,\\beta)$ as the fixed pair that minimizes $\\mathrm{E}[\\varepsilon^2]$. It then writes the first-order conditions as derivatives in $\\alpha$ and $\\beta$, on which $\\mathrm{E}[\\varepsilon^2]$, as defined, does not depend. The values $\\partial\\varepsilon/\\partial\\alpha=-1$ and $\\partial\\varepsilon/\\partial\\beta=-X$ used in the next display and in footnote 19 require $\\varepsilon=Y-(\\alpha+\\beta X)$ with $\\alpha$ and $\\beta$ treated as variables. The proof means to differentiate in $a$ and $b$ and to evaluate at $(a,b)=(\\alpha,\\beta)$. In the second line of footnote 19, the $X$ in $-2\\mathrm{E}[\\varepsilon X]$ is set in bold where the scalar $X$ is meant.", "what_survives": "Theorem 2.2.21, the formulas for $\\alpha$ and $\\beta$, the first-order equations read at the minimizer and the second-order calculation of footnote 19 are unaffected. Only the letters change." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-N006", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "historical_lean_attribution", "exhibits": [ { "alt": "Screenshot of the last line of footnote 20 and the four lines of footnote 21 at the foot of page 78. Two red boxes enclose the words: the BLP is the only linear function that satisfies, and, on the next line, the equation Cov[X, g(X)] = Cov[X, Y].", "caption": "Footnote 21 on p. 78 (PDF p. 96). The red boxes mark the claim that the BLP is the only linear function satisfying the covariance equation.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-E001.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T44", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T44" }, { "id": "P001-T45", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T45" } ], "headline": "The best linear predictor is the only linear function $g$ that satisfies both $\\operatorname{E}[g(X)]=\\operatorname{E}[Y]$ and $\\operatorname{Cov}[X,g(X)]=\\operatorname{Cov}[X,Y]$; the covariance equation alone fixes the slope and leaves the intercept free.", "historical_evidence_type": "lean_proof", "id": "P001-C02-E001", "location": "Footnote 21, p. 78 (PDF p. 96), first sentence", "pdf_page": 96, "printed_page": "78", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T44", "P001-T45" ], "text": { "as_printed": "The BLP also demonstrates that covariance and correlation indeed measure the \\emph{linear} relationship between $X$ and $Y$ (as in Section 2.2.2): the BLP is \\hl{the only} linear function that satisfies $\\operatorname{Cov}[X,g(X)]=\\operatorname{Cov}[X,Y]$.", "corrected": "The BLP also demonstrates that covariance and correlation indeed measure the \\emph{linear} relationship between $X$ and $Y$ (as in Section 2.2.2): under the condition $\\operatorname{V}[X]>0$ of Theorem 2.2.21, the BLP is the only linear function $g$ satisfying \\hl{both} $\\hl{\\operatorname{E}[g(X)]=\\operatorname{E}[Y]}$ \\hl{and} $\\operatorname{Cov}[X,g(X)]=\\operatorname{Cov}[X,Y]$.", "evidence": "The evidence is an exact counterexample to the printed sentence and a derivation of the corrected one. Let $X$ and $Y$ have finite second moments, and write $\\mu=\\operatorname{E}[X]$, $\\nu=\\operatorname{E}[Y]$, $v=\\operatorname{V}[X]>0$ and $k=\\operatorname{Cov}[X,Y]$. For all real $a$ and $b$, linearity of expectation gives\n\\[\n\\operatorname{Cov}[X,a+bX]=a\\mu+b\\operatorname{E}[X^2]-\\mu(a+b\\mu)=bv .\n\\]\nThe covariance equation is therefore equivalent to $b=k/v$, and it puts no condition on $a$. The mean equation $\\operatorname{E}[g(X)]=\\operatorname{E}[Y]$ reads $a+b\\mu=\\nu$. Combining the two equations, I obtain\n\\[\nb=\\frac{k}{v},\\qquad a=\\nu-\\frac{k}{v}\\mu ,\n\\]\nwhich are the coefficients $\\beta$ and $\\alpha$ of Theorem 2.2.21. Next, set $\\epsilon=Y-\\alpha-\\beta X$, $u=a-\\alpha$ and $t=b-\\beta$. Then $\\operatorname{E}[\\epsilon]=0$ and $\\operatorname{E}[X\\epsilon]=0$. Expanding the square, I find\n\\[\n\\operatorname{E}\\bigl[(Y-a-bX)^2\\bigr]=\\operatorname{E}[\\epsilon^2]+(u+t\\mu)^2+t^2v .\n\\]\nSince $v>0$, the last two terms vanish only when $t=0$ and $u=0$. This implies that $\\alpha+\\beta X$ is the unique linear function of minimum MSE. A function that satisfies the covariance equation has $t=0$, and its MSE exceeds the minimum by $u^2$.\n\nThe table lists two counterexamples to the printed sentence. In the first, $X=Y$ takes the values $-1$ and $1$ with probability $1/2$ each, so that $\\operatorname{E}[Y]=0$ and $\\operatorname{V}[X]=\\operatorname{Cov}[X,Y]=1$; the function $x$ has mean $0$ and the function $x+1$ has mean $1$. This counterexample is formalized as a Lean proof. The second is the book's Example 2.2.23, where $X\\sim U(0,1)$ and $Y=10X^2+W$ with $W\\sim N(0,1)$ independent of $X$, so that $\\operatorname{V}[X]=1/12$ and $\\operatorname{Cov}[X,Y]=5/6$; its entries are computed in exact arithmetic. In each case both functions satisfy the covariance equation, and only the first is the BLP.\n\n\\begin{audittable}{llrr}{Two linear functions with the same covariance with $X$ and different mean squared errors, in each of two settings.}\nSetting & $g(x)$ & $\\operatorname{Cov}[X,g(X)]$ & MSE \\\\\n$X=Y=\\pm1$ & $x$ (the BLP) & $1$ & $0$ \\\\\n$X=Y=\\pm1$ & $x+1$ & $1$ & $1$ \\\\\nExample 2.2.23 & $-5/3+10x$ (the BLP) & $5/6$ & $14/9$ \\\\\nExample 2.2.23 & $10x$ & $5/6$ & $13/3$ \\\\\n\\end{audittable}", "problem": "In the book a linear function has the form $g(X)=a+bX$, with a free intercept (p.~76). The footnote presents the covariance equation as identifying both coefficients of the BLP, but the equation determines the slope and leaves the intercept free. For all real $a$ and $b$, I have $\\operatorname{Cov}[X,a+bX]=b\\operatorname{V}[X]$. The covariance equation therefore holds exactly when $b=\\operatorname{Cov}[X,Y]/\\operatorname{V}[X]$, whatever the value of $a$. Infinitely many linear functions satisfy it, and one of them is the BLP. Each of the others differs from the BLP by a nonzero constant $u$, and its mean squared error (MSE) exceeds that of the BLP by $u^2$. The equation that fixes the intercept is $\\operatorname{E}[g(X)]=\\operatorname{E}[Y]$. The proof of Theorem 2.2.21 uses this equation, and the footnote omits it.", "what_survives": "Theorem 2.2.21 with its intercept and slope formulas (pp.~76--78), the properties of deviations from the BLP in Theorem 2.2.22 (p.~79), the BLP $-5/3+10X$ of Example 2.2.23 (pp.~79--81) and the regression coefficient formulas of Section 4.1.1 (p.~144) are unaffected. The second sentence of footnote 21 needs the almost-sure qualification in \\fid{P001-C02-E004}. A search of the book's text finds no later passage that uses the covariance equation alone to characterize the BLP." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-E001", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Screenshot of the seven lines of Example 2.2.24, BLP under Nonlinearity. A red box encloses the term X squared in the equation E[Y|X] = X squared on the third line of the example.", "caption": "Example 2.2.24 on p. 82 (PDF p. 100). The red box marks the right-hand side $X^2$ of the conditional expectation.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-E003.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T46", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T46" } ], "headline": "The conditional expectation function in Example 2.2.24 is $10X^2$, not $X^2$.", "id": "P001-C02-E003", "location": "Example 2.2.24, p. 82 (PDF p. 100), second sentence", "pdf_page": 100, "printed_page": "82", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T46" ], "text": { "as_printed": "Suppose that, as in Example 2.2.23, $Y=10X^2+W$, $W\\sim N(0,1)$ and $X\\perp\\!\\!\\!\\perp W$. If $X$ is uniformly distributed between zero and one, the CEF of $Y$ given $X$ is the same as above: $\\operatorname{E}[Y\\mid X]=\\hl{X^2}$.", "corrected": "Suppose that, as in Example 2.2.23, $Y=10X^2+W$, $W\\sim N(0,1)$ and $X\\perp\\!\\!\\!\\perp W$. If $X$ is uniformly distributed between zero and one, the CEF of $Y$ given $X$ is the same as above: $\\operatorname{E}[Y\\mid X]=\\hl{10X^2}$.", "evidence": "The evidence is a derivation. Since $W$ is independent of $X$ and $\\operatorname{E}[W]=0$, I have\n\\[\n\\operatorname{E}[Y\\mid X]=10X^2+\\operatorname{E}[W\\mid X]=10X^2 .\n\\]\nAt $X=1/2$ the CEF equals $5/2$ and the printed formula equals $1/4$.", "problem": "The conditional expectation drops the factor of ten in the stated outcome equation. Example 2.2.23 derives $\\operatorname{E}[Y\\mid X]=10X^2$ for the same model on p.~79, and the words ``the same as above'' refer to that formula.", "what_survives": "Everything else is unaffected. The BLP $-5/3+10X$ derived on pp.~80--81 and the black curve of Figures 2.2.1 and 2.2.2 use $10X^2$." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-E003", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Screenshot of the last two bullets of Theorem 2.2.25. The bullet reads: If g is a function of X and h is a function of Y, then. A red box encloses the first sub-bullet, E[g(Y) | h(X)] = E[g(Y)]. The second sub-bullet reads: The BLP of h(Y) given g(X) is E[h(Y)].", "caption": "Theorem 2.2.25, continued on p. 83 (PDF p. 101). The red box marks the first sub-bullet, where $g$ is applied to $Y$ and $h$ to $X$; the sub-bullet below it uses $h(Y)$ and $g(X)$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-E006.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T47", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T47" } ], "headline": "In the last part of Theorem 2.2.25 the names of the two functions are interchanged: the identity is $\\operatorname{E}[h(Y)\\mid g(X)]=\\operatorname{E}[h(Y)]$.", "id": "P001-C02-E006", "location": "Theorem 2.2.25, p. 83 (PDF p. 101), fourth bullet, first sub-bullet", "pdf_page": 101, "printed_page": "83", "reaches": [], "related_entries": [], "revisions": [ { "classification_after": { "class": "error", "severity": "Minor" }, "classification_before": { "class": "error", "severity": "Minor" }, "note": "The measurability scope was revised; the classification is unchanged (error, Minor). How this came to light: the Lean work showed that measurability of the composed variables on an arbitrary sample space alone does not ensure that transformations preserve independence. Why it changed: the corrected identity needs a qualification on the transformations under the laws of the variables to which they are applied. The wording now requires agreement almost surely with Borel measurable functions under those laws, retaining measurable compositions and the required moments. Global Borel measurability is a sufficient special case. The function-name correction is unchanged. P. M. Aronow delegated this decision to the model session that runs the audit, which adopted the amendment after two checks; this note was written by that session." } ], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T47" ], "text": { "as_printed": "If $X$ and $Y$ are independent random variables, then [\\ldots] If $g$ is a function of $X$ and $h$ is a function of $Y$, then $\\operatorname{E}\\bigl[\\hl{g(Y)}\\bigm|\\hl{h(X)}\\bigr]=\\operatorname{E}\\bigl[\\hl{g(Y)}\\bigr]$. The BLP of $h(Y)$ given $g(X)$ is $\\operatorname{E}\\bigl[h(Y)\\bigr]$.", "corrected": "If $X$ and $Y$ are independent random variables, $g$ and $h$ agree almost surely under the respective laws of $X$ and $Y$ with Borel measurable functions, $g(X)$ and $h(Y)$ are random variables, and the displayed moments exist, then $\\operatorname{E}\\bigl[\\hl{h(Y)}\\bigm|\\hl{g(X)}\\bigr]=\\operatorname{E}\\bigl[\\hl{h(Y)}\\bigr]$, and the BLP of $h(Y)$ given $g(X)$ is $\\operatorname{E}\\bigl[h(Y)\\bigr]$.", "evidence": "The evidence is an exact counterexample to the printed expression. Let $X$ be uniform on $\\{0,1\\}$ and $Y$ uniform on $\\{2,3\\}$, with $X$ and $Y$ independent, and let $g$ and $h$ be the identity functions defined on $\\{0,1\\}$ and on $\\{2,3\\}$. Here $g$ is a function of $X$ and $h$ is a function of $Y$, but $g(Y)$ and $h(X)$ are undefined at every outcome. The corrected expression is defined, and $\\operatorname{E}[h(Y)\\mid g(X)]=5/2=\\operatorname{E}[h(Y)]$. If $g$ and $h$ agree almost surely under the respective laws of $X$ and $Y$ with Borel measurable functions, $g(X)$ and $h(Y)$ are random variables, and the displayed moments exist, then $g(X)$ and $h(Y)$ are independent when $X$ and $Y$ are, and the conditional mean of $h(Y)$ given $g(X)$ equals its mean.", "problem": "The theorem declares $g$ a function of $X$ and $h$ a function of $Y$, and the first sub-bullet then applies $g$ to $Y$ and $h$ to $X$. By Definition 1.2.2 (p.~16), a function of $X$ need only be defined on a set that contains the range of $X$, and hence $g(Y)$ and $h(X)$ need not be defined. The next sub-bullet uses $h(Y)$ and $g(X)$, as declared. When $g(Y)$ and $h(X)$ are random variables with the required moments and $g$ and $h$ agree almost surely with Borel measurable functions under the laws of $Y$ and $X$, respectively, the printed identity is true, and the error lies in the names.", "what_survives": "The conclusions of Theorem 2.2.25 as intended and the BLP sub-bullet that follows are unaffected. The printed identity also holds whenever $g(Y)$ and $h(X)$ are random variables with the required moments and $g$ and $h$ agree almost surely with Borel measurable functions under the laws of $Y$ and $X$, respectively." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-E006", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Screenshot of panel (a) of Figure 2.2.2. A black parabola with minimum at zero runs across the whole plot from x = -1 to x = 1, and a magenta straight line rises to the right. The density strip under the plot is filled only between 0 and 1. A red box encloses the part of the parabola to the left of x = 0.", "caption": "Figure 2.2.2, panel (a), on p. 83 (PDF p. 101). The red box marks the stretch of the black curve over negative values of $x$, where the density of $X$ shown in the strip below is zero.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-N007.png" } ], "formal_release": null, "formal_statements": [], "headline": "In panel (a) of Figure 2.2.2 the black curve is the CEF only on the support of $X$; over $[-1,0)$ it continues the formula $10x^2$.", "id": "P001-C02-N007", "location": "Figure 2.2.2, panel (a), p. 83 (PDF p. 101), with the sentences of Example 2.2.24 on p. 82 (PDF p. 100) that describe it", "pdf_page": 101, "printed_page": "83", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "However, we see in panel (a) of Figure 2.2.2 that the BLP approximates the CEF reasonably well only over certain areas. Additionally, if the distribution of $X$ changes, the BLP changes. Figure 2.2.2 demonstrates how, with \\hl{the same (nonlinear) CEF (in black)}, the BLP (in red) depends on the distribution of $X$.\n\nFIGURE 2.2.2. \\emph{Plotting the CEF and the BLP over Different Distributions of $X$}", "corrected": "Caption: FIGURE 2.2.2. \\emph{Plotting the CEF and the BLP over Different Distributions of $X$.} \\hl{The black curve is} $\\hl{10x^2}$\\hl{; it equals the CEF on the support of} $\\hl{X}$ \\hl{and shows a polynomial continuation elsewhere.}\n\nExample 2.2.24: Figure 2.2.2 demonstrates how, with \\hl{the same CEF formula on each support} (in black), the BLP (in red) depends on the distribution of $X$.", "intended_meaning": "The black curve is the function $10x^2$ given by the model $Y=10X^2+W$, drawn on a common axis while the distribution of $X$ changes from panel to panel. The book defines the CEF only on the support of $X$ (Definition 2.2.15, p.~69). In panel (a) $X$ is uniform on $[0,1]$. At $x=-1$ the density of $X$ is zero, and the ratio that defines the conditional PDF (Definition 1.3.14, p.~38) is $0/0$. The stretch of the curve over $[-1,0)$ is therefore the same formula continued, not a conditional expectation. On the support the figure is correct. For $x\\in[0,1]$ the conditional density of $Y$ given $X=x$ is the $N(10x^2,1)$ density, and the CEF is $10x^2$. The BLP is $-5/3+10x$, as derived on p.~81. The comparison across panels also holds, since for $X$ uniform on $[-1,1]$ the BLP in the same model is the constant $10/3$. A search of the book's text finds no later use of Example 2.2.24 or of Figure 2.2.2." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-N007", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "recalculation_from_source", "exhibits": [ { "alt": "Screenshot of the four panels of Figure 2.2.2 with its caption. Each panel shows a black parabola, a magenta straight line and a density strip below. A red box encloses the horizontal magenta line of panel (b), which sits near 3.4, and another encloses the descending magenta line of panel (c), which runs from about 6 at x = -1 to about -2 at x = 1.", "caption": "Figure 2.2.2 on p. 83 (PDF p. 101). The red boxes mark the BLP lines of panels (b) and (c), which are drawn slightly away from $10/3$ and from $2-4x$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-N008.png" } ], "formal_release": null, "formal_statements": [], "headline": "In panels (b) and (c) of Figure 2.2.2 the red lines for the BLP lie 2 to 3 percent away from the exact BLPs, $10/3$ and $2-4x$, for the densities drawn beneath them.", "id": "P001-C02-N008", "location": "Figure 2.2.2, panels (b) and (c), p. 83 (PDF p. 101), the red lines", "pdf_page": 101, "printed_page": "83", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "FIGURE 2.2.2. \\emph{Plotting the CEF and the BLP over Different Distributions of $X$}", "corrected": "For the densities drawn, the red line of panel (b) is the horizontal line at height $\\hl{10/3}$, and the red line of panel (c) is $\\hl{2-4x}$. Alternatively, the caption can say that the plotted lines are numerical approximations.", "evidence": "The evidence is a recalculation from the published figure. The endpoints of the two red lines were read from the vector drawing in the PDF and converted to the units of the axes with the tick marks of each panel. For the exact BLPs, let $W$ be independent of $X$ with mean zero. Then the slope and the intercept of the BLP are\n\\[\n\\beta=\\frac{10\\bigl(\\operatorname{E}[X^3]-\\operatorname{E}[X]\\operatorname{E}[X^2]\\bigr)}{\\operatorname{E}[X^2]-\\operatorname{E}[X]^2},\\qquad \\alpha=10\\operatorname{E}[X^2]-\\beta\\operatorname{E}[X] .\n\\]\nIn panel (b) the moments $\\operatorname{E}[X]$, $\\operatorname{E}[X^2]$, $\\operatorname{E}[X^3]$ are $0$, $1/3$, $0$, and I obtain $\\beta=0$ and $\\alpha=10/3$. In panel (c) they are $-1/3$, $1/3$, $-1/5$, so that $\\operatorname{V}[X]=2/9$ and $\\operatorname{Cov}[X,X^2]=-4/45$, and I obtain $\\beta=-4$ and $\\alpha=2$.\n\n\\begin{audittable}{llll}{The red lines of panels (b) and (c) as drawn and the exact BLPs for the densities drawn beneath them.}\nPanel & Density of $X$ on $[-1,1]$ & Line as drawn & Exact BLP \\\\\n(b) & $1/2$ & $3.396$ & $10/3$ \\\\\n(c) & $(1-x)/2$ & $2.037-3.879x$ & $2-4x$ \\\\\n\\end{audittable}\n\nThe figure's code is not available, so the distributions actually used and the cause of the difference are not established. An equally weighted grid of 101 points on $[-1,1]$ yields the constant $17/5=3.4$ for panel (b), which is consistent with a numerical approximation, but the check does not show that such a grid was used.", "problem": "The two panels show the BLP of $Y$ given $X$ in the model $Y=10X^2+W$ for two distributions of $X$ that the text does not name. The density strips show the uniform density $1/2$ on $[-1,1]$ in panel (b) and the density $(1-x)/2$ on $[-1,1]$ in panel (c). For these densities the BLP is the constant $10/3$ in panel (b) and the line $2-4x$ in panel (c). The red lines as drawn are approximately $3.40$ and $2.04-3.88x$. The constant of panel (b) is off by 1.87 percent, and the intercept and slope of panel (c) are off by 1.84 and 3.01 percent. In panel (c) the red line differs from the exact BLP by $-0.084$ at $x=-1$ and by $0.157$ at $x=1$. The text reports no coefficient for either panel.", "what_survives": "The CEF, the formula for the BLP, the worked calculation for $X$ uniform on $[0,1]$ and the point of the figure, that the BLP changes with the distribution of $X$, are unaffected. Only the positions of the two red lines change." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-N008", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Screenshot of Definition 2.3.1, Covariance Matrix, with the displayed K by K matrix of variances and covariances. A red box encloses the symbols for all i, j in the second line, which follow an entry written with the indices k and k prime.", "caption": "Definition 2.3.1 on p. 84 (PDF p. 102). The red box marks the quantifier over $i,j$; the entry before it and the displayed matrix use the indices of the components.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-E008.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T48", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T48" } ], "headline": "The quantifier in Definition 2.3.1 runs over $k,k'$, the indices of the entry, not over $i,j$.", "id": "P001-C02-E008", "location": "Definition 2.3.1, p. 84 (PDF p. 102), first sentence", "pdf_page": 102, "printed_page": "84", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T48" ], "text": { "as_printed": "For a random vector $\\mathbf{X}$ of length $K$, the \\emph{covariance matrix} $\\operatorname{V}[\\mathbf{X}]$ is a matrix whose $(k,k')^{\\text{th}}$ entry is $\\operatorname{Cov}[X_{[k]},X_{[k']}]$, $\\forall\\,\\hl{i,j}\\in\\{1,2,\\ldots,K\\}$.", "corrected": "For a random vector $\\mathbf{X}$ of length $K$, the \\emph{covariance matrix} $\\operatorname{V}[\\mathbf{X}]$ is a matrix whose $(k,k')^{\\text{th}}$ entry is $\\operatorname{Cov}[X_{[k]},X_{[k']}]$, $\\forall\\,\\hl{k,k'}\\in\\{1,2,\\ldots,K\\}$.", "evidence": "The evidence is a check of the page. The displayed matrix and the sentences after the definition, on symmetry and on the diagonal entries, index the components as intended.", "problem": "The quantifier uses $i,j$ although the entries of the covariance matrix are indexed by $k,k'$. The letters $i$ and $j$ appear nowhere else in the definition.", "what_survives": "Everything else is unaffected; the displayed matrix fixes the meaning." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-E008", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Screenshot of Definition 2.3.6, BLP (Multivariate Case). Three red boxes enclose the ellipses in the sums b0 + b1 X[1] + b2 X[2] ... + bK X[K], in the same sum with beta coefficients, and in the sum inside the arg min display; each ellipsis directly follows X[2] with no plus sign.", "caption": "Definition 2.3.6 on p. 86 (PDF p. 104). The red boxes mark the three ellipses that have no plus sign before them.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-N009.png" }, { "alt": "Screenshot of the one-line proof of Theorem 2.3.7. In the numerator of the right-hand fraction a red box encloses the ellipsis in beta0 + beta1 X[1] + beta2 X[2] + ... betaK X[K], which is followed directly by betaK.", "caption": "Proof of Theorem 2.3.7 on p. 87 (PDF p. 105). The red box marks the ellipsis that has no plus sign after it.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-N009-p87.png" }, { "alt": "Screenshot of Theorem 2.3.2, Multivariate Variance Rule. Two red boxes enclose the italic letters V at the start of the first and second expressions of the displayed equation.", "caption": "Theorem 2.3.2 on p. 84 (PDF p. 102). The red boxes mark the two variance symbols set in italic.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-N009-p84.png" }, { "alt": "Screenshot of one line of text and the displayed fraction for beta that follows it. The line reads: X is distributed U(0,1), so its PDF is f sub X of x = 1, for all x in [0,1], and f(x) = 0 otherwise, so. A red box encloses f(x).", "caption": "Example 2.2.23 on p. 80 (PDF p. 98). The red box marks the density written without its subscript.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-N009-p80.png" } ], "formal_release": null, "formal_statements": [], "headline": "Four formulas carry small misprints: a plus sign is missing beside an ellipsis in Definition 2.3.6 and in the proof of Theorem 2.3.7, the variance in Theorem 2.3.2 is set as an italic $V$, and a density in Example 2.2.23 has lost its subscript.", "id": "P001-C02-N009", "location": "Definition 2.3.6, p. 86 (PDF p. 104), the three sums; also the proof of Theorem 2.3.7, p. 87 (PDF p. 105); Theorem 2.3.2, p. 84 (PDF p. 102); Example 2.2.23, p. 80 (PDF p. 98)", "pdf_page": 104, "printed_page": "86", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\begin{itemize}\n\\item Definition 2.3.6, p.~86, in the two formulas for $g(\\mathbf{X})$ and again inside the arg min: $g(\\mathbf{X})=b_0+b_1X_{[1]}+b_2X_{[2]}\\hl{\\ldots}+b_KX_{[K]}$ and $g(\\mathbf{X})=\\beta_0+\\beta_1X_{[1]}+\\beta_2X_{[2]}\\hl{\\ldots}+\\beta_KX_{[K]}$.\n\\item Proof of Theorem 2.3.7, p.~87, in the numerator of the derivative: $\\beta_0+\\beta_1X_{[1]}+\\beta_2X_{[2]}+\\hl{\\ldots}\\beta_KX_{[K]}$.\n\\item Theorem 2.3.2, p.~84, with both variances set in italic: $\\hl{V}\\bigl[X_{[1]}+X_{[2]}+\\cdots+X_{[K]}\\bigr]=\\hl{V}\\Bigl[\\sum_{k=1}^{K}X_{[k]}\\Bigr]=\\sum_{k=1}^{K}\\sum_{k'=1}^{K}\\operatorname{Cov}\\bigl[X_{[k]},X_{[k']}\\bigr]$.\n\\item Example 2.2.23, p.~80: its PDF is $f_X(x)=1$, $\\forall x\\in[0,1]$, and $\\hl{f(x)}=0$ otherwise.\n\\end{itemize}", "corrected": "\\begin{itemize}\n\\item Definition 2.3.6, in all three places: $g(\\mathbf{X})=b_0+b_1X_{[1]}+b_2X_{[2]}\\mathbin{\\hl{+}}\\cdots+b_KX_{[K]}$ and $g(\\mathbf{X})=\\beta_0+\\beta_1X_{[1]}+\\beta_2X_{[2]}\\mathbin{\\hl{+}}\\cdots+\\beta_KX_{[K]}$.\n\\item Proof of Theorem 2.3.7: $\\beta_0+\\beta_1X_{[1]}+\\beta_2X_{[2]}+\\cdots\\mathbin{\\hl{+}}\\beta_KX_{[K]}$.\n\\item Theorem 2.3.2: $\\hl{\\operatorname{V}}\\bigl[X_{[1]}+X_{[2]}+\\cdots+X_{[K]}\\bigr]=\\hl{\\operatorname{V}}\\Bigl[\\sum_{k=1}^{K}X_{[k]}\\Bigr]=\\sum_{k=1}^{K}\\sum_{k'=1}^{K}\\operatorname{Cov}\\bigl[X_{[k]},X_{[k']}\\bigr]$.\n\\item Example 2.2.23: its PDF is $f_X(x)=1$, $\\forall x\\in[0,1]$, and $\\hl{f_X(x)}=0$ otherwise.\n\\end{itemize}", "evidence": "The evidence is a check of pp.~80, 84, 86 and 87. The paragraph after Definition 2.3.6 prints the same sums with every plus sign, and Definition 2.3.1 on p.~84 uses the upright $\\operatorname{V}$.", "problem": "Each ellipsis continues a sum, and the plus sign before it (p.~86) or after it (p.~87) is missing. In Theorem 2.3.2 the variance is set as an italic $V$, where the book otherwise uses the upright $\\operatorname{V}$. On p.~80 the second $f$ denotes the same density $f_X$ and has lost its subscript. Each formula keeps its evident meaning.", "what_survives": "Everything else is unaffected; nothing depends on these formulas as misprinted." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-N009", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Screenshot of five lines of text from the top of page 87. Four red boxes enclose, in each of the two expansions beta0 + beta1 x[1] + beta2 x[2] ... + betaK x[k], the ellipsis that follows x[2] and the lower-case subscript [k] of the last term. The argument lists of g on the same lines end in x[K] with a capital K.", "caption": "First paragraph of p. 87 (PDF p. 105). The red boxes mark, in each of the two expansions of $g$, the ellipsis with no plus sign before it and the lower-case subscript $[k]$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C02-N010.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T49", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T49" } ], "headline": "In the two expansions of $g$ on p. 87 the last term is $\\beta_Kx_{[K]}$, not $\\beta_Kx_{[k]}$, and a plus sign belongs before each ellipsis.", "id": "P001-C02-N010", "location": "p. 87 (PDF p. 105), first paragraph, the two expansions of $g$", "pdf_page": 105, "printed_page": "87", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T49" ], "text": { "as_printed": "[\\ldots] so that $g(x_{[1]},x_{[2]},\\ldots,x_{[K]})=\\beta_0+\\beta_1x_{[1]}+\\beta_2x_{[2]}\\hl{\\ldots}+\\beta_K\\hl{x_{[k]}}$, and $g(x_{[1]}+1,x_{[2]},\\ldots,x_{[K]})=\\beta_0+\\beta_1(x_{[1]}+1)+\\beta_2x_{[2]}\\hl{\\ldots}+\\beta_K\\hl{x_{[k]}}$.", "corrected": "[\\ldots] so that $g(x_{[1]},x_{[2]},\\ldots,x_{[K]})=\\beta_0+\\beta_1x_{[1]}+\\beta_2x_{[2]}\\mathbin{\\hl{+}}\\cdots+\\beta_K\\hl{x_{[K]}}$, and $g(x_{[1]}+1,x_{[2]},\\ldots,x_{[K]})=\\beta_0+\\beta_1(x_{[1]}+1)+\\beta_2x_{[2]}\\mathbin{\\hl{+}}\\cdots+\\beta_K\\hl{x_{[K]}}$.", "evidence": "The evidence is a check of the page. Definition 2.3.6 on p.~86 and the derivative display on p.~87 end the same sum in $[K]$. The difference stated in the next sentence, $g(x_{[1]}+1,x_{[2]},\\ldots,x_{[K]})-g(x_{[1]},x_{[2]},\\ldots,x_{[K]})=\\beta_1$, makes the intended sums clear.", "problem": "The last subscript of each expansion is the lower-case $[k]$, although the argument lists of $g$ on the same lines end in $x_{[K]}$. Each sum also lacks the plus sign before its ellipsis.", "what_survives": "Everything else is unaffected, including the conclusion that the difference equals $\\beta_1$." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C02-N010", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "recalculation_from_source", "exhibits": [ { "alt": "Screenshot of the two bullets that end Example 3.1.3: E[X] = 5 and V[X] = 17/2. A red box encloses the first expression after V[X] =, which reads E[(x - E[X])], with a lower-case x and no exponent.", "caption": "Example 3.1.3 (continued) on p. 94 (PDF p. 112). The red box marks the first expression for the variance, which has a lower-case $x$ and no square.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-E003.png" }, { "alt": "Screenshot of the four-line display that begins V[X] = T_V(F). The third line is the integral of x squared dF(x) minus E[X] squared. In the fourth line a red box encloses a plus sign between the integral of x squared dF(x) and the square of the integral of x dF(x).", "caption": "The display for the variance functional on p. 118 (PDF p. 136). The red box marks the plus sign in the last line; the line above it has the minus sign.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-E003-p118.png" }, { "alt": "Screenshot of the end of Example 3.3.4 and the whole of Example 3.3.5. One red box encloses the summand x in the sum from i = 1 to n that is set equal to X bar. A second red box encloses the plus sign in V[X] = T_V(F) = integral of x squared dF(x) + (integral of x dF(x)) squared. The last display, for the plug-in estimator, has a minus sign in the same position.", "caption": "Examples 3.3.4 and 3.3.5 on p. 119 (PDF p. 137). The red boxes mark the unindexed summand $x$ in the last line of Example 3.3.4 and the plus sign in the first display of Example 3.3.5; the plug-in display below it has the minus sign.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-E003-p119.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T50", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T50" } ], "headline": "Three displays misstate quantities that are defined correctly beside them: the variance on p. 94 is $\\operatorname{E}[(X-\\operatorname{E}[X])^2]$, the variance functional on pp. 118–119 subtracts the squared mean, and the last sum for the sample mean on p. 119 is over $X_i$.", "id": "P001-C03-E003", "location": "Example 3.1.3 (continued), p. 94 (PDF p. 112), variance bullet; the display for the variance functional, p. 118 (PDF p. 136), repeated in Example 3.3.5, p. 119 (PDF p. 137); Example 3.3.4, p. 119 (PDF p. 137), last line", "pdf_page": 112, "printed_page": "94", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T50" ], "text": { "as_printed": "Example 3.1.3 (continued), p.~94:\n\\[\n\\operatorname{V}[X]=\\operatorname{E}\\bigl[\\hl{(x-\\operatorname{E}[X])}\\bigr]=\\sum_{x}\\bigl(x-\\operatorname{E}[X]\\bigr)^2f(x)=\\sum_{x\\in\\mathcal{X}}(x-\\mu)^2f_{FP}(x)=\\sigma^2=\\frac{17}{2}.\n\\]\nThe variance functional, p.~118, whose last expression is repeated in Example 3.3.5 on p.~119:\n\\[\n\\begin{aligned}\n\\operatorname{V}[X]&=T_{\\mathrm{V}}(F)\\\\\n&=\\int_{-\\infty}^{\\infty}\\bigl(x-\\operatorname{E}[X]\\bigr)^2dF(x)\\\\\n&=\\int_{-\\infty}^{\\infty}x^2dF(x)-\\operatorname{E}[X]^2\\\\\n&=\\int_{-\\infty}^{\\infty}x^2dF(x)\\mathbin{\\hl{+}}\\Bigl(\\int_{-\\infty}^{\\infty}x\\,dF(x)\\Bigr)^2.\n\\end{aligned}\n\\]\nExample 3.3.4, p.~119, last two lines of the display:\n\\[\n=\\frac{1}{n}\\sum_{x}x\\,(\\text{number of observations }i\\text{ with outcome }X_i=x)=\\frac{1}{n}\\sum_{i=1}^{n}\\hl{x}=\\overline{X},\n\\]", "corrected": "Example 3.1.3 (continued), p.~94:\n\\[\n\\operatorname{V}[X]=\\operatorname{E}\\bigl[(\\hl{X}-\\operatorname{E}[X])^{\\hl{2}}\\bigr]=\\sum_{x}\\bigl(x-\\operatorname{E}[X]\\bigr)^2f(x)=\\sum_{x\\in\\mathcal{X}}(x-\\mu)^2f_{FP}(x)=\\sigma^2=\\frac{17}{2}.\n\\]\nThe variance functional, p.~118, and the first display of Example 3.3.5, p.~119:\n\\[\n\\begin{aligned}\n\\operatorname{V}[X]&=T_{\\mathrm{V}}(F)\\\\\n&=\\int_{-\\infty}^{\\infty}\\bigl(x-\\operatorname{E}[X]\\bigr)^2dF(x)\\\\\n&=\\int_{-\\infty}^{\\infty}x^2dF(x)-\\operatorname{E}[X]^2\\\\\n&=\\int_{-\\infty}^{\\infty}x^2dF(x)\\mathbin{\\hl{-}}\\Bigl(\\int_{-\\infty}^{\\infty}x\\,dF(x)\\Bigr)^2.\n\\end{aligned}\n\\]\nExample 3.3.4, p.~119, last two lines of the display:\n\\[\n=\\frac{1}{n}\\sum_{x}x\\,(\\text{number of observations }i\\text{ with outcome }X_i=x)=\\frac{1}{n}\\sum_{i=1}^{n}\\hl{X_i}=\\overline{X},\n\\]", "evidence": "The evidence is a recalculation with the population $(3,4,3,10)$ of Example 3.1.3. I have $\\operatorname{E}[X]=5$ and $\\operatorname{E}[X^2]=(9+16+9+100)/4=67/2$, so that\n\\[\n\\operatorname{V}[X]=\\frac{67}{2}-25=\\frac{17}{2},\n\\]\nwhich is the value printed on p.~94. For the same population $\\operatorname{E}[X-\\operatorname{E}[X]]=0$, and the expression with the plus sign equals $67/2+25=117/2$. For the sample mean, I group the observations by their value and obtain\n\\[\n\\frac{1}{n}\\sum_{x}x\\,\\#\\{i:X_i=x\\}=\\frac{1}{n}\\sum_{i=1}^{n}X_i .\n\\]\nFor the sample $(1,3)$ both sides equal $2$. The third line of the display on p.~118 and the plug-in display of Example 3.3.5 on p.~119 have the minus sign.", "problem": "On p.~94 the first expression for $\\operatorname{V}[X]$ is the expectation of the centered variable without the square, and it has the lower-case $x$ where the random variable $X$ belongs. Since the expectation of $X-\\operatorname{E}[X]$ is zero, the first equality of the line is false as printed. On p.~118 the last line of the variance functional adds the squared mean instead of subtracting it, against the line above it, and Example 3.3.5 repeats the plus sign. In the last line of Example 3.3.4 the sum runs over $i$ and its summand is an unindexed $x$, which makes the sum equal to $x$ and not to the sample mean. In each case the adjacent expressions state the intended quantity.", "what_survives": "The value $\\sigma^2=17/2$ on p.~94, the remaining expressions of each display and everything else are unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-E003", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Seven lines of running text that explain convergence in probability. A red box marks the expression 'p in [0,1]', with a closed bracket at 1, in the sentence that begins 'More specifically'.", "caption": "Paragraph after Definition 3.2.6, p. 99 (PDF p. 117). The marked interval includes the endpoint $p=1$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-E005.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T51", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T51" } ], "headline": "The verbal description of convergence in probability holds for every $p\\in[0,1)$; at the endpoint $p=1$ it is false.", "id": "P001-C03-E005", "location": "Paragraph after Definition 3.2.6, p. 99 (PDF p. 117), sentence beginning “More specifically”", "pdf_page": 117, "printed_page": "99", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T51" ], "text": { "as_printed": "More specifically, for any $p\\in[0,1\\hl{]}$ and any $\\epsilon>0$, there is always an $N$ large enough that, for every $n\\ge N$, the probability that $T_{(n)}$ will be within $\\epsilon$ of $c$ is at least $p$.", "corrected": "More specifically, for any $p\\in[0,1\\hl{)}$ and any $\\epsilon>0$, there is always an $N$ large enough that, for every $n\\ge N$, the probability that $T_{(n)}$ will be within $\\epsilon$ of $c$ is at least $p$.", "evidence": "The evidence is an exact counterexample to the case $p=1$ and a derivation of the case $p<1$. Let $T_{(n)}$ take the value $1$ with probability $1/(n+1)$ and the value $0$ otherwise, and set $c=0$. For $0<\\epsilon\\le1$, I have $\\Pr[|T_{(n)}|\\ge\\epsilon]=1/(n+1)$, which tends to $0$, and for $\\epsilon>1$ this probability is $0$. I conclude that $T_{(n)}\\xrightarrow{p}0$. For $\\epsilon=1/2$, I have $\\Pr[|T_{(n)}|<\\epsilon]=n/(n+1)<1$ for all $n$, and there is no $N$ such that the sentence holds with $p=1$.\n\nNow fix $p<1$ and $\\epsilon>0$, and set $q_n=\\Pr[|T_{(n)}-c|<\\epsilon]$. Since $q_n\\to1$, there exists $N$ such that $|q_n-1|<1-p$ for all $n\\ge N$, which implies that $q_n>p$. The sentence holds for every $p\\in[0,1)$ with no further hypothesis.", "problem": "By Definition 3.2.6, the statement $T_{(n)}\\xrightarrow{p}c$ means that $\\Pr[|T_{(n)}-c|<\\epsilon]$ tends to $1$ as $n\\to\\infty$, for all $\\epsilon>0$. A sequence of probabilities that tends to $1$ is eventually larger than any fixed $p<1$, but it need not equal $1$ for any finite $n$. For $p=1$ the sentence asserts that the probability equals $1$ for every $n\\ge N$, and convergence in probability does not imply this.", "what_survives": "Definition 3.2.6, the results on convergence in probability and their proofs are unchanged. Only the bracket in this sentence changes." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-E005", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Two lines of text and one displayed formula. A red box marks 'by the WLLN, X-squared-bar converges in probability to E[X squared]'. The display below applies the continuous mapping theorem and ends with V[X] and a proof-end square.", "caption": "End of the proof of Theorem 3.2.19, p. 107 (PDF p. 125). The marked step applies the WLLN to the sample mean of the squares.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-E006.png" }, { "alt": "The statement of Theorem 3.2.8, Weak Law of Large Numbers (WLLN), in a shaded box. Red boxes mark the words 'finite variance V[X] >' at the end of the first line and the '0' that begins the second line. The conclusion is that the sample mean converges in probability to E[X].", "caption": "Theorem 3.2.8, p. 100 (PDF p. 118). The marked hypothesis is finite variance of the variables that are averaged.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-E006-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T53", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T53" }, { "id": "P001-T54", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T54" } ], "headline": "The proof of Theorem 3.2.19 applies the weak law of large numbers for variables with finite variance (Theorem 3.2.8) to the squares $X_i^2$, which need not have finite variance; the step follows from the weak law for integrable variables, and the theorem holds as stated.", "id": "P001-C03-E006", "location": "Proof of Theorem 3.2.19, p. 107 (PDF p. 125), sentence beginning “Now, by the WLLN”; Theorem 3.2.8, p. 100 (PDF p. 118)", "pdf_page": 125, "printed_page": "107", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T53", "P001-T54" ], "text": { "as_printed": "Now, \\hl{by the WLLN}, $\\overline{X^2}\\xrightarrow{p}\\mathrm E[X^2]$ and $\\overline X\\xrightarrow{p}\\mathrm E[X]$. Let $g(u,v)=u-v^2$. Then, by the CMT,\n\\[\n\\overline{X^2}-\\overline X^{\\,2}=g\\bigl(\\overline{X^2},\\overline X\\bigr)\\xrightarrow{p}g\\bigl(\\mathrm E[X^2],\\mathrm E[X]\\bigr)=\\mathrm E[X^2]-\\mathrm E[X]^2=\\mathrm V[X].\\quad\\square\n\\]", "corrected": "Since $\\mathrm E[X^2]<\\infty$, \\hl{the WLLN for i.i.d. integrable random variables} gives $\\overline{X^2}\\xrightarrow{p}\\mathrm E[X^2]$. \\hl{Theorem 3.2.8} gives $\\overline X\\xrightarrow{p}\\mathrm E[X]$. Let $g(u,v)=u-v^2$. Then, by the CMT,\n\\[\n\\overline{X^2}-\\overline X^{\\,2}=g\\bigl(\\overline{X^2},\\overline X\\bigr)\\xrightarrow{p}g\\bigl(\\mathrm E[X^2],\\mathrm E[X]\\bigr)=\\mathrm E[X^2]-\\mathrm E[X]^2=\\mathrm V[X].\\quad\\square\n\\]\nThe law used in the first sentence is the following statement, which the book does not state. Let $Y_1,Y_2,\\ldots,Y_n$ be i.i.d. random variables with $\\mathrm E|Y|<\\infty$, and let $\\overline Y_{(n)}=n^{-1}\\sum_{i=1}^nY_i$. Then $\\overline Y_{(n)}\\xrightarrow{p}\\mathrm E[Y]$.\n\nTheorem 3.2.19 itself is unchanged: for i.i.d. random variables $X_1,X_2,\\ldots,X_n$ with finite variance $\\mathrm V[X]>0$,\n\\[\n\\mathrm E\\bigl[\\overline{X^2}-\\overline X^{\\,2}\\bigr]=\\frac{n-1}{n}\\,\\mathrm V[X]\\qquad\\text{and}\\qquad\\overline{X^2}-\\overline X^{\\,2}\\xrightarrow{p}\\mathrm V[X].\n\\]", "evidence": "The evidence is an exact example in which the cited theorem does not apply, and a derivation of the limit without a fourth moment. Let $\\Pr[X=2^k]=7/8^{k+1}$ for $k=0,1,2,\\ldots$. Summing geometric series, I obtain $\\mathrm E[X]=7/6$, $\\mathrm E[X^2]=7/4$ and $\\mathrm V[X]=7/18>0$. The partial sums of the fourth moment are\n\\[\n\\sum_{k=0}^{N}\\frac{7}{8^{k+1}}\\,2^{4k}=\\frac78\\bigl(2^{N+1}-1\\bigr),\n\\]\nwhich tend to infinity, so that $\\mathrm V[X^2]=\\infty$. This distribution satisfies the assumption of Theorem 3.2.19, and its squares do not satisfy the assumption of Theorem 3.2.8.\n\nFor the limit, let $Y_1,Y_2,\\ldots$ be i.i.d. nonnegative random variables with $m=\\mathrm E[Y]<\\infty$. Fix $M>0$ and set $U_i=\\min(Y_i,M)$, $R_i=Y_i-U_i$, $m_M=\\mathrm E[U_i]$ and $r_M=\\mathrm E[R_i]=m-m_M$. I write $\\overline Y$, $\\overline U$ and $\\overline R$ for the sample means of the first $n$ terms. By monotone convergence, $r_M\\to0$ as $M\\to\\infty$. Since $0\\le U_i\\le M$, I have $\\mathrm V[U_i]\\le M^2$. By Chebyshev's inequality, I obtain $\\Pr[|\\overline U-m_M|\\ge\\epsilon/2]\\le4M^2/(n\\epsilon^2)$ for all $\\epsilon>0$. Markov's inequality implies that $\\Pr[\\overline R\\ge\\epsilon/4]\\le4r_M/\\epsilon$. Since $|\\overline Y-m|\\le|\\overline U-m_M|+\\overline R+r_M$, for all $M$ such that $r_M\\le\\epsilon/4$ I find\n\\[\n\\Pr\\bigl[|\\overline Y-m|\\ge\\epsilon\\bigr]\\le\\frac{4M^2}{n\\epsilon^2}+\\frac{4r_M}{\\epsilon}.\n\\]\nGiven $\\eta>0$, I choose $M$ such that $r_M<\\min(\\epsilon/4,\\eta\\epsilon/8)$ and then $n_0>8M^2/(\\eta\\epsilon^2)$. For all $n\\ge n_0$ the right side is less than $\\eta$, which implies that $\\overline Y\\xrightarrow{p}m$. For integrable variables of either sign, I apply the same argument to the positive and negative parts. By the bound $X^2\\le2(X-\\mathrm E[X])^2+2\\mathrm E[X]^2$, I have $\\mathrm E[X^2]<\\infty$ under the assumption of Theorem 3.2.19, and the argument applies to $Y_i=X_i^2$.", "problem": "Theorem 3.2.19 assumes i.i.d. random variables $X_1,X_2,\\ldots,X_n$ with finite variance $\\mathrm V[X]>0$. The weak law of large numbers stated in the book is Theorem 3.2.8 (p.~100), which the glossary (p.~287) gives as the meaning of the abbreviation WLLN. It assumes that the variables being averaged have finite variance. The proof applies it to the squares $X_i^2$, and this requires $\\mathrm V[X^2]<\\infty$, that is, a finite fourth moment of $X$, which Theorem 3.2.19 does not assume. The cited theorem does not cover the step. Note that the limit $\\overline{X^2}\\xrightarrow{p}\\mathrm E[X^2]$ is true under the stated assumption: the proof has a gap, and the theorem is not false. The consistency parts of Theorems 3.2.21, 3.2.33 and 3.2.35 are deduced from this step, and the interval of Example 3.4.3 (p.~127) and the variance estimator of Section 3.5.2 (p.~140) use it through them.", "what_survives": "Theorem 3.2.19 holds as stated, with no fourth moment and no added assumption, and the computation of the expectation in its proof needs no repair. With the repaired step, the consistency parts of Theorems 3.2.21, 3.2.33 and 3.2.35, the interval of Example 3.4.3 and the consistency claim of Section 3.5.2 hold under their stated conditions. No estimator, estimand or number changes. The simulations of Example 3.4.7 use uniform and Bernoulli distributions, which are bounded, and the gap does not concern them." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-E006", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Three lines of running text that announce an informal statement of a key result (Wasserman 2006, Theorem 2.27), followed by Theorem 3.3.6 in a shaded box: 'If T is well behaved at F and theta-hat = T(F-hat), then theta-hat is asymptotically normal.' A red box marks the words 'is asymptotically normal.'", "caption": "Theorem 3.3.6, p. 120 (PDF p. 138), with the sentence that introduces it. The marked conclusion is stated with no condition on the limiting variance.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-E002.png" }, { "alt": "The seven lines of footnote 31. Red boxes mark its first sentence: 'We say T is well behaved if it is Hadamard differentiable at F with respect to the Kolmogorov distance.'", "caption": "Footnote 31, p. 120 (PDF p. 138). The marked sentence is the definition of “well behaved”.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-E002-c.png" }, { "alt": "Definition 3.2.28, Asymptotic Normality, in a shaded box: an estimator is asymptotically normal if root n times (theta-hat minus theta) converges in distribution to N(0, phi squared), for some finite phi greater than 0. Red boxes mark the words 'for some' and 'finite phi > 0.'", "caption": "Definition 3.2.28, p. 112 (PDF p. 130). The marked condition requires a positive limiting standard deviation.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-E002-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T56", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T56" }, { "id": "P001-T57", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T57" }, { "id": "P001-T118", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T118" } ], "headline": "Theorem 3.3.6 needs a positive limiting variance: Hadamard differentiability of $T$ at $F$ allows a degenerate limit, which is not asymptotic normality as Definition 3.2.28 defines it.", "id": "P001-C03-E002", "location": "Theorem 3.3.6 and footnote 31, p. 120 (PDF p. 138); Definition 3.2.28, p. 112 (PDF p. 130)", "pdf_page": 138, "printed_page": "120", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T118", "P001-T56", "P001-T57" ], "text": { "as_printed": "Theorem 3.3.6. Asymptotic Normality of Plug-In Estimators. If $T$ is well behaved$^{31}$ at $F$ and $\\hat\\theta=T(\\hat F)$, then $\\hat\\theta$ \\hl{is asymptotically normal}.\n\nFootnote 31, first sentence: We say $T$ is well behaved if it is Hadamard differentiable at $F$ with respect to the Kolmogorov distance.\n\nDefinition 3.2.28. Asymptotic Normality. An estimator $\\hat\\theta$ is \\emph{asymptotically normal} if $\\sqrt n(\\hat\\theta-\\theta)\\xrightarrow{d}N(0,\\phi^2)$, for some finite $\\phi>0$.", "corrected": "Theorem 3.3.6. Asymptotic Normality of Plug-In Estimators. For i.i.d. observations with CDF $F$, let $\\hat\\theta=T(\\hat F)$. If $T$ is Hadamard differentiable at $F$ with respect to the Kolmogorov distance, with derivative $\\dot T_F$, \\hl{and $0<\\operatorname{Var}[\\dot T_F(\\mathbb B_F)]<\\infty$, where $\\mathbb B_F$ is the limiting empirical-CDF process}, then $\\hat\\theta$ is asymptotically normal in the sense of Definition 3.2.28. \\hl{Without the positive-variance condition the delta-method limit may be degenerate.}\n\nHere $\\mathbb B_F$ is the centered Gaussian limit of $\\sqrt n(\\hat F-F)$, with covariance $F(\\min(x,y))-F(x)F(y)$. A derivative $\\dot T_F$ that is not zero does not suffice: the variance of $\\dot T_F(\\mathbb B_F)$ must be positive. The book omits the proof of Theorem 3.3.6, and I do not give a proof of the corrected statement.", "evidence": "The evidence is two exact counterexamples. Let $X_1,X_2,\\ldots,X_n$ be independent with $\\Pr[X_i=0]=\\Pr[X_i=1]=1/2$, and let $F$ be their CDF. Both functionals below are defined by their formulas on every bounded function $G$. Let $t_j\\to0$ with $t_j\\ne0$, and let $h_j\\to h$ uniformly.\n\nFirst, set $T_0(G)=7$ for every $G$. For all $j$, I have $(T_0(F+t_jh_j)-T_0(F))/t_j=0$, and $T_0$ is Hadamard differentiable at $F$ with derivative zero. Since $\\theta=\\hat\\theta=7$, it follows that $\\sqrt n(\\hat\\theta-\\theta)=0$ for all $n$. In particular, $\\Pr[\\sqrt n(\\hat\\theta-\\theta)\\le0]=1$ for every $n$, while the CDF of $N(0,\\phi^2)$ is continuous and equals $1/2$ at $0$ for every $\\phi>0$.\n\nSecond, set $T_1(G)=(G(0)-1/2)^2$, which is not constant. Since $F(0)=1/2$, I have\n\\[\n\\frac{T_1(F+t_jh_j)-T_1(F)}{t_j}=t_jh_j(0)^2\\longrightarrow0,\n\\]\nbecause $|h_j(0)|\\le\\lVert h\\rVert_\\infty+1$ for all large $j$. This implies that $T_1$ is Hadamard differentiable at $F$ with derivative zero. Let $K_n$ be the number of observations equal to $0$, and set $S_n=2K_n-n$. Then $\\theta=0$, $\\hat\\theta=(K_n/n-1/2)^2$ and $Y_n=\\sqrt n\\,\\hat\\theta=S_n^2/(4n^{3/2})$. Since $S_n$ is a sum of $n$ independent signs, $\\mathrm E[S_n^4]=3n^2-2n$, and I obtain\n\\[\n\\mathrm E[Y_n^2]=\\frac{3n-2}{16n^2}\\le\\frac{3}{16n}.\n\\]\nSince $\\Pr[|Y_n|\\ge a]\\le\\mathrm E[Y_n^2]/a^2$ for every $a>0$, I conclude that $Y_n\\xrightarrow{p}0$. Further, since $Y_n\\ge0$, I have $\\Pr[Y_n\\le-1]=0$ for all $n$, and $Y_n$ does not converge in distribution to $N(0,\\phi^2)$ for any $\\phi>0$. Summing over the binomial distribution in rational arithmetic for $n=1,\\ldots,64$, I find the same values of $\\mathrm E[Y_n^2]$, for example $1/16$ at $n=1$, $11/512$ at $n=8$ and $95/32768$ at $n=64$.", "problem": "Definition 3.2.28 requires that $\\sqrt n(\\hat\\theta-\\theta)$ converge in distribution to $N(0,\\phi^2)$ for some finite $\\phi>0$, and Definition 2.1.19 (p.~55) defines a normal distribution with positive standard deviation. The hypothesis of Theorem 3.3.6 is that $T$ is Hadamard differentiable at $F$ with respect to the Kolmogorov distance. This does not exclude a derivative that vanishes on the limit of $\\sqrt n(\\hat F-F)$. In that case $\\sqrt n(\\hat\\theta-\\theta)$ converges to the point mass at zero, and the conclusion fails in the sense that the book defines. The paragraph before the theorem calls it an informal statement, but neither that paragraph nor footnote 31 adds a condition on the limiting variance.", "what_survives": "The definitions of the empirical CDF and of plug-in estimators, and the cases in which the limiting variance is positive, are unchanged. Theorem 3.4.2 (p.~125) and Theorem 3.4.6 (p.~129) assume asymptotic normality separately, and the counterexamples do not bear on them. The usual plug-in regularity conditions of Section 3.3.1 are invoked for normal-approximation inference in the bootstrap discussion (p.~131), for regression (pp.~152--154), for the bounds with missing data (p.~212), for doubly robust estimation with missing data (p.~231), for the causal bounds (p.~244) and for doubly robust causal estimation (p.~270). Where these passages need a normal limit with positive variance, the positive-variance condition has to be added. No reported estimate, interval or simulation is shown to be wrong, and I did not examine each of these uses. The sentence about the population mean in the paragraph after the theorem is the subject of \\fid{P001-C03-E001}." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-E002", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "A paragraph of eight lines that begins 'We omit the proof, which requires empirical process theory.' Red boxes mark the clause 'our assumption about finite variance was sufficient to imply that the population mean is well behaved.', which runs over the third and fourth lines.", "caption": "Section 3.3.1, paragraph after Theorem 3.3.6, p. 120 (PDF p. 138). The marked clause says that finite variance makes the population mean well behaved.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-E001.png" }, { "alt": "The seven lines of footnote 31. Red boxes mark its first sentence, 'We say T is well behaved if it is Hadamard differentiable at F with respect to the Kolmogorov distance.', and the expression 'sup over x of |F(x) - G(x)|' in its second sentence.", "caption": "Footnote 31, p. 120 (PDF p. 138). The marked sentence defines “well behaved” as Hadamard differentiability with respect to the Kolmogorov distance, and the marked expression is that distance.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-E001-b.png" }, { "alt": "Two line charts with a common vertical scale from 0 to 1.2 and the index n = 2, 3, 5, 7, 11 on the horizontal axis. Left panel, 'Differentiability at F': blue circles for the size of the direction fall from 0.375 to 0.091, and orange squares for the difference quotient of the mean stay at 1. Right panel, 'Continuity at F': blue circles for the Kolmogorov distance fall from 0.234 to 0.008, and orange squares for the change in the mean stay at 1. Each panel is labeled 'stays at 1' beside the orange line and 'tends to 0' beside the blue line.", "caption": "Exact values for the two families of perturbations of $F$ described in the Evidence section above, at $n=2,3,5,7,11$. Left: the size $\\lVert h_n\\rVert_\\infty$ of the direction tends to $0$ and the difference quotient of the mean equals $1$. Right: the Kolmogorov distance $d_K(F,\\widetilde G_n)$ tends to $0$ and the mean changes by $1$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-E001-quotient.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T1", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T1" }, { "id": "P001-T2", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T2" }, { "id": "P001-T118", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T118" } ], "headline": "Finite variance does not make the population mean well behaved in the sense of footnote 31; the asymptotic normality of the sample mean follows from the Central Limit Theorem (Theorem 3.2.24), not from Hadamard differentiability with respect to the Kolmogorov distance.", "id": "P001-C03-E001", "location": "Section 3.3.1, paragraph after Theorem 3.3.6, p. 120 (PDF p. 138), read with footnote 31", "pdf_page": 138, "printed_page": "120", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P001-T1", "P001-T118", "P001-T2" ], "text": { "as_printed": "We omit the proof, which requires empirical process theory. The conditions for $T$ to be well behaved at $F$ are often implied by other assumptions; for instance, \\hl{our assumption about finite variance was sufficient to imply that the population mean is well behaved}. We will refer to these assumptions, as they apply to the functional under consideration, as the \\emph{usual plug-in regularity conditions}. The usual plug-in regularity conditions can be satisfied by assuming (and even verifying, in some cases) that the functional of interest will be well behaved in the neighborhoods of plausible population distributions.\n\nFootnote 31, first two sentences: We say $T$ is well behaved if it is Hadamard differentiable at $F$ with respect to the Kolmogorov distance. Formally, the Kolmogorov distance between two distributions $F$ and $G$ is given by the value of the largest divergence between their two CDFs, $\\sup_x|F(x)-G(x)|$.", "corrected": "We omit the proof, which requires empirical process theory. The conditions for $T$ to be well behaved at $F$ are often implied by other assumptions. \\hl{For the sample mean, our finite-variance assumption was sufficient for asymptotic normality by the scalar central limit theorem (Theorem 3.2.24). This does not imply that the population-mean functional is Hadamard differentiable with respect to the Kolmogorov distance of footnote 31.} We will refer to \\hl{such differentiability assumptions}, as they apply to the functional under consideration, as the \\emph{usual plug-in regularity conditions}. The usual plug-in regularity conditions can be satisfied by assuming (and even verifying, in some cases) that the functional of interest will be well behaved in the neighborhoods of plausible population distributions.\n\nTheorem 3.2.24 is the statement that replaces the clause. If $X_1,X_2,\\ldots,X_n$ are i.i.d. random variables with finite $\\mathrm E[X]=\\mu$ and finite $\\mathrm V[X]=\\sigma^2>0$, then\n\\[\n\\sqrt n\\,(\\overline X-\\mu)\\xrightarrow{d}N(0,\\sigma^2).\n\\]\nAn argument for the mean through Theorem 3.3.6 would need a topology and a set of directions that control first moments, with the derivative and the convergence of the empirical process established in that setting. I do not give one.", "evidence": "The evidence is an exact counterexample, with a derivation that holds for every $n\\ge2$, and a check of the definition in Wasserman (2006), to which the book refers, and in van der Vaart (1998). Let $F$ put mass $2^{-(k+1)}$ on each integer $k\\ge0$, so that $\\mathrm E_F[X]=1$, $\\mathrm E_F[X^2]=3$ and $\\operatorname{Var}_F X=2$. For an integer $L\\ge1$ and $00$ tending to $0$ and all $h_n\\to h$ uniformly such that $F+t_nh_n$ is a CDF with finite mean. For $h=0$ the limit is $A(0)=0$. Set $p_n=n^{-3}$, $L_n=n$, $t_n=n^{-2}$, $G_n=(1-p_n)F+p_nH_{L_n}$ and $h_n=(G_n-F)/t_n$. Substituting into the display, I obtain\n\\[\n\\begin{aligned}\n\\lVert h_n\\rVert_\\infty&=\\frac{1-2^{-n}}{n}\\longrightarrow0,\\qquad\\operatorname{Var}_{G_n}X=2+n^{-1}-n^{-4}\\longrightarrow2,\\\\\n\\frac{T_{\\mathrm E}(F+t_nh_n)-T_{\\mathrm E}(F)}{t_n}&=\\frac{n^{-2}}{n^{-2}}=1.\n\\end{aligned}\n\\]\nThe difference quotient equals $1$ for all $n$ and does not tend to $0$. I conclude that no such derivative exists. Note that each $G_n$ has finite variance, the same support as $F$, and second moment at most $4$.\n\\begin{audittable}{rrrrr}{The perturbations $G_n$ of $F$: exact values.}\n$n$ & $d_K(F,G_n)$ & $\\lVert h_n\\rVert_\\infty$ & $\\operatorname{Var}_{G_n}X$ & Difference quotient \\\\\n2 & $3/32$ & $3/8$ & $39/16$ & $1$ \\\\\n3 & $7/216$ & $7/24$ & $188/81$ & $1$ \\\\\n5 & $31/4000$ & $31/160$ & $1374/625$ & $1$ \\\\\n7 & $127/43904$ & $127/896$ & $5144/2401$ & $1$ \\\\\n11 & $2047/2725888$ & $2047/22528$ & $30612/14641$ & $1$ \\\\\n\\end{audittable}\nThe mean is not continuous at $F$ either. Set $\\widetilde G_n=(1-n^{-2})F+n^{-2}H_{n^2}$. Then $d_K(F,\\widetilde G_n)=n^{-2}(1-2^{-n^2})\\to0$ and $\\operatorname{Var}_{\\widetilde G_n}X=n^2+1<\\infty$, while $T_{\\mathrm E}(\\widetilde G_n)=2$ and $T_{\\mathrm E}(F)=1$ for every $n$.\n\nThe book gives no formal definition of Hadamard differentiability and refers to Wasserman (2006, Ch.~2). That chapter uses the distance $\\sup_x|F(x)-G(x)|$ over all distribution functions (Theorem 2.27 and Section 2.6), with no restriction on support, moments or directions under which the mean would be differentiable. Its appendix states the definition less precisely than the standard one and does not explicitly require the derivative to be continuous. The counterexample does not refute the clause if that weaker formula is read literally. Under the standard definition, which requires a continuous linear derivative (van der Vaart 1998, Section 20.2), the clause is false.", "problem": "The marked clause concerns the functional $T_{\\mathrm E}(F)=\\int x\\,dF(x)$. Read with footnote 31, it says that finite variance of $F$ makes $T_{\\mathrm E}$ Hadamard differentiable at $F$ with respect to the Kolmogorov distance. Under the standard definition of Hadamard differentiability, which requires a continuous linear derivative, this is false. The Kolmogorov distance does not control the tails of a distribution. A mixture that moves a small fraction of the mass far to the right changes the CDF uniformly by a small amount, and it can move the mean by a fixed amount. Finite variance of $F$ does not prevent this, and neither does finite variance of every perturbed distribution. The clause is the one concrete case that the book offers for the usual plug-in regularity conditions, and finite variance does not ensure that the mean satisfies them. Theorem 3.3.6 cannot be used, on the strength of finite variance alone, to justify the asymptotic normality of the sample mean. Under i.i.d. sampling with finite positive variance, the sample mean is asymptotically normal by Theorem 3.2.24 (p.~109).", "what_survives": "By Theorem 3.2.24 (p.~109), the sample mean is asymptotically normal under i.i.d. sampling with finite positive variance, and that theorem requires no differentiability of the mean functional. Example 3.3.4 (p.~119), which identifies the sample mean as a plug-in estimator, is unchanged, and so are the results on the consistency, unbiasedness and sampling variance of the sample mean. No estimator, estimand, theorem or reported number changes, and no simulation or application estimate was recalculated for this point. The clause is withdrawn. Where a later argument relies on the usual plug-in regularity conditions, they have to be established for the functional under consideration, and no later result is shown to be false. The missing condition in Theorem 3.3.6 itself is the subject of \\fid{P001-C03-E002}." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-E001", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of three lines of text and the shaded box of Theorem 3.3.8, Consistency of the Kernel Density Estimator (Rectangular Kernel). A red box marks the quantifier 'for all x in R' in the statement: For i.i.d. continuous random variables X1, X2, ..., Xn, for all x in R, if K(x) is one half times the indicator that x lies between -1 and 1 (that is, the kernel is rectangular), and if h tends to 0 and nh tends to infinity as n tends to infinity, then f-hat sub K (x) converges in probability to f(x).", "caption": "Theorem 3.3.8, p. 122 (PDF p. 140). The marked quantifier covers every real $x$; the box does not repeat the condition that the PDF is continuous.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-A001.png" }, { "alt": "Crop of four lines of text. Red boxes mark the words 'as long as the PDF really is continuous' in the sentence 'Then, as long as the PDF really is continuous, kernel density estimation will be consistent.'", "caption": "P. 121 (PDF p. 139), last paragraph. The marked words state the condition under which the estimator is consistent; Theorem 3.3.8, on the next page, uses it.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-A001-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T127", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T127" } ], "headline": "Theorem 3.3.8 holds at each point $x$ at which the density $f$ is continuous; the box says “$\\forall x\\in\\mathbb{R}$” and does not repeat the continuity condition stated on p. 121.", "id": "P001-C03-A001", "location": "Theorem 3.3.8, p. 122 (PDF p. 140), boxed statement; the continuity condition is in the last paragraph of p. 121 (PDF p. 139)", "pdf_page": 140, "printed_page": "122", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [ "P001-T127" ], "text": { "as_printed": "P.~121: ``Then, as long as the PDF really is continuous, kernel density estimation will be consistent.''\n\nP.~122: ``Theorem 3.3.8. \\emph{Consistency of the Kernel Density Estimator (Rectangular Kernel)}. For i.i.d.\\ continuous random variables $X_1, X_2, \\ldots, X_n$, $\\hl{\\forall x\\in\\mathbb{R}}$, if $K(x)=\\frac{1}{2}\\mathrm{I}(-1\\le x\\le 1)$ (that is, the kernel is rectangular), and if $h\\to 0$ and $nh\\to\\infty$ as $n\\to\\infty$, then $\\hat f_K(x)\\xrightarrow{p}f(x)$.''", "corrected": "Theorem 3.3.8. For i.i.d.\\ continuous random variables $X_1, X_2, \\ldots, X_n$ with common PDF $f$, \\hl{and for each $x\\in\\mathbb{R}$ at which $f$ is continuous}, if $K(x)=\\frac{1}{2}\\mathrm{I}(-1\\le x\\le 1)$ (that is, the kernel is rectangular), and if $h\\to 0$ and $nh\\to\\infty$ as $n\\to\\infty$, then $\\hat f_K(x)\\xrightarrow{p}f(x)$.\n\nHere $f$ is the density, $h$ the bandwidth and $n$ the sample size. To keep the conclusion for every real $x$, assume that $f$ is continuous on $\\mathbb{R}$.", "intended_meaning": "The theorem is a pointwise statement, at each fixed $x$, and it uses the condition stated in the sentence before it on p.~121: the PDF is continuous. Under that condition the theorem is correct. Fix $x$, and set $q_h=\\Pr[x-h\\le X_i\\le x+h]$ and $a_h=q_h/(2h)$. The count $2nh\\hat f_K(x)$ has the binomial distribution with parameters $n$ and $q_h$, so $\\hat f_K(x)$ has expectation $a_h$ and variance $a_h(1-q_h)/(2nh)$. If $f$ is continuous at $x$, then $a_h\\to f(x)$ as $h\\to 0$, and the variance tends to zero since $nh\\to\\infty$. Chebyshev's inequality then implies that $\\hat f_K(x)\\xrightarrow{p}f(x)$. The limit $a_h\\to f(x)$ is the last display of the proof sketch on p.~123, and it holds whenever $F'(x)$ exists and equals $f(x)$.\n\nRead without p.~121, the box covers every continuous random variable in the sense of Definition 1.2.14 and every real $x$, and the statement fails at a point where the density jumps. For the standard uniform distribution of Example 1.2.18, I have $f(0)=f(1)=1$. For $0 0', and in the sentence 'Then, for all i, j, j-prime, Cov[X ij, X ij-prime] = rho[X ij, X ij-prime] sigma[X ij] sigma[X ij-prime] = rho sigma squared, so'. The first bulleted assumption reads 'for all i, j, V[X ij] = sigma squared'.", "caption": "Example 3.5.8, p. 140 (PDF p. 158). The two marked quantifiers cover every pair $j,j'$, including $j=j'$; the common correlation and the covariance $\\rho\\sigma^2$ are meant for $j\\neq j'$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-E004.png" }, { "alt": "Crop of a numbered list of four steps and the paragraph after it. A red box marks the letter m in step 1: 'Take a with replacement sample of m clusters from our sample, keeping clusters intact.' Step 2 reads 'Calculate our would-be estimate using this collected n observations in our block bootstrap sample.' The paragraph after the list reads 'Since we drew a sample of l i.i.d. clusters to begin with, we are again replicating the sampling process, taking with-replacement samples of l clusters from our sample.'", "caption": "Block bootstrap, p. 141 (PDF p. 159). The marked symbol in step 1 is the number of clusters to resample; the paragraph after the four steps says “samples of $l$ clusters”.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-E004-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T64", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T64" } ], "headline": "In Example 3.5.8 the common correlation $\\rho$ applies to distinct observations of a cluster ($j\\neq j'$), and step 1 of the block bootstrap resamples $l$ clusters, not $m$.", "id": "P001-C03-E004", "location": "Example 3.5.8, p. 140 (PDF p. 158), second assumption and the sentence after the two assumptions; block bootstrap, step 1, p. 141 (PDF p. 159)", "pdf_page": 158, "printed_page": "140", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T64" ], "text": { "as_printed": "P.~140, Example 3.5.8: ``Suppose that:\n\\begin{itemize}\n\\item $\\forall i,j$, $\\mathrm{V}[X_{ij}]=\\sigma^2$.\n\\item $\\hl{\\forall i,j,j'}$, $\\rho[X_{ij},X_{ij'}]=\\rho>0$.\n\\end{itemize}\nIn other words, the variance of each observation is identical, and every unit within a cluster shares the same correlation. Then, $\\hl{\\forall i,j,j'}$, $\\mathrm{Cov}[X_{ij},X_{ij'}]=\\rho[X_{ij},X_{ij'}]\\sigma[X_{ij}]\\sigma[X_{ij'}]=\\rho\\sigma^2$, so \\ldots''\n\nP.~141, the block bootstrap, steps 1 and 2:\n\\begin{enumerate}\n\\item ``Take a \\emph{with replacement} sample of $\\hl{m}$ clusters from our sample, keeping clusters intact.\n\\item Calculate our would-be estimate using this collected $n$ observations in our block bootstrap sample.''\n\\end{enumerate}", "corrected": "P.~140, Example 3.5.8: ``Suppose that:\n\\begin{itemize}\n\\item $\\forall i,j$, $\\mathrm{V}[X_{ij}]=\\sigma^2$.\n\\item $\\forall i,j,j'$ \\hl{with $j\\neq j'$}, $\\rho[X_{ij},X_{ij'}]=\\rho>0$.\n\\end{itemize}\nIn other words, the variance of each observation is identical, and every unit within a cluster shares the same correlation. Then, $\\forall i,j,j'$ \\hl{with $j\\neq j'$}, $\\mathrm{Cov}[X_{ij},X_{ij'}]=\\rho[X_{ij},X_{ij'}]\\sigma[X_{ij}]\\sigma[X_{ij'}]=\\rho\\sigma^2$, so\n\\[\n\\begin{aligned}\n\\mathrm{V}\\bigl[\\overline{X}\\bigr]&=\\frac{1}{lm^2}\\sum_{j=1}^{m}\\sum_{j'=1}^{m}\\mathrm{Cov}[X_{ij},X_{ij'}]\\\\\n&=\\frac{1}{lm^2}\\bigl(m\\sigma^2+m(m-1)\\rho\\sigma^2\\bigr)\\\\\n&=\\frac{\\sigma^2}{lm}\\bigl(1+\\rho(m-1)\\bigr).\\text{''}\n\\end{aligned}\n\\]\n\nP.~141, the block bootstrap, steps 1 and 2:\n\\begin{enumerate}\n\\item ``Take a \\emph{with replacement} sample of $\\hl{l}$ clusters from our sample, keeping clusters intact.\n\\item Calculate our would-be estimate using this collected $n$ observations in our block bootstrap sample.''\n\\end{enumerate}\n\nHere $l$ is the number of clusters, $m$ is the number of observations in each cluster and $n=lm$.", "evidence": "The evidence is a derivation and a check of pp.~135--136 and 140--141. With the restriction $j\\neq j'$, Theorem 3.5.5 implies that\n\\[\n\\mathrm{V}\\bigl[\\overline{X}\\bigr]=\\frac{1}{lm^2}\\sum_{j=1}^{m}\\sum_{j'=1}^{m}\\mathrm{Cov}[X_{ij},X_{ij'}]=\\frac{1}{lm^2}\\bigl(m\\sigma^2+m(m-1)\\rho\\sigma^2\\bigr)=\\frac{\\sigma^2}{lm}\\bigl(1+\\rho(m-1)\\bigr),\n\\]\nwhich is the printed formula. At $\\rho=0$ it equals $\\sigma^2/(lm)$, and at $\\rho=1$ it equals $\\sigma^2/l$. As a check, let $m=3$, $l=5$, $\\sigma^2=2$ and $\\rho=1/4$. The covariance matrix of a cluster has diagonal elements $2$ and off-diagonal elements $1/2$. Computing directly from this matrix, I find $\\mathrm{V}[\\overline{X}]=1/5$, which agrees with the formula.\n\nFor the block bootstrap, let $w_1,w_2,\\ldots,w_l$ be the observed cluster averages, with mean $\\overline{w}$, and set $v=(1/l)\\sum_{i=1}^{l}(w_i-\\overline{w})^2$. A bootstrap sample of $k$ intact clusters holds $km$ observations, and its mean is the average of $k$ independent draws from $w_1,w_2,\\ldots,w_l$. Conditional on the data, the variance of this mean is $v/k$. For the intended choice $k=l$, the sample holds $n$ observations and the variance is $v/l$. For the printed choice $k=m$, the sample holds $m^2$ observations and the variance is $v/m$, which is $l/m$ times the intended variance. For $l=100$ and $m=2$, the printed step collects $4$ observations in place of $200$, and the variance is $50$ times the intended one.", "problem": "On p.~140, the condition of a common correlation within clusters lacks the restriction to distinct observations. As printed, the condition and the covariance statement after it hold for all $j$ and $j'$, including $j=j'$. An observation with positive variance has correlation $1$ with itself and covariance $\\sigma^2$ with itself, so the condition as printed forces $\\rho=1$, and the covariance statement puts $\\rho\\sigma^2$ on the diagonal in place of $\\sigma^2$. The calculation after the statement uses the intended condition: it takes $\\sigma^2$ for the $m$ terms with $j=j'$ and $\\rho\\sigma^2$ for the $m(m-1)$ terms with $j\\neq j'$.\n\nIn step 1 of the block bootstrap on p.~141, the number of resampled clusters is mislabeled. Recall that Section 3.5 has $l$ clusters of size $m$ and total sample size $n=lm$. A sample of $m$ clusters holds $m^2$ observations, which equals the $n=lm$ observations of step 2 only if $m=l$, and the standard deviation in step 4 is then that of an estimate from $m$ clusters. The paragraph after the steps states the intended procedure: ``taking with-replacement samples of $l$ clusters from our sample''.", "what_survives": "The formula $\\mathrm{V}[\\overline{X}]=\\sigma^2(1+\\rho(m-1))/(lm)$, its two extreme cases and the block bootstrap as described in the paragraph after the steps are correct as printed. The example assumes $\\rho>0$ and then considers $\\rho=0$ as an extreme case; this extends the formula to the boundary and is not an error. The condition may be stated as $0\\le\\rho\\le 1$ to cover both extreme cases." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-E004", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Screenshot of the last six lines of Section 3.1.1. A red box encloses the words Dubins and Freedman 1979 inside the parenthesis that follows: Some technical finesse is required to formalize this intuition.", "caption": "Last sentence of Section 3.1.1 on p. 94 (PDF p. 112). The red box marks the citation of Dubins and Freedman (1979).", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N001.png" } ], "formal_release": null, "formal_statements": [], "headline": "For the approximation of sampling without replacement from a large population by i.i.d. sampling, the fitting citation is Diaconis and Freedman (1980), not Dubins and Freedman (1979).", "id": "P001-C03-N001", "location": "Section 3.1.1, p. 94 (PDF p. 112), last sentence", "pdf_page": 112, "printed_page": "94", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "Some technical finesse is required to formalize this intuition (e.g., \\hl{Dubins and Freedman 1979}), but it implies that sampling units at random from a hypothetical, infinite \\emph{superpopulation} yields data that can be treated as i.i.d.", "corrected": "Some technical finesse is required to formalize this intuition (e.g., \\hl{Diaconis and Freedman 1980}), but it implies that sampling units at random from a hypothetical, infinite \\emph{superpopulation} yields data that can be treated as i.i.d.\n\nThe reference to add is: Diaconis, P., and Freedman, D. (1980). Finite exchangeable sequences. \\emph{Annals of Probability} 8(4): 745--764. Dubins and Freedman (1979) can be kept only as a reference for the caveat that de Finetti's representation can fail on general measurable spaces.", "intended_meaning": "The sentence points to the technical work behind the preceding claim, that draws without replacement from a large population are approximately independent and identically distributed. It is not false, and nothing in the book depends on the citation. The paper cited, Dubins and Freedman (1979), ``Exchangeable processes need not be mixtures of independent, identically distributed random variables'', constructs spaces on which de Finetti's representation fails; its summary (p.~115) states: ``This paper demonstrates the existence of a separable metric space for which the conclusion fails.'' It does not give the approximation for finite populations. That approximation is the subject of Diaconis and Freedman (1980), ``Finite exchangeable sequences'', whose abstract states: ``The main tool is a bound on the variation distance between sampling with and without replacement.'' I checked the published 1979 paper and the abstract of the 1980 paper, not the body of the 1980 paper." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-N001", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Footnote 12, one line, followed by the three lines of footnote 13. A red box marks 'for all D in R' in footnote 12, printed with the membership sign.", "caption": "Footnotes 12 and 13, p. 102 (PDF p. 120). The marked quantifier makes $D$ an element of $\\mathbb R$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N004.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T52", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T52" } ], "headline": "In footnote 12 the quantifier ranges over sets $D\\subseteq\\mathbb R$; the footnote prints $D\\in\\mathbb R$.", "id": "P001-C03-N004", "location": "Footnote 12, p. 102 (PDF p. 120)", "pdf_page": 120, "printed_page": "102", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T52" ], "text": { "as_printed": "Note that for any random variable $X$, $\\forall D\\hl{\\in}\\mathbb R$, $\\mathrm E[\\mathrm I(X\\in D)]=\\Pr[X\\in D]$.", "corrected": "Note that for any random variable $X$ \\hl{and any measurable set} $D\\hl{\\subseteq}\\mathbb R$, $\\mathrm E[\\mathrm I(X\\in D)]=\\Pr[X\\in D]$.", "evidence": "The evidence is a check of the page: the printed symbol is $\\in$ and not $\\subseteq$. For a measurable set $D\\subseteq\\mathbb R$, the indicator $\\mathrm I(X\\in D)$ takes the value $1$ with probability $\\Pr[X\\in D]$ and the value $0$ otherwise, so that\n\\[\n\\mathrm E[\\mathrm I(X\\in D)]=1\\cdot\\Pr[X\\in D]+0\\cdot\\Pr[X\\notin D]=\\Pr[X\\in D].\n\\]\nThe proof of Theorem 3.2.9 on the same page computes this expectation for the event $X\\le x$.", "problem": "The event $X\\in D$ is defined when $D$ is a set of real numbers. The footnote quantifies over $D\\in\\mathbb R$, which makes $D$ a real number. Theorem 1.3.17 (p.~39) writes the corresponding quantifier as $\\forall D,E\\subseteq\\mathbb R$.", "what_survives": "The identity is correct for sets. Theorem 3.2.9 and its proof use only the event $X_i\\le x$ and are unchanged." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-N004", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "The three lines of footnote 19. A red box marks the word 'Sterling's' in the phrase 'primarily relies on Sterling's approximation'.", "caption": "Footnote 19, p. 109 (PDF p. 127). The marked name is misspelled.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N005.png" } ], "formal_release": null, "formal_statements": [], "headline": "The approximation named in footnote 19 is Stirling’s approximation.", "id": "P001-C03-N005", "location": "Footnote 19, p. 109 (PDF p. 127)", "pdf_page": 127, "printed_page": "109", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Full proofs of the Central Limit Theorem invariably require fairly advanced mathematics, though the proof of a special case, the De Moivre--Laplace Theorem, is relatively accessible and primarily relies on \\hl{Sterling's} approximation. See, e.g., Feller (1968, Sections II.9, VII.3).", "corrected": "\\ldots\\ is relatively accessible and primarily relies on \\hl{Stirling's} approximation.", "evidence": "The evidence is a check of the page.", "problem": "The approximation $n!\\sim\\sqrt{2\\pi n}\\,(n/e)^n$ for the factorial, which the proof of the De Moivre--Laplace Theorem uses, is Stirling's approximation. The footnote prints the name as ``Sterling's''.", "what_survives": "The content of the footnote and the statement of the Central Limit Theorem are unchanged." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-N005", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "The heading '3.2.5 Asymptotic Estimation Theory' and five lines of text. A red box marks the words 'in probability' in the first line, which reads 'We noted above that when a sample statistic T(n) converges in probability to a'.", "caption": "Opening of Section 3.2.5, p. 111 (PDF p. 129): the first sentence, with “in probability” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N006.png" }, { "alt": "Four lines of text. A red box marks the words 'in distribution' in the sentence 'The distribution of the random variable T to which a sequence of random variables T(n) converges in distribution is referred to as the limit distribution (or asymptotic distribution) of T(n).' The sentence after it, in parentheses, reads 'This notion will be important in Section 3.2.5.'", "caption": "Paragraph after Definition 3.2.22, p. 109 (PDF p. 127): the passage that the sentence on p. 111 recalls, with “in distribution” marked. The sentence after it refers ahead to Section 3.2.5.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N006-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "The first sentence of Section 3.2.5 should say “converges in distribution”, as does the passage on p. 109 that it recalls; it is printed with “converges in probability”.", "id": "P001-C03-N006", "location": "Section 3.2.5, first sentence, p. 111 (PDF p. 129)", "pdf_page": 129, "printed_page": "111", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "We noted above that when a sample statistic $T_{(n)}$ converges \\hl{in probability} to a random variable $T$, the distribution of $T$ is known as the limit distribution (or asymptotic distribution) of $T_{(n)}$.", "corrected": "We noted above that when a sample statistic $T_{(n)}$ converges \\hl{in distribution} to a random variable $T$, the distribution of $T$ is known as the limit distribution (or asymptotic distribution) of $T_{(n)}$.", "evidence": "The evidence is a check of the pages and a derivation. The sentence is true as printed, since convergence in probability implies convergence in distribution. Let $F_{(n)}$ and $F$ be the CDFs of $T_{(n)}$ and $T$, and suppose that $T_{(n)}$ converges in probability to $T$. For every continuity point $t$ of $F$ and every $\\delta>0$, I have\n\\[\nF(t-\\delta)-\\Pr\\bigl[|T_{(n)}-T|>\\delta\\bigr]\\le F_{(n)}(t)\\le F(t+\\delta)+\\Pr\\bigl[|T_{(n)}-T|>\\delta\\bigr].\n\\]\nLetting $n\\to\\infty$ and then $\\delta\\to0$, I find that $F_{(n)}(t)\\to F(t)$, which is the convergence required by Definition 3.2.22. The printed sentence states a sufficient condition, and the passage on p.~109 covers more cases. The standardized sample mean $Z$ of Theorem 3.2.24 is one of them: when each sample consists of the first $n$ terms of one i.i.d.\\ sequence, $Z$ converges in distribution to $N(0,1)$ and does not converge in probability to any random variable.", "problem": "The sentence recalls the paragraph after Definition 3.2.22 (p.~109), which names the limit distribution for a sequence that converges in distribution and refers ahead to Section 3.2.5. The two passages use the same symbols and the same two names, but different modes of convergence. The Central Limit Theorem (p.~109) and Definitions 3.2.26, 3.2.28, 3.2.29, 3.2.30 and 3.2.32 (pp.~112--113) are stated with convergence in distribution. The word ``probability'' on p.~111 is a misprint for ``distribution''.", "what_survives": "No result changes. The sentence is true as printed, and the passage on p.~109, the Central Limit Theorem and the definitions of Section 3.2.5 already use convergence in distribution. The correction changes one word and adds no assumption." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-N006", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "The first half of Definition 3.2.32 in a shaded box. The sentence above the display names the sampling variance estimator V-hat of theta-hat. A red box marks 'n V-hat[theta]', with no hat on theta, at the left of the displayed convergence to V[T].", "caption": "Definition 3.2.32, p. 113 (PDF p. 131), first display. The introducing sentence writes $\\hat{\\mathrm V}[\\hat\\theta]$, and the marked term has no hat on $\\theta$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N007.png" }, { "alt": "The second half of Definition 3.2.32. The sentence names the standard error estimator, the square root of V-hat of theta-hat. In the display below, a red box marks 'V-hat[theta]' under the square root, with no hat on theta.", "caption": "Definition 3.2.32, continued, p. 114 (PDF p. 132), second display. The marked term has no hat on $\\theta$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N007-b.png" }, { "alt": "The two lines of footnote 23. Red boxes mark 'n V-hat[theta]' in the first equivalent expression and 'V-hat[theta]' under the square root in the second.", "caption": "Footnote 23, p. 114 (PDF p. 132). Both marked terms have no hat on $\\theta$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N007-c.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T55", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T55" }, { "id": "P001-T128", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T128" } ], "headline": "In the two displays of Definition 3.2.32 and in footnote 23, the sampling variance estimator is $\\hat{\\mathrm V}[\\hat\\theta]$; the hat on $\\theta$ is missing in four places.", "id": "P001-C03-N007", "location": "Definition 3.2.32, pp. 113–114 (PDF pp. 131–132), both displays; footnote 23, p. 114 (PDF p. 132)", "pdf_page": 131, "printed_page": "113–114", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T128", "P001-T55" ], "text": { "as_printed": "For an estimator $\\hat\\theta$ such that $\\sqrt n(\\hat\\theta-\\theta)\\xrightarrow{d}T$, a sampling variance estimator $\\hat{\\mathrm V}[\\hat\\theta]$ is \\emph{consistent} if\n\\[\nn\\hat{\\mathrm V}[\\hl{\\theta}]\\xrightarrow{p}\\mathrm V[T]\n\\]\nand a standard error estimator $\\sqrt{\\hat{\\mathrm V}[\\hat\\theta]}$ is \\emph{consistent} if\n\\[\n\\sqrt n\\sqrt{\\hat{\\mathrm V}[\\hl{\\theta}]}\\xrightarrow{p}\\sqrt{\\mathrm V[T]}.\n\\]\nFootnote 23: Note that it is equivalent to define these as $n\\hat{\\mathrm V}[\\hl{\\theta}]-\\mathrm V[T]\\xrightarrow{p}0$ and $\\sqrt n\\sqrt{\\hat{\\mathrm V}[\\hl{\\theta}]}-\\sqrt{\\mathrm V[T]}\\xrightarrow{p}0$, respectively.", "corrected": "For an estimator $\\hat\\theta$ such that $\\sqrt n(\\hat\\theta-\\theta)\\xrightarrow{d}T$, a sampling variance estimator $\\hat{\\mathrm V}[\\hat\\theta]$ is \\emph{consistent} if\n\\[\nn\\hat{\\mathrm V}[\\hl{\\hat\\theta}]\\xrightarrow{p}\\mathrm V[T]\n\\]\nand a standard error estimator $\\sqrt{\\hat{\\mathrm V}[\\hat\\theta]}$ is \\emph{consistent} if\n\\[\n\\sqrt n\\sqrt{\\hat{\\mathrm V}[\\hl{\\hat\\theta}]}\\xrightarrow{p}\\sqrt{\\mathrm V[T]}.\n\\]\nFootnote 23: Note that it is equivalent to define these as $n\\hat{\\mathrm V}[\\hl{\\hat\\theta}]-\\mathrm V[T]\\xrightarrow{p}0$ and $\\sqrt n\\sqrt{\\hat{\\mathrm V}[\\hl{\\hat\\theta}]}-\\sqrt{\\mathrm V[T]}\\xrightarrow{p}0$, respectively.", "evidence": "The evidence is a check of the two pages. Theorem 3.4.2 (p.~125), which uses this definition, writes $\\hat{\\mathrm V}[\\hat\\theta]$ throughout.", "problem": "The sentence that introduces each display names $\\hat{\\mathrm V}[\\hat\\theta]$, the estimator of the sampling variance of the estimator $\\hat\\theta$. In both displays and in both expressions of footnote 23 the argument is printed as $\\theta$, the fixed target, which has no sampling variance.", "what_survives": "The definition as intended and Theorem 3.4.2 are unchanged." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-N007", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "The last line of a proof and a paragraph of six lines. Red boxes mark the phrase 'the sampling variance of the sample mean' in the first line of the paragraph and the phrase 'limit distribution of the sample mean' in its third line.", "caption": "Paragraph after the proof of Theorem 3.2.33, p. 115 (PDF p. 133). The two marked phrases name the sampling variance where its estimator is meant, and the sample mean where its centered and scaled version is meant.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N008.png" } ], "formal_release": null, "formal_statements": [], "headline": "The sentence after Theorem 3.2.33 describes the estimator of the sampling variance of the sample mean, and the limit distribution it refers to is that of $\\sqrt n(\\overline X-\\mathrm E[X])$.", "id": "P001-C03-N008", "location": "Paragraph after the proof of Theorem 3.2.33, p. 115 (PDF p. 133)", "pdf_page": 133, "printed_page": "115", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "The first property states that \\hl{the sampling variance of the sample mean} is unbiased, and the second is equivalent to stating that it is consistent (as the \\hl{limit distribution of the sample mean} has variance $\\mathrm V[X]$).", "corrected": "The first property states that \\hl{the estimator of} the sampling variance of the sample mean is unbiased, and the second is equivalent to stating that it is consistent (as the limit distribution of $\\hl{\\sqrt n(\\overline X-\\mathrm E[X])}$ has variance $\\mathrm V[X]$).", "intended_meaning": "The two properties in Theorem 3.2.33 concern the estimator $\\hat{\\mathrm V}[\\overline X]=\\hat{\\mathrm V}[X]/n$: its expectation is $\\mathrm V[\\overline X]$, and $n\\hat{\\mathrm V}[\\overline X]-n\\mathrm V[\\overline X]\\xrightarrow{p}0$. Unbiasedness and consistency are properties of this estimator. The sampling variance $\\mathrm V[\\overline X]=\\mathrm V[X]/n$ is a fixed number. The second property is consistency in the sense of Definition 3.2.32, which refers to the limit distribution of $\\sqrt n(\\overline X-\\mathrm E[X])$. By the Central Limit Theorem (Theorem 3.2.24), this limit is $N(0,\\mathrm V[X])$, with variance $\\mathrm V[X]$. The sample mean without centering and scaling converges in probability to $\\mathrm E[X]$, and its limit distribution has variance zero. Theorem 3.2.33 and Definition 3.2.32, directly above the sentence, give the intended meaning, which is correct. The theorem, its proof and the standard error formulas are unchanged." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-N008", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "The end of Example 3.3.5: the words 'the plug-in estimator of V[X] is', a displayed formula, and a closing sentence. A red box marks the symbol 'V-hat[X]' at the left of the display, which is set equal to T sub V of F-hat, to a difference of two integrals, and to X-squared-bar minus X-bar squared.", "caption": "Example 3.3.5, p. 119 (PDF p. 137), second display. The marked symbol denotes the plug-in sample variance here.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N009.png" }, { "alt": "Definition 3.2.20, Unbiased Sample Variance, in a shaded box. A red box marks the symbol 'V-hat[X]' at the left of the display, which is set equal to n over (n minus 1) times the difference of X-squared-bar and X-bar squared.", "caption": "Definition 3.2.20, p. 107 (PDF p. 125). The same symbol is defined as the unbiased sample variance.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N009-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "Example 3.3.5 writes $\\hat{\\mathrm V}[X]$ for the plug-in sample variance, and Definition 3.2.20 fixes the same symbol as the unbiased sample variance; the two differ by the factor $n/(n-1)$.", "id": "P001-C03-N009", "location": "Example 3.3.5, p. 119 (PDF p. 137), second display; Definition 3.2.20, p. 107 (PDF p. 125)", "pdf_page": 137, "printed_page": "119", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "the plug-in estimator of $\\mathrm V[X]$ is\n\\[\n\\hl{\\hat{\\mathrm V}[X]}=T_{\\mathrm V}(\\hat F)=\\int_{-\\infty}^{\\infty}x^2\\,d\\hat F(x)-\\left(\\int_{-\\infty}^{\\infty}x\\,d\\hat F(x)\\right)^2=\\overline{X^2}-\\overline X^{\\,2}.\n\\]", "corrected": "the plug-in estimator of $\\mathrm V[X]$ is\n\\[\n\\hl{\\hat{\\mathrm V}_{\\mathrm{PI}}[X]}=T_{\\mathrm V}(\\hat F)=\\int_{-\\infty}^{\\infty}x^2\\,d\\hat F(x)-\\left(\\int_{-\\infty}^{\\infty}x\\,d\\hat F(x)\\right)^2=\\overline{X^2}-\\overline X^{\\,2}.\n\\]", "intended_meaning": "The example derives the plug-in sample variance, which its title and its last sentence name, and the formula is correct. The symbol $\\hat{\\mathrm V}[X]$ is defined in Definition 3.2.20 as the unbiased sample variance, $n(\\overline{X^2}-\\overline X^{\\,2})/(n-1)$, and p.~114 uses it with that meaning. In the example the symbol denotes $(n-1)/n$ times that quantity. The later formulas for the standard error of the sample mean use $\\hat{\\mathrm V}[X]$ with the meaning of Definition 3.2.20. With the plug-in sample variance in its place, the standard error would be smaller by the factor $\\sqrt{(n-1)/n}$. No calculation in the book is shown to use the plug-in sample variance where the unbiased one is meant." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-N009", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Three lines of running text. A red box marks the words 'estimate theta', with no hat on theta, in the sentence 'Then, as n grows large, the estimate theta tends to more and more closely approximate theta'.", "caption": "Section 3.3, p. 119 (PDF p. 137). The marked estimate is printed as $\\theta$ without its hat.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N010.png" }, { "alt": "Three lines of running text. A red box marks 'i = 1,2,...N' in the phrase 'a finite population U consisting of N units, indexed i = 1,2,...N'.", "caption": "Section 3.1.1, p. 92 (PDF p. 110). The marked list has no comma after the ellipsis.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N010-b.png" }, { "alt": "Three lines of running text. A red box marks the words 'allow to us to' at the start of the second line.", "caption": "Section 3.2.2, p. 103 (PDF p. 121). The marked phrase has a misplaced “to”.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N010-c.png" }, { "alt": "Three lines of running text. A red box marks the words 'defined it in' in the sentence 'Indeed, this is the same MSE that we defined it in Definition 2.1.22'.", "caption": "Section 3.2.2, p. 104 (PDF p. 122). The marked phrase has an extra “it”.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N010-d.png" }, { "alt": "Four lines of running text. A red box marks 'Section 3.2.2).' at the end of the sentence 'We begin with a simple case: the asymptotic variant of the unbiasedness property from Section 3.2.2).' The parenthesis in the sentence before it is opened and closed.", "caption": "Section 3.2.5, p. 111 (PDF p. 129). The marked closing parenthesis has no opening parenthesis in its sentence.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N010-e.png" }, { "alt": "Four lines of running text. A red box marks the words 'a way to a represent' in the sentence 'It is simply a way to a represent a feature of a random variable'.", "caption": "Section 3.3, p. 118 (PDF p. 136). The marked phrase has an extra “a”.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N010-f.png" }, { "alt": "Three lines of running text followed by the heading of Theorem 3.3.8, Consistency of the Kernel Density Estimator (Rectangular Kernel), in a shaded box. A red box marks the word 'example' in 'The following example shows how'.", "caption": "Section 3.3.2, p. 122 (PDF p. 140). The marked word introduces a result that is labeled Theorem 3.3.8.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N010-g.png" }, { "alt": "The heading and first three lines of Theorem 3.4.2, Normal Approximation-Based Confidence Interval, in a shaded box. A red box marks the words 'an consistent' at the end of the first line of the statement.", "caption": "Theorem 3.4.2, p. 125 (PDF p. 143). The marked article should be “a”.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N010-h.png" }, { "alt": "Three lines of running text. A red box marks the words 'an result' in 'we would infrequently encounter an result as extreme as the one that we saw'.", "caption": "Section 3.4.2, p. 128 (PDF p. 146). The marked article should be “a”.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N010-i.png" } ], "formal_release": null, "formal_statements": [], "headline": "The running text of Chapter 3 has nine typographical slips; none changes a meaning.", "id": "P001-C03-N010", "location": "Running text at nine places: p. 92 (PDF p. 110); p. 103 (PDF p. 121); p. 104 (PDF p. 122); p. 111 (PDF p. 129); p. 118 (PDF p. 136); p. 119 (PDF p. 137); p. 122 (PDF p. 140); Theorem 3.4.2, p. 125 (PDF p. 143); p. 128 (PDF p. 146)", "pdf_page": 110, "printed_page": "92", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\begin{itemize}\n\\item p.~92: indexed $i=1,2,\\hl{\\ldots N}$.\n\\item p.~103: Our usual definitions that allow us to characterize random variables also \\hl{allow to us to} characterize the distributions of estimators.\n\\item p.~104: Indeed, this is the same MSE that we \\hl{defined it} in Definition 2.1.22, just applied to estimators.\n\\item p.~111: We begin with a simple case: the asymptotic variant of the unbiasedness property from Section 3.2.2\\hl{)}.\n\\item p.~118: It is simply a way \\hl{to a represent} a feature of a random variable.\n\\item p.~119: Then, as $n$ grows large, the estimate $\\hl{\\theta}$ tends to more and more closely approximate $\\theta$.\n\\item p.~122: The following \\hl{example} shows how, for continuous $X$, the kernel density estimate (with a rectangular kernel) will be consistent for the true PDF. The proof itself leans on the plug-in principle.\n\\item p.~125, Theorem 3.4.2: Let $\\hat\\theta$ be an asymptotically normal estimator of $\\theta$, $\\hat{\\mathrm V}[\\hat\\theta]$ be \\hl{an} consistent estimator of the sampling variance, and $\\alpha\\in(0,1)$.\n\\item p.~128: we would infrequently encounter \\hl{an} result as extreme as the one that we saw.\n\\end{itemize}", "corrected": "\\begin{itemize}\n\\item p.~92: indexed $i=1,2,\\ldots\\hl{,}N$.\n\\item p.~103: Our usual definitions that allow us to characterize random variables also \\hl{allow us to} characterize the distributions of estimators.\n\\item p.~104: Indeed, this is the same MSE that we \\hl{defined} in Definition 2.1.22, just applied to estimators.\n\\item p.~111: We begin with a simple case: the asymptotic variant of the unbiasedness property from Section 3.2.2\\hl{.}\n\\item p.~118: It is simply a way \\hl{to represent} a feature of a random variable.\n\\item p.~119: Then, as $n$ grows large, the estimate $\\hl{\\hat\\theta}$ tends to more and more closely approximate $\\theta$.\n\\item p.~122: The following \\hl{theorem} shows how, for continuous $X$, the kernel density estimate (with a rectangular kernel) will be consistent for the true PDF. The proof itself leans on the plug-in principle.\n\\item p.~125, Theorem 3.4.2: Let $\\hat\\theta$ be an asymptotically normal estimator of $\\theta$, $\\hat{\\mathrm V}[\\hat\\theta]$ be \\hl{a} consistent estimator of the sampling variance, and $\\alpha\\in(0,1)$.\n\\item p.~128: we would infrequently encounter \\hl{a} result as extreme as the one that we saw.\n\\end{itemize}", "evidence": "The evidence is a check of the nine pages. On p.~119 the estimator is defined in the sentences directly above, and on p.~122 the result that follows is labeled Theorem 3.3.8.", "problem": "Each item is a slip of typesetting or copy-editing. On p.~92 the comma after the ellipsis is missing. On p.~103 a ``to'' is misplaced, p.~104 has an extra ``it'', and p.~118 has an extra ``a''. On p.~111 a closing parenthesis has no opening one. On p.~119 the estimate is printed as $\\theta$ without its hat. On p.~122 the result that the sentence introduces is Theorem 3.3.8 and not an example. On pp.~125 and 128 the article ``an'' stands before a consonant.", "what_survives": "No statement, formula or result changes. The intended reading is immediate in each case." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-N010", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the shaded box of Definition 3.3.7, Kernel Density Estimator. A red box marks the clause 'Let K : R to R be an even function satisfying integral from minus infinity to infinity of K(x) dx = 1'. The rest of the definition gives the rescaled kernel K sub h and the estimator f-hat sub K (x), the average of K sub h (x - X i) over the n observations.", "caption": "Definition 3.3.7, p. 121 (PDF p. 139). The marked clause holds the only conditions placed on the kernel $K$: it is even and it integrates to one. The definition does not require $K\\ge 0$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N011.png" }, { "alt": "Crop of a paragraph and the shaded box of Definition 3.3.9, Kernel Plug-In Estimator. Red boxes mark the words 'and thus we can apply any statistical functional to that estimate' in the paragraph, and the expression T(F-hat sub K) in the displayed formula theta-hat sub K = T(F-hat sub K). The last line of the definition reads: where F-hat sub K (x) = integral from minus infinity to x of f-hat(u) du.", "caption": "P. 123 (PDF p. 141), the sentence before Definition 3.3.9 and the definition. The marked words apply a statistical functional $T$, which Definition 3.3.2 defines on CDFs, to the integrated estimate $\\hat F_K$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N011-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T58", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T58" }, { "id": "P001-T59", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T59" } ], "headline": "The kernel plug-in estimator of Definition 3.3.9 requires a kernel that is nonnegative almost everywhere: for a kernel with negative values, which Definition 3.3.7 admits, the integrated estimate $\\hat F_K$ need not be a CDF.", "id": "P001-C03-N011", "location": "Definition 3.3.7, p. 121 (PDF p. 139), the conditions on the kernel $K$; read with Definition 3.3.9, p. 123 (PDF p. 141), and Definition 3.3.2, p. 118 (PDF p. 136)", "pdf_page": 139, "printed_page": "121", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T58", "P001-T59" ], "text": { "as_printed": "Definition 3.3.7 (p.~121): ``Let $X_1, X_2, \\ldots, X_n$ be i.i.d.\\ continuous random variables with common PDF $f$. Let $K:\\mathbb{R}\\to\\mathbb{R}$ be \\hl{an even function}$^{32}$ satisfying $\\hl{\\int_{-\\infty}^{\\infty}K(x)dx=1}$, and let $K_h(x)=\\frac{1}{h}K(\\frac{x}{h})$, $\\forall x\\in\\mathbb{R}$ and $h>0$. Then a \\emph{kernel density estimator} of $f(x)$ is\n\\[\n\\hat f_K(x)=\\frac{1}{n}\\sum_{i=1}^{n}K_h(x-X_i),\\ \\forall x\\in\\mathbb{R}.\\text{''}\n\\]\n\nP.~123, before Definition 3.3.9: ``We can integrate our estimate of the PDF to obtain an estimate of the full CDF, and thus we can apply any statistical functional to that estimate.''\n\nDefinition 3.3.9 (p.~123): ``The \\emph{kernel plug-in estimator} of $\\theta=T(F)$ is\n\\[\n\\hat\\theta_K=T\\bigl(\\hat F_K\\bigr),\n\\]\nwhere $\\hat F_K(x)=\\int_{-\\infty}^{x}\\hat f(u)du$.''\n\nDefinition 3.3.2 (p.~118): ``A \\emph{statistical functional} is a function $T:\\mathcal{F}\\to\\mathbb{R}$, where $\\mathcal{F}$ is the set of all possible CDFs.''", "corrected": "Definition 3.3.9. Let $X_1, X_2, \\ldots, X_n$ be i.i.d.\\ continuous random variables with common PDF $f$ and CDF $F$, let $K:\\mathbb{R}\\to\\mathbb{R}$ be an even function satisfying $\\int_{-\\infty}^{\\infty}K(x)\\,dx=1$, let $h>0$, and let $\\hat f_K$ be the kernel density estimator of Definition 3.3.7. \\hl{For kernel plug-in estimation, assume additionally that $K$ is nonnegative almost everywhere.} The \\emph{kernel plug-in estimator} of $\\theta=T(F)$ is\n\\[\n\\hat\\theta_K=T\\bigl(\\hat F_K\\bigr),\n\\]\nwhere $\\hat F_K(x)=\\int_{-\\infty}^{x}\\hat f_K(u)\\,du$.\n\nThe subscript $K$ on the integrand is the correction of \\fid{P001-C03-N012}. The condition may instead be added to Definition 3.3.7, which then excludes kernels with negative values from density estimation as well.", "evidence": "The evidence is an exact counterexample, computed in rational arithmetic, and a derivation for the repair. Define\n\\[\nK(t)=\\begin{cases}1,&|t|\\le 1,\\\\ -1/2,&1<|t|\\le 2,\\\\ 0,&|t|>2.\\end{cases}\n\\]\nSince $K$ is even and $\\int_{-\\infty}^{\\infty}K(t)\\,dt=-1/2+2-1/2=1$, the function $K$ satisfies the conditions of Definition 3.3.7. Set $H(t)=\\int_{-\\infty}^{t}K(u)\\,du$. Integrating piece by piece, I find $H(-2)=0$, $H(-1)=-1/2$, $H(1)=3/2$ and $H(2)=1$. Let $n=1$ and $h=1$, and let $a$ be the value of the observation $X_1$. Then $\\hat f_K(x)=K(x-a)$ and $\\hat F_K(x)=H(x-a)$, and I have\n\\[\n\\hat F_K(a-2)=0>\\hat F_K(a-1)=-\\frac{1}{2},\\qquad \\hat F_K(a+1)=\\frac{3}{2}.\n\\]\nNote that this holds for all real $a$.\n\nThe estimator can also be undefined for every sample. Let $T(G)=\\sqrt{G(0)}$, which is defined for any CDF $G$. Let $f(a)=4(1-4|a-3/2|)$ for $5/4\\le a\\le 7/4$ and $f(a)=0$ otherwise, so that $f$ is a continuous density and $T(F)=0$. For $1\\le a\\le 2$, I have $H(-a)=a/2-1$. For $h=1$, every $n$ and every sample from $f$, I obtain\n\\[\n\\hat F_K(0)=\\frac{1}{n}\\sum_{i=1}^{n}H(-X_i)\\in\\Bigl[-\\frac{3}{8},-\\frac{1}{8}\\Bigr].\n\\]\nFor such a sample, $T(\\hat F_K)$ is the square root of a negative number.\n\nSuppose now that $K\\ge 0$ almost everywhere. Substituting $v=(x-X_i)/h$, I find that each function $x\\mapsto K_h(x-X_i)$ is nonnegative almost everywhere and integrates to one, and the same holds for their average $\\hat f_K$. This implies that $\\hat F_K$ is nondecreasing and continuous, with limit $0$ at $-\\infty$ and limit $1$ at $+\\infty$. I conclude that $\\hat F_K$ is a CDF for every sample and every $h>0$.", "problem": "Definition 3.3.7 admits every even function $K$ with integral one, including a function $K$ that takes negative values. Recall that Definition 3.3.2 defines a statistical functional $T$ on the set of CDFs only. If $K$ takes negative values, then the integrated estimate $\\hat F_K$ of Definition 3.3.9 can decrease, can be negative and can exceed one. In that case $\\hat F_K$ is not a CDF and $T(\\hat F_K)$ is not defined. The four kernels of Figure 3.3.2 are nonnegative, but the definitions are not restricted to them.", "what_survives": "The four kernels of Figure 3.3.2 (rectangular, normal, triangular and Epanechnikov) are nonnegative, and the missing condition does not affect Theorem 3.3.8, which concerns the rectangular kernel. Plug-in estimation with the empirical CDF is unaffected, as is plug-in estimation by maximum likelihood, which Section 5.2.4 (p.~197) introduces by analogy with Definition 3.3.9. A kernel with negative values can still be used to estimate a density alone. A search of the text of the book finds no simulation, empirical estimate or reported number computed with such a kernel. What changes is the claim on p.~123 that any statistical functional can be applied to the integrated estimate, together with the general plug-in discussion on p.~124 that continues it. Both need the estimated distribution to be a CDF, or a stated extension of the functional to functions that are not CDFs." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-N011", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the shaded box of Definition 3.3.9, Kernel Plug-In Estimator. A red box marks the symbol f-hat, printed with no subscript, in the last line: where F-hat sub K (x) = integral from minus infinity to x of f-hat(u) du.", "caption": "Definition 3.3.9, p. 123 (PDF p. 141). The marked integrand is printed as $\\hat f$, without the subscript $K$ of the kernel density estimator $\\hat f_K$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N012.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T60", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T60" } ], "headline": "The integrand in Definition 3.3.9 is the kernel density estimator $\\hat f_K$; the subscript $K$ is missing.", "id": "P001-C03-N012", "location": "Definition 3.3.9, p. 123 (PDF p. 141), last line", "pdf_page": 141, "printed_page": "123", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T60" ], "text": { "as_printed": "``The \\emph{kernel plug-in estimator} of $\\theta=T(F)$ is\n\\[\n\\hat\\theta_K=T\\bigl(\\hat F_K\\bigr),\n\\]\nwhere $\\hat F_K(x)=\\int_{-\\infty}^{x}\\hl{\\hat f}(u)du$.''", "corrected": "``The \\emph{kernel plug-in estimator} of $\\theta=T(F)$ is\n\\[\n\\hat\\theta_K=T\\bigl(\\hat F_K\\bigr),\n\\]\nwhere $\\hat F_K(x)=\\int_{-\\infty}^{x}\\hl{\\hat f_K}(u)du$.''", "evidence": "The evidence is a check of pp.~119, 121 and 123.", "problem": "The integrand is printed as $\\hat f$. The kernel density estimator is written $\\hat f_K$ in Definition 3.3.7 and in the proof sketch on the same page, and $\\hat f$ without a subscript denotes the empirical PMF in Example 3.3.4 (p.~119).", "what_survives": "The definition as intended is unchanged, and nothing depends on the misprint. A condition on the kernel $K$ in the same definition is the subject of \\fid{P001-C03-N011}." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-N012", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of seven lines of text. Red boxes mark the word 'implies' and the expression 'p converges in distribution to U(0,1)' in the sentence 'Loosely speaking, however, asymptotic validity in the context of p-values implies that, under the null hypothesis that theta = theta 0, as n tends to infinity, p converges in distribution to U(0,1).'", "caption": "First paragraph after Theorem 3.4.6, p. 130 (PDF p. 148). The marked word presents the uniform limit, also marked, as a consequence of asymptotic validity.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N014.png" } ], "formal_release": null, "formal_statements": [], "headline": "The limit $p\\xrightarrow{d}U(0,1)$ under the null hypothesis is a property of the three $p$-values of Theorem 3.4.6; asymptotic validity does not imply it in general.", "id": "P001-C03-N014", "location": "First paragraph after Theorem 3.4.6, p. 130 (PDF p. 148), sentence beginning “Loosely speaking”", "pdf_page": 148, "printed_page": "130", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "``We omit the proof of this theorem, as well as a formal definition of asymptotic validity. Loosely speaking, however, asymptotic validity in the context of $p$-values \\hl{implies} that, under the null hypothesis that $\\theta=\\theta_0$, as $n\\to\\infty$, $p\\xrightarrow{d}U(0,1)$.''", "corrected": "``Loosely speaking, \\hl{under the null hypothesis that $\\theta=\\theta_0$, the $p$-values in Theorem 3.4.6 satisfy $p\\xrightarrow{d}U(0,1)$ as $n\\to\\infty$}.''", "intended_meaning": "The sentence describes the $p$-values of Theorem 3.4.6, and for them it is correct. Under the null hypothesis that $\\theta=\\theta_0$, the $t$-statistic satisfies $t\\xrightarrow{d}Z$ with $Z$ standard normal. The three $p$-values are $\\Phi(t)$, $1-\\Phi(t)$ and $2(1-\\Phi(|t|))$, and they converge in distribution to $\\Phi(Z)$, $1-\\Phi(Z)$ and $2(1-\\Phi(|Z|))$. Each limit is uniform on $(0,1)$; for the third, I have $\\Pr[2(1-\\Phi(|Z|))\\le u]=\\Pr[|Z|\\ge\\Phi^{-1}(1-u/2)]=u$ for $00$,\n\\[\n\\lim_{K\\to\\infty}\\limsup_{n\\to\\infty}\\Pr\\Bigl[\\mathrm{E}_*\\bigl[(Z_n^*)^2\\mathbf{1}\\{|Z_n^*|>K\\}\\bigr]>\\varepsilon\\Bigr]=0.\\qquad\\text{(M)}\n\\]\nThen $\\sqrt n\\sqrt{\\mathrm{V}_*[\\hat\\theta^*]}\\xrightarrow{p}\\phi$.\n\nCondition (M) holds with no further assumption when $\\hat\\theta$ is the sample mean of a distribution with finite variance. When $\\hat\\theta$ is the sample median and $F$ has a positive continuous density near its unique median, (M) holds if $\\mathrm{E}|X|^{\\eta}<\\infty$ for some $\\eta>0$. If the standard deviation is computed from finitely many bootstrap samples, the conclusion also requires that $n$ times the error of the Monte Carlo variance tends to zero in probability.", "evidence": "The evidence is an exact counterexample and a derivation. Let $X_1,\\ldots,X_n$ be i.i.d.\\ with CDF $F$, and suppose that $\\sqrt n(\\hat\\theta-\\theta)\\xrightarrow{d}N(0,\\phi^2)$ with $0<\\phi<\\infty$. Let $\\hat\\theta^*$ be the estimate computed from a sample of size $n$ drawn with replacement from $X_1,\\ldots,X_n$. I write $\\mathrm{E}_*$ and $\\mathrm{V}_*$ for the expectation and the variance conditional on $X_1,\\ldots,X_n$, and I set $Z_n^*=\\sqrt n(\\hat\\theta^*-\\hat\\theta)$. At the scale of Definition 3.2.32, the printed sentence states that $n\\mathrm{V}_*[\\hat\\theta^*]\\xrightarrow{p}\\phi^2$.\n\nFor the counterexample, let $U$ be uniform on $(0,1)$ and independently choose one of three components: let $X$ be uniform on $[-1,1]$ with probability $1/2$, equal to $2^{1/U^2}$ with probability $1/4$, and equal to $-2^{1/U^2}$ with probability $1/4$. The distribution of $X$ is symmetric, its unique median is $0$, its density equals $1/4$ on $[-1,1]$, and $\\Pr[X\\ge 2^{j^2}]=1/(4j)$ for all integers $j\\ge 1$. The median is Hadamard differentiable at $F$ along the directions that are continuous at $0$, which are the usual directions for a quantile. The sample median $M_n$ satisfies $\\sqrt n M_n\\xrightarrow{d}N(0,4)$, and the conditional distribution of $\\sqrt n(M_n^*-M_n)$ converges to $N(0,4)$ in probability. The finite-sample median has infinite second moment here, so it has no finite sampling variance; the example concerns the finite variance of its limiting law, which Definition 3.2.32 uses. Fix an odd $n\\ge 5$, and let $A_n$ be the event that the sample contains an observation $a\\ge 2^{n^2}$ and an observation $b\\in[-1,1]$. By Bonferroni's inequality, I have\n\\[\n\\Pr[A_n]\\ge\\frac{7}{32}+\\frac{1}{32n}-2^{-n}\\ge\\frac{3}{16}.\n\\]\nOn $A_n$, the bootstrap sample that repeats $a$ in all $n$ draws has probability $n^{-n}$ and median $a$, and the bootstrap sample that repeats $b$ has the same probability and median $b$. Using $n\\le 2^{n-1}$, I obtain\n\\[\n\\mathrm{V}_*[M_n^*]\\ge n^{-2n}\\bigl(2^{n^2}-1\\bigr)^2\\ge 2^{2n-2}.\n\\]\nI conclude that $\\Pr\\bigl[n\\mathrm{V}_*[M_n^*]>5\\bigr]\\ge 3/16$ for all odd $n\\ge 5$, while the target is $4$.\n\nA second functional shows that the failure does not depend on the directions allowed in the derivative. Let $T_0(G)=G(0)-1/2$ if $1/4\\le G(0)\\le 3/4$, and let $T_0(G)$ be the median of $G$ otherwise. The functional $T_0$ is affine on the CDFs within Kolmogorov distance $1/4$ of $F$, and it is Hadamard differentiable at $F$ in every direction. For the same distribution of $X$, I have $\\sqrt n\\,T_0(\\hat F)\\xrightarrow{d}N(0,1/4)$, the conditional distribution of the bootstrap statistic has the same normal limit, and the bound above holds with $T_0$ in place of the median. Its target is $1/4$.\n\nFor the repair, suppose that the conditional distribution of $Z_n^*$ converges weakly to $N(0,\\phi^2)$ in probability and that condition (M) of the corrected statement below holds. The functions $z\\mapsto\\min(z^2,K^2)$ and $z\\mapsto\\max(-K,\\min(z,K))$ are bounded and continuous, so their conditional expectations at $Z_n^*$ converge in probability to the corresponding expectations under $N(0,\\phi^2)$. Removing the truncation changes these conditional expectations by at most $\\mathrm{E}_*[(Z_n^*)^2\\mathbf{1}\\{|Z_n^*|>K\\}]$ and $K^{-1}$ times this quantity, respectively. Letting $n\\to\\infty$ and then $K\\to\\infty$, and using (M), I find that $\\mathrm{E}_*[Z_n^*]\\xrightarrow{p}0$ and $\\mathrm{E}_*[(Z_n^*)^2]\\xrightarrow{p}\\phi^2$, which implies that $n\\mathrm{V}_*[\\hat\\theta^*]\\xrightarrow{p}\\phi^2$. Conversely, if the weak limit holds and the conditional second moments are finite, then this convergence implies (M).\n\nFor the sample median, the abstract of Ghosh, Parr, Singh and Babu, ``A note on bootstrapping the sample median'', \\emph{Annals of Statistics} 12 (1984), 1130--1135 (\\url{https://doi.org/10.1214/aos/1176346731}), says that the conjecture of almost-sure convergence of the bootstrap variance estimator to the asymptotic variance is ``false unless a tail condition is imposed on $F$''.", "problem": "Section 3.3.1 (p.~120) calls ``the usual plug-in regularity conditions'' the conditions under which the functional $T$ is well behaved at $F$, and footnote 31 defines ``well behaved'' as Hadamard differentiable at $F$ with respect to the Kolmogorov distance. These conditions concern the estimand $\\theta=T(F)$. They control limiting distributions: with positive limiting variance, they give asymptotic normality of $\\hat\\theta=T(\\hat F)$ as in the correction of \\fid{P001-C03-E002}. They do not control the second moment of the bootstrap distribution. The standard deviation in step 4 is consistent, at the scale of Definition 3.2.32, only if that second moment converges, and this requires a moment condition. Without one, a few extreme bootstrap samples can keep the bootstrap variance far from its target while the bootstrap distribution converges to the correct normal limit.", "what_survives": "The four bootstrap steps are unchanged, as is the bootstrap approximation to the sampling distribution. Bootstrap standard errors of sample means and of cluster means with finite variance are consistent with no further assumption. The moment condition also holds for the uniform and Bernoulli designs of Section 3.4.4, and for the sample means of bounded variables that give the bounds in Sections 6.3 and 7.5. Theorems 3.4.2 and 3.4.6 assume a consistent variance estimator and stand as stated. Percentile intervals use only quantiles of the bootstrap distribution, and this point does not affect them. No standard error, estimate or conclusion in the book is shown to be wrong. What changes is the general assurance on p.~131 and the statements that use the same rationale: the block bootstrap for an arbitrary estimator (p.~141), and the bootstrap standard errors for regression (pp.~152 and 154) and for maximum likelihood (p.~203). Each needs the moment condition. For the estimators in the applications that refit a regression, a propensity score or a match in every bootstrap sample, the book does not give enough to decide whether the condition holds." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-N016", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the four lines of footnote 42. A red box marks the words 'Example 7.2.4 (Matching Estimation)' in the last sentence, 'We will briefly discuss a practical situation in Example 7.2.4 (Matching Estimation) where the bootstrap may also fail.'", "caption": "Footnote 42, p. 131 (PDF p. 149). The marked reference gives Example 7.2.4 the title “Matching Estimation”; the example is titled “Propensity Score Matching” and the section “Matching”.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N017.png" } ], "formal_release": null, "formal_statements": [], "headline": "Footnote 42 should refer to Section 7.2.4 (Matching): Example 7.2.4 is titled “Propensity Score Matching”, and the remark on the bootstrap is in the text of the section after the example.", "id": "P001-C03-N017", "location": "Footnote 42, p. 131 (PDF p. 149), last sentence", "pdf_page": 149, "printed_page": "131", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "``We will briefly discuss a practical situation in \\hl{Example 7.2.4 (\\emph{Matching Estimation})} where the bootstrap may also fail.''", "corrected": "``We will briefly discuss a practical situation in \\hl{Section 7.2.4 (\\emph{Matching})} where the bootstrap may also fail.''", "evidence": "The evidence is a check of pp.~131, 262 and 264. A search of the text of the book finds ``Matching Estimation'' only in this footnote.", "problem": "No item of the book is titled ``Matching Estimation''. On p.~262, Section 7.2.4 is titled ``Matching'' and Example 7.2.4 is titled ``Propensity Score Matching''. The statement that the usual methods for constructing confidence intervals, including the bootstrap, do not generally work is on p.~264, in the text of Section 7.2.4 after the example.", "what_survives": "The rest of the footnote is unchanged, and the printed reference still points to the right pages." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-N017", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of four lines of text. A red box marks 'Theorem 3.2.34' in the sentence 'Theorem 3.2.34 establishes that the standard error of the sample mean is sigma[X-bar] = the square root of V[X]/n.'", "caption": "Section 3.4.4, p. 132 (PDF p. 150), second paragraph. The marked reference calls item 3.2.34 a theorem; it is a definition, and the formula also uses Theorem 3.2.4.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N018.png" }, { "alt": "Crop of four lines of text. A red box marks 'Theorem 3.2.34' in the sentence 'The second column reports the true standard error (which can be exactly derived via Theorem 3.2.34).'", "caption": "Example 3.4.7, p. 133 (PDF p. 151), last paragraph of the page. The same reference recurs in the description of the second column of Table 3.4.1.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N018-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "Item 3.2.34 is a definition, and the formula $\\sigma[\\overline{X}]=\\sqrt{\\mathrm{V}[X]/n}$ follows from Theorem 3.2.4 together with Definition 3.2.34.", "id": "P001-C03-N018", "location": "Section 3.4.4, p. 132 (PDF p. 150), second paragraph; repeated in Example 3.4.7, p. 133 (PDF p. 151), last paragraph of the page", "pdf_page": 150, "printed_page": "132", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "P.~132: ``\\hl{Theorem 3.2.34} establishes that the standard error of the sample mean is $\\sigma[\\overline{X}]=\\sqrt{\\mathrm{V}[X]/n}$.''\n\nP.~133: ``The second column reports the true standard error (which can be exactly derived via \\hl{Theorem 3.2.34}).''", "corrected": "P.~132: ``\\hl{Theorem 3.2.4 together with Definition 3.2.34} establishes that the standard error of the sample mean is $\\sigma[\\overline{X}]=\\sqrt{\\mathrm{V}[X]/n}$.''\n\nP.~133: ``The second column reports the true standard error (which can be exactly derived via \\hl{Theorem 3.2.4 and Definition 3.2.34}).''", "evidence": "The evidence is a check of pp.~98, 115, 132 and 133.", "problem": "Item 3.2.34 (p.~115) is Definition 3.2.34, \\emph{Standard Error of the Sample Mean}, which states that $\\sigma[\\overline{X}]=\\sqrt{\\mathrm{V}[\\overline{X}]}$. The identity $\\mathrm{V}[\\overline{X}]=\\mathrm{V}[X]/n$ is Theorem 3.2.4 (p.~98). Substituting the second into the first, I obtain the printed formula, which is correct.", "what_survives": "The formula and the second column of Table 3.4.1 are unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-N018", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of a paragraph of seven lines. A red box marks the symbol E[X] in the sentence 'Alternatively, we can think about the estimator as the average of i.i.d. random variables, with mean E[X] and common variance V[W].' The rest of the paragraph says that a version of the WLLN still holds for the cluster sample mean when the number of clusters l tends to infinity with m not growing in n.", "caption": "Paragraph before Theorem 3.5.6, p. 138 (PDF p. 156). The marked symbol is the mean of the i.i.d. random variables that the sentence describes; the same sentence gives their variance as $\\mathrm{V}[W]$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C03-N019.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T63", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T63" } ], "headline": "The i.i.d. random variables averaged by the cluster sample mean are the cluster averages $W_i$, and their mean is $\\mathrm{E}[W]$, not $\\mathrm{E}[X]$.", "id": "P001-C03-N019", "location": "Paragraph between the proof of Theorem 3.5.5 and Theorem 3.5.6, p. 138 (PDF p. 156), first sentence", "pdf_page": 156, "printed_page": "138", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T63" ], "text": { "as_printed": "``Alternatively, we can think about the estimator as the average of i.i.d.\\ random variables, with mean $\\mathrm{E}[\\hl{X}]$ and common variance $\\mathrm{V}[W]$.''", "corrected": "``Alternatively, we can think about the estimator as the average of i.i.d.\\ random variables, with mean $\\mathrm{E}[\\hl{W}]$ and common variance $\\mathrm{V}[W]$.''\n\nHere the estimator is the cluster sample mean $\\overline{X}$ of Definition 3.5.2, and the i.i.d.\\ random variables are the cluster averages $W_1,W_2,\\ldots,W_l$ of Definition 3.5.1.", "evidence": "The evidence is a check of pp.~136--138 and a short derivation. Definition 3.5.1 sets $W_i=(1/m)\\sum_{j=1}^{m}X_{ij}$, so that\n\\[\n\\mathrm{E}[W_i]=\\frac{1}{m}\\sum_{j=1}^{m}\\mathrm{E}[X_{ij}].\n\\]\nTheorem 3.5.4 states that $\\mathrm{E}[\\overline{X}]=\\mathrm{E}[W]$, and Theorem 3.5.6 states that $\\overline{X}\\xrightarrow{p}\\mathrm{E}[W]$. An example shows that the coordinates need not have a common mean. Let $m=2$, let $U_1,U_2,\\ldots,U_l$ be independent Bernoulli random variables with parameter $1/2$, and set $(X_{i1},X_{i2})=(U_i,U_i+2)$. Then $W_i=U_i+1$ and $\\mathrm{E}[W]=3/2$, while $\\mathrm{E}[X_{i1}]=1/2$ and $\\mathrm{E}[X_{i2}]=5/2$.", "problem": "The estimator in the sentence is the cluster sample mean $\\overline{X}$, and Theorem 3.5.3 shows that $\\overline{X}$ equals the sample mean $\\overline{W}$ of the cluster averages $W_1,W_2,\\ldots,W_l$. The cluster averages are the i.i.d.\\ random variables that the sentence describes, and their common mean is $\\mathrm{E}[W]$. The symbol $\\mathrm{E}[X]$ has no meaning in the cluster model: the random vectors $(X_{i1},X_{i2},\\ldots,X_{im})$ are i.i.d.\\ across clusters $i$, but the coordinates $X_{ij}$ need not have the same distribution across $j$.", "what_survives": "Theorems 3.5.4 and 3.5.6 are stated with $\\mathrm{E}[W]$, and the rest of the paragraph is unchanged. Nothing depends on the misprint." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C03-N019", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of one line of text, 'We can stack all n of these e i's to form a system of equations', and three displayed equations separated by vertical dots: e 1 = Y 1 - (...), e 2 = Y 2 - (...), and e n = Y 3 - (b 0 + b 1 X[1]n + b 2 X[2]n + ... b K X[K]n). A red box marks the symbol Y 3 in the last equation.", "caption": "Proof of Theorem 4.1.4, p. 148 (PDF p. 166), the stacked system of equations. The marked outcome in the equation for $e_n$ has the subscript $3$; every regressor in that equation has the subscript $n$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-E005.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T65", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T65" } ], "headline": "The last equation of the stacked system is the residual of observation $n$, and its outcome is $Y_n$, not $Y_3$.", "id": "P001-C04-E005", "location": "Proof of Theorem 4.1.4, p. 148 (PDF p. 166), last line of the stacked system of equations", "pdf_page": 166, "printed_page": "148", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T65" ], "text": { "as_printed": "\\[\ne_n=Y_{\\hl{3}}-\\bigl(b_0+b_1X_{[1]n}+b_2X_{[2]n}+\\ldots b_KX_{[K]n}\\bigr).\n\\]", "corrected": "\\[\ne_n=Y_{\\hl{n}}-\\bigl(b_0+b_1X_{[1]n}+b_2X_{[2]n}+\\ldots+b_KX_{[K]n}\\bigr).\n\\]\n\nThe plus sign before the last term is the correction of \\fid{P001-C04-N002}.", "evidence": "The evidence is a check of p.~148. The proof defines $e_i=Y_i-\\bigl(b_0+b_1X_{[1]i}+b_2X_{[2]i}+\\ldots+b_KX_{[K]i}\\bigr)$ for every observation $i$. Substituting $i=n$, I obtain the outcome $Y_n$. The matrix equation directly below the stacked system has $Y_n$ in the last row of the vector of outcomes.", "problem": "On p.~148, the last equation of the stacked system uses $Y_3$ where the last row of a sample of size $n$ requires $Y_n$. The equation is the residual of observation $n$: its left side is $e_n$, and every regressor on its right side has the subscript $n$.", "what_survives": "The matrix equation $\\mathbf{e}=\\mathbf{Y}-\\mathbb{X}\\mathbf{b}$, the normal equations and the OLS solution of Theorem 4.1.4 are unaffected, since the proof continues from the matrix equation, which is correct." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C04-E005", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Four lines of text and a shaded box with Definition 4.3.7. The definition displays PDA with subscript X[k] as the partial derivative of E[Y|X] with respect to X[k], followed by a vertical bar. A red box marks the subscript of the bar, X = E[X].", "caption": "Definition 4.3.7, p. 167 (PDF p. 185), with the paragraph that introduces it: the point of evaluation $\\mathbf{X}=\\mathrm{E}[\\mathbf{X}]$ is marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-E002.png" }, { "alt": "A paragraph of eight lines that compares the average partial derivative with the partial derivative at the average. A red box marks the words 'has a well-defined meaning' at the start of the fourth line; the sentence continues 'it is limited to just one particular point on the probability distribution of the explanatory variables'.", "caption": "First paragraph of p. 168 (PDF p. 186): the words “has a well-defined meaning” are marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-E002-p168.png" }, { "alt": "A line chart titled 'The same distribution of (Y, X) gives a partial derivative at the average of 0 or of 1, depending on the extension', with ticks from -2 to 2 on the horizontal axis, labeled as a value of the explanatory variable X, and the conditional mean of Y on the vertical axis. Two shaded bands, from -2 to -1 and from 1 to 2, are labeled 'X has density 1/2', and a thick black horizontal line at height 0 runs across each band. Between -1 and 1, labeled 'no observations', a solid blue line stays at 0 and a dashed orange curve dips to about -0.29 near x = -0.45, passes through 0 at x = 0 and rises to about 0.29 near x = 0.45 before returning to 0 at x = 1. A dotted vertical line marks E[X] = 0. Labels read 'slope of m1 at E[X]: 1' and 'slope of m0 at E[X]: 0'.", "caption": "The counterexample to Definition 4.3.7 (p. 167). The explanatory variable $X$ has density $1/2$ on $[-2,-1]\\cup[1,2]$ (shaded), and $\\mathrm{E}[Y\\mid X=x]=0$ there (black). Two continuously differentiable extensions across the interval with no observations, $m_0$ (blue, solid) and $m_1$ (orange, dashed), have derivatives $0$ and $1$ at $\\mathrm{E}[X]=0$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-E002-extensions.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T3", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T3" }, { "id": "P001-T4", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T4" }, { "id": "P001-T102", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T102" } ], "headline": "Definition 4.3.7 evaluates the derivative of the CEF at $\\mathrm{E}[\\mathbf{X}]$ without requiring that the derivative be defined there and determined by the distribution of $(Y,\\mathbf{X})$; a sufficient condition is that $\\mathrm{E}[\\mathbf{X}]$ have an open neighborhood inside the support of $\\mathbf{X}$ on which the CEF is continuous, and that the CEF be differentiable at $\\mathrm{E}[\\mathbf{X}]$.", "id": "P001-C04-E002", "location": "Definition 4.3.7, p. 167 (PDF p. 185); first paragraph of p. 168 (PDF p. 186)", "pdf_page": 185, "printed_page": "167", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T102", "P001-T3", "P001-T4" ], "text": { "as_printed": "\\textbf{Definition 4.3.7.} \\emph{Partial Derivative at the Average}\nFor a random variable $Y$ and random vector $\\mathbf{X}$, the \\emph{partial derivative at the average} of the CEF (of $Y$ given $\\mathbf{X}$) with respect to $X_{[k]}$ is\n\\[\nPDA_{X_{[k]}}=\\left.\\frac{\\partial\\mathrm{E}[Y\\mid\\mathbf{X}]}{\\partial X_{[k]}}\\right|_{\\hl{\\scriptstyle\\mathbf{X}=\\mathrm{E}[\\mathbf{X}]}}.\n\\]\n\n[p.~168] While the partial derivative at the average also \\hl{has a well-defined meaning}, it is limited to just one particular point on the probability distribution of the explanatory variables.", "corrected": "\\textbf{Definition 4.3.7.} \\emph{Partial Derivative at the Average}\nFor a random variable $Y$ and random vector $\\mathbf{X}$, write $m(\\mathbf{x})=\\mathrm{E}[Y\\mid\\mathbf{X}=\\mathbf{x}]$ for the CEF. Suppose that $\\boldsymbol{\\mu}=\\mathrm{E}[\\mathbf{X}]$ exists and \\hl{has an open neighborhood $U$ contained in $\\operatorname{Supp}[\\mathbf{X}]$}, and that \\hl{$m$ has a version that is continuous on $U$ and differentiable with respect to $x_{[k]}$ at $\\boldsymbol{\\mu}$}. Then the \\emph{partial derivative at the average} of the CEF (of $Y$ given $\\mathbf{X}$) with respect to $X_{[k]}$ is\n\\[\nPDA_{X_{[k]}}=\\frac{\\partial m}{\\partial x_{[k]}}(\\boldsymbol{\\mu}).\n\\]\n\n[p.~168] While the partial derivative at the average also has a well-defined meaning \\hl{under these conditions}, it is limited to just one particular point on the probability distribution of the explanatory variables. \\hl{A full open neighborhood is not necessary in every case. The derivative at $\\mathrm{E}[\\mathbf{X}]$ of a chosen extrapolating predictor can still be reported, but is not in general determined by the distribution of $(Y,\\mathbf{X})$.}", "evidence": "The evidence is an exact counterexample to the printed definition and a derivation of the corrected one.\n\nLet $X$ have density $1/2$ on $S=[-2,-1]\\cup[1,2]$, and let $Y\\sim U(-1,1)$ be independent of $X$. Then $\\mathrm{E}[X]=0$, which is not in $S$, and the interval $(-1,1)$ has probability zero. For $x\\in S$, I have $\\mathrm{E}[Y\\mid X=x]=0$. Define $m_0(x)=0$ for all $x$, and define $m_1(x)=x(1-x^2)^2$ for $|x|<1$ and $m_1(x)=0$ for $|x|\\ge1$. Both functions equal the CEF at every point of $S$. Set $h(x)=x(1-x^2)^2=x-2x^3+x^5$, so that $h'(x)=1-6x^2+5x^4$. Since $h(\\pm1)=0$ and $h'(\\pm1)=0$, the function $m_1$ is continuously differentiable on $\\mathbb{R}$. At the mean,\n\\[\nm_0'(0)=0,\\qquad m_1'(0)=h'(0)=1,\n\\]\nand replacing $h$ by $ah$ gives the value $a$ for any real $a$. The distribution of $(Y,X)$ is the same whichever extension is used, and each extension has average partial derivative $0$ and mean squared prediction error $\\mathrm{E}[Y^2]=1/3$. A statistic of an i.i.d.\\ sample that converged in probability to both $0$ and $1$ would lie within $1/3$ of each with probability tending to one, which is impossible.\n\nFor a vector, let $(X_1,X_2)$ be uniform on $[-2,2]^2\\setminus(-1,1)^2$, and let $Y\\sim U(-1,1)$ be independent. The support is connected, each coordinate is supported on $[-2,2]$, and the mean $(0,0)$ lies in the hole. The CEF is $0$ on the support. The function equal to $x_1(1-x_1^2)^2(1-x_2^2)^2$ on $(-1,1)^2$ and to $0$ elsewhere is continuously differentiable, and its partial derivative in $x_1$ at $(0,0)$ is $1$; the zero extension gives $0$.\n\nUnder the corrected definition the derivative is determined by the distribution. Let $m$ and $\\tilde m$ be two versions of the CEF that are continuous on the open set $U$. Then $m(\\mathbf{X})=\\tilde m(\\mathbf{X})$ almost surely. If $m$ and $\\tilde m$ differed at a point of $U$, they would differ on a ball around it, and this ball has positive probability because $U\\subseteq\\operatorname{Supp}[\\mathbf{X}]$. Therefore $m=\\tilde m$ on $U$, the difference quotients of $m$ and $\\tilde m$ at $\\boldsymbol{\\mu}$ coincide, and so do the derivatives.", "problem": "The book defines the conditional expectation function (CEF) only at points of the support of $\\mathbf{X}$ (Definitions 2.2.10 and 2.2.15 on pp.~67 and 69, and Definitions 2.3.3 and 2.3.4 on p.~85). Footnote 13 on p.~166 adds that $\\mathbf{X}$ is continuous and that the CEF is differentiable at every value of $\\mathbf{X}$ with positive support. None of these conditions places $\\mathrm{E}[\\mathbf{X}]$ in the support. When $\\mathrm{E}[\\mathbf{X}]$ lies in a region where $\\mathbf{X}$ has no probability, the CEF has no value there, and a derivative at $\\mathrm{E}[\\mathbf{X}]$ is a derivative of whichever extension of the CEF one chooses. Different differentiable extensions give different derivatives, while the distribution of $(Y,\\mathbf{X})$ is the same. In that case the quantity in Definition 4.3.7 is not a property of the distribution of $(Y,\\mathbf{X})$, and the data cannot determine it. The sentence on p.~168, which says that the partial derivative at the average ``has a well-defined meaning'' at a point ``on the probability distribution of the explanatory variables'', also needs a derivative defined and determined by the distribution. A gap in the support of one explanatory variable is not the only way this can happen: the support of $\\mathbf{X}$ can be connected, with every explanatory variable supported on a full interval, and still omit a neighborhood of $\\mathrm{E}[\\mathbf{X}]$.", "what_survives": "Example 4.3.8 is unaffected: its mean $(1/2,1/2)$ is an interior point of the support $[0,1]^2$, and $PDA_{X_{[1]}}=-1$ and $APD_{X_{[1]}}=1$ hold. Definition 4.3.6 does not evaluate the CEF at the mean and is unchanged. The values in the PDA row of Table 4.4.5 (p.~173) are derivatives of the fitted specifications at the sample means, and no number there is shown to be wrong. What changes is their interpretation as estimates of a partial derivative at the average of the CEF, which needs a derivative defined and determined by the distribution; the corrected definition gives sufficient conditions. Since the data for the application are not available, I cannot say whether these conditions hold there. Outside this discussion and the application of Section 4.4, I found no substantive use of the partial derivative at the average and no theorem that depends on it." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C04-E002", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "The last part of a shaded definition box: the display of beta-hat as an arg min over b in R to the power J_n + 1, and below it the line 'where J_n tends to infinity and (1/n) J_n tends to 0', with the two conditions in a red box. Below the box, a paragraph of ten lines. Two further red boxes, one at the end of a line and one at the start of the next, mark the words 'will be consistent across regions of the data with nonzero probability mass'.", "caption": "End of Definition 4.3.9 and the paragraph that follows it, p. 169 (PDF p. 187): the two conditions on the degree and the claim of consistency are marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-E001.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T5", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T5" }, { "id": "P001-T6", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T6" }, { "id": "P001-T70", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T70" }, { "id": "P001-T119", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T119" } ], "headline": "The conditions $J_n\\to\\infty$ and $J_n/n\\to0$ do not imply that the polynomial least squares sieve estimator is consistent; the claim holds under a stronger condition on $J_n$, which for uniform $X$ reads $(J_n+1)^3/n\\to0$.", "id": "P001-C04-E001", "location": "Definition 4.3.9, pp. 168–169 (PDF pp. 186–187), and the paragraph that follows it on p. 169", "pdf_page": 186, "printed_page": "168–169", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P001-T119", "P001-T5", "P001-T6", "P001-T70" ], "text": { "as_printed": "\\textbf{Definition 4.3.9.} \\emph{Polynomial Least Squares Sieve Estimator}\nFor i.i.d. random vectors $(Y_1,X_1)$, $(Y_2,X_2)$, \\ldots, $(Y_n,X_n)$, the \\emph{polynomial least squares sieve estimator} of the conditional expectation function is\n\\[\n\\hat{\\mathrm{E}}_n[Y\\mid X=x]=\\sum_{k=0}^{J_n}\\hat{\\beta}_kx^k,\n\\]\nwhere\n\\[\n\\hat{\\boldsymbol{\\beta}}=\\underset{\\mathbf{b}\\in\\mathbb{R}^{J_n+1}}{\\arg\\min}\\sum_{i=1}^n\\left(Y_i-\\sum_{k=0}^{J_n}b_kX_i^k\\right),\n\\]\nwhere $J_n\\to\\infty$ and $\\hl{\\tfrac{1}{n}J_n\\to0}$.\n\n[p.~169, next paragraph] As long as the CEF is smooth with appropriate support across $X$, then $\\hat{\\mathrm{E}}[Y\\mid X=x]$ \\hl{will be consistent across regions of the data with nonzero probability mass} (by the logic of Theorem 4.3.3, \\emph{Polynomial Approximation of the CEF}).", "corrected": "\\textbf{Definition 4.3.9.} \\emph{Polynomial Least Squares Sieve Estimator}\nFor i.i.d. random vectors $(Y_1,X_1)$, $(Y_2,X_2)$, \\ldots, $(Y_n,X_n)$, the \\emph{polynomial least squares sieve estimator} of the conditional expectation function is\n\\[\n\\hat{\\mathrm{E}}_n[Y\\mid X=x]=\\sum_{k=0}^{J_n}\\hat{\\beta}_kx^k,\n\\]\nwhere\n\\[\n\\hat{\\boldsymbol{\\beta}}=\\underset{\\mathbf{b}\\in\\mathbb{R}^{J_n+1}}{\\arg\\min}\\sum_{i=1}^n\\left(Y_i-\\sum_{k=0}^{J_n}b_kX_i^k\\right)^2,\n\\]\nwhere $J_n\\to\\infty$ and $\\hl{\\zeta_{J_n+1}^2\\,(J_n+1)/n\\to0}$. Here, for each $K\\ge1$, $p_K=(p_0,\\ldots,p_{K-1})^T$ is a basis of the polynomials of degree at most $K-1$ such that $\\mathrm{E}[p_K(X)p_K(X)^T]=I_K$, and $\\zeta_K^2=\\sup_{x\\in\\operatorname{Supp}[X]}\\|p_K(x)\\|^2$. If $X$ is uniform on a compact interval, then $\\zeta_K^2=K^2$, and the condition reads \\hl{$(J_n+1)^3/n\\to0$}.\n\n[p.~169, next paragraph] As long as the CEF is \\hl{continuous}, \\hl{$\\operatorname{Supp}[X]$ is a compact interval} and \\hl{$\\mathrm{E}[(Y-\\mathrm{E}[Y\\mid X])^2]<\\infty$}, then $\\hat{\\mathrm{E}}[Y\\mid X=x]$ will be consistent \\hl{in mean square}, that is,\n\\[\n\\int\\bigl(\\hat{\\mathrm{E}}[Y\\mid X=x]-\\mathrm{E}[Y\\mid X=x]\\bigr)^2\\,dF(x)\\to0\\quad\\text{in probability,}\n\\]\nwhere $F$ is the CDF of $X$, and so it will be consistent across regions of the data with nonzero probability mass. \\hl{Theorem 4.3.3, \\emph{Polynomial Approximation of the CEF}, bounds the error of approximation, and the condition on $J_n$ bounds the sampling error.}\n\nThe square in the objective is the correction \\fid{P001-C04-E003}.", "evidence": "The evidence is an exact counterexample to the printed conditions and a derivation of the corrected ones.\n\nLet $(X_i,Y_i)$ be i.i.d.\\ with $X_i\\sim U[0,1]$ and $Y_i\\sim N(0,1)$ independent of $X_i$, so that $m(x)=\\mathrm{E}[Y\\mid X=x]=0$ and a polynomial of any degree represents $m$ exactly. Set $J_n=\\lfloor n^{3/4}\\rfloor$. Then $J_n\\to\\infty$ and $J_n/n\\le n^{-1/4}\\to0$. Consider $n=t^4$ for integers $t\\ge2$, so that the degree is $d=t^3$, and write $\\hat m_n$ for the least squares fit. Set $a=1/(2n)$, and let $B_t$ be the event that $X_i\\ge a$ for all $i$. By a union bound, I have $\\Pr(B_t)\\ge1-na=1/2$. Let $T_d$ be the Chebyshev polynomial of degree $d$, and define\n\\[\nr_t(x)=T_d\\!\\left(\\frac{2x-1-a}{1-a}\\right).\n\\]\nFor $x\\in[a,1]$, I have $|r_t(x)|\\le1$, so that $\\sum_ir_t(X_i)^2\\le n$ on $B_t$. For $x\\in[0,a/4]$, the bound $T_d(1+u)\\ge d^6u^3/360$, valid for $u\\ge0$ and $d\\ge3$, implies that $|r_t(x)|\\ge3t^6/2560$. Fix $\\delta\\in(0,1]$ and define\n\\[\nZ=\\left(\\int_0^\\delta r_t(x)^2\\,dx\\right)^{-1/2}\\int_0^\\delta\\hat m_n(x)\\,r_t(x)\\,dx.\n\\]\nConditional on $X_1,\\ldots,X_n$, the coefficients of $\\hat m_n$ are jointly normal with mean zero, and so is $Z$. By the Cauchy--Schwarz inequality for the matrix of sums of squares and cross products of the regressors, I have on $B_t$, for all $t$ such that $a/4\\le\\delta$,\n\\[\n\\operatorname{Var}(Z\\mid X_1,\\ldots,X_n)\\ge\\frac{\\int_0^\\delta r_t(x)^2\\,dx}{\\sum_ir_t(X_i)^2}\\ge\\frac{1}{n}\\cdot\\frac{a}{4}\\left(\\frac{3t^6}{2560}\\right)^2=\\frac{9t^4}{52428800}=V_t.\n\\]\nBy the Cauchy--Schwarz inequality for integrals, I have $|Z|\\le\\bigl(\\int_0^\\delta\\hat m_n(x)^2\\,dx\\bigr)^{1/2}$. Further, a normal variable with mean zero and variance $V$ lies in $[-\\eta,\\eta]$ with probability at most $2\\eta/\\sqrt{2\\pi V}$. Since $V_t\\to\\infty$, I obtain, for every $\\varepsilon>0$,\n\\[\n\\liminf_{t\\to\\infty}\\Pr\\left(\\int_0^\\delta\\hat m_{t^4}(x)^2\\,dx>\\varepsilon\\right)\\ge\\frac{1}{2}.\n\\]\n\n\\begin{audittable}{rrrrrr}{Exact lower bounds $V_t$ on the conditional variance of $Z$ in the counterexample, along $n=t^4$ with $J_n=t^3$. The printed condition holds, since $J_n/n\\to0$, and the bound grows without limit.}\n$t$ & $n=t^4$ & $J_n=t^3$ & $J_n/n$ & $V_t=9t^4/52428800$ & $V_t$ as a decimal \\\\\n\\hline\n$2$ & $16$ & $8$ & $1/2$ & $9/3276800$ & $2.75\\times10^{-6}$ \\\\\n$4$ & $256$ & $64$ & $1/4$ & $9/204800$ & $4.39\\times10^{-5}$ \\\\\n$8$ & $4{,}096$ & $512$ & $1/8$ & $9/12800$ & $7.03\\times10^{-4}$ \\\\\n$16$ & $65{,}536$ & $4{,}096$ & $1/16$ & $9/800$ & $0.01125$ \\\\\n$32$ & $1{,}048{,}576$ & $32{,}768$ & $1/32$ & $9/50$ & $0.18$ \\\\\n$64$ & $16{,}777{,}216$ & $262{,}144$ & $1/64$ & $72/25$ & $2.88$ \\\\\n$128$ & $268{,}435{,}456$ & $2{,}097{,}152$ & $1/128$ & $1152/25$ & $46.08$ \\\\\n$256$ & $4{,}294{,}967{,}296$ & $16{,}777{,}216$ & $1/256$ & $18432/25$ & $737.28$ \\\\\n$1024$ & $1{,}099{,}511{,}627{,}776$ & $1{,}073{,}741{,}824$ & $1/1024$ & $4718592/25$ & $188{,}743.68$ \\\\\n\\end{audittable}\n\nThe constants in $V_t$ come from crude bounds. The argument uses only the divergence of $V_t$, not its size at a given $n$. A second derivation with other constants gives $\\Pr\\bigl(\\int_0^{1/2}\\hat m_n(x)^2\\,dx>1\\bigr)\\ge1/48$ for every $n\\ge372827023$, and it applies also when $Y_i=\\pm1$ with probability $1/2$ each. In the same example the average of $\\hat m_n(X_i)^2$ over the observations has conditional expectation $(J_n+1)/n$ and tends to zero in probability. The fit is consistent in average squared error at the observed points, while population mean-square consistency fails. The counterexample does not show that the fit fails at every interior point, and it does not identify the fastest growth of $J_n$ that preserves consistency.\n\nFor the corrected conditions, set $K=J_n+1$ and $\\hat G=n^{-1}\\sum_ip_K(X_i)p_K(X_i)^T$, with $p_K$ and $\\zeta_K$ as in the corrected definition below. Since $\\mathrm{E}[p_K(X)p_K(X)^T]=I_K$, I have\n\\[\n\\mathrm{E}\\bigl\\|\\hat G-I_K\\bigr\\|_F^2=\\frac{1}{n}\\Bigl(\\mathrm{E}\\|p_K(X)\\|^4-K\\Bigr)\\le\\frac{\\zeta_K^2K}{n}\\to0,\n\\]\nwhich implies that the smallest eigenvalue of $\\hat G$ is at least $1/2$ with probability tending to one. Fix $\\varepsilon>0$ and a polynomial $q$ such that $\\sup_x|m(x)-q(x)|\\le\\varepsilon$, which exists by the Weierstrass Approximation Theorem. On that event, for all large $n$,\n\\[\n\\int\\bigl(\\hat m_n(x)-m(x)\\bigr)^2\\,dF(x)\\le9\\varepsilon^2+12\\|w\\|^2,\\qquad w=\\frac{1}{n}\\sum_{i=1}^np_K(X_i)\\bigl(Y_i-m(X_i)\\bigr),\n\\]\nwhere $F$ is the CDF of $X$, and $\\mathrm{E}\\|w\\|^2\\le\\zeta_K^2\\sigma^2/n\\to0$ with $\\sigma^2=\\mathrm{E}[(Y-m(X))^2]$. Since $\\varepsilon$ is arbitrary, the integral tends to zero in probability. For $X$ uniform on $[0,1]$, the polynomials $p_k(x)=\\sqrt{2k+1}\\,P_k(2x-1)$, where $P_k$ is the Legendre polynomial of degree $k$, satisfy $\\mathrm{E}[p_j(X)p_k(X)]=\\mathbf{1}\\{j=k\\}$ and $\\zeta_K^2=\\sum_{k=0}^{K-1}(2k+1)=K^2$. The condition is sufficient; I do not claim that it is necessary.", "problem": "The paragraph presents the two conditions on $J_n$, with a smooth CEF and suitable support, as enough for the estimated CEF to be consistent over regions of nonzero probability. They are not enough. Theorem 4.3.3, to which the text appeals, concerns the best polynomial predictor in the population: it states that $\\mathrm{E}[(\\mathrm{E}[Y\\mid X]-g(X,X^2,\\ldots,X^K))^2]$ is small for large $K$, where $g$ is the BLP of $Y$ given $X,X^2,\\ldots,X^K$. It does not bound the sampling error of a least squares fit whose number of coefficients grows with $n$, and this error need not vanish when $J_n/n\\to0$. Let $X$ be uniform on $[0,1]$, let $Y$ be standard normal and independent of $X$, so that the CEF is zero, and set $J_n=\\lfloor n^{3/4}\\rfloor$. Both printed conditions hold. For every $\\delta\\in(0,1]$, the integral of the squared fit over $[0,\\delta]$, a region of probability $\\delta$, does not converge to zero in probability. The sentence on p.~168 that the estimator is consistent if flexibility grows ``slowly'' enough is correct; the error is to present $J_n/n\\to0$ as slow enough. (The printed objective also lacks a square, which is the subject of \\fid{P001-C04-E003}. I restore the square throughout.)", "what_survives": "Theorem 4.3.3, the consistency of OLS with a fixed specification, and the statement on p.~168 that the estimator is consistent when flexibility grows slowly enough are unchanged. If the CEF is continuous, the support is compact and the conditional variance is bounded, then under the printed conditions the average squared error at the observed values of $X$ still converges to zero in probability. The regressions of Section 4.4 use fixed specifications with $n=184$, and I found no later displayed theorem invoking Definition 4.3.9 or its conditions on $J_n$. What changes is the condition on $J_n$ in the definition and the hypotheses of the claim of consistency that follows it." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C04-E001", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "The top of a shaded definition box. The display reads beta-hat equals the arg min over b in R to the power J_n + 1 of the sum over i from 1 to n of (Y_i minus the sum over k from 0 to J_n of b_k X_i to the power k). A red box marks the parenthesized residual, which carries no exponent. The next line reads 'where J_n tends to infinity and (1/n) J_n tends to 0'.", "caption": "Definition 4.3.9 (continued), p. 169 (PDF p. 187): the residual in the objective is marked; no exponent follows the closing parenthesis.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-E003.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T5", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T5" }, { "id": "P001-T70", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T70" }, { "id": "P001-T119", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T119" } ], "headline": "The objective in Definition 4.3.9 is the sum of squared residuals; the printed display omits the square.", "id": "P001-C04-E003", "location": "Definition 4.3.9 (continued), p. 169 (PDF p. 187), display that defines $\\hat{\\boldsymbol{\\beta}}$", "pdf_page": 187, "printed_page": "169", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T119", "P001-T5", "P001-T70" ], "text": { "as_printed": "\\[\n\\hat{\\boldsymbol{\\beta}}=\\underset{\\mathbf{b}\\in\\mathbb{R}^{J_n+1}}{\\arg\\min}\\sum_{i=1}^n\\hl{\\left(Y_i-\\sum_{k=0}^{J_n}b_kX_i^k\\right)},\n\\]", "corrected": "\\[\n\\hat{\\boldsymbol{\\beta}}=\\underset{\\mathbf{b}\\in\\mathbb{R}^{J_n+1}}{\\arg\\min}\\sum_{i=1}^n\\left(Y_i-\\sum_{k=0}^{J_n}b_kX_i^k\\right)^{\\hl{\\scriptstyle2}},\n\\]", "evidence": "The evidence is a derivation and a check of the page. Set $b_1=\\cdots=b_{J_n}=0$. Since $X_i^0=1$, the printed objective equals $\\sum_{i=1}^nY_i-nb_0$, which tends to $-\\infty$ as $b_0\\to\\infty$, for every $n\\ge1$. The objective is unbounded below, and the arg min does not exist. The corresponding displays of Definition 4.1.2 (p.~147) and Definition 4.3.10 (p.~170) print the square.", "problem": "The printed objective sums the residuals without squaring them. It is then not a least squares criterion, and it has no minimizer. The name of the estimator and the text around the definition show that the sum of squared residuals is meant.", "what_survives": "The polynomial least squares estimator that the text describes is unchanged. Restoring the square does not repair the conditions on $J_n$ in the same definition, which are corrected in \\fid{P001-C04-E001}." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C04-E003", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Three lines at the foot of a page: the heading 'Definition 4.3.10. Lasso Estimator' and two lines of text. Red boxes mark 'X =' at the end of the second line and '(1, X[1], X[2], ..., X[K])' at the start of the third.", "caption": "Start of Definition 4.3.10, p. 169 (PDF p. 187): the vector $\\mathbf{X}=(1,X_{[1]},X_{[2]},\\ldots,X_{[K]})$, which has $K+1$ entries, is marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-E004-p169.png" }, { "alt": "The top of a shaded definition box. The display reads beta-hat-lambda equals the arg min over b in R to the power K of (1/n) times the sum of (Y_i minus X_i b) squared, plus lambda times the norm of b with subscript 1. A red box marks 'b in R^K' under the arg min. A second red box, in the line below, marks 'norm of b with subscript 1 equals the sum from k = 1 to K of the absolute value of b_k'.", "caption": "Definition 4.3.10 (continued), p. 170 (PDF p. 188): the domain $\\mathbf{b}\\in\\mathbb{R}^K$ and the line that defines $\\|\\mathbf{b}\\|_1$ are marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-E004.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T71", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T71" } ], "headline": "In Definition 4.3.10 the minimization is over $\\mathbb{R}^{K+1}$, and the penalty is the $\\ell^1$ norm of the slope coefficients, without the intercept.", "id": "P001-C04-E004", "location": "Definition 4.3.10, pp. 169–170 (PDF pp. 187–188): the vector of explanatory variables, the arg min display and the line that defines the norm", "pdf_page": 187, "printed_page": "169–170", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T71" ], "text": { "as_printed": "[p.~169] For i.i.d. random vectors $(Y_1,\\mathbf{X}_1)$, $(Y_2,\\mathbf{X}_2)$, \\ldots, $(Y_n,\\mathbf{X}_n)$ where $\\mathbf{X}=(1,X_{[1]},X_{[2]},\\ldots,X_{[K]})$, the \\emph{lasso estimator} with \\emph{penalty parameter} $\\lambda$ is\n\n[p.~170]\n\\[\n\\hat{\\boldsymbol{\\beta}}_\\lambda=\\underset{\\mathbf{b}\\in\\hl{\\scriptstyle\\mathbb{R}^K}}{\\arg\\min}\\left(\\frac{1}{n}\\sum_{i=1}^n(Y_i-\\mathbf{X}_i\\mathbf{b})^2+\\lambda\\|\\mathbf{b}\\|_1\\right),\n\\]\nwhere $\\|\\cdot\\|_1$ is the $\\ell^1$ norm, that is, $\\hl{\\textstyle\\|\\mathbf{b}\\|_1=\\sum_{k=1}^K|b_k|}$ (the sum of the absolute value of the coefficients).", "corrected": "[p.~169] For i.i.d. random vectors $(Y_1,\\mathbf{X}_1)$, $(Y_2,\\mathbf{X}_2)$, \\ldots, $(Y_n,\\mathbf{X}_n)$ where $\\mathbf{X}=(1,X_{[1]},X_{[2]},\\ldots,X_{[K]})$, the \\emph{lasso estimator} with \\emph{penalty parameter} $\\lambda$ is\n\n[p.~170]\n\\[\n\\hat{\\boldsymbol{\\beta}}_\\lambda=\\underset{\\mathbf{b}\\in\\hl{\\scriptstyle\\mathbb{R}^{K+1}}}{\\arg\\min}\\left(\\frac{1}{n}\\sum_{i=1}^n(Y_i-\\mathbf{X}_i\\mathbf{b})^2+\\lambda\\hl{\\sum_{k=1}^K|b_k|}\\right),\n\\]\nwhere \\hl{$\\mathbf{b}=(b_0,b_1,\\ldots,b_K)^T$ and $b_0$, the coefficient of the constant, is not penalized}. The penalty $\\sum_{k=1}^K|b_k|$ is the $\\ell^1$ norm of \\hl{the slope coefficients $(b_1,\\ldots,b_K)$} (the sum of the absolute value of the \\hl{slope} coefficients).", "evidence": "The evidence is a derivation and a check of the two pages. For $K=1$, the row $\\mathbf{X}_i=(1,X_{[1]i})$ has two entries and cannot multiply a vector in $\\mathbb{R}^1$. For $\\mathbf{b}=(1,0,\\ldots,0)^T\\in\\mathbb{R}^{K+1}$, the printed sum equals $0$, whereas $\\|\\mathbf{b}\\|_1=1$. A norm does not vanish at a nonzero vector, and the sum is not a norm on $\\mathbb{R}^{K+1}$. Definition 4.1.2 (p.~147), with the same vector of explanatory variables, takes its arg min over $\\mathbb{R}^{K+1}$.", "problem": "The vector $\\mathbf{X}=(1,X_{[1]},\\ldots,X_{[K]})$ has $K+1$ entries, and the arg min is taken over $\\mathbf{b}\\in\\mathbb{R}^K$. The product $\\mathbf{X}_i\\mathbf{b}$ is then not defined. Once $\\mathbf{b}$ has $K+1$ entries, the sum $\\sum_{k=1}^K|b_k|$ omits the coefficient of the constant and is not the $\\ell^1$ norm of $\\mathbf{b}$. The text before the definition says that the penalty typically excludes the intercept. The sum is the intended penalty, and the name is wrong.", "what_survives": "The lasso with an unpenalized intercept, its equality with OLS at $\\lambda=0$ and the discussion of the choice of $\\lambda$ are unchanged. Only the domain of the arg min and the description of the penalty change." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C04-E004", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Part of a figure with two three-dimensional surface plots of infant mortality against access to clean water and electricity. Above the plots are the labels (5) OLS: alpha plus beta X plus gamma Z plus omega X Z, and (6) OLS: alpha plus beta X plus gamma Z plus a double sum, over kappa from 1 to 2 and l from 1 to 2, of omega kappa l times X to the kappa times Z to the l. Below the plots are the labels (7) OLS and (8) Lasso, with the same formula and kappa running from 1 to 3. Red boxes mark the terms beta X plus gamma Z in labels (6), (7) and (8).", "caption": "Figure 4.4.4, p. 176 (PDF p. 194), lower part: the labels of panels (5) and (6), and panels (7) and (8) with their labels. In labels (6), (7) and (8) the marked terms $\\beta X+\\gamma Z$ are the only nonconstant terms that do not contain both variables; the pure powers $X^2$, $X^3$ and $Z^2$ do not appear.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-E006.png" }, { "alt": "Two formula labels. On the left, (3) OLS: alpha plus beta X plus gamma Z. On the right, (4) OLS: alpha plus the sum over kappa from 1 to 2 of beta sub kappa times X to the kappa, plus the sum over kappa from 1 to 2 of gamma kappa Z to the kappa. A red box marks gamma kappa, where the kappa is printed at full size on the line and not as a subscript.", "caption": "Figure 4.4.4, p. 176 (PDF p. 194), labels of panels (3) and (4). In label (4) the marked $\\kappa$ after $\\gamma$ is set on the line, where the subscript of $\\gamma_\\kappa$ is meant.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-E006-b.png" }, { "alt": "Upper part of the regression table for cross-national infant mortality, with columns (1) to (8) and rows Intercept, Water, Water squared, Water cubed, Elec and Elec squared, standard errors in parentheses. Red boxes mark, in columns (6), (7) and (8), the Water squared coefficients -0.006, -0.076 and 0.000; in columns (7) and (8), the Water cubed coefficients 0.001 and 0.000; and in columns (6), (7) and (8), the Elec squared coefficients -0.004, -0.010 and -0.003.", "caption": "Table 4.4.5, p. 173 (PDF p. 191), upper rows. The marked coefficients on Water$^2$, Water$^3$ and Elec$^2$ in columns (6)–(8) belong to pure powers that the labels of Figure 4.4.4 omit.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-E006-c.png" }, { "alt": "Text of Example 4.3.5, Overfitting with a Nonlinear CEF. It supposes that Y equals g of X plus epsilon for a nonlinear function g, with X uniform on minus 10 to 10, epsilon normal with mean 0 and variance 25, and X independent of epsilon. A red box marks g(X), with a capital X, in the sentence: The CEF of Y given X is thus E[Y|X = x] = g(X). A second red box marks Figure 4.3.4 in the sentence: Let g be the function shown in black in Figure 4.3.4. The next sentence begins: As in Figure 4.3.4, panel (a) Figure 4.3.5 plots ten random draws.", "caption": "Example 4.3.5, p. 163 (PDF p. 181). The marked $g(X)$ should be $g(x)$, and the marked reference should be to Figure 4.3.5.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-E006-d.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T68", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T68" } ], "headline": "The labels of panels (6)–(8) of Figure 4.4.4 omit the pure powers of $X$ and $Z$ that Table 4.4.5 reports for the same regressions; label (4) has a misplaced subscript, and Example 4.3.5 should read $g(x)$ and refer to Figure 4.3.5.", "id": "P001-C04-E006", "location": "Figure 4.4.4, labels of panels (4) and (6)–(8), p. 176 (PDF p. 194), against Table 4.4.5, columns (4) and (6)–(8), p. 173 (PDF p. 191); Example 4.3.5, p. 163 (PDF p. 181)", "pdf_page": 194, "printed_page": "176", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T68" ], "text": { "as_printed": "Figure 4.4.4, p.~176, labels of panels (4), (6), (7) and (8):\n\\[\n\\begin{aligned}\n&\\text{(4) OLS: }\\alpha+\\sum_{\\kappa=1}^{2}\\beta_\\kappa X^\\kappa+\\sum_{\\kappa=1}^{2}\\hl{\\gamma\\kappa}Z^\\kappa\\\\\n&\\text{(6) OLS: }\\alpha+\\hl{\\beta X+\\gamma Z}+\\sum_{\\kappa=1}^{2}\\sum_{l=1}^{2}\\omega_{\\kappa l}X^\\kappa Z^l\\\\\n&\\text{(7) OLS: }\\alpha+\\hl{\\beta X+\\gamma Z}+\\sum_{\\kappa=1}^{3}\\sum_{l=1}^{2}\\omega_{\\kappa l}X^\\kappa Z^l\\\\\n&\\text{(8) Lasso: }\\alpha+\\hl{\\beta X+\\gamma Z}+\\sum_{\\kappa=1}^{3}\\sum_{l=1}^{2}\\omega_{\\kappa l}X^\\kappa Z^l\n\\end{aligned}\n\\]\nExample 4.3.5, p.~163:\n\nSuppose we had a random vector $(X,Y)$ such that\n\\[\nY=g(X)+\\varepsilon,\n\\]\nfor some (nonlinear) function $g$, where $X\\sim U(-10,10)$, $\\varepsilon\\sim N(0,25)$, and $X\\perp\\!\\!\\!\\perp\\varepsilon$. The CEF of $Y$ given $X$ is thus $\\mathrm{E}[Y|X=x]=\\hl{g(X)}$. Let $g$ be the function shown in black in \\hl{Figure 4.3.4}.", "corrected": "Figure 4.4.4, labels of panels (4), (6), (7) and (8), with $X$ the percentage with access to clean water and $Z$ the percentage with access to electricity:\n\\[\n\\begin{aligned}\n&\\text{(4) OLS: }\\alpha+\\sum_{\\kappa=1}^{2}\\beta_\\kappa X^\\kappa+\\sum_{\\kappa=1}^{2}\\hl{\\gamma_\\kappa}Z^\\kappa\\\\\n&\\text{(6) OLS: }\\alpha+\\hl{\\sum_{\\kappa=1}^{2}\\beta_\\kappa X^\\kappa+\\sum_{l=1}^{2}\\gamma_lZ^l}+\\sum_{\\kappa=1}^{2}\\sum_{l=1}^{2}\\omega_{\\kappa l}X^\\kappa Z^l\\\\\n&\\text{(7) OLS: }\\alpha+\\hl{\\sum_{\\kappa=1}^{3}\\beta_\\kappa X^\\kappa+\\sum_{l=1}^{2}\\gamma_lZ^l}+\\sum_{\\kappa=1}^{3}\\sum_{l=1}^{2}\\omega_{\\kappa l}X^\\kappa Z^l\\\\\n&\\text{(8) Lasso: }\\alpha+\\hl{\\sum_{\\kappa=1}^{3}\\beta_\\kappa X^\\kappa+\\sum_{l=1}^{2}\\gamma_lZ^l}+\\sum_{\\kappa=1}^{3}\\sum_{l=1}^{2}\\omega_{\\kappa l}X^\\kappa Z^l\n\\end{aligned}\n\\]\nThe highlighted sums replace the terms $\\beta X+\\gamma Z$; they are not added to them.\n\nExample 4.3.5, p.~163:\n\nSuppose we had a random vector $(X,Y)$ such that\n\\[\nY=g(X)+\\varepsilon,\n\\]\nfor some (nonlinear) function $g$, where $X\\sim U(-10,10)$, $\\varepsilon\\sim N(0,25)$, and $X\\perp\\!\\!\\!\\perp\\varepsilon$. The CEF of $Y$ given $X$ is thus $\\mathrm{E}[Y|X=x]=\\hl{g(x)}$. Let $g$ be the function shown in black in \\hl{Figure 4.3.5}.", "evidence": "The evidence is a check of the pages and an exact calculation from the printed coefficients. For any coefficients, the formula in labels (6)--(8) reduces at $X=0$ to $h(z)=\\alpha+\\gamma z$, and the second difference $h(0)-2h(1)+h(2)$ of a linear function is zero. For $h(z)=a+bz+cz^2$ the same difference is $2c$. From the coefficients printed in Table 4.4.5, I obtain at $X=0$ the polynomials $T(0,z)$ in the table below. Each has a nonzero second difference, and I conclude that no choice of $\\alpha$, $\\beta$, $\\gamma$ and $\\omega_{\\kappa l}$ in the labels reproduces the polynomial of the table.\n\\begin{audittable}{llrrrr}{Table 4.4.5 at $X=0$, computed exactly from the printed coefficients. Every formula of the printed form has second difference $0$.}\nColumn & $T(0,z)$ & $T(0,0)$ & $T(0,1)$ & $T(0,2)$ & $T(0,0)-2T(0,1)+T(0,2)$ \\\\\n(6) & $67.385+0.137z-0.004z^2$ & $67.385$ & $67.518$ & $67.643$ & $-0.008$ \\\\\n(7) & $50.900+0.990z-0.010z^2$ & $50.900$ & $51.880$ & $52.840$ & $-0.020$ \\\\\n(8) & $68.631+0.062z-0.003z^2$ & $68.631$ & $68.690$ & $68.743$ & $-0.006$ \\\\\n\\end{audittable}\nThe pure powers of $X$ are missing in the same way. At $Z=0$ the polynomial of the table is $67.385-0.432x-0.006x^2$ in column (6) and $50.900+1.808x-0.076x^2+0.001x^3$ in column (7). Its second difference in $x$ at $0,1,2$ is $-0.012$ in column (6) and $-0.146$ in column (7), and its third difference at $0,1,2,3$ in column (7) is $0.006$. Each of these differences is zero for the labels. In column (8) the table prints $0.000$ for Water$^2$ and Water$^3$, which does not show that these terms are active; for that label the mismatch follows from the coefficient $-0.003$ on Elec$^2$. For label (4), the literal reading requires the coefficient on Elec$^2$ to be $2\\times0.157=0.314$, and the table prints $-0.004$. For Example 4.3.5, since $\\varepsilon$ is independent of $X$ and has mean zero, I have\n\\[\n\\mathrm{E}[Y|X=x]=\\mathrm{E}[g(X)|X=x]+\\mathrm{E}[\\varepsilon|X=x]=g(x)+\\mathrm{E}[\\varepsilon]=g(x).\n\\]\nI did not refit the regressions, because the data are not available.", "problem": "Here $X$ is the percentage with access to clean water and $Z$ the percentage with access to electricity, and the six panels of Figure 4.4.4 show the fits of regressions (3)--(8) of Table 4.4.5. In the labels of panels (6)--(8), every term of the double sum has positive powers of both $X$ and $Z$, and none is a pure power $X^2$, $X^3$ or $Z^2$. Table 4.4.5 reports coefficients on these pure powers in the same columns: Water$^2$ and Elec$^2$ in column (6), and Water$^2$, Water$^3$ and Elec$^2$ in columns (7) and (8). Then the labels describe different regressors from the table, and someone who fits the regressions as labelled does not fit the ones that the table reports. The specifications to follow are those of the table. In label (4), the $\\kappa$ after $\\gamma$ is set on the line and not as a subscript; read literally, $\\gamma\\kappa Z^\\kappa$ makes the coefficient on $Z^2$ twice the coefficient on $Z$, which the table contradicts. In Example 4.3.5, the conditional expectation at a fixed $x$ is written as $g(X)$, a random variable. The example also says that $g$ is shown in Figure 4.3.4, but that figure shows the linear CEF of Example 4.3.4, and the nonlinear $g$ is the black curve of Figure 4.3.5.", "what_survives": "These are corrections to labels, a subscript, a letter and a figure number. No coefficient, plotted surface or estimate of the APD or PDA is shown to be wrong, and the discussion of Table 4.4.5 on p.~175 rests on the table, which I have not recomputed. The description of regression (6) on p.~174 lists the interactions that it includes and needs no change. In Example 4.3.5, the next sentence, which begins ``As in Figure 4.3.4'', is a correct comparison and stays as printed." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C04-E006", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of three lines of displayed mathematics. The first line reads g(X[1], X[2], ..., X[K]) = beta 0 + beta 1 X[1] + beta 2 X[2] + ... + beta K X[K] such that. The next lines define (beta 0, beta 1, ..., beta K) as the argmin over (b 0, b 1, ..., b K) of E[(Y - (b 0 + b 1 X[1] + b 2 X[2] + ... b K X[K])) squared]. A red box marks the ellipsis and b K, which are printed with no plus sign between them.", "caption": "Definition of the multivariate BLP, p. 146 (PDF p. 164). In the marked place the ellipsis is followed by $b_K$ with no plus sign; the first line of the crop prints the same sum with the sign, “$+\\ldots+\\beta_KX_{[K]}$”.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N002.png" }, { "alt": "Crop of the end of the shaded box of Definition 4.1.2 and two lines of text below it. Red boxes mark two places. The first is the ellipsis and b K in the displayed argmin of one over n times the sum over i of (Y i - (b 0 + b 1 X[1]i + b 2 X[2]i + ... b K X[K]i)) squared. The second is the ellipsis and beta-hat K in the sentence 'The quantity e i = Y i - (beta-hat 0 + beta-hat 1 X[1]i + beta-hat 2 X[2]i + ... beta-hat K X[K]i) is known as the i-th residual.' In both places no plus sign follows the ellipsis.", "caption": "Definition 4.1.2 and the sentence after it, p. 147 (PDF p. 165). The two marked places lack the plus sign before the last term; the first line of the crop has it.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N002-b.png" }, { "alt": "Crop of four displayed equations and one line of text. The first equation reads e i = Y i - (b 0 + b 1 X[1]i + b 2 X[2]i + ... + b K X[K]i). After the sentence 'We can stack all n of these e i's to form a system of equations', three equations follow, for e 1, e 2 and e n. A red box in each marks the ellipsis and b K, printed with no plus sign between them. The last equation begins e n = Y 3.", "caption": "Proof of Theorem 4.1.4, p. 148 (PDF p. 166). The three equations of the stacked system lack the plus sign in the marked places; the definition of $e_i$ in the first line of the crop has it.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N002-c.png" } ], "formal_release": null, "formal_statements": [], "headline": "A plus sign is missing before the last term of the linear predictor in six formulas on pp. 146–148.", "id": "P001-C04-N002", "location": "Display defining the multivariate BLP, p. 146 (PDF p. 164); Definition 4.1.2 and the sentence after it, p. 147 (PDF p. 165); proof of Theorem 4.1.4, stacked system of equations, p. 148 (PDF p. 166)", "pdf_page": 164, "printed_page": "146", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "P.~146:\n\\[\n(\\beta_0,\\beta_1,\\ldots,\\beta_K)=\\operatorname*{arg\\,min}_{(b_0,b_1,\\ldots,b_K)\\in\\mathbb{R}^{K+1}}\\mathrm{E}\\Bigl[\\bigl(Y-(b_0+b_1X_{[1]}+b_2X_{[2]}+\\hl{\\ldots b_K}X_{[K]})\\bigr)^2\\Bigr].\n\\]\n\nP.~147, Definition 4.1.2:\n\\[\n\\hat{\\boldsymbol\\beta}=(\\hat\\beta_0,\\hat\\beta_1,\\ldots,\\hat\\beta_K)=\\operatorname*{arg\\,min}_{(b_0,b_1,\\ldots,b_K)\\in\\mathbb{R}^{K+1}}\\frac{1}{n}\\sum_{i=1}^{n}\\bigl(Y_i-(b_0+b_1X_{[1]i}+b_2X_{[2]i}+\\hl{\\ldots b_K}X_{[K]i})\\bigr)^2.\n\\]\n\nP.~147, after Definition 4.1.2: ``The quantity $e_i=Y_i-(\\hat\\beta_0+\\hat\\beta_1X_{[1]i}+\\hat\\beta_2X_{[2]i}+\\hl{\\ldots\\hat\\beta_K}X_{[K]i})$ is known as the $i^{\\text{th}}$ \\emph{residual}.''\n\nP.~148, proof of Theorem 4.1.4:\n\\[\n\\begin{aligned}\ne_1&=Y_1-\\bigl(b_0+b_1X_{[1]1}+b_2X_{[2]1}+\\hl{\\ldots b_K}X_{[K]1}\\bigr)\\\\\ne_2&=Y_2-\\bigl(b_0+b_1X_{[1]2}+b_2X_{[2]2}+\\hl{\\ldots b_K}X_{[K]2}\\bigr)\\\\\n&\\ \\ \\vdots\\\\\ne_n&=Y_3-\\bigl(b_0+b_1X_{[1]n}+b_2X_{[2]n}+\\hl{\\ldots b_K}X_{[K]n}\\bigr).\n\\end{aligned}\n\\]", "corrected": "P.~146:\n\\[\n(\\beta_0,\\beta_1,\\ldots,\\beta_K)=\\operatorname*{arg\\,min}_{(b_0,b_1,\\ldots,b_K)\\in\\mathbb{R}^{K+1}}\\mathrm{E}\\Bigl[\\bigl(Y-(b_0+b_1X_{[1]}+b_2X_{[2]}+\\ldots\\hl{+}b_KX_{[K]})\\bigr)^2\\Bigr].\n\\]\n\nP.~147, Definition 4.1.2:\n\\[\n\\hat{\\boldsymbol\\beta}=(\\hat\\beta_0,\\hat\\beta_1,\\ldots,\\hat\\beta_K)=\\operatorname*{arg\\,min}_{(b_0,b_1,\\ldots,b_K)\\in\\mathbb{R}^{K+1}}\\frac{1}{n}\\sum_{i=1}^{n}\\bigl(Y_i-(b_0+b_1X_{[1]i}+b_2X_{[2]i}+\\ldots\\hl{+}b_KX_{[K]i})\\bigr)^2.\n\\]\n\nP.~147, after Definition 4.1.2: ``The quantity $e_i=Y_i-(\\hat\\beta_0+\\hat\\beta_1X_{[1]i}+\\hat\\beta_2X_{[2]i}+\\ldots\\hl{+}\\hat\\beta_KX_{[K]i})$ is known as the $i^{\\text{th}}$ \\emph{residual}.''\n\nP.~148, proof of Theorem 4.1.4:\n\\[\n\\begin{aligned}\ne_1&=Y_1-\\bigl(b_0+b_1X_{[1]1}+b_2X_{[2]1}+\\ldots\\hl{+}b_KX_{[K]1}\\bigr)\\\\\ne_2&=Y_2-\\bigl(b_0+b_1X_{[1]2}+b_2X_{[2]2}+\\ldots\\hl{+}b_KX_{[K]2}\\bigr)\\\\\n&\\ \\ \\vdots\\\\\ne_n&=Y_n-\\bigl(b_0+b_1X_{[1]n}+b_2X_{[2]n}+\\ldots\\hl{+}b_KX_{[K]n}\\bigr).\n\\end{aligned}\n\\]\n\nThe outcome $Y_n$ in the last equation is the correction of \\fid{P001-C04-E005}.", "evidence": "I checked pp.~146--148 at high magnification. The six formulas are the argmin display on p.~146, the argmin display and the residual on p.~147, and the three equations of the stacked system on p.~148. The signs are absent from the printed page and not only from text extracted from the PDF.", "problem": "In each of the six formulas the ellipsis is followed directly by the last term of the sum, $b_KX_{[K]}$, $b_KX_{[K]i}$ or $\\hat\\beta_KX_{[K]i}$. The sum is the linear predictor, and its last term is added like the others. The same sum is printed with the plus sign in the BLP function on p.~146, in the fitted function at the top of p.~147, and in the definition of $e_i$ before the stacked system on p.~148.", "what_survives": "The linear predictor, the OLS regression estimator and the proof of Theorem 4.1.4 are unchanged. Nothing depends on the missing signs." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C04-N002", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of four lines of text. Red boxes mark the bold symbol beta k and the ordinal k-th in the sentence 'the robust standard error estimator for beta k is the square root of the k-th diagonal element of V-hat[beta-hat].'", "caption": "Paragraph after Definition 4.2.1, p. 153 (PDF p. 171). The marked coefficient is set in bold, and the marked ordinal counts diagonal elements from $1$ while the coefficients are numbered from $0$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N003.png" }, { "alt": "Crop of five lines of text. A red box marks the ordinal k-th in the sentence 'The classical standard error estimator for beta k is then just defined as square root of the k-th diagonal element of V-hat C[beta-hat].'", "caption": "Sentence after Definition 4.2.3, p. 155 (PDF p. 173). The marked ordinal is the same as on p. 153.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N003-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T66", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T66" } ], "headline": "The standard error estimator for $\\hat\\beta_k$ is the square root of the $(k+1)^{\\text{th}}$ diagonal element of the estimated covariance matrix, since the coefficients are numbered from $0$; on p. 153 the scalar coefficient is also set in bold.", "id": "P001-C04-N003", "location": "Paragraph after Definition 4.2.1, p. 153 (PDF p. 171); sentence after Definition 4.2.3, p. 155 (PDF p. 173)", "pdf_page": 171, "printed_page": "153", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T66" ], "text": { "as_printed": "P.~153: ``the robust standard error estimator for $\\hl{\\boldsymbol\\beta_k}$ is the square root of the $\\hl{k^{\\text{th}}}$ diagonal element of $\\hat{\\mathrm{V}}[\\hat{\\boldsymbol\\beta}]$.''\n\nP.~155: ``The classical standard error estimator for $\\beta_k$ is then just defined as square root of the $\\hl{k^{\\text{th}}}$ diagonal element of $\\hat{\\mathrm{V}}_{\\mathrm{C}}[\\hat{\\boldsymbol\\beta}]$.''", "corrected": "P.~153: ``the robust standard error estimator for $\\hl{\\hat\\beta_k}$ is the square root of the $\\hl{(k+1)^{\\text{th}}}$ diagonal element of $\\hat{\\mathrm{V}}[\\hat{\\boldsymbol\\beta}]$\\hl{, for $k=0,1,\\ldots,K$}.''\n\nP.~155: ``The classical standard error estimator for $\\hl{\\hat\\beta_k}$ is then just defined as square root of the $\\hl{(k+1)^{\\text{th}}}$ diagonal element of $\\hat{\\mathrm{V}}_{\\mathrm{C}}[\\hat{\\boldsymbol\\beta}]$\\hl{, for $k=0,1,\\ldots,K$}.''\n\nHere $\\hat{\\boldsymbol\\beta}=(\\hat\\beta_0,\\hat\\beta_1,\\ldots,\\hat\\beta_K)$ is the OLS coefficient estimator, and $\\hat{\\mathrm{V}}[\\hat{\\boldsymbol\\beta}]$ and $\\hat{\\mathrm{V}}_{\\mathrm{C}}[\\hat{\\boldsymbol\\beta}]$ are the $(K+1)\\times(K+1)$ matrices of Definitions 4.2.1 and 4.2.3.", "evidence": "The evidence is a derivation from the ordering in Definitions 4.1.2, 4.2.1 and 4.2.3, with an illustration that is not taken from the book. Let $K=1$, and suppose that the estimated covariance matrix has diagonal elements $4$ and $9$. The standard error estimate for the slope $\\hat\\beta_1$ is $3$, the square root of the second diagonal element. Read literally, the sentence takes the first diagonal element, which gives $2$, the standard error estimate for the intercept $\\hat\\beta_0$.", "problem": "Recall that Definition 4.1.2 orders the coefficients as $\\hat{\\boldsymbol\\beta}=(\\hat\\beta_0,\\hat\\beta_1,\\ldots,\\hat\\beta_K)$, and Definitions 4.2.1 and 4.2.3 order the regressors as $\\mathbf{X}=(1,X_{[1]},X_{[2]},\\ldots,X_{[K]})$. The estimated covariance matrix is therefore $(K+1)\\times(K+1)$, with the variance estimate for $\\hat\\beta_0$ in the first diagonal position and the variance estimate for $\\hat\\beta_k$ in position $k+1$. Read literally, ``the $k^{\\text{th}}$ diagonal element'' is the variance estimate for $\\hat\\beta_{k-1}$, and for $k=0$ there is no such element. On p.~153 the scalar coefficient is also printed in bold, the typeface of the vector $\\boldsymbol\\beta$.", "what_survives": "The robust and classical covariance estimators of Definitions 4.2.1 and 4.2.3 are correct as printed, and their interpretation is unchanged. Only the position of the diagonal element for a given coefficient changes. Note that the phrase ``standard error estimator for $\\beta_k$'', without a hat, is common shorthand and not an error; the corrected sentences write $\\hat\\beta_k$." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C04-N003", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of two lines of text and a display of three lines. The first line of the display reads n V[beta-hat] is approximately (E[X transpose X]) inverse times E[epsilon squared X transpose X] times (E[X i transpose X]) inverse. A red box marks X i transpose X in the last factor, where the first X carries a subscript i. The second line reads (E[X transpose X]) inverse times (sigma squared E[X transpose X]) times (E[X transpose X]) inverse, and the third reads sigma squared (E[X transpose X]) inverse.", "caption": "Section 4.2.3, p. 155 (PDF p. 173), the display for $n\\mathrm{V}[\\hat{\\boldsymbol\\beta}]$. The marked factor carries a subscript $i$; the first factor of the same line and both inverse factors of the second line have none.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N005.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T67", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T67" } ], "headline": "The last factor in the first line of the display is $(\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}])^{-1}$; the subscript $i$ on $\\mathbf{X}^T$ is a misprint.", "id": "P001-C04-N005", "location": "Section 4.2.3, p. 155 (PDF p. 173), first line of the display for $n\\mathrm{V}[\\hat{\\boldsymbol\\beta}]$, last factor", "pdf_page": 173, "printed_page": "155", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T67" ], "text": { "as_printed": "\\[\n\\begin{aligned}\nn\\mathrm{V}[\\hat{\\boldsymbol\\beta}]&\\approx\\bigl(\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}]\\bigr)^{-1}\\mathrm{E}[\\varepsilon^2\\mathbf{X}^T\\mathbf{X}]\\bigl(\\mathrm{E}[\\mathbf{X}_{\\hl{i}}^T\\mathbf{X}]\\bigr)^{-1}\\\\\n&=\\bigl(\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}]\\bigr)^{-1}\\bigl(\\sigma^2\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}]\\bigr)\\bigl(\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}]\\bigr)^{-1}\\\\\n&=\\sigma^2\\bigl(\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}]\\bigr)^{-1},\n\\end{aligned}\n\\]", "corrected": "Suppose that $\\mathrm{E}[\\varepsilon\\mid X_{[1]},X_{[2]},\\ldots,X_{[K]}]=0$ and $\\mathrm{V}[\\varepsilon\\mid X_{[1]},X_{[2]},\\ldots,X_{[K]}]=\\sigma^2$ for all possible values of $X_{[1]},X_{[2]},\\ldots,X_{[K]}$, and let $\\mathbf{X}=(1,X_{[1]},X_{[2]},\\ldots,X_{[K]})$. Then\n\\[\n\\begin{aligned}\nn\\mathrm{V}[\\hat{\\boldsymbol\\beta}]&\\approx\\bigl(\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}]\\bigr)^{-1}\\mathrm{E}[\\varepsilon^2\\mathbf{X}^T\\mathbf{X}]\\bigl(\\mathrm{E}[\\hl{\\mathbf{X}^T}\\mathbf{X}]\\bigr)^{-1}\\\\\n&=\\bigl(\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}]\\bigr)^{-1}\\bigl(\\sigma^2\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}]\\bigr)\\bigl(\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}]\\bigr)^{-1}\\\\\n&=\\sigma^2\\bigl(\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}]\\bigr)^{-1}.\n\\end{aligned}\n\\]", "evidence": "The evidence is a check of pp.~152 and 155 and a derivation. Set $Q=\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}]$, and assume that $Q$ is nonsingular. Under the two assumptions of Section 4.2.3, I have $\\mathrm{E}[\\varepsilon^2\\mid\\mathbf{X}]=\\sigma^2$. By the Law of Iterated Expectations,\n\\[\n\\mathrm{E}[\\varepsilon^2\\mathbf{X}^T\\mathbf{X}]=\\mathrm{E}\\bigl[\\mathbf{X}^T\\mathbf{X}\\,\\mathrm{E}[\\varepsilon^2\\mid\\mathbf{X}]\\bigr]=\\sigma^2Q.\n\\]\nSubstituting this into $Q^{-1}\\mathrm{E}[\\varepsilon^2\\mathbf{X}^T\\mathbf{X}]Q^{-1}$, I obtain $Q^{-1}(\\sigma^2Q)Q^{-1}=\\sigma^2Q^{-1}$, which is the third line of the display. The reduction requires the same matrix $Q$ in both inverse factors.", "problem": "The last factor of the first line carries an observation subscript $i$ on $\\mathbf{X}^T$ and none on $\\mathbf{X}$. The matrix meant is $\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}]$, the second-moment matrix of the regressor row $\\mathbf{X}=(1,X_{[1]},X_{[2]},\\ldots,X_{[K]})$. It is printed without a subscript in the first factor of the same line, in both inverse factors of the second line, and in the same expression in Section 4.2.2 (p.~152).", "what_survives": "The simplification $n\\mathrm{V}[\\hat{\\boldsymbol\\beta}]\\approx\\sigma^2(\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}])^{-1}$ under the two assumptions and Definition 4.2.3 are unaffected. Nothing depends on the misprint." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C04-N005", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of a paragraph of eight lines. A red box marks the words 'If homoskedasticity does hold,' in the sentence 'If homoskedasticity does hold, then in large samples, the classical and robust sampling variance estimates will agree.' The next sentence reads 'With large n and i.i.d. sampling, there is little reason not to use robust standard errors or the bootstrap to characterize the uncertainty of a regression fit.'", "caption": "Last paragraph of Section 4.2.3, p. 156 (PDF p. 174). The marked condition names homoskedasticity alone.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N006.png" }, { "alt": "Crop of six lines of text. Red boxes mark the two assumptions in the sentence 'Suppose we assume that E[epsilon | X[1], X[2], ..., X[K]] = 0 and V[epsilon | X[1], X[2], ..., X[K]] = sigma squared for all possible values of X[1], X[2], ..., X[K].' The next sentence reads 'The first assumption is equivalent to linearity of E[Y | X[1], X[2], ..., X[K]], and the second assumption is known as (conditional) homoskedasticity.'", "caption": "Opening of Section 4.2.3, p. 155 (PDF p. 173). The two marked assumptions are both in force in the section: the first is linearity of the CEF, and the second is homoskedasticity.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N006-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "The classical and robust sampling variance estimates agree in large samples when both assumptions of Section 4.2.3 hold, linearity of the CEF and homoskedasticity; the closing sentence of the section names only homoskedasticity.", "id": "P001-C04-N006", "location": "Section 4.2.3, last paragraph, p. 156 (PDF p. 174); the two assumptions of the section are on p. 155 (PDF p. 173)", "pdf_page": 174, "printed_page": "156", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "P.~155: ``Suppose we \\emph{assume} that $\\mathrm{E}[\\varepsilon\\mid X_{[1]},X_{[2]},\\ldots,X_{[K]}]=0$ and $\\mathrm{V}[\\varepsilon\\mid X_{[1]},X_{[2]},\\ldots,X_{[K]}]=\\sigma^2$ for all possible values of $X_{[1]},X_{[2]},\\ldots,X_{[K]}$. The first assumption is equivalent to linearity of $\\mathrm{E}[Y\\mid X_{[1]},X_{[2]},\\ldots,X_{[K]}]$, and the second assumption is known as (conditional) \\emph{homoskedasticity}.''\n\nP.~156: ``\\hl{If homoskedasticity \\emph{does} hold}, then in large samples, the classical and robust sampling variance estimates will agree.''", "corrected": "``If \\hl{both the linearity and homoskedasticity assumptions} hold, then in large samples, the classical and robust sampling variance estimates will agree.''", "intended_meaning": "The sentence closes a section that assumes both $\\mathrm{E}[\\varepsilon\\mid\\mathbf{X}]=0$ and $\\mathrm{V}[\\varepsilon\\mid\\mathbf{X}]=\\sigma^2$, and the derivation on p.~155 obtains $\\mathrm{E}[\\varepsilon^2\\mid\\mathbf{X}]=\\sigma^2$ ``by the linearity and homoskedasticity assumptions''. With both assumptions in force the sentence is correct. Since $\\mathrm{E}[\\varepsilon^2\\mid\\mathbf{X}]=\\sigma^2$, I have $\\mathrm{E}[\\varepsilon^2\\mathbf{X}^T\\mathbf{X}]=\\sigma^2\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}]$ and $\\mathrm{E}[\\varepsilon^2]=\\sigma^2$. The limit of $n$ times the robust estimate of the sampling variance is $(\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}])^{-1}\\mathrm{E}[\\varepsilon^2\\mathbf{X}^T\\mathbf{X}](\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}])^{-1}$, and the limit of $n$ times the classical one is $\\mathrm{E}[\\varepsilon^2](\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}])^{-1}$. Both equal $\\sigma^2(\\mathrm{E}[\\mathbf{X}^T\\mathbf{X}])^{-1}$.\n\nRead alone, with homoskedasticity as its only premise, the sentence is too broad. The error $\\varepsilon$ is the deviation from the BLP, and\n\\[\n\\mathrm{E}[\\varepsilon^2\\mid\\mathbf{X}]=\\mathrm{V}[\\varepsilon\\mid\\mathbf{X}]+\\bigl(\\mathrm{E}[\\varepsilon\\mid\\mathbf{X}]\\bigr)^2,\n\\]\nwhich can vary with $\\mathbf{X}$ when the CEF is not linear. This occurs in Example 4.3.1 on the same page. Let $X\\sim U(0,1)$ and $Y=10X^2+W$, with $W\\sim N(0,1)$ independent of $X$, and regress $Y$ on $(1,X)$. The BLP is $-5/3+10X$, so that $\\mathrm{V}[\\varepsilon\\mid X]=1$ for all $X$ and $\\mathrm{E}[\\varepsilon\\mid X]=10X^2-10X+5/3$. An exact calculation gives the limit $472/21$ for $n$ times the robust estimate of the sampling variance of the slope, and $56/3$ for $n$ times the classical one. I found no calculation that uses the broader reading: the derivation of Section 4.2.3 uses both assumptions, and Table 4.4.5 reports bootstrap standard errors." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C04-N006", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Five lines of text from Example 4.3.1. A red box marks the words 'slope of the CEF' in the sentence 'With bivariate regression, the estimated slope of the CEF of Y with respect to X was beta-hat-1.'", "caption": "Example 4.3.1, p. 157 (PDF p. 175): the sentence on bivariate regression, with “slope of the CEF” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N007.png" } ], "formal_release": null, "formal_statements": [], "headline": "Bivariate regression estimates the slope of the BLP, which this sentence calls the slope of the CEF.", "id": "P001-C04-N007", "location": "Example 4.3.1, p. 157 (PDF p. 175), paragraph before the display of the derivative", "pdf_page": 175, "printed_page": "157", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "With bivariate regression, the estimated slope of the \\hl{CEF} of $Y$ with respect to $X$ was $\\hat{\\beta}_1$.", "corrected": "With bivariate regression, the estimated slope of the \\hl{BLP} of $Y$ \\hl{given} $X$ was $\\hat{\\beta}_1$.", "intended_meaning": "The sentence refers to the regression of $Y$ on $X$ alone. The coefficient $\\hat{\\beta}_1$ estimates the slope of the best linear predictor (BLP) of $Y$ given $X$, and the preceding page says that OLS regression estimates the BLP. Under this reading the sentence is correct. Read as a statement about the derivative of the conditional expectation function (CEF), it would be false. In the example, $X\\sim U(0,1)$ and $\\mathrm{E}[Y\\mid X]=10X^2$. A direct computation gives $\\operatorname{Cov}[X,10X^2]=5/6$ and $\\mathrm{V}[X]=1/12$. The BLP is $-5/3+10X$, with slope $10$. The derivative of the CEF is $20x$, which equals $10$ only at $x=1/2$. The formula $\\beta_1+2\\beta_2X$ that follows in the text, and its estimate $\\hat{\\beta}_1+2\\hat{\\beta}_2x$, are correct." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C04-N007", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Two lines of text that end 'For example,' and a displayed equation in two lines for E[Y | X[1], X[2]]. The second line of the equation reads: plus beta-6 X[1] squared X[1], plus beta-7 X[1] X[2] squared, plus beta-8 X[1] squared X[2] squared. A red box marks the second factor, X[1], of the beta-6 term.", "caption": "Section 4.3.3, p. 165 (PDF p. 183): the display that combines interactions with higher-order polynomial terms, with the second factor of the $\\beta_6$ term marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N009.png" }, { "alt": "Three lines of text: 'Regression (6) includes higher-order interaction terms: in addition to an interaction term for X and Z, the specification accounts for the interaction of X and Z squared, Z and X squared, and X squared and Z squared. Regression (7) includes even more'. A red box marks the words 'Z and X squared'.", "caption": "Section 4.4, p. 174 (PDF p. 192): the description of regression (6), with the interaction of $Z$ and $X^2$ marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N009-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "In the display on p. 165 that combines interactions with higher-order polynomial terms, the $\\beta_6$ term should be $\\beta_6X_{[1]}^2X_{[2]}$; it is printed as $\\beta_6X_{[1]}^2X_{[1]}$.", "id": "P001-C04-N009", "location": "Section 4.3.3, p. 165 (PDF p. 183), display that combines interactions with higher-order polynomial terms, second line, the $\\beta_6$ term", "pdf_page": 183, "printed_page": "165", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "For example,\n\\[\n\\begin{aligned}\n\\mathrm{E}\\bigl[Y\\mid X_{[1]},X_{[2]}\\bigr]&=\\beta_0+\\beta_1X_{[1]}+\\beta_2X_{[2]}+\\beta_3X_{[1]}X_{[2]}+\\beta_4X_{[1]}^2+\\beta_5X_{[2]}^2\\\\\n&\\quad+\\beta_6X_{[1]}^2X_{[\\hl{\\scriptscriptstyle 1}]}+\\beta_7X_{[1]}X_{[2]}^2+\\beta_8X_{[1]}^2X_{[2]}^2.\n\\end{aligned}\n\\]", "corrected": "For example,\n\\[\n\\begin{aligned}\n\\mathrm{E}\\bigl[Y\\mid X_{[1]},X_{[2]}\\bigr]&=\\beta_0+\\beta_1X_{[1]}+\\beta_2X_{[2]}+\\beta_3X_{[1]}X_{[2]}+\\beta_4X_{[1]}^2+\\beta_5X_{[2]}^2\\\\\n&\\quad+\\beta_6X_{[1]}^2X_{[\\hl{\\scriptscriptstyle 2}]}+\\beta_7X_{[1]}X_{[2]}^2+\\beta_8X_{[1]}^2X_{[2]}^2.\n\\end{aligned}\n\\]", "evidence": "The evidence is a check of the pages and an exact calculation. I write each regressor as $X_{[1]}^aX_{[2]}^b$. The printed display has eight of the nine pairs $(a,b)$ with $a,b\\in\\{0,1,2\\}$, lacks $(2,1)$, and has $(3,0)$. With $(2,1)$ in place of $(3,0)$, the nine pairs are those of the regressors in column (6) of Table 4.4.5. Subtracting the corrected right-hand side from the printed one, I obtain $\\beta_6X_{[1]}^2(X_{[1]}-X_{[2]})$. When $X_{[1]}$ and $X_{[2]}$ are independent and $U(0,1)$, neither list of regressors reproduces the term that only the other has: the BLP of $X_{[1]}^2X_{[2]}$ given the nine printed regressors has mean squared error $1/34560$, and the BLP of $X_{[1]}^3$ given the nine corrected regressors has mean squared error $1/2800$.", "problem": "As printed, the $\\beta_6$ term equals $\\beta_6X_{[1]}^3$, a cubic in $X_{[1]}$ alone. The other eight terms are products of one of $1$, $X_{[1]}$, $X_{[1]}^2$ with one of $1$, $X_{[2]}$, $X_{[2]}^2$, and the only such product that the display lacks is $X_{[1]}^2X_{[2]}$. The nine such products are the regressors of regression (6) in Section 4.4, with $X$ in place of $X_{[1]}$ and $Z$ in place of $X_{[2]}$. The description of that regression on p.~174 lists the interactions of $X$ and $Z$, of $X$ and $Z^2$, of $Z$ and $X^2$, and of $X^2$ and $Z^2$, and column (6) of Table 4.4.5 (p.~173) has an estimate in the row ``Elec $\\times$ Water$^2$'' and none in the row ``Water$^3$''. The second factor $X_{[1]}$ of the $\\beta_6$ term is a misprint for $X_{[2]}$. A cubic term is a legitimate regressor, and the display as printed is a possible specification. It is a different specification from the corrected one.", "what_survives": "The discussion of interactions and of partial derivatives in Section 4.3.3, footnote 11, and the statement that OLS consistently estimates the BLP for the specification chosen are unchanged. Table 4.4.5 and the description of regression (6) already give the corrected list of regressors, and no reported estimate is shown to change. The code for Section 4.4 is not available, and I could not check which regressors it used. A fit that copies the display as printed has $X_{[1]}^3$ in place of $X_{[1]}^2X_{[2]}$, and its predictions and partial derivatives can differ from those of the corrected specification." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C04-N009", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "recalculation_from_source", "exhibits": [ { "alt": "A shaded box with Definition 4.3.6, which displays APD with subscript X[k] equal to the expectation of the partial derivative of E[Y|X] with respect to X[k]. Below it, a sentence beginning 'A natural plug-in estimator for', in which a red box marks APD with subscript X[k]i. Below the sentence, the display of the estimator, written APD-hat with subscript X[k].", "caption": "Definition 4.3.6 and the sentence after it, p. 166 (PDF p. 184): the label $APD_{X_{[k]i}}$ is marked. The definition above it and the estimator below it both write $APD_{X_{[k]}}$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N010.png" }, { "alt": "Two displayed equations from Example 4.3.8, each introduced by a line of text. In the first, for APD with subscript X[1], a red box marks X[2]i inside the second probability, Pr[X[2]i > .8]; the first probability reads Pr[X[2] <= .8]. In the second, a red box marks the left side, PDA with subscript X[k], which is set equal to the partial derivative of minus X[1] with respect to X[1], equal to minus 1.", "caption": "Example 4.3.8, p. 167 (PDF p. 185): the index $i$ in the second probability and the label $PDA_{X_{[k]}}$ are marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N010-p167.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T69", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T69" } ], "headline": "Three subscripts are misprinted: the estimator after Definition 4.3.6 targets $APD_{X_{[k]}}$, the second probability in Example 4.3.8 is $\\Pr[X_{[2]}>.8]$, and the last display of the example computes $PDA_{X_{[1]}}$.", "id": "P001-C04-N010", "location": "Sentence after Definition 4.3.6, p. 166 (PDF p. 184); Example 4.3.8, p. 167 (PDF p. 185), the displays for $APD_{X_{[1]}}$ and for the PDA", "pdf_page": 184, "printed_page": "166", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T69" ], "text": { "as_printed": "[p.~166] A natural plug-in estimator for $APD_{X_{[k]\\hl{i}}}$ is obtained by replacing the expected value with a sample mean and the CEF with the estimated CEF:\n\n[p.~167] Then, by the Law of Iterated Expectations,\n\\[\n\\begin{aligned}\nAPD_{X_{[1]}}&=\\frac{\\partial\\bigl(-X_{[1]}\\bigr)}{\\partial X_{[1]}}\\Pr\\bigl[X_{[2]}\\le.8\\bigr]+\\frac{\\partial\\bigl(9X_{[1]}\\bigr)}{\\partial X_{[1]}}\\Pr\\bigl[X_{[2]\\hl{i}}>.8\\bigr]\\\\\n&=-1\\times.8+9\\times.2=1.\n\\end{aligned}\n\\]\nBut the partial derivative at the average gives a very different result:\n\\[\nPDA_{X_{[\\hl{\\scriptscriptstyle k}]}}=\\frac{\\partial\\bigl(-X_{[1]}\\bigr)}{\\partial X_{[1]}}=-1.\n\\]", "corrected": "[p.~166] A natural plug-in estimator for $APD_{\\hl{\\scriptstyle X_{[k]}}}$ is obtained by replacing the expected value with a sample mean and the CEF with the estimated CEF:\n\n[p.~167] Then, by the Law of Iterated Expectations,\n\\[\n\\begin{aligned}\nAPD_{X_{[1]}}&=\\frac{\\partial\\bigl(-X_{[1]}\\bigr)}{\\partial X_{[1]}}\\Pr\\bigl[X_{[2]}\\le.8\\bigr]+\\frac{\\partial\\bigl(9X_{[1]}\\bigr)}{\\partial X_{[1]}}\\Pr\\bigl[\\hl{X_{[2]}}>.8\\bigr]\\\\\n&=-1\\times.8+9\\times.2=1.\n\\end{aligned}\n\\]\nBut the partial derivative at the average gives a very different result:\n\\[\nPDA_{X_{[\\hl{\\scriptscriptstyle 1}]}}=\\frac{\\partial\\bigl(-X_{[1]}\\bigr)}{\\partial X_{[1]}}=-1.\n\\]", "evidence": "The evidence is a check of the two pages and a recalculation of the example. In Example 4.3.8, $X_{[1]}$ and $X_{[2]}$ are independent and $U(0,1)$, and the CEF is $-X_{[1]}$ when $X_{[2]}\\le.8$ and $9X_{[1]}$ otherwise. The partial derivative with respect to $X_{[1]}$ is $-1$ with probability $4/5$ and $9$ with probability $1/5$, and I obtain\n\\[\nAPD_{X_{[1]}}=(-1)(4/5)+9(1/5)=1.\n\\]\nThe mean of $(X_{[1]},X_{[2]})$ is $(1/2,1/2)$, which lies in the region $X_{[2]}\\le.8$, where the derivative is $-1$. I conclude that $PDA_{X_{[1]}}=-1$. Both printed values are correct. Note that the probability printed with the index $i$ also has the right value, since each observation of $X_{[2]}$ has the $U(0,1)$ distribution.", "problem": "Definition 4.3.6 and the estimator displayed below it write the average partial derivative with respect to the $k$th variable as $APD_{X_{[k]}}$. The sentence between them adds an observation index $i$, which a population quantity does not carry. In Example 4.3.8 the second probability has the same index, although the example involves the random variable $X_{[2]}$ and no sample. Since the last display of the example differentiates with respect to $X_{[1]}$, its label is $PDA_{X_{[1]}}$ and not the generic $PDA_{X_{[k]}}$.", "what_survives": "Definitions 4.3.6 and 4.3.7, the two plug-in estimators and the comparison of $APD_{X_{[1]}}=1$ with $PDA_{X_{[1]}}=-1$ in Example 4.3.8 are unchanged. Only the three labels change." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C04-N010", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "The heading '4.4 Application: Access to Clean Water and Infant Mortality' and the first four lines of the section. A red box marks the italic words 'Quality of Governance' at the end of the first line, before 'data (Teorell et al. 2016)'.", "caption": "Opening of Section 4.4, p. 170 (PDF p. 188): the name “Quality of Governance” is marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N012.png" } ], "formal_release": null, "formal_statements": [], "headline": "The dataset is the Quality of Government data; the text calls it the “Quality of Governance” data.", "id": "P001-C04-N012", "location": "Section 4.4, p. 170 (PDF p. 188), first sentence", "pdf_page": 188, "printed_page": "170", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "We now consider an application of regression using the \\emph{Quality of \\hl{Governance}} data (Teorell et al. 2016), a dataset that compiles country-level information from a variety of sources.", "corrected": "We now consider an application of regression using the \\emph{Quality of \\hl{Government}} data (Teorell et al. 2016), a dataset that compiles country-level information from a variety of sources.", "evidence": "The evidence is a check of the page against the reference list of the book. The entry on p.~292 reads ``Teorell, Jan et al. 2016. The Quality of Government standard dataset, version Jan 16. University of Gothenburg: The Quality of Government Institute.''", "problem": "The work cited is the Quality of Government standard dataset. The text writes ``Governance'' for ``Government''.", "what_survives": "The citation identifies the dataset, and nothing else changes. The same name is misprinted in Section 6.3; see \\fid{P001-C06-N006}." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C04-N012", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Bottom of a regression table with eight columns. In the last column the rows for the estimated APD and PDA of water give the estimates -0.345 and -0.394, and red boxes mark the standard errors (0.075) and (0.087) printed beneath them. The two coefficient rows above show dashes in the last column where columns six and seven have standard errors. The note below the table says that regression (8) reports lasso coefficients and that bootstrap standard error estimates are reported in parentheses under each estimate.", "caption": "Table 4.4.5, p. 173 (PDF p. 191), last rows and note. The rightmost column is regression (8), the lasso fit. The marked entries are the bootstrap standard errors printed for its estimated APD and PDA; the coefficient rows of the same column have dashes.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N015.png" }, { "alt": "Paragraph of text. A red outline marks the sentence: However, as we increase the complexity of the specification, the standard errors grow quickly for both the estimated APD and PDA, suggesting overfitting. The sentence before it says that in columns (3) through (8) the estimate of the APD is largely unchanged by introducing nonlinear specifications.", "caption": "p. 175 (PDF p. 193), the paragraph below Figure 4.4.3. The marked sentence compares the standard errors of the estimated APD and PDA across the columns of Table 4.4.5.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N015-b.png" }, { "alt": "Footnote 14. It says that the lasso has a sparsity property and that most coefficients are set to exactly zero. A red outline marks the words: This sparsity property implies that standard errors cannot generally be computed for the lasso coefficients without very strong assumptions. The footnote ends: though this is a topic of ongoing inquiry. See Leeb and Pötscher (2006).", "caption": "Footnote 14, p. 174 (PDF p. 192). The marked words are the book’s own caution about standard errors for lasso coefficients.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N015-c.png" } ], "formal_release": null, "formal_statements": [], "headline": "Column (8) of Table 4.4.5 prints bootstrap standard errors, $0.075$ and $0.087$, for the lasso estimates of the APD and PDA, with no argument that the bootstrap is valid for a lasso fit; they should be described as unvalidated, and neither is shown to be wrong.", "id": "P001-C04-N015", "location": "Table 4.4.5, column (8), rows $\\widehat{APD}_{\\text{Water}}$ and $\\widehat{PDA}_{\\text{Water}}$, p. 173 (PDF p. 191); footnote 14, p. 174 (PDF p. 192); the comparison of standard errors on p. 175 (PDF p. 193)", "pdf_page": 191, "printed_page": "173", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\begin{audittable}{lr}{Table 4.4.5, column (8), p.~173 (excerpt).}\n & (8) \\\\\n$\\widehat{APD}_{\\text{Water}}$ & $-0.345$ \\\\\n & \\hl{$(0.075)$} \\\\\n$\\widehat{PDA}_{\\text{Water}}$ & $-0.394$ \\\\\n & \\hl{$(0.087)$} \\\\\n\\end{audittable}\nThe note to the table:\n\nRegressions (1)--(7) report OLS coefficients and regression (8) reports lasso coefficients. $n=184$ for all regressions. The final two rows report the estimated average partial derivatives and partial derivatives at the average with respect to access to clean water. Bootstrap standard error estimates are reported in parentheses under each estimate.\n\nFootnote 14, p.~174:\n\nNote that the lasso has a ``sparsity'' property that is manifest here---most coefficients are set to exactly zero. This sparsity property implies that standard errors cannot generally be computed for the lasso coefficients without very strong assumptions, though this is a topic of ongoing inquiry. See Leeb and Pötscher (2006).\n\np.~175:\n\nHowever, as we increase the complexity of \\hl{the specification}, the standard errors grow quickly for both the estimated APD and PDA, suggesting overfitting.", "corrected": "Note to Table 4.4.5, last sentence:\n\nBootstrap standard error estimates are reported in parentheses under each estimate. \\hl{For regression (8), the values in parentheses describe the variability of the two estimates across bootstrap resamples; the bootstrap has not been shown to be valid for these functions of a lasso fit.}\n\np.~175:\n\nHowever, as we increase the complexity of the specification \\hl{in the OLS regressions (3) through (7)}, the standard errors grow quickly for both the estimated APD and PDA, suggesting overfitting.", "evidence": "The evidence is a published result and a derivation; it is not a recalculation of the table. It was shown in Camponovo (2015, \\emph{Biometrika} 102(4), 981--987) that the naive pairs bootstrap is not in general a valid method for approximating the distribution of the lasso estimator. The following example shows that the same failure can occur for the APD and PDA. Let $X\\sim U[1,2]$, let $\\varepsilon$ equal $1$ or $-1$ with probability $1/2$ each, independently of $X$, and let $Y=X\\varepsilon$, so that $\\mathbb{E}[Y\\mid X=x]=0$. The lasso fit of $f_b(x)=bx$ minimizes $n^{-1}\\sum_{i=1}^n(Y_i-X_ib)^2+\\lambda_n|b|$ over $b$. Since $\\partial_xf_b(x)=b$ for every $x$, the estimated APD and the estimated PDA both equal the fitted slope $\\widehat b_n$. Set $q=\\mathbb{E}[X^2]=7/3$ and $V=\\operatorname{Var}(X^2\\varepsilon)=31/5$, and define $v(z)=\\operatorname{Var}(s(z+G))$, where $G\\sim N(0,1)$ and $s(u)=\\operatorname{sign}(u)(|u|-1)_+$. With $\\lambda_n=2\\sqrt{V}/\\sqrt{n}$, I have\n\\[\n\\frac{nq^2}{V}\\operatorname{Var}(\\widehat b_n)\\longrightarrow v(0)=4\\Phi(-1)-2\\phi(1)\\approx 0.1507,\n\\]\nwhere $\\Phi$ and $\\phi$ are the standard normal distribution function and density. With pairs $(X_i,Y_i)$ resampled with replacement and the same deterministic penalty used in each resample, the bootstrap variance, scaled in the same way, converges in distribution to $v(Z)$ with $Z\\sim N(0,1)$. Note that $v$ is not constant: $v(3)\\approx 0.9602$, and $v(z)\\to1$ as $z\\to\\infty$. This implies that the ratio of the bootstrap standard error to the true standard error has a nondegenerate limit and does not converge to one. With the smaller penalty $\\lambda_n=2\\sqrt{V}/n$ in the same population, both scaled variances converge to $1$, and the bootstrap standard error is consistent in ratio. Neither case uses cross-validation, and the data behind the table are not available. The derivation does not determine whether $0.075$ and $0.087$ are valid.", "problem": "Column (8) reports a lasso fit whose penalty $\\lambda$ is chosen by leave-one-out cross-validation (p.~174). The table reports no standard errors for the coefficients of this fit, and footnote 14 says why: most coefficients are set to exactly zero, and inference for such estimates is difficult. For two functions of those coefficients, the estimated APD and PDA, the table does print bootstrap standard errors, and the text on p.~175 reads the standard errors across the columns as a sign of overfitting. In Section 3.4.3 (p.~131), the book states that the bootstrap consistently estimates the standard error under the usual regularity conditions for plug-in estimation, and it does not argue that these two functions of the lasso coefficients satisfy them. A function of the lasso coefficients can be smooth in the coefficients and still fail these conditions. The book also does not say how the bootstrap was carried out for column (8), in particular whether $\\lambda$ was chosen again in each resample.", "what_survives": "No point estimate is shown to be wrong, and neither standard error is shown to be wrong: for some contrasts and some penalties the bootstrap is valid. The comparison of the APD point estimates across columns (3)--(8), the OLS columns (1)--(7) and the definitions of the APD and PDA do not depend on the two standard errors. What changes is the support for the remark on p.~175 that the standard errors grow with the complexity of the specification: it can rest on the OLS columns only. Justifying the two standard errors requires an argument covering these functions of the coefficients, penalty selection and resampling; checking their computation also requires the data and the resampling procedure." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C04-N015", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Paragraph on kernel regression and locally weighted regression. A red box marks the words: 2006 provides a clear treatment. They follow a semicolon after the words closer to the point of prediction, and no author's name stands before the year. The next sentence recommends James et al. (2014) and Hastie, Tibshirani, and Friedman (2009).", "caption": "Section 4.5, p. 177 (PDF p. 195). The marked clause begins with a year; the author’s name that should stand before it is missing.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C04-N016.png" } ], "formal_release": null, "formal_statements": [], "headline": "The author’s name has dropped out of a citation in Section 4.5, which now reads “; 2006 provides a clear treatment”; the name should be restored, and the page does not show which work was meant.", "id": "P001-C04-N016", "location": "Section 4.5 (Further Readings), p. 177 (PDF p. 195), the sentence on kernel and locally weighted regression", "pdf_page": 195, "printed_page": "177", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "These can be thought of as locally weighted regression fits of observations near a given set of values for the explanatory variables, with larger weights assigned to observations closer to the point of prediction; \\hl{2006} provides a clear treatment.", "corrected": "Restore the author's name before the year from the manuscript. If the name cannot be recovered, delete the clause and end the sentence earlier:\n\nThese can be thought of as locally weighted regression fits of observations near a given set of values for the explanatory variables, with larger weights assigned to observations closer to the point of prediction\\hl{.}", "evidence": "The evidence is a check of the page and of the reference list. The reference list has several entries dated 2006. One of them, Wasserman (2006), \\emph{All of Nonparametric Statistics}, is recommended on p.~141 for the theory of statistical estimation and is a plausible match, but neither the page nor the reference list establishes that it is the work meant here. The manuscript was not consulted.", "problem": "The clause after the semicolon is the remainder of an author--year citation that recommends a treatment of kernel or locally weighted regression. No name stands between the semicolon and the year, and the sentence does not identify the work.", "what_survives": "The description of kernel and locally weighted regression and the other references of Section 4.5 are unaffected. One pointer to further reading is lost." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C04-N016", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Paragraph that follows the display Y = X beta + epsilon. Red boxes mark the words independent of X and the citation (Theorem 2.2.19) in the sentence: The fact that the deviations from the linear component are mean zero and independent of X should be somewhat familiar; these are the properties of the deviations from the CEF (Theorem 2.2.19).", "caption": "Section 5.1.1, p. 181 (PDF p. 199), the paragraph after the display $Y=\\mathbf{X}\\boldsymbol{\\beta}+\\varepsilon$. The marked words attribute independence from $\\mathbf{X}$ to deviations from the CEF and cite Theorem 2.2.19 for it.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E004.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T73", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T73" }, { "id": "P001-T74", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T74" } ], "headline": "Independence from $\\mathbf{X}$ is a property of the errors of the classical linear model and not of deviations from the CEF in general; Theorem 2.2.19, which the sentence cites, states only conditional mean zero and zero covariance.", "id": "P001-C05-E004", "location": "Section 5.1.1, p. 181 (PDF p. 199), the sentence before Theorem 5.1.5 that begins “The fact that the deviations”", "pdf_page": 199, "printed_page": "181", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T73", "P001-T74" ], "text": { "as_printed": "The fact that the deviations from the linear component are mean zero and \\hl{independent of} $\\mathbf{X}$ should be somewhat familiar; these are the properties of the deviations from the CEF \\hl{(Theorem 2.2.19)}.", "corrected": "\\hl{In the classical linear model,} the deviations from the linear component are mean zero and independent of $\\mathbf{X}$. \\hl{More generally, deviations from the CEF have conditional mean zero and are uncorrelated with functions of the covariates whenever the relevant expectations exist} (Theorem 2.2.19)\\hl{; independence is not a property of deviations from the CEF in general.}\n\nThe smallest change with the same effect replaces ``independent of'' by ``uncorrelated with'' in the printed sentence.", "evidence": "The evidence is an exact counterexample. Let $(X,Y)$ take the four pairs of values in the table below with probability $1/4$ each, and let $\\varepsilon=Y-\\mathrm{E}[Y|X]$.\n\\begin{audittable}{rrrrr}{A joint distribution with a linear CEF whose deviation is not independent of $X$.}\n$X$ & $Y$ & $\\mathrm{E}[Y|X]$ & $\\varepsilon$ & Probability \\\\\n$0$ & $1$ & $2$ & $-1$ & $1/4$ \\\\\n$0$ & $3$ & $2$ & $1$ & $1/4$ \\\\\n$1$ & $0$ & $3$ & $-3$ & $1/4$ \\\\\n$1$ & $6$ & $3$ & $3$ & $1/4$ \\\\\n\\end{audittable}\nThen $\\mathrm{E}[Y|X=x]=2+x$, which is linear in $x$. The deviation $\\varepsilon$ has all five properties of Theorem 2.2.19. Indeed, $\\mathrm{E}[\\varepsilon|X=0]=\\mathrm{E}[\\varepsilon|X=1]=0$, and $\\mathrm{E}[g(X)\\varepsilon]=0$ for every function $g$, which implies that $\\operatorname{Cov}[g(X),\\varepsilon]=0$. Further, $\\mathrm{V}[\\varepsilon|X=0]=\\mathrm{V}[Y|X=0]=1$ and $\\mathrm{V}[\\varepsilon|X=1]=\\mathrm{V}[Y|X=1]=9$, and $\\mathrm{V}[\\varepsilon]=5=\\mathrm{E}[\\mathrm{V}[Y|X]]$. However,\n\\[\n\\Pr(X=0,\\ |\\varepsilon|=1)=\\frac12\\neq\\frac14=\\Pr(X=0)\\Pr(|\\varepsilon|=1),\n\\]\nwhich shows that $\\varepsilon$ and $X$ are dependent. In the classical linear model with $\\sigma>0$, the conditional density of $\\varepsilon=Y-\\mathbf{X}\\boldsymbol{\\beta}$ given $\\mathbf{X}=\\mathbf{x}$ is $\\phi(t;(0,\\sigma^2))$ for every $\\mathbf{x}$. Since this density does not depend on $\\mathbf{x}$, the error is independent of $\\mathbf{X}$ in that model.", "problem": "Under the classical linear model, the errors $\\varepsilon=Y-\\mathbf{X}\\boldsymbol{\\beta}$ are independent of $\\mathbf{X}$, and the first clause is correct. The clause after the semicolon says that mean zero and independence are the properties of deviations from the CEF, and it cites Theorem 2.2.19. For $\\varepsilon=Y-\\mathrm{E}[Y|X]$, that theorem states that $\\mathrm{E}[\\varepsilon|X]=0$, that $\\mathrm{E}[\\varepsilon]=0$, that $\\operatorname{Cov}[g(X),\\varepsilon]=0$ for any function $g$ of $X$, that $\\mathrm{V}[\\varepsilon|X]=\\mathrm{V}[Y|X]$, and that $\\mathrm{V}[\\varepsilon]=\\mathrm{E}[\\mathrm{V}[Y|X]]$. It does not state independence, and independence does not hold in general, since by the fourth property the conditional variance of the deviation can depend on $X$. In the classical linear model, independence comes from the conditional normal density with constant variance that Definition 5.1.4 assumes.", "what_survives": "Theorem 2.2.19, Definition 5.1.4 and Theorem 5.1.5 are unaffected. The proof of Theorem 5.1.5 uses the conditional normal density directly and does not use this sentence. The statement that the errors of the classical linear model are independent of $\\mathbf{X}$ stands. Only the comparison with deviations from the CEF in general is weakened, from independence to conditional mean zero and zero covariance." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C05-E004", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "End of Definition 5.1.4 in a shaded box. It says that beta is the vector of beta 0 to beta K, that phi of y with parameters mu and sigma squared denotes the PDF of the normal distribution with mean mu and variance sigma squared, and that the parameter space Theta is the set of pairs beta, sigma in R to the K plus 2 with sigma greater than or equal to 0. A red box marks the condition sigma greater than or equal to 0.", "caption": "Definition 5.1.4, last lines, p. 181 (PDF p. 199). The marked condition $\\sigma\\ge0$ should be $\\sigma>0$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E009.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T72", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T72" } ], "headline": "The parameter space of the classical linear model should require $\\sigma>0$; at $\\sigma=0$ the normal density that defines the model does not exist.", "id": "P001-C05-E009", "location": "Definition 5.1.4 (Classical Linear Model), last line, p. 181 (PDF p. 199); the definition begins on p. 180 (PDF p. 198)", "pdf_page": 199, "printed_page": "181", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T72" ], "text": { "as_printed": "For a random variable $Y$ and random vector $\\mathbf{X}=(1,X_{[1]},X_{[2]},\\ldots,X_{[K]})$, the \\emph{classical linear model} is a parametric model with\n\\[\ng\\bigl(y,\\mathbf{x};(\\boldsymbol{\\beta},\\sigma)\\bigr)=\\phi\\bigl(y;(\\mathbf{x}\\boldsymbol{\\beta},\\sigma^2)\\bigr),\n\\]\nwhere $\\boldsymbol{\\beta}=(\\beta_0,\\beta_1,\\ldots,\\beta_K)^T$, $\\phi(y;(\\mu,\\sigma^2))$ denotes the PDF of the normal distribution with mean $\\mu$ and variance $\\sigma^2$, and the parameter space is $\\boldsymbol{\\Theta}=\\{(\\boldsymbol{\\beta},\\sigma)\\in\\mathbb{R}^{K+2}:\\sigma\\hl{\\ge}0\\}$.", "corrected": "For a random variable $Y$ and random vector $\\mathbf{X}=(1,X_{[1]},X_{[2]},\\ldots,X_{[K]})$, the \\emph{classical linear model} is a parametric model with\n\\[\ng\\bigl(y,\\mathbf{x};(\\boldsymbol{\\beta},\\sigma)\\bigr)=\\phi\\bigl(y;(\\mathbf{x}\\boldsymbol{\\beta},\\sigma^2)\\bigr),\n\\]\nwhere $\\boldsymbol{\\beta}=(\\beta_0,\\beta_1,\\ldots,\\beta_K)^T$, $\\phi(y;(\\mu,\\sigma^2))$ denotes the PDF of the normal distribution with mean $\\mu$ and variance $\\sigma^2$, and the parameter space is $\\boldsymbol{\\Theta}=\\{(\\boldsymbol{\\beta},\\sigma)\\in\\mathbb{R}^{K+2}:\\sigma\\hl{>}0\\}$.", "evidence": "The evidence is a check of the pages. In Definition 2.1.19, the normal PDF is\n\\[\nf(x)=\\frac{1}{\\sigma\\sqrt{2\\pi}}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}},\\qquad\\forall x\\in\\mathbb{R},\n\\]\nfor constants $\\mu,\\sigma\\in\\mathbb{R}$ with $\\sigma>0$. At $\\sigma=0$ both denominators vanish. The distribution at that boundary is a point mass at $\\mathbf{x}\\boldsymbol{\\beta}$, which has no density with respect to Lebesgue measure.", "problem": "The model is defined through the normal PDF $\\phi(y;(\\mathbf{x}\\boldsymbol{\\beta},\\sigma^2))$, and Definition 2.1.19 (p.~55) defines that PDF for $\\sigma>0$ only, since its formula divides by $\\sigma$. The printed parameter space includes $\\sigma=0$, where the density is undefined. The inequality should be strict.", "what_survives": "The model with $\\sigma>0$, Theorem 5.1.5 and the likelihood calculations that follow are unaffected. One could treat $\\sigma=0$ separately as a point mass, but the correction does not need it." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C05-E009", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Theorem 5.1.8, CEF of a Binary Choice Model, in a shaded box. Its display reads E[Y|X = x] = h(X beta), for all x in the support of X. A red box marks h(X beta), which is printed with a capital bold X although the left side conditions on X = x.", "caption": "Theorem 5.1.8, p. 183 (PDF p. 201). In the marked term the argument of $h$ is the random vector $\\mathbf{X}$, where the fixed value $\\mathbf{x}$ is meant.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E010.png" }, { "alt": "Definition 5.1.7, Binary Choice Model, in a shaded box. The display defines g(y, X; beta) as 1 minus h(X beta) when y = 0, as h(X beta) when y = 1, and as 0 otherwise. Red boxes mark g(y, X; beta) and the two occurrences of h(X beta), each printed with a capital bold X.", "caption": "Definition 5.1.7, p. 182 (PDF p. 200). The three marked terms have the random vector $\\mathbf{X}$ where the argument $\\mathbf{x}$ is meant.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E010-b.png" }, { "alt": "A four-line display that begins: beta hat ML = arg max of log L(b | Y, X). Under each of the three arg max operators and under the final arg min operator stands the domain b in R to the K, and each of the four is marked with a red box. A fifth red box marks L(b | Y, X) in the first line; the sentence above the display writes log L((b, s) | Y, X). The second line contains s in the terms 1 over s root 2 pi and 2 s squared.", "caption": "Example 5.2.4, p. 191 (PDF p. 209), the display for $\\hat{\\boldsymbol{\\beta}}_{ML}$. The four marked domains read $\\mathbf{b}\\in\\mathbb{R}^{K}$, where $\\mathbb{R}^{K+1}$ is meant, and the marked likelihood in the first line omits $s$ from its argument.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E010-c.png" }, { "alt": "Paragraph on l1-penalized ML estimation. A red box marks the word flexibility in the sentence: Mirroring the language of Section 4.3.6, it is a way of flexibility estimating the conditional density function when we have a very large number of parameters to estimate.", "caption": "Section 5.2.6, p. 202 (PDF p. 220). The marked word should be “flexibly”.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E010-d.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T75", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T75" }, { "id": "P001-T76", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T76" } ], "headline": "Three local slips: Theorem 5.1.8 should read $h(\\mathbf{x}\\boldsymbol{\\beta})$, the maximization in Example 5.2.4 is over $\\mathbf{b}\\in\\mathbb{R}^{K+1}$, and “flexibility estimating” on p. 202 should read “flexibly estimating”.", "id": "P001-C05-E010", "location": "Theorem 5.1.8, p. 183 (PDF p. 201), with Definition 5.1.7, p. 182 (PDF p. 200); Example 5.2.4, the display for $\\hat{\\boldsymbol{\\beta}}_{ML}$, p. 191 (PDF p. 209); Section 5.2.6, second paragraph, p. 202 (PDF p. 220)", "pdf_page": 201, "printed_page": "183", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T75", "P001-T76" ], "text": { "as_printed": "(a) Theorem 5.1.8, p.~183:\n\nFor a binary random variable $Y$ and random vector $\\mathbf{X}=(1,X_{[1]},X_{[2]},\\ldots,X_{[K]})$, if the binary choice model with mean function $h$ is true, the CEF of $Y$ given $\\mathbf{X}$ is\n\\[\n\\mathrm{E}[Y|\\mathbf{X}=\\mathbf{x}]=h(\\hl{\\mathbf{X}}\\boldsymbol{\\beta}),\\ \\forall\\mathbf{x}\\in\\operatorname{Supp}[\\mathbf{X}].\n\\]\nDefinition 5.1.7, p.~182, the display:\n\\[\ng(y,\\hl{\\mathbf{X}};\\boldsymbol{\\beta})=\n\\begin{cases}\n1-h(\\hl{\\mathbf{X}}\\boldsymbol{\\beta}) & :\\ y=0\\\\\nh(\\hl{\\mathbf{X}}\\boldsymbol{\\beta}) & :\\ y=1\\\\\n0 & :\\ \\text{otherwise,}\n\\end{cases}\n\\]\n(b) Example 5.2.4, p.~191:\n\nWe want to find the $\\mathbf{b}$ that maximizes $\\log\\mathcal{L}((\\mathbf{b},s)\\,|\\,\\mathbf{Y},\\mathbb{X})$:\n\\[\n\\begin{aligned}\n\\hat{\\boldsymbol{\\beta}}_{ML}&=\\operatorname*{arg\\,max}_{\\mathbf{b}\\in\\hl{\\scriptstyle\\mathbb{R}^{K}}}\\bigl(\\log\\mathcal{L}(\\hl{\\mathbf{b}}\\,|\\,\\mathbf{Y},\\mathbb{X})\\bigr)\\\\\n&=\\operatorname*{arg\\,max}_{\\mathbf{b}\\in\\hl{\\scriptstyle\\mathbb{R}^{K}}}\\sum_{i=1}^{n}\\left(\\log\\left(\\frac{1}{s\\sqrt{2\\pi}}\\right)+\\frac{-(Y_i-\\mathbf{X}_i\\mathbf{b})^2}{2s^2}\\right)\\\\\n&=\\operatorname*{arg\\,max}_{\\mathbf{b}\\in\\hl{\\scriptstyle\\mathbb{R}^{K}}}\\sum_{i=1}^{n}\\bigl(-(Y_i-\\mathbf{X}_i\\mathbf{b})^2\\bigr)\\\\\n&=\\operatorname*{arg\\,min}_{\\mathbf{b}\\in\\hl{\\scriptstyle\\mathbb{R}^{K}}}\\sum_{i=1}^{n}(Y_i-\\mathbf{X}_i\\mathbf{b})^2.\n\\end{aligned}\n\\]\n(c) Section 5.2.6, p.~202:\n\nMirroring the language of Section 4.3.6, it is a way of \\hl{flexibility} estimating the conditional density function when we have a very large number of parameters to estimate.", "corrected": "(a) Theorem 5.1.8:\n\nFor a binary random variable $Y$ and random vector $\\mathbf{X}=(1,X_{[1]},X_{[2]},\\ldots,X_{[K]})$, if the binary choice model with mean function $h$ is true, the CEF of $Y$ given $\\mathbf{X}$ is\n\\[\n\\mathrm{E}[Y|\\mathbf{X}=\\mathbf{x}]=h(\\hl{\\mathbf{x}}\\boldsymbol{\\beta}),\\ \\forall\\mathbf{x}\\in\\operatorname{Supp}[\\mathbf{X}].\n\\]\nDefinition 5.1.7:\n\nFor a binary random variable $Y$ and random vector $\\mathbf{X}=(1,X_{[1]},X_{[2]},\\ldots,X_{[K]})$, a \\emph{binary choice model} is a parametric model with\n\\[\ng(y,\\hl{\\mathbf{x}};\\boldsymbol{\\beta})=\n\\begin{cases}\n1-h(\\hl{\\mathbf{x}}\\boldsymbol{\\beta}) & :\\ y=0\\\\\nh(\\hl{\\mathbf{x}}\\boldsymbol{\\beta}) & :\\ y=1\\\\\n0 & :\\ \\text{otherwise,}\n\\end{cases}\n\\]\nwhere $\\boldsymbol{\\beta}=(\\beta_0,\\beta_1,\\ldots,\\beta_K)^T$, the mean function $h:\\mathbb{R}\\to[0,1]$ is known, and the parameter space is $\\boldsymbol{\\Theta}=\\mathbb{R}^{K+1}$.\n\n(b) Example 5.2.4, with $s>0$ held fixed:\n\\[\n\\begin{aligned}\n\\hat{\\boldsymbol{\\beta}}_{ML}&=\\operatorname*{arg\\,max}_{\\mathbf{b}\\in\\hl{\\scriptstyle\\mathbb{R}^{K+1}}}\\bigl(\\log\\mathcal{L}(\\hl{(\\mathbf{b},s)}\\,|\\,\\mathbf{Y},\\mathbb{X})\\bigr)\\\\\n&=\\operatorname*{arg\\,max}_{\\mathbf{b}\\in\\hl{\\scriptstyle\\mathbb{R}^{K+1}}}\\sum_{i=1}^{n}\\left(\\log\\left(\\frac{1}{s\\sqrt{2\\pi}}\\right)+\\frac{-(Y_i-\\mathbf{X}_i\\mathbf{b})^2}{2s^2}\\right)\\\\\n&=\\operatorname*{arg\\,max}_{\\mathbf{b}\\in\\hl{\\scriptstyle\\mathbb{R}^{K+1}}}\\sum_{i=1}^{n}\\bigl(-(Y_i-\\mathbf{X}_i\\mathbf{b})^2\\bigr)\\\\\n&=\\operatorname*{arg\\,min}_{\\mathbf{b}\\in\\hl{\\scriptstyle\\mathbb{R}^{K+1}}}\\sum_{i=1}^{n}(Y_i-\\mathbf{X}_i\\mathbf{b})^2.\n\\end{aligned}\n\\]\n(c) Section 5.2.6:\n\nMirroring the language of Section 4.3.6, it is a way of \\hl{flexibly} estimating the conditional density function when we have a very large number of parameters to estimate.", "evidence": "The evidence is a check of the pages and a short derivation. For a fixed $\\mathbf{x}$, under the binary choice model I have\n\\[\n\\mathrm{E}[Y|\\mathbf{X}=\\mathbf{x}]=\\sum_{y\\in\\{0,1\\}}y\\,g(y,\\mathbf{x};\\boldsymbol{\\beta})=0\\cdot\\bigl(1-h(\\mathbf{x}\\boldsymbol{\\beta})\\bigr)+1\\cdot h(\\mathbf{x}\\boldsymbol{\\beta})=h(\\mathbf{x}\\boldsymbol{\\beta}).\n\\]\nThis is a number. The printed $h(\\mathbf{X}\\boldsymbol{\\beta})$ is in general random. The book writes $h(\\mathbf{x}\\hat{\\boldsymbol{\\beta}}_{ML})$ when it applies the theorem on p.~198. For (b), p.~190 defines $\\mathbf{b}$ with the entries $b_0,\\ldots,b_K$ and $\\mathbb{X}$ with $K+1$ columns. The product $\\mathbf{X}_i\\mathbf{b}$ then requires $\\mathbf{b}\\in\\mathbb{R}^{K+1}$, and the parallel display for the probit model on p.~193 has $\\mathbb{R}^{K+1}$. At any fixed $s>0$, the log-likelihood equals\n\\[\n-n\\log\\bigl(s\\sqrt{2\\pi}\\bigr)-\\frac{1}{2s^2}\\sum_{i=1}^{n}(Y_i-\\mathbf{X}_i\\mathbf{b})^2,\n\\]\nand its maximizers in $\\mathbf{b}$ are the minimizers of the sum of squared residuals. The chain of equalities holds once the domain is corrected.", "problem": "(a) The left side of the display in Theorem 5.1.8 conditions on $\\mathbf{X}=\\mathbf{x}$ and is a number for each $\\mathbf{x}$, but the right side is written with the random vector $\\mathbf{X}$. Definition 5.1.7 has the same slip in the function $g$, whose covariate argument is the lower-case $\\mathbf{x}$ in Definitions 5.1.1 and 5.1.4. (b) The vector $\\mathbf{b}=(b_0,b_1,\\ldots,b_K)^T$ has $K+1$ entries, one for each column of the matrix $\\mathbb{X}$, whose first column consists of ones, but the four domains are printed as $\\mathbb{R}^{K}$. The first line also drops $s$ from the argument of the likelihood, which the sentence above the display writes as $\\mathcal{L}((\\mathbf{b},s)\\,|\\,\\mathbf{Y},\\mathbb{X})$, although $s$ appears in the second line. (c) ``Flexibility'' is printed for ``flexibly''. Section 4.3.6, whose language the sentence mirrors, speaks of ``a way of flexibly estimating the CEF''.", "what_survives": "The CEF of the binary choice model, the result of Example 5.2.4 that the ML estimator of $\\boldsymbol{\\beta}$ minimizes the sum of squared residuals whenever a positive-scale ML solution exists, and the discussion of penalized ML estimation are unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C05-E010", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Footnote 5: Formally, we would say that the likelihood function is itself a random function, script L from Omega to script L, where script L is the space of functions L from R to the interval [0, 1]. A red box marks L : R to [0, 1].", "caption": "Footnote 5, p. 186 (PDF p. 204). The marked type gives the realized likelihood a scalar argument and values in $[0,1]$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E006.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T77", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T77" } ], "headline": "In footnote 5 the realized likelihood has the type $L:\\mathbb{R}\\to[0,1]$; in general its domain is the parameter space $\\boldsymbol{\\Theta}$, and its values are nonnegative but need not be at most one.", "id": "P001-C05-E006", "location": "Footnote 5, p. 186 (PDF p. 204)", "pdf_page": 204, "printed_page": "186", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T77" ], "text": { "as_printed": "Formally, we would say that the likelihood function is itself a random function, $\\mathcal{L}:\\Omega\\to\\mathscr{L}$, where $\\mathscr{L}$ is the space of functions $L:\\hl{\\mathbb{R}}\\to\\hl{[0,1]}$.", "corrected": "Formally, we would say that the likelihood function is itself a random function, $\\mathcal{L}:\\Omega\\to\\mathscr{L}$, where $\\mathscr{L}$ is the space of functions $L:\\hl{\\boldsymbol{\\Theta}}\\to\\hl{[0,\\infty)}$.\n\nA finite-valued likelihood is a nonnegative function on its parameter space $\\boldsymbol{\\Theta}$; its values need not be bounded by one.", "evidence": "The evidence is an exact counterexample and a check of the pages. Definition 5.1.1 (p.~179) takes $\\boldsymbol{\\Theta}\\subseteq\\mathbb{R}^{\\tau}$, and in the normal model of Example 5.2.4 (p.~190) the parameter $(\\mathbf{b},s)$ has $K+2$ coordinates. In that model, one observation with $Y_i=\\mathbf{X}_i\\mathbf{b}$ and $s=0.1$ has likelihood\n\\[\n\\phi\\bigl(0;(0,0.1^2)\\bigr)=\\frac{1}{0.1\\sqrt{2\\pi}}\\approx3.989>1.\n\\]", "problem": "The likelihood of Definition 5.2.1 is evaluated at a parameter value $\\mathbf{t}\\in\\boldsymbol{\\Theta}\\subseteq\\mathbb{R}^{\\tau}$, which is a vector unless $\\tau=1$. For a continuous outcome its value is a density, and a density can exceed one. A discrete outcome ensures that likelihood values lie in $[0,1]$, but density values need not obey this bound. The description of the likelihood as a random function is correct.", "what_survives": "The definition of the likelihood in the body of the page and the likelihood calculations that follow are unaffected, and nothing uses the footnote. A fully general statement would also specify measurability and allow for densities that take the value $+\\infty$; the correction below is for finite-valued likelihoods." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C05-E006", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Footnote 10: We do not need to estimate sigma here, but simple algebraic manipulations show that its ML estimator will equal 1 over n times the sum over i from 1 to n of (Y i minus X i beta hat) squared, or the plug-in sample variance of the residuals. Red boxes mark the symbol sigma in the first line and the closing parenthesis after the final period. The first two lines of footnote 11 follow.", "caption": "Footnote 10, p. 191 (PDF p. 209). The marked $\\sigma$ should be $\\sigma^2$, and the marked closing parenthesis has no partner.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E005.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T78", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T78" } ], "headline": "The expression in footnote 10 is the ML estimator of $\\sigma^2$, not of $\\sigma$; the footnote should begin “We do not need to estimate $\\sigma^2$ here”, and its final parenthesis is unmatched.", "id": "P001-C05-E005", "location": "Footnote 10, Example 5.2.4, p. 191 (PDF p. 209)", "pdf_page": 209, "printed_page": "191", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T78" ], "text": { "as_printed": "We do not need to estimate \\hl{$\\sigma$} here, but simple algebraic manipulations show that its ML estimator will equal $\\frac{1}{n}\\sum_{i=1}^{n}(Y_i-\\mathbf{X}_i\\hat{\\boldsymbol{\\beta}})^2$, or the plug-in sample variance of the residuals.\\hl{)}", "corrected": "We do not need to estimate \\hl{$\\sigma^2$} here, but \\hl{when the sum of squared residuals is positive,} simple algebraic manipulations show that its ML estimator will equal $\\frac{1}{n}\\sum_{i=1}^{n}(Y_i-\\mathbf{X}_i\\hat{\\boldsymbol{\\beta}})^2$, or the plug-in sample variance of the residuals.\n\nThe unmatched parenthesis is deleted. If the sum of squared residuals is zero, no maximum exists with $s>0$. If the footnote is to keep $\\sigma$, the estimator is the square root of the displayed expression, and ``sample variance'' becomes ``sample standard deviation''.", "evidence": "The evidence is a derivation. Set $S=\\sum_{i=1}^{n}(Y_i-\\mathbf{X}_i\\hat{\\boldsymbol{\\beta}})^2$ and suppose that $S>0$. As a function of the scale $s>0$, the log-likelihood at $\\mathbf{b}=\\hat{\\boldsymbol{\\beta}}$ and its derivative are\n\\[\n\\ell(s)=-n\\log s-\\frac{S}{2s^2}-\\frac{n}{2}\\log(2\\pi),\\qquad\n\\ell'(s)=-\\frac{n}{s}+\\frac{S}{s^3}=\\frac{S-ns^2}{s^3}.\n\\]\nThe derivative is positive for $s<\\sqrt{S/n}$ and negative for $s>\\sqrt{S/n}$. Then the unique maximum is at $\\hat\\sigma=\\sqrt{S/n}$, and the ML estimator of the variance is $\\hat\\sigma^2=S/n$, which is the expression in the footnote. For the two observations $0$ and $4$ and a model with an intercept only, the fitted mean is $2$ and $S=8$. I obtain $\\hat\\sigma=2$, while the expression in the footnote equals $4$; the log-likelihood is $-4.224$ at $s=2$ and $-4.860$ at $s=4$.", "problem": "In Example 5.2.4, $\\sigma$ is the standard deviation of the errors: the model has $\\varepsilon\\sim N(0,\\sigma^2)$, and the likelihood is written with the scale $s$ and the variance $s^2$. The expression in the footnote is the ML estimator of the variance $\\sigma^2$, as the footnote's own gloss, ``the plug-in sample variance of the residuals'', says. When the sum of squared residuals is positive, the ML estimator of $\\sigma$ is its square root. The footnote also ends with a closing parenthesis that has no opening one.", "what_survives": "No later calculation uses the footnote. The result of Example 5.2.4 that the ML estimator of $\\boldsymbol{\\beta}$ minimizes the sum of squared residuals whenever a positive-scale ML solution exists is unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C05-E005", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the top of the page. The first line reads 'Let L(t) = E[log g(Y, X; t)] = E[log L(t | Y, X)], so that we can write the above statement as', followed by the display 'L_n(t) converges in probability to L(t)'. In the sentence below it, red boxes mark the word 'likelihood' in 'the sample mean of the likelihood of each observation' and again in 'the expected likelihood of a single observation'. The crop ends with the lemma 'L(t) is at most L(theta)'.", "caption": "Section 5.2.2, p. 194 (PDF p. 212), the sentence after the display $L_n(\\mathbf t)\\xrightarrow{p}L(\\mathbf t)$. The two marked words stand for the log-likelihood: the line above defines $L(\\mathbf t)$ as the expectation of $\\log g$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E008.png" }, { "alt": "Crop of the seven-line paragraph that begins 'What this proof sketch illustrates is that the ML estimator is itself a type of plug-in estimator.' A red box marks the word 'likelihood' at the end of the fifth line, in 'the parameter values that maximize the sample mean of the likelihood function'. Two lines above, the paragraph reads 'the parameter values that maximize the expected value of the log-likelihood function'.", "caption": "Section 5.2.2, p. 194 (PDF p. 212), the paragraph after the proof sketch. The marked word stands for the log-likelihood; the preceding sentence of the paragraph speaks of “the expected value of the log-likelihood function”.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E008-2.png" } ], "formal_release": null, "formal_statements": [], "headline": "In three places on p. 194, “likelihood” means the log-likelihood, as the adjacent definitions of $L_n$ and $L$ show.", "id": "P001-C05-E008", "location": "Section 5.2.2, p. 194 (PDF p. 212): the sentence after the display $L_n(\\mathbf t)\\xrightarrow{p}L(\\mathbf t)$, and the paragraph after the proof sketch", "pdf_page": 212, "printed_page": "194", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "Let $L(\\mathbf t)=\\mathrm{E}[\\log g(Y,\\mathbf X;\\mathbf t)]=\\mathrm{E}[\\log\\mathcal L(\\mathbf t|Y,\\mathbf X)]$, so that we can write the above statement as\n\\[\nL_n(\\mathbf t)\\xrightarrow{p}L(\\mathbf t),\n\\]\nthat is, the sample mean of the \\hl{likelihood} of each observation given parameter values $\\mathbf t$ converges to the expected \\hl{likelihood} of a single observation given parameter values $\\mathbf t$.\n\nIn the paragraph after the proof sketch:\n\nTo estimate this population quantity, we use the sample analog: the parameter values that maximize the sample mean of the \\hl{likelihood} function.", "corrected": "that is, the sample mean of the \\hl{log-likelihood} of each observation given parameter values $\\mathbf t$ converges to the expected \\hl{log-likelihood} of a single observation given parameter values $\\mathbf t$.\n\nTo estimate this population quantity, we use the sample analog: the parameter values that maximize the sample mean of the \\hl{log-likelihood} function.", "intended_meaning": "The quantities in these sentences are averages of logarithms. On p.~193 the book sets $L_n(\\mathbf t)=n^{-1}\\sum_{i=1}^n\\log g(Y_i,\\mathbf X_i;\\mathbf t)$, and on p.~194 it sets $L(\\mathbf t)=\\mathrm{E}[\\log g(Y,\\mathbf X;\\mathbf t)]$. Between the two marked passages, the text speaks of ``the expected value of the log-likelihood function''. The intended meaning is the sample mean and the expectation of the log-likelihood, and with this meaning these sentences agree with the displayed criteria. The difference matters only if the prose is read apart from the equations: for the coin data $(1,0,1)$ of the chapter, the mean of the three likelihoods is $(1+p)/3$, which is largest at $p=1$, and the mean log-likelihood is largest at the ML estimate $p=2/3$." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C05-E008", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the box of Theorem 5.2.8, The ML Estimator Is Consistent for the Minimum KL Divergence Predictor. A red box marks the expression 'KL[ f_{Y|X}( . | x) || g( . , x; t) ]' in the display that defines theta-tilde as the arg min over t in Theta; the x in it is lower case and bold. Below the display, the theorem concludes that theta-hat-ML converges in probability to theta-tilde. Under the box, a paragraph says the proof is omitted and compares the result with OLS, using KL divergence rather than MSE.", "caption": "Theorem 5.2.8, p. 195 (PDF p. 213). The marked divergence is evaluated at a fixed value $\\mathbf x$ of the explanatory variables, with no expectation over $\\mathbf X$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E001.png" }, { "alt": "Line chart. The horizontal axis is Pr[X = 1], with the values 1/4, 1/2 and 3/4. The vertical axis is the parameter value t, with ticks from 1/4 to 3/4. Two horizontal dashed orange lines, labeled 'as printed, at x = 1: t = 3/4' and 'as printed, at x = 0: t = 1/4', stay level across the chart. A solid blue line with three round markers rises from t = 3/8 at Pr[X = 1] = 1/4, through t = 1/2 at 1/2, to t = 5/8 at 3/4. The legend calls the dashed lines the target as printed, the minimizer of the divergence at a fixed x, and the blue line the limit of the ML estimator, the minimizer of the divergence averaged over X. Every blue point lies strictly between the two dashed lines.", "caption": "The counterexample to Theorem 5.2.8 (p. 195) as printed, for three distributions of $X$ and the same conditional distributions of $Y$ given $X$. The dashed lines are the minimizers at a fixed $x$, which are $1/4$ at $x=0$ and $3/4$ at $x=1$. The points are the limit of the ML estimator, which is $3/8$, $1/2$ and $5/8$ for $\\Pr[X=1]=1/4$, $1/2$ and $3/4$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E001-targets.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T7", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T7" }, { "id": "P001-T8", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T8" }, { "id": "P001-T120", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T120" } ], "headline": "In Theorem 5.2.8 the limit of the ML estimator is the minimizer of the Kullback–Leibler divergence averaged over the distribution of $\\mathbf X$, not the minimizer at a fixed value $\\mathbf x$.", "id": "P001-C05-E001", "location": "Theorem 5.2.8, p. 195 (PDF p. 213), the display that defines $\\tilde{\\boldsymbol\\theta}$; the explanation continues on p. 196 (PDF p. 214)", "pdf_page": 213, "printed_page": "195", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T120", "P001-T7", "P001-T8" ], "text": { "as_printed": "\\textbf{Theorem 5.2.8.} \\emph{The ML Estimator Is Consistent for the Minimum KL Divergence Predictor}\n\nLet $Y$ be a random variable and $\\mathbf X$ be a random vector. Let $(Y_1,\\mathbf X_1)$, $(Y_2,\\mathbf X_2)$, \\ldots, $(Y_n,\\mathbf X_n)$ be i.i.d. observations of $(Y,\\mathbf X)$. Then the ML estimate of $f_{Y|\\mathbf X}(y|\\mathbf x)$, $g(y,\\mathbf x;\\hat{\\boldsymbol\\theta}_{ML})$ consistently estimates the minimum Kullback--Leibler divergence approximation of $f_{Y|\\mathbf X}(y|\\mathbf x)$ among all probability distributions of the form $g(y,\\mathbf x;\\mathbf t)$. That is, assuming that there exists a unique\n\\[\n\\tilde{\\boldsymbol\\theta}=\\operatorname*{arg\\,min}_{\\mathbf t\\in\\boldsymbol\\Theta}\\mathrm{KL}\\big[f_{Y|\\mathbf X}(\\cdot\\,|\\hl{\\mathbf x})\\,\\big\\|\\,g(\\cdot,\\hl{\\mathbf x};\\mathbf t)\\big],\n\\]\nthen\n\\[\n\\hat{\\boldsymbol\\theta}_{ML}\\xrightarrow{p}\\tilde{\\boldsymbol\\theta}.\n\\]", "corrected": "\\textbf{Theorem 5.2.8.} \\emph{The ML Estimator Is Consistent for the Minimum KL Divergence Predictor}\n\nLet $Y$ be a random variable and $\\mathbf X$ be a random vector. Let $(Y_1,\\mathbf X_1)$, $(Y_2,\\mathbf X_2)$, \\ldots, $(Y_n,\\mathbf X_n)$ be i.i.d. observations of $(Y,\\mathbf X)$. Let $L_n(\\mathbf t)=n^{-1}\\sum_{i=1}^n\\log g(Y_i,\\mathbf X_i;\\mathbf t)$ and $L(\\mathbf t)=\\mathrm{E}[\\log g(Y,\\mathbf X;\\mathbf t)]$, and suppose that $L(\\mathbf t)$ is finite for every $\\mathbf t\\in\\boldsymbol\\Theta$ and that the following conditions hold.\n\\begin{itemize}\n\\item $\\sup_{\\mathbf t\\in\\boldsymbol\\Theta}|L_n(\\mathbf t)-L(\\mathbf t)|\\xrightarrow{p}0$.\n\\item $L$ has a unique maximizer $\\tilde{\\boldsymbol\\theta}\\in\\boldsymbol\\Theta$, and $L(\\tilde{\\boldsymbol\\theta})-\\sup_{\\|\\mathbf t-\\tilde{\\boldsymbol\\theta}\\|\\ge\\epsilon}L(\\mathbf t)>0$ for every $\\epsilon>0$ such that the set $\\{\\mathbf t\\in\\boldsymbol\\Theta:\\|\\mathbf t-\\tilde{\\boldsymbol\\theta}\\|\\ge\\epsilon\\}$ is not empty.\n\\item $L_n(\\hat{\\boldsymbol\\theta}_{ML})\\ge\\sup_{\\mathbf t\\in\\boldsymbol\\Theta}L_n(\\mathbf t)-\\eta_n$, where $\\eta_n\\ge0$ and $\\eta_n\\xrightarrow{p}0$. An exact maximizer has $\\eta_n=0$.\n\\end{itemize}\nThen\n\\[\n\\hat{\\boldsymbol\\theta}_{ML}\\xrightarrow{p}\\tilde{\\boldsymbol\\theta}.\n\\]\nIf, in addition, $\\mathrm{E}|\\log f_{Y|\\mathbf X}(Y|\\mathbf X)|<\\infty$ and $\\mathrm{E}|\\log g(Y,\\mathbf X;\\mathbf t)|<\\infty$ for each $\\mathbf t\\in\\boldsymbol\\Theta$, then\n\\[\n\\tilde{\\boldsymbol\\theta}=\\operatorname*{arg\\,min}_{\\mathbf t\\in\\boldsymbol\\Theta}\\hl{\\mathrm{E}_{\\mathbf X}}\\Big[\\mathrm{KL}\\big[f_{Y|\\mathbf X}(\\cdot\\,|\\hl{\\mathbf X})\\,\\big\\|\\,g(\\cdot,\\hl{\\mathbf X};\\mathbf t)\\big]\\Big].\n\\]\nThe parameter $\\tilde{\\boldsymbol\\theta}$ gives the closest fit averaged over the distribution of $\\mathbf X$. It need not give the closest fit at each fixed value $\\mathbf x$.\n\nOn p.~196, ``closest'' (in terms of KL divergence) and the population KL divergence in the paragraph on the idea behind the proof refer to this averaged divergence.", "evidence": "I give an exact counterexample to the theorem as printed, with a derivation of the corrected target. Let $X$ take the values $0$ and $1$ with probability $1/2$ each, and let\n\\[\nY\\mid X=0\\sim\\operatorname{Bernoulli}(1/4),\\qquad Y\\mid X=1\\sim\\operatorname{Bernoulli}(3/4).\n\\]\nConsider the model $g(y,x;t)=t^y(1-t)^{1-y}$ for $y\\in\\{0,1\\}$, with $t\\in\\Theta=[1/10,9/10]$. The parameter space is compact, and each observation's log-likelihood is continuous in $t$ and at most $\\log 10$ in absolute value. For $0p$, the function $K_p$ has the unique minimizer $t=p$ on $\\Theta$ when $p\\in\\Theta$. The target printed in Theorem 5.2.8 is then $\\tilde\\theta=1/4$ at $x=0$ and $\\tilde\\theta=3/4$ at $x=1$, and it is unique in both cases.\n\nLet $\\bar Y_n=n^{-1}\\sum_{i=1}^nY_i$. The average log-likelihood is $L_n(t)=\\bar Y_n\\log t+(1-\\bar Y_n)\\log(1-t)$, with derivative $(\\bar Y_n-t)/(t(1-t))$, and the ML estimator is\n\\[\n\\hat\\theta_{ML}=\\min\\{9/10,\\max\\{1/10,\\bar Y_n\\}\\}.\n\\]\nI have $\\mathrm{E}[Y]=1/2$ and $\\mathrm{V}[\\bar Y_n]=1/(4n)$, and restricting $\\bar Y_n$ to $\\Theta$ does not increase its distance from $1/2$. By Chebyshev's inequality, I have\n\\[\n\\Pr\\big[|\\hat\\theta_{ML}-1/2|\\ge\\epsilon\\big]\\le\\frac{1}{4n\\epsilon^2}\n\\]\nfor every $\\epsilon>0$, so that $\\hat\\theta_{ML}\\xrightarrow{p}1/2$. Taking $\\epsilon=1/8$, I obtain $\\Pr[|\\hat\\theta_{ML}-1/4|>1/8]\\ge1-16/n$, and the same bound holds with $3/4$ in place of $1/4$. The ML estimator converges to neither printed target.\n\nThe averaged divergence is $\\bar K(t)=K_{1/4}(t)/2+K_{3/4}(t)/2$, with $\\bar K'(t)=(t-1/2)/(t(1-t))$. Its unique minimizer is $1/2$, the limit of the ML estimator, and $\\bar K(1/4)-\\bar K(1/2)=\\bar K(3/4)-\\bar K(1/2)=\\log(4/3)/2>0$. If $\\Pr[X=1]=q$ and the conditional distributions are unchanged, the minimizer of the averaged divergence is $1/4+q/2$. It equals $3/8$ for $q=1/4$ and $5/8$ for $q=3/4$, and the minimizers at a fixed $x$ remain $1/4$ and $3/4$.\n\nFor the corrected target, suppose that $\\mathrm{E}|\\log f_{Y|\\mathbf X}(Y|\\mathbf X)|<\\infty$ and $\\mathrm{E}|\\log g(Y,\\mathbf X;\\mathbf t)|<\\infty$ for each $\\mathbf t$. By the Law of Iterated Expectations,\n\\[\n\\mathrm{E}_{\\mathbf X}\\Big[\\mathrm{KL}\\big[f_{Y|\\mathbf X}(\\cdot\\,|\\mathbf X)\\,\\big\\|\\,g(\\cdot,\\mathbf X;\\mathbf t)\\big]\\Big]=\\mathrm{E}\\big[\\log f_{Y|\\mathbf X}(Y|\\mathbf X)\\big]-L(\\mathbf t).\n\\]\nThe first term on the right does not depend on $\\mathbf t$. Minimizing the averaged divergence is then the same as maximizing $L(\\mathbf t)$, and it is not in general the same as minimizing the divergence at each $\\mathbf x$. Consistency under the three conditions of the corrected statement follows from a comparison of $L_n$ and $L$. Set $D_n=\\sup_{\\mathbf t\\in\\boldsymbol\\Theta}|L_n(\\mathbf t)-L(\\mathbf t)|$ and $\\delta_\\epsilon=L(\\tilde{\\boldsymbol\\theta})-\\sup_{\\|\\mathbf t-\\tilde{\\boldsymbol\\theta}\\|\\ge\\epsilon}L(\\mathbf t)$. On the event $\\|\\hat{\\boldsymbol\\theta}_{ML}-\\tilde{\\boldsymbol\\theta}\\|\\ge\\epsilon$, I have\n\\[\nL(\\tilde{\\boldsymbol\\theta})-D_n-\\eta_n\\le L_n(\\tilde{\\boldsymbol\\theta})-\\eta_n\\le L_n(\\hat{\\boldsymbol\\theta}_{ML})\\le L(\\hat{\\boldsymbol\\theta}_{ML})+D_n\\le L(\\tilde{\\boldsymbol\\theta})-\\delta_\\epsilon+D_n,\n\\]\nwhich implies that $\\delta_\\epsilon\\le2D_n+\\eta_n$. Since $\\delta_\\epsilon>0$, and $D_n$ and $\\eta_n$ tend to zero in probability, the probability of this event tends to zero. The counterexample satisfies these conditions, with $D_n\\le|\\bar Y_n-1/2|\\log 9$.", "problem": "By Definition 5.2.7 the Kullback--Leibler (KL) divergence integrates over the outcome alone. The displayed quantity therefore compares the true conditional distribution of $Y$ with the model at one fixed value $\\mathbf x$ of the explanatory variables, and the display contains no expectation over $\\mathbf X$. The ML estimator fits a single parameter to the whole sample, and under the usual consistency conditions its limit is the maximizer of the expected log-likelihood $L(\\mathbf t)=\\mathrm{E}[\\log g(Y,\\mathbf X;\\mathbf t)]$ of p.~194. This maximizer minimizes the divergence averaged over the distribution of $\\mathbf X$, and it can change when that distribution changes. A minimizer at a fixed $\\mathbf x$ need not equal it, even when that minimizer is unique at every $\\mathbf x$, the parameter space is compact, the likelihood is continuous, and the sample and population maxima are unique. Read as printed, the theorem names the wrong parameter, and it promises a best approximation at each value of $\\mathbf x$ separately, which ML estimation does not provide. The expected log-likelihood on p.~194 and the comparison with OLS on pp.~195--196 point to the intended averaged target, but the printed formula does not state it. The two targets do not differ in every misspecified model with explanatory variables: there exist false models in which the minimizers at each $\\mathbf x$ and the minimizer of the averaged divergence coincide.", "what_survives": "The definition of the likelihood, the ML estimates themselves, the equivalence of OLS and ML in the classical linear model, and Theorem 5.2.6 with its regularity conditions are unchanged, and no number in the book is shown to depend on the printed target. Examples 5.2.9 and 5.2.11 and Figures 5.2.2--5.2.5 have no explanatory variables, and the correction does not alter their targets; I did not recompute those figures. The claim that ML estimation consistently estimates a best-fitting distribution in the parametric family survives, with ``best'' defined by the averaged divergence, and what is given up is the closest fit at each value of $\\mathbf x$ separately. The reference in Section 5.2.6 (p.~202) to this predictor should also use the averaged target (see \\fid{P001-C05-N006} for its wording), and Section 7.2.3 (pp.~261--262) assumes that the parametric model is true and is not affected." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C05-E001", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the paragraph that begins 'The idea behind the proof of Theorem 5.2.8 is again related to the plug-in principle.' Red boxes mark the phrase 'KL divergence of the hypothesized distribution from the observed data', which runs from the end of the fifth line to the sixth, in the sentence 'It can be shown that the ML estimate is equal to the sample analog: the parameter values that minimize the KL divergence of the hypothesized distribution from the observed data.' The paragraph ends 'the ML estimate converges to the value that minimizes the KL divergence of the hypothesized distribution from the true population distribution.'", "caption": "Section 5.2.3, p. 196 (PDF p. 214), the paragraph on the idea behind the proof of Theorem 5.2.8. The marked phrase describes the sample objective of ML estimation as a KL divergence from the observed data.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E003.png" } ], "formal_release": null, "formal_statements": [], "headline": "The sample analog on p. 196 is the minimizer of the empirical cross-entropy, that is, the maximizer of the average log-likelihood; it minimizes a KL divergence from the observed data only when the outcome is discrete.", "id": "P001-C05-E003", "location": "Section 5.2.3, p. 196 (PDF p. 214), second paragraph, the sentence that begins “It can be shown”", "pdf_page": 214, "printed_page": "196", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "It can be shown that the ML estimate is equal to the sample analog: the parameter values that minimize the \\hl{KL divergence of the hypothesized distribution from the observed data}. So, since the distribution of the observed data converges to the population distribution as $n$ grows large, the ML estimate converges to the value that minimizes the KL divergence of the hypothesized distribution from the true population distribution.", "corrected": "It can be shown that the ML estimate is equal to the sample analog: the parameter values that minimize the \\hl{empirical cross-entropy}\n\\[\n\\hl{H_n(\\mathbf t)=-\\frac{1}{n}\\sum_{i=1}^n\\log g(Y_i,\\mathbf X_i;\\mathbf t)},\n\\]\n\\hl{the sample analog of the part of the population KL divergence that depends on $\\mathbf t$, when the population entropy and expected log-density are finite}. So, since \\hl{$H_n(\\mathbf t)$ converges to $-\\mathrm{E}[\\log g(Y,\\mathbf X;\\mathbf t)]$} as $n$ grows large, the ML estimate converges, \\hl{under the requisite regularity conditions}, to the value that minimizes the KL divergence of the hypothesized distribution from the true population distribution.", "intended_meaning": "The sample analog meant here is the maximizer of the average log-likelihood $L_n(\\mathbf t)$ of p.~193, which is the minimizer of the empirical cross-entropy $-L_n(\\mathbf t)$, and with this meaning the sentence is correct. For a discrete outcome, such as the coin flip with $y=(1,0,1)$ on the same page, minimizing $-L_n$ is the same as minimizing the KL divergence from the empirical distribution, since the two criteria differ by a term that does not depend on $\\mathbf t$. For a continuous model the description is not literal: the empirical distribution has no density, Definition 5.2.7 does not apply to it, and under the general definition the divergence of a model with a density from the empirical distribution is infinite for every $\\mathbf t$. With explanatory variables, the population criterion is the divergence averaged over the distribution of $\\mathbf X$ (see \\fid{P001-C05-E001}), which differs from $-\\mathrm{E}[\\log g(Y,\\mathbf X;\\mathbf t)]$ by a constant when both are finite. The estimator and the calculations of the chapter are unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C05-E003", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of the box of Definition 5.2.10, Maximum Likelihood Plug-In Estimator, to the end of p. 197. In the display for F-hat-ML(y, x), the indicator reads 'I(X_[1]i <= x_[1]i, X_[2] <= x_[2]i, ..., X_[K]i <= x_[x])', followed by the sum over v <= y of g(v, X_i; theta-hat-ML). One red box marks 'x_[1]i, X_[2] <= x_[2]i' and a second red box marks 'x_[x]'.", "caption": "Definition 5.2.10, p. 197 (PDF p. 215), the display for a PMF. The marked subscripts are wrong: the thresholds $x_{[1]i}$ and $x_{[2]i}$ carry an observation index, $X_{[2]}$ lacks one, and the last threshold is printed $x_{[x]}$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E007.png" }, { "alt": "Crop of the end of the box of Definition 5.2.10 at the top of p. 198. The first line reads 'if f_{Y|X}(y|x) is a PMF, and'. In the display for F-hat-ML(y, x), the indicator reads 'I(X_[1]i <= x_[1]i, X_[2] <= x_[2]i, ..., X_[K]i <= x_[x])', followed by the integral from minus infinity to y of g(v, X_i; theta-hat-ML) dv. One red box marks 'x_[1]i, X_[2] <= x_[2]i' and a second red box marks 'x_[x]'. The last line reads 'if f_{Y|X}(y|x) is a PDF.'", "caption": "Definition 5.2.10, continued on p. 198 (PDF p. 216), the display for a PDF. The indicator repeats the subscripts of the first display.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E007-2.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T79", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T79" } ], "headline": "In both displays of Definition 5.2.10 the indicator should read $\\mathrm{I}(X_{[1]i}\\le x_{[1]},X_{[2]i}\\le x_{[2]},\\ldots,X_{[K]i}\\le x_{[K]})$.", "id": "P001-C05-E007", "location": "Definition 5.2.10, pp. 197–198 (PDF pp. 215–216), the indicator in both displays for $\\hat F_{ML}$", "pdf_page": 215, "printed_page": "197–198", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T79" ], "text": { "as_printed": "\\textbf{Definition 5.2.10.} \\emph{Maximum Likelihood Plug-In Estimator}\n\nFor i.i.d. random vectors $(Y_1,\\mathbf X_1)$, $(Y_2,\\mathbf X_2)$, \\ldots, $(Y_n,\\mathbf X_n)$ with common CDF $F$, the \\emph{ML plug-in estimator} of $\\psi=T(F)$ is\n\\[\n\\hat\\psi_{ML}=T\\big(\\hat F_{ML}\\big),\n\\]\nwhere\n\\[\n\\hat F_{ML}(y,\\mathbf x)=\\frac{1}{n}\\sum_{i=1}^n\\left(\\mathrm{I}\\big(X_{[1]i}\\le\\hl{x_{[1]i}},\\hl{X_{[2]}}\\le\\hl{x_{[2]i}},\\ldots,X_{[K]i}\\le\\hl{x_{[x]}}\\big)\\sum_{v\\le y}g(v,\\mathbf X_i;\\hat{\\boldsymbol\\theta}_{ML})\\right)\n\\]\nif $f_{Y|\\mathbf X}(y|\\mathbf x)$ is a PMF, and\n\\[\n\\hat F_{ML}(y,\\mathbf x)=\\frac{1}{n}\\sum_{i=1}^n\\left(\\mathrm{I}\\big(X_{[1]i}\\le\\hl{x_{[1]i}},\\hl{X_{[2]}}\\le\\hl{x_{[2]i}},\\ldots,X_{[K]i}\\le\\hl{x_{[x]}}\\big)\\int_{-\\infty}^{y}g(v,\\mathbf X_i;\\hat{\\boldsymbol\\theta}_{ML})dv\\right)\n\\]\nif $f_{Y|\\mathbf X}(y|\\mathbf x)$ is a PDF.", "corrected": "\\textbf{Definition 5.2.10.} \\emph{Maximum Likelihood Plug-In Estimator}\n\nFor i.i.d. random vectors $(Y_1,\\mathbf X_1)$, $(Y_2,\\mathbf X_2)$, \\ldots, $(Y_n,\\mathbf X_n)$ with common CDF $F$, the \\emph{ML plug-in estimator} of $\\psi=T(F)$ is\n\\[\n\\hat\\psi_{ML}=T\\big(\\hat F_{ML}\\big),\n\\]\nwhere\n\\[\n\\hat F_{ML}(y,\\mathbf x)=\\frac{1}{n}\\sum_{i=1}^n\\left(\\mathrm{I}\\big(X_{[1]i}\\le\\hl{x_{[1]}},\\hl{X_{[2]i}}\\le\\hl{x_{[2]}},\\ldots,X_{[K]i}\\le\\hl{x_{[K]}}\\big)\\sum_{v\\le y}g(v,\\mathbf X_i;\\hat{\\boldsymbol\\theta}_{ML})\\right)\n\\]\nif $f_{Y|\\mathbf X}(y|\\mathbf x)$ is a PMF, and\n\\[\n\\hat F_{ML}(y,\\mathbf x)=\\frac{1}{n}\\sum_{i=1}^n\\left(\\mathrm{I}\\big(X_{[1]i}\\le\\hl{x_{[1]}},\\hl{X_{[2]i}}\\le\\hl{x_{[2]}},\\ldots,X_{[K]i}\\le\\hl{x_{[K]}}\\big)\\int_{-\\infty}^{y}g(v,\\mathbf X_i;\\hat{\\boldsymbol\\theta}_{ML})dv\\right)\n\\]\nif $f_{Y|\\mathbf X}(y|\\mathbf x)$ is a PDF. Here $\\mathbf x=(x_{[1]},\\ldots,x_{[K]})$ is the point at which the CDF is evaluated, and $X_{[k]i}$ is the $k$th component of $\\mathbf X_i$.", "evidence": "The evidence is a derivation, with a check of both pages. Let $A_{\\mathbf x}$ be the event that $X_{[1]}\\le x_{[1]},\\ldots,X_{[K]}\\le x_{[K]}$. By the Law of Iterated Expectations, which the book invokes below the definition, I have\n\\[\nF(y,\\mathbf x)=\\mathrm{E}\\big[\\mathrm{I}(A_{\\mathbf x})\\,\\mathrm{I}(Y\\le y)\\big]=\\mathrm{E}\\big[\\mathrm{I}(A_{\\mathbf x})\\Pr[Y\\le y\\,|\\,\\mathbf X]\\big].\n\\]\nSubstituting the empirical distribution of $\\mathbf X$ for its population distribution, and the fitted model for the conditional distribution of $Y$, I obtain\n\\[\n\\hat F_{ML}(y,\\mathbf x)=\\frac{1}{n}\\sum_{i=1}^n\\mathrm{I}\\big(X_{[1]i}\\le x_{[1]},\\ldots,X_{[K]i}\\le x_{[K]}\\big)\\,G(y|\\mathbf X_i),\n\\]\nwhere $G(y|\\mathbf X_i)$ is $\\sum_{v\\le y}g(v,\\mathbf X_i;\\hat{\\boldsymbol\\theta}_{ML})$ for a PMF and $\\int_{-\\infty}^yg(v,\\mathbf X_i;\\hat{\\boldsymbol\\theta}_{ML})dv$ for a PDF. Every summand uses the $i$th observation in each coordinate and the same fixed thresholds. As a check, take $K=2$ and $n=2$, with $\\mathbf X_1=(0,2)$ and $\\mathbf X_2=(2,1)$, a fitted Bernoulli model with success probability $1/4$ at $\\mathbf X_1$ and $3/4$ at $\\mathbf X_2$, $y=0$ and $\\mathbf x=(1,3)$. Only $\\mathbf X_1$ satisfies both inequalities, and the corrected formula gives $\\hat F_{ML}(0,(1,3))=(1-1/4)/2=3/8$.", "problem": "The indicator should compare the $i$th observation with the fixed point $\\mathbf x=(x_{[1]},\\ldots,x_{[K]})$ in every coordinate. As printed in both displays, the thresholds $x_{[1]i}$ and $x_{[2]i}$ carry an observation index although they are coordinates of the fixed point, $X_{[2]}$ lacks the index $i$, and the last threshold is $x_{[x]}$ where $x_{[K]}$ is meant. The printed expression is not defined in the notation of the book and cannot be evaluated as written.", "what_survives": "The ML plug-in estimator, its construction from the Law of Iterated Expectations and the plug-in estimates of the CEF on p.~198 are unaffected. Only the subscripts of the two indicators change." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C05-E007", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "historical_lean_attribution", "exhibits": [ { "alt": "Crop of the top of p. 199 with the running head 'Parametric Models' and the page number 199. The first paragraph reads 'where theta = (p_1, theta_1, p_2, theta_2, ..., p_{M-1}, theta_{M-1}, theta_M). Then, as before, the product over i from 1 to n of g(Y_i, X_i; theta) can be maximized to find the ML estimate of theta. (This can sometimes be computationally tricky.)' A red box marks the words 'can be maximized to find the ML estimate of theta'. The second paragraph begins 'If any subset of the models' and ends 'our estimates should be interpreted explicitly as an approximation.'", "caption": "Section 5.2.5, p. 199 (PDF p. 217). The marked clause says that the likelihood of the mixture can be maximized; for the mixture of Example 5.2.11 it has no maximum on a sample of positive observations.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E002.png" }, { "alt": "Crop of the end of Example 5.2.11. After the line 'Consider a mixture model that incorporates all three models:', the display gives g(y; theta) as p_N times the normal density, plus p_LN times the log-normal density, plus (1 - p_N - p_LN) times the exponential density. Small red boxes mark sigma_N in the denominator of the normal term and sigma_LN in the denominator of the log-normal term. Below the display, the text reads 'where theta = (p_N, p_LN, mu_N, sigma_N, mu_LN, sigma_LN, lambda_E). Figure 5.2.5 visualizes the ML estimate of this mixture model, which provides a markedly better approximation.' Red boxes mark the words 'Figure 5.2.5 visualizes the ML estimate of this mixture model'. The next paragraph begins 'Technically, mixture models are parametric'.", "caption": "Example 5.2.11, p. 200 (PDF p. 218). The marked scale parameters $\\sigma_N$ and $\\sigma_{LN}$ are free. On positive samples, the likelihood grows without bound if either component is centered at an observation and its scale tends to zero while another component stays fixed with positive weight. The marked sentence presents the curve of Figure 5.2.5 as the ML estimate of this mixture.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-E002-2.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T9", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T9" }, { "id": "P001-T10", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T10" }, { "id": "P001-T103", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T103" } ], "headline": "On every sample of positive observations, the likelihood of the normal, log-normal and exponential mixture of Example 5.2.11 is unbounded and the ML estimate of the unrestricted mixture does not exist; Figure 5.2.5 cannot show a global ML estimate of that mixture.", "historical_evidence_type": "lean_proof", "id": "P001-C05-E002", "location": "Section 5.2.5, p. 199 (PDF p. 217), “can be maximized to find the ML estimate”; Example 5.2.11, p. 200 (PDF p. 218); Figure 5.2.5, p. 201 (PDF p. 219)", "pdf_page": 217, "printed_page": "199", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P001-T10", "P001-T103", "P001-T9" ], "text": { "as_printed": "Section 5.2.5, p.~199:\n\nwhere $\\boldsymbol\\theta=(p_1,\\boldsymbol\\theta_1,p_2,\\boldsymbol\\theta_2,\\ldots,p_{M-1},\\boldsymbol\\theta_{M-1},\\boldsymbol\\theta_M)$. Then, as before, $\\prod_{i=1}^ng(Y_i,\\mathbf X_i;\\boldsymbol\\theta)$ \\hl{can be maximized} to find the ML estimate of $\\boldsymbol\\theta$. (This can sometimes be computationally tricky.)\n\nIf any subset of the models in $(g_1(y,\\mathbf x;\\boldsymbol\\theta_1),\\ldots,g_M(y,\\mathbf x;\\boldsymbol\\theta_M))$ jointly characterize the conditional distribution of $Y$, then this estimator is consistent and asymptotically efficient. When the models are not valid (even jointly), our estimates should be interpreted explicitly as an approximation.\n\nExample 5.2.11, p.~200:\n\nConsider a mixture model that incorporates all three models:\n\\[\n\\begin{aligned}\ng(y;\\boldsymbol\\theta)&=p_Ng_N\\big(y;(\\mu_N,\\sigma_N)\\big)+p_{LN}g_{LN}\\big(y;(\\mu_{LN},\\sigma_{LN})\\big)\\\\\n&\\quad+(1-p_N-p_{LN})g_E(y;\\lambda_E)\\\\\n&=p_N\\frac{1}{\\sigma_N\\sqrt{2\\pi}}e^{-\\frac{(y-\\mu_N)^2}{2\\sigma_N^2}}+p_{LN}\\frac{1}{y\\sigma_{LN}\\sqrt{2\\pi}}e^{-\\frac{(\\log y-\\mu_{LN})^2}{2\\sigma_{LN}^2}}\\\\\n&\\quad+(1-p_N-p_{LN})\\lambda_Ee^{-\\lambda_Ey},\n\\end{aligned}\n\\]\nwhere $\\boldsymbol\\theta=(p_N,p_{LN},\\mu_N,\\sigma_N,\\mu_{LN},\\sigma_{LN},\\lambda_E)$. Figure 5.2.5 visualizes the \\hl{ML estimate} of this mixture model, which provides a markedly better approximation. $\\triangle$\n\nFigure 5.2.5, p.~201, has the plot title ``Mixture \\hl{ML} Fit'' and the caption:\n\nFIGURE 5.2.5. \\emph{Mixture Model \\hl{Maximum Likelihood} Approximation}", "corrected": "Section 5.2.5, p.~199:\n\nThen, as before, $\\prod_{i=1}^ng(Y_i,\\mathbf X_i;\\boldsymbol\\theta)$ can be maximized to find the ML estimate of $\\boldsymbol\\theta$\\hl{, provided that a maximum exists}. (This can sometimes be computationally tricky. \\hl{A maximum need not exist: when a component has a free scale parameter, as the normal and log-normal components of Example 5.2.11 do, the likelihood can be unbounded. The parameters must then be restricted to ensure an attained likelihood maximum, for example to a nonempty compact subset of the admissible parameter space. Alternatively, a penalty must ensure an attained maximum of the penalized log-likelihood before a penalized ML estimate is defined.})\n\nThe sentence on consistency and efficiency that follows applies only when a maximum exists and the regularity conditions hold; its correction is given in \\fid{P001-C05-N005}.\n\nExample 5.2.11, p.~200:\n\n\\hl{Figure 5.2.5 shows a fitted mixture of this form}, which provides a markedly better approximation.\n\nThe example should state the objective and the restrictions used for the curve: for example, a population (minimum KL divergence) approximation as in Example 5.2.9, or a fit to a sample with the scale parameters restricted. Until then the title and caption of Figure 5.2.5 should describe a fitted mixture approximation, not a maximum likelihood estimate.\n\nThe mathematical statement is as follows. Consider the mixture of Example 5.2.11 with $p_N,p_{LN}\\ge0$, $p_N+p_{LN}\\le1$, $\\mu_N,\\mu_{LN}\\in\\mathbb{R}$ and $\\sigma_N,\\sigma_{LN},\\lambda_E>0$, and let $y_1,\\ldots,y_n$ be positive observations with $n\\ge1$. Then $\\sup_{\\boldsymbol\\theta}\\prod_{i=1}^ng(y_i;\\boldsymbol\\theta)=\\infty$, and no parameter value attains the supremum. If the parameters are restricted to a nonempty compact set on which $\\sigma_N$, $\\sigma_{LN}$ and $\\lambda_E$ are bounded below by a positive constant, then the likelihood of the sample attains its maximum on that set. This restriction does not by itself give a unique maximizer or asymptotic efficiency, and when the model is not true it can change the fitted density and the KL divergence target. A fixed $\\ell^1$ penalty on the seven parameters, as in Section 5.2.6, does not restore a maximum.", "evidence": "The evidence is a Lean proof of the counterexample on the sample $(1,2,3)$, with a derivation that extends it to every sample of positive observations. I use the parameter space described above. For $\\sigma>0$, I set, in the printed order of the parameters,\n\\[\n\\boldsymbol\\theta_\\sigma=(1/2,\\,1/4,\\,1,\\,\\sigma,\\,0,\\,1,\\,1).\n\\]\nThe three weights are $1/2$, $1/4$ and $1/4$, and for $y>0$ the density is\n\\[\ng(y;\\boldsymbol\\theta_\\sigma)=\\frac{e^{-(y-1)^2/(2\\sigma^2)}}{2\\sigma\\sqrt{2\\pi}}+\\frac{e^{-(\\log y)^2/2}}{4y\\sqrt{2\\pi}}+\\frac{e^{-y}}{4}.\n\\]\nEach term is nonnegative, so that $g(1;\\boldsymbol\\theta_\\sigma)\\ge1/(2\\sigma\\sqrt{2\\pi})$, $g(2;\\boldsymbol\\theta_\\sigma)\\ge e^{-2}/4$ and $g(3;\\boldsymbol\\theta_\\sigma)\\ge e^{-3}/4$. Multiplying these bounds, I obtain\n\\[\ng(1;\\boldsymbol\\theta_\\sigma)\\,g(2;\\boldsymbol\\theta_\\sigma)\\,g(3;\\boldsymbol\\theta_\\sigma)\\ge\\frac{e^{-5}}{32\\sqrt{2\\pi}\\,\\sigma}.\n\\]\nThe right side exceeds any given real number when $\\sigma$ is small enough. Since the likelihood is finite at every admissible parameter value, no admissible parameter value maximizes it. The Lean proof establishes these two statements for the sample $(1,2,3)$. Using $e<3$ and $\\sqrt{2\\pi}<3$, I find that the likelihood at $\\sigma=1/k$ exceeds $k/23328$ for every positive integer $k$. The table lists five values of this bound.\n\\begin{audittable}{rr}{Likelihood of the sample $(1,2,3)$ at $\\boldsymbol\\theta_\\sigma=(1/2,1/4,1,\\sigma,0,1,1)$: an exact lower bound as $\\sigma_N=\\sigma$ decreases.}\n$\\sigma_N$ & The likelihood exceeds \\\\\n\\hline\n$1/23328$ & $1$ \\\\\n$1/46656$ & $2$ \\\\\n$1/256608$ & $11$ \\\\\n$1/2356128$ & $101$ \\\\\n$1/23328023328$ & $1000001$ \\\\\n\\end{audittable}\nFor positive observations $y_1,\\ldots,y_n$, the same weights with $\\mu_N=y_1$, $\\sigma_N=\\sigma$, $\\mu_{LN}=0$, $\\sigma_{LN}=1$ and $\\lambda_E=1$ give\n\\[\n\\prod_{i=1}^ng(y_i;\\boldsymbol\\theta)\\ge\\frac{e^{-\\sum_{i=2}^ny_i}}{2\\cdot4^{n-1}\\sqrt{2\\pi}\\,\\sigma},\n\\]\nwhich tends to infinity as $\\sigma\\to0$. A sample that contains negative values and at least one positive value is treated in the same way, with the log-normal component concentrated at a positive observation and the normal component kept as a positive background. The likelihood is not unbounded on every real sample: on the sample $(-1,-2)$ only the normal component is positive, and the likelihood is at most $2/(\\pi e)$, attained at $p_N=1$, $\\mu_N=-3/2$ and $\\sigma_N=1/2$. The failure does not come from misspecification. If the data are drawn independently from this mixture with $\\boldsymbol\\theta=(1/2,1/4,0,1,0,1,1)$, then $\\Pr[Y\\le0]=1/4$, and a sample of size $n$ contains a positive observation, and has an unbounded likelihood, with probability $1-4^{-n}$.\n\nFor the restricted statement in the correction, the likelihood of a sample of positive observations is a continuous function of $\\boldsymbol\\theta$ where $\\sigma_N$, $\\sigma_{LN}$ and $\\lambda_E$ are positive, and a continuous function attains its maximum on a nonempty compact set. Along $\\boldsymbol\\theta_\\sigma$ the $\\ell^1$ norm of the seven parameters is $15/4+\\sigma$, which stays bounded while the log-likelihood diverges.", "problem": "Definition 5.2.2 defines the ML estimate as the global maximizer of the likelihood of the sample. For the mixture of Example 5.2.11 the example states no restriction on the parameters beyond those of the three distributions: nonnegative $p_N$ and $p_{LN}$ with $p_N+p_{LN}\\le1$ and remaining weight $1-p_N-p_{LN}$, real $\\mu_N$ and $\\mu_{LN}$, and positive $\\sigma_N$, $\\sigma_{LN}$ and $\\lambda_E$. On this parameter space the likelihood has no global maximizer when every observation is positive. Keep the normal and exponential weights fixed and positive, let $\\mu_N$ equal one observation, and let $\\sigma_N$ tend to zero. Then the density at that observation grows like $1/\\sigma_N$, while the exponential component keeps the density at every other observation bounded away from zero, and the likelihood exceeds every bound. Since the likelihood is finite at every admissible parameter value, no parameter value attains the supremum. For this mixture the product on p.~199 cannot ``be maximized'' as printed, the consistency and efficiency claimed for ``this estimator'' have no object until the parameter space is restricted, and a curve with finite parameter values, such as the curve in Figure 5.2.5, cannot be the global ML estimate of the unrestricted mixture from a sample of positive observations. The ML estimate fails to exist here, which is different from being hard to compute.\n\nTheorem 5.2.6 is not contradicted: it requires regularity conditions, a compact parameter space and a unique maximum among them, and the glossary's convention for $\\arg\\max$ assumes that a unique maximum exists (p.~285). The mixture of the example does not satisfy these conditions as printed. No data or code for computing Figure 5.2.5 accompany the source available to me, and the book does not state the objective or the restrictions used to compute it. Example 5.2.11 refers back to Example 5.2.9, which describes an asymptotic ML estimate, and the curve may be a population (minimum KL divergence) approximation, or a restricted or local fit. The evidence does not show that the curve is wrong.", "what_survives": "Theorem 5.2.6 with its regularity conditions and the use of mixtures to approximate a distribution are unaffected. The curve of Figure 5.2.5 is not shown to be wrong, and its closer fit to the histogram is not in question; what changes is its description as the ML estimate of the unrestricted mixture, and the consistency and efficiency attached to that estimator. The later references to mixtures for binary propensity models (pp.~226 and 264) are not affected, since every factor of a binary likelihood is at most one, and the propensity model in the application of Chapter 6 is a logit model (p.~233). No reported number is shown to change." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C05-E002", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Paragraph of text. A red box marks Section 7.2.3 in the sentence: In settings where respecting the bounds of predictions is important (we will have such a case in Section 7.2.3), we will generally prefer to use a binary choice model rather than an approach (like regression) that might yield nonsensical predictions.", "caption": "The paragraph after Theorem 5.1.8, p. 183 (PDF p. 201). The marked forward reference should be to Section 7.2.5.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-N002.png" } ], "formal_release": null, "formal_statements": [], "headline": "The forward reference to Section 7.2.3 should point to Section 7.2.5, where propensity scores are estimated with a binary choice model so that the estimates stay in $[0,1]$.", "id": "P001-C05-N002", "location": "p. 183 (PDF p. 201), the paragraph after Theorem 5.1.8", "pdf_page": 201, "printed_page": "183", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "In settings where respecting the bounds of predictions is important (we will have such a case in \\hl{Section 7.2.3}), we will generally prefer to use a binary choice model rather than an approach (like regression) that might yield nonsensical predictions.", "corrected": "In settings where respecting the bounds of predictions is important (we will have such a case in \\hl{Section 7.2.5}), we will generally prefer to use a binary choice model rather than an approach (like regression) that might yield nonsensical predictions.\n\nThe reference could also be given as ``Sections 6.2.4 and 7.2.5''.", "evidence": "The evidence is a check of the pages of Sections 7.2.3, 7.2.5 and 6.2.4. Both Section 7.2.5 and Section 6.2.4 fit the sentence, and the intended target is not certain. Section 7.2.5 is the one that refers back to Section 5.1.2.", "problem": "Section 7.2.3 (pp.~261--262) discusses conditional outcome models and causal effects in general terms. It contains no case in which predictions must respect bounds and no binary choice model. The case that the sentence promises is in Section 7.2.5 (p.~264), which says that a linear approximation could predict propensity score estimates ``greater than one or less than zero'' and turns to a binary choice model, with a reference back to Section 5.1.2. Section 6.2.4 (pp.~225--226) treats the same point for response propensities.", "what_survives": "Nothing else depends on the reference." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C05-N002", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Text of Example 5.2.3 with two displays of the likelihood of p hat given Y. Red boxes mark g(Y i | p hat) in two places: at the start of the line g(Y i | p hat) = 1 minus p hat, so, and at the start of the line g(Y i | p hat) = p hat, so.", "caption": "Example 5.2.3, p. 188 (PDF p. 206). In the two marked terms a bar separates $\\hat p$ from the argument of $g$, where the book’s convention is a semicolon.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-N003.png" }, { "alt": "Display: L((b, s) | Y, X) equals the product over i from 1 to n of phi(Y i, (X i b, s squared)), which equals the product over i of 1 over s root 2 pi times e to the power minus (Y i minus X i b) squared over 2 s squared. A red box marks Y i and the comma that follows it inside phi.", "caption": "Example 5.2.4, the likelihood display, p. 190 (PDF p. 208). The marked comma after $Y_i$ stands where Definition 5.1.4 has a semicolon.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-N003-b.png" }, { "alt": "Start of Example 5.2.5, Maximum Likelihood Estimation of the Probit Model. Its second display reads: f of Y given X, evaluated at (Y|X), = g(Y, X; beta) = Phi(X beta) to the power Y times (1 minus Phi(X beta)) to the power 1 minus Y. A red box marks g(Y, X; beta), where the X is italic and not bold, unlike the other occurrences of X in the display.", "caption": "Example 5.2.5, p. 192 (PDF p. 210). In the marked term the covariate vector is an italic $X$; it is bold elsewhere in the display.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-N003-c.png" }, { "alt": "Paragraph of text. A red box marks an italic X in the sentence: If we have estimates of the full distribution of Y conditional on X, we can use that information to construct a standard error estimate.", "caption": "Section 5.2.7, p. 203 (PDF p. 221). The marked $X$ is italic, where the bold vector $\\mathbf{X}$ is meant.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-N003-d.png" } ], "formal_release": null, "formal_statements": [], "headline": "Four notational inconsistencies: a bar for the semicolon in $g(Y_i;\\hat p)$, twice; a comma for the semicolon in $\\phi(Y_i;(\\mathbf{X}_i\\mathbf{b},s^2))$; and the covariate vector $\\mathbf{X}$ set without bold face in two places.", "id": "P001-C05-N003", "location": "Example 5.2.3, p. 188 (PDF p. 206); Example 5.2.4, the likelihood display, p. 190 (PDF p. 208); Example 5.2.5, p. 192 (PDF p. 210); Section 5.2.7, p. 203 (PDF p. 221)", "pdf_page": 206, "printed_page": "188", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "(a) Example 5.2.3, p.~188, in two places:\n\\[\n\\mathcal{L}(\\hat p\\,|\\,Y_i)=g(Y_i\\hl{|}\\hat p)=1-\\hat p\n\\qquad\\text{and}\\qquad\n\\mathcal{L}(\\hat p\\,|\\,Y_i)=g(Y_i\\hl{|}\\hat p)=\\hat p\n\\]\n(b) Example 5.2.4, p.~190:\n\\[\n\\mathcal{L}\\bigl((\\mathbf{b},s)\\,|\\,\\mathbf{Y},\\mathbb{X}\\bigr)=\\prod_{i=1}^{n}\\phi\\bigl(Y_i\\hl{,}(\\mathbf{X}_i\\mathbf{b},s^2)\\bigr)\n\\]\n(c) Example 5.2.5, p.~192:\n\\[\nf_{Y|\\mathbf{X}}(Y|\\mathbf{X})=g(Y,\\hl{X};\\boldsymbol{\\beta})=\\Phi(\\mathbf{X}\\boldsymbol{\\beta})^{Y}\\bigl(1-\\Phi(\\mathbf{X}\\boldsymbol{\\beta})\\bigr)^{1-Y}.\n\\]\n(d) Section 5.2.7, p.~203:\n\nIf we have estimates of the full distribution of $Y$ conditional on \\hl{$X$}, we can use that information to construct a standard error estimate.", "corrected": "(a) $g(Y_i\\hl{;}\\hat p)$, in both places.\n\n(b) $\\phi\\bigl(Y_i\\hl{;}(\\mathbf{X}_i\\mathbf{b},s^2)\\bigr)$.\n\n(c) $g(Y,\\hl{\\mathbf{X}};\\boldsymbol{\\beta})$.\n\n(d) If we have estimates of the full distribution of $Y$ conditional on \\hl{$\\mathbf{X}$}, we can use that information to construct a standard error estimate.", "evidence": "The evidence is a check of the pages. Footnote 1 reads ``We separate $\\boldsymbol{\\theta}$ from the arguments of $g$ with a semicolon.'' On p.~192 the italic $X$ stands in the same display as the bold $\\mathbf{X}$.", "problem": "Footnote 1 on p.~179 fixes the convention that $\\boldsymbol{\\theta}$ is separated from the arguments of $g$ by a semicolon, and Definition 5.1.4 writes the normal PDF as $\\phi(y;(\\mu,\\sigma^2))$. Items (a) and (b) depart from these conventions. In (c) and (d) the covariate vector, which is the bold $\\mathbf{X}$ elsewhere, is set in italic. In each case the intended meaning is clear, and no mathematics is affected.", "what_survives": "Nothing depends on these four places. I do not count as an error the sentence on p.~195 that writes $P$ and $Q$ for the distributions with densities $p(\\cdot)$ and $q(\\cdot)$. The letters are not introduced there, but the reading is standard and nothing false follows from it." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C05-N003", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of Figure 5.2.2. The left panel, titled 'Histogram of Y', shows a histogram on the range 0 to 5 with its tallest bar, of height about 0.74, just above Y = 1. The right panel, titled 'Normal ML Fit', shows the same histogram with a red bell-shaped curve that peaks at a height of about 0.52 near Y = 1.6. Red boxes mark the title 'Normal ML Fit' and the caption 'FIGURE 5.2.2. Minimum KL Divergence Approximation of an Unusual Distribution'.", "caption": "Figure 5.2.2, p. 197 (PDF p. 215). The marked title and caption present the red curve as the normal ML fit to the histogram; the book gives no data, bin heights or fitted mean and standard deviation for it.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-N004.png" } ], "formal_release": null, "formal_statements": [], "headline": "The fitted curves in Figures 5.2.2–5.2.5 illustrate the text and cannot be reproduced exactly, since the book reports neither the distribution behind the histogram nor the fitted parameters.", "id": "P001-C05-N004", "location": "Figures 5.2.2–5.2.5, pp. 197–201 (PDF pp. 215–219), with Example 5.2.9, p. 196 (PDF p. 214), and Example 5.2.11, pp. 199–200 (PDF pp. 217–218)", "pdf_page": 215, "printed_page": "197–201", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "Example 5.2.9, p.~196:\n\nSuppose that the true distribution of $Y$ were given by the histogram on the lefthand side of Figure 5.2.2. [\\ldots] Then the asymptotic ML estimate of the distribution would be \\hl{as shown in red} on the righthand side of the figure.\n\nFIGURE 5.2.2. \\emph{Minimum KL Divergence Approximation of an Unusual Distribution}\n\nExample 5.2.11, p.~200:\n\nThe ML estimates from each of these models are \\hl{visualized in Figure 5.2.4}. [\\ldots] \\hl{Figure 5.2.5 visualizes the ML estimate} of this mixture model, which provides a markedly better approximation.\n\nFIGURE 5.2.4. \\emph{Normal, Log-Normal, and Exponential Maximum Likelihood Approximations}\n\nFIGURE 5.2.5. \\emph{Mixture Model Maximum Likelihood Approximation}", "corrected": "No wording has to change. To make the figures reproducible, Example 5.2.9 could add: ``\\hl{The histogram and the fitted curves in Figures 5.2.2--5.2.5 are illustrations; the underlying distribution, the fitted parameter values and the fitting procedure are not reported here.}'' A full account would give the distribution or data behind the histogram, the fitted parameters of each curve, and the fitting procedure, including any restriction placed on the mixture.", "intended_meaning": "These figures are illustrations. A normal distribution, and then a log-normal distribution, an exponential distribution and a mixture of the three, are fitted to the positive distribution drawn as a histogram, and the mixture follows its shape most closely. This reading is clear, and no curve is shown to be wrong. The book gives no observations, bin heights or generating distribution for the histogram, no fitted parameter values and no account of the fitting, and no replication files were available to me. The curves therefore cannot be recomputed exactly, and I did not check that each is the best fit in its family. For Figure 5.2.5 the objective and the restrictions on the parameters also matter (see \\fid{P001-C05-E002})." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C05-N004", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of the first two paragraphs of p. 199 and the opening of Example 5.2.11. The first paragraph defines theta = (p_1, theta_1, p_2, theta_2, ..., p_{M-1}, theta_{M-1}, theta_M) and says that the product of the g(Y_i, X_i; theta) can be maximized to find the ML estimate of theta. In the second paragraph, 'If any subset of the models in (g_1(y, x; theta_1), ..., g_M(y, x; theta_M)) jointly characterize the conditional distribution of Y, then this estimator is consistent and asymptotically efficient.', red boxes mark the words 'this estimator is consistent and asymptotically efficient'. Below, the heading 'Example 5.2.11. Mixture Model Approximation' and its first two lines.", "caption": "Section 5.2.5, p. 199 (PDF p. 217). The marked clause promises consistency and asymptotic efficiency for the ML estimate of the full vector $\\boldsymbol\\theta$ of the first line whenever a subset of the component models characterizes the distribution.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C05-N005.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T11", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T11" }, { "id": "P001-T12", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T12" }, { "id": "P001-T103", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T103" }, { "id": "P001-T121", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T121" } ], "headline": "When only some of the component models of a mixture are needed, the parameters of the unused components are not identified, and the ML estimator of the full parameter vector is consistent and asymptotically efficient only under identification and regularity conditions that the sentence does not state.", "id": "P001-C05-N005", "location": "Section 5.2.5, p. 199 (PDF p. 217), second paragraph", "pdf_page": 217, "printed_page": "199", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P001-T103", "P001-T11", "P001-T12", "P001-T121" ], "text": { "as_printed": "where $\\boldsymbol\\theta=(p_1,\\boldsymbol\\theta_1,p_2,\\boldsymbol\\theta_2,\\ldots,p_{M-1},\\boldsymbol\\theta_{M-1},\\boldsymbol\\theta_M)$. Then, as before, $\\prod_{i=1}^ng(Y_i,\\mathbf X_i;\\boldsymbol\\theta)$ can be maximized to find the ML estimate of $\\boldsymbol\\theta$. (This can sometimes be computationally tricky.)\n\nIf any subset of the models in $(g_1(y,\\mathbf x;\\boldsymbol\\theta_1),\\ldots,g_M(y,\\mathbf x;\\boldsymbol\\theta_M))$ jointly characterize the conditional distribution of $Y$, then \\hl{this estimator is consistent and asymptotically efficient}. When the models are not valid (even jointly), our estimates should be interpreted explicitly as an approximation.", "corrected": "If any subset of the models in $(g_1(y,\\mathbf x;\\boldsymbol\\theta_1),\\ldots,g_M(y,\\mathbf x;\\boldsymbol\\theta_M))$ jointly characterize the conditional distribution of $Y$, \\hl{then the mixture model is true. This alone does not make the ML estimator of $\\boldsymbol\\theta$ consistent or asymptotically efficient, since a component with weight zero leaves its own parameters unidentified. If in addition the true value of $\\boldsymbol\\theta$ is identified and lies in the interior of the parameter space, and the regularity conditions for ML estimation hold, among them a unique maximizer of the expected log-likelihood and a nonsingular information matrix,} then this estimator is consistent and asymptotically efficient. \\hl{A component with weight zero must not retain free parameters in the vector to which this conclusion is applied.}\n\nA sufficient set of conditions is as follows. The observations are i.i.d., the dimension of $\\boldsymbol\\theta$ is fixed, and $f_{Y|\\mathbf X}(y|\\mathbf x)=g(y,\\mathbf x;\\boldsymbol\\theta^0)$ for some $\\boldsymbol\\theta^0\\in\\boldsymbol\\Theta$. \\hl{A measurable global maximizer of the sample likelihood exists and is used as the ML estimator.} The parameter space $\\boldsymbol\\Theta$ is compact, $L(\\mathbf t)=\\mathrm{E}[\\log g(Y,\\mathbf X;\\mathbf t)]$ is continuous on $\\boldsymbol\\Theta$ and uniquely maximized at $\\boldsymbol\\theta^0$, and $\\sup_{\\mathbf t\\in\\boldsymbol\\Theta}|L_n(\\mathbf t)-L(\\mathbf t)|\\xrightarrow{p}0$, where $L_n(\\mathbf t)=n^{-1}\\sum_{i=1}^n\\log g(Y_i,\\mathbf X_i;\\mathbf t)$. The point $\\boldsymbol\\theta^0$ is interior to $\\boldsymbol\\Theta$. Near $\\boldsymbol\\theta^0$ the density $g$ is positive on a common support and twice differentiable in $\\mathbf t$, the identity $\\int g(y,\\mathbf x;\\mathbf t)\\,dy=1$ (with a sum in place of the integral for a PMF) may be differentiated twice under the integral, the score $s=\\nabla_{\\mathbf t}\\log g$ has finite second moments, the information matrix $I=\\mathrm{E}[ss^{T}]$ at $\\boldsymbol\\theta^0$ is positive definite, and the averages of the second derivatives of $\\log g$ converge uniformly near $\\boldsymbol\\theta^0$ to their expectations, which are continuous. Then $\\hat{\\boldsymbol\\theta}_{ML}\\xrightarrow{p}\\boldsymbol\\theta^0$, and $\\sqrt n(\\hat{\\boldsymbol\\theta}_{ML}-\\boldsymbol\\theta^0)$ converges in distribution to $N(0,I^{-1})$.\n\nFor the mixture of Example 5.2.11 these conditions hold when the distribution of $Y$ is this mixture with all three weights positive, and the fit is restricted to a compact set on which the three weights are at least some $\\epsilon>0$, the parameters $\\mu_N$ and $\\mu_{LN}$ are bounded, and $\\sigma_N$, $\\sigma_{LN}$ and $\\lambda_E$ are bounded above and below by positive constants, with the true parameter in the interior of that set. They fail when a weight is zero and the parameters of that component are left free. A different repair changes the target from $\\boldsymbol\\theta$ to the fitted conditional distribution, which requires conditions of its own.", "evidence": "The evidence is an exact counterexample, with a derivation that holds for every sample size. For the general mixture, let $p_j=0$ and let $\\boldsymbol\\theta^A$ and $\\boldsymbol\\theta^B$ differ only in the parameter of component $j$. Then\n\\[\ng(y,\\mathbf x;\\boldsymbol\\theta^A)-g(y,\\mathbf x;\\boldsymbol\\theta^B)=p_j\\big\\{g_j(y,\\mathbf x;\\boldsymbol\\theta_j^A)-g_j(y,\\mathbf x;\\boldsymbol\\theta_j^B)\\big\\}=0,\n\\]\nwhich implies that the two vectors give the same likelihood for every sample.\n\nFor a bounded example, let $Y$ take values in $\\{0,1,2,3\\}$, with no explanatory variables. Writing a distribution as its vector of probabilities at $y=0,1,2,3$, consider the two component models\n\\[\ng_1(\\cdot\\,;a)=(1-a,\\;a,\\;0,\\;0),\\qquad g_2(\\cdot\\,;b)=(0,\\;0,\\;1-b,\\;b),\n\\]\nand the mixture $g=p\\,g_1+(1-p)\\,g_2$ with $\\boldsymbol\\theta=(p,a,b)\\in[0,1]\\times[0,1]\\times[1/4,3/4]$. The mixture has probabilities $q=(p(1-a),\\,pa,\\,(1-p)(1-b),\\,(1-p)b)$. Since the parameter space is compact and the likelihood is continuous and at most one, a maximum exists for every sample. For $00$ or $z<1$'' is ``$z\\le0$ and $z\\ge1$'', which no real number $z$ satisfies. The printed condition therefore holds for every real $z$, including the valid probability $z=1/2$.", "problem": "The sentence has two slips. First, since the support of a random variable is a set (Definition 1.2.20, p.~31), the support of $\\mathrm{E}[R_i|\\mathbf{X}_i]$ is a subset of $[0,1]$ and not an element of it. The chapter writes $\\operatorname{Supp}[Y_i]\\subseteq[a,b]$ on p.~209. Second, every real number is greater than zero or less than one, so the parenthesis as printed describes all predictions, including valid ones. The impossible values of a probability are those greater than one or less than zero. The parallel sentence on p.~264 has the thresholds in this order.", "what_survives": "The reason for using a binary choice model and the estimators that follow are unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C06-E002", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "The statement of Theorem 6.2.6 in a shaded box: three lines of text and a displayed equation. A red box marks the symbol R i in 'Then, if R i is MAR conditional on X i'.", "caption": "Theorem 6.2.6, p. 226 (PDF p. 244): the statement of the theorem, with $R_i$ marked in the hypothesis.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C06-E003.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T83", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T83" } ], "headline": "In Theorem 6.2.6 on p. 226, the missing-at-random assumption is attributed to the response indicator $R_i$ instead of the outcome $Y_i$; the hypothesis should read “if $Y_i$ is MAR conditional on $\\mathbf{X}_i$”.", "id": "P001-C06-E003", "location": "Theorem 6.2.6 (Reweighting under MAR), p. 226 (PDF p. 244), hypothesis", "pdf_page": 244, "printed_page": "226", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T83" ], "text": { "as_printed": "\\textbf{Theorem 6.2.6.} \\emph{Reweighting under MAR}\n\nLet $Y_i$ and $R_i$ be random variables with $\\operatorname{Supp}[R_i]=\\{0,1\\}$. Let $Y_i^*=Y_iR_i+(-99)(1-R_i)$ and let $\\mathbf{X}_i$ be a random vector. Then, if $\\hl{R_i}$ is MAR conditional on $\\mathbf{X}_i$,\n\\[\n\\mathrm{E}[Y_i]=\\mathrm{E}\\left[\\frac{Y_i^*R_i}{p_R(\\mathbf{X}_i)}\\right].\n\\]", "corrected": "\\textbf{Theorem 6.2.6.} \\emph{Reweighting under MAR}\n\nLet $Y_i$ and $R_i$ be random variables with $\\operatorname{Supp}[R_i]=\\{0,1\\}$. Let $Y_i^*=Y_iR_i+(-99)(1-R_i)$ and let $\\mathbf{X}_i$ be a random vector. Then, if $\\hl{Y_i}$ is MAR conditional on $\\mathbf{X}_i$,\n\\[\n\\mathrm{E}[Y_i]=\\mathrm{E}\\left[\\frac{Y_i^*R_i}{p_R(\\mathbf{X}_i)}\\right].\n\\]", "evidence": "A check of the pages (Definition 6.1.8, the theorem and its proof), and a derivation of the identity under the corrected hypothesis, which follows the printed proof. By the definition of $Y_i^*$, I have $Y_i^*R_i=Y_iR_i$. By the two conditions of Definition 6.1.8,\n\\[\n\\mathrm{E}\\left[\\frac{Y_i^*R_i}{p_R(\\mathbf{X}_i)}\\,\\Big|\\,\\mathbf{X}_i\\right]\n=\\frac{\\mathrm{E}[Y_iR_i|\\mathbf{X}_i]}{p_R(\\mathbf{X}_i)}\n=\\frac{\\mathrm{E}[Y_i|\\mathbf{X}_i]\\,\\mathrm{E}[R_i|\\mathbf{X}_i]}{p_R(\\mathbf{X}_i)}\n=\\mathrm{E}[Y_i|\\mathbf{X}_i].\n\\]\nIn the second equality, I used the conditional independence of $Y_i$ and $R_i$, and in the third, I used that $\\mathrm{E}[R_i|\\mathbf{X}_i]=\\Pr[R_i=1|\\mathbf{X}_i]=p_R(\\mathbf{X}_i)>0$. Then the Law of Iterated Expectations implies that $\\mathrm{E}[Y_i]=\\mathrm{E}[Y_i^*R_i/p_R(\\mathbf{X}_i)]$.", "problem": "The chapter defines MAR as a property of the outcome. Definition 6.1.8 (p.~215) says that $Y_i$ is missing at random conditional on $\\mathbf{X}_i$ if $Y_i$ is independent of $R_i$ conditional on $\\mathbf{X}_i$ and there exists $\\epsilon>0$ such that $\\Pr[R_i=1|\\mathbf{X}_i]>\\epsilon$. Theorems 6.1.9, 6.1.10 and 6.1.12 write ``if $Y_i$ is MAR conditional on $\\mathbf{X}_i$''. The phrase ``$R_i$ is MAR'' is not defined, and $R_i$ is observed for every unit. The proof on p.~227 uses the intended hypothesis: ``Then, since MAR holds, $Y_i\\perp\\!\\!\\!\\perp R_i|\\mathbf{X}_i$''.", "what_survives": "The theorem and its proof hold under the intended hypothesis, and nothing else in the chapter changes." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C06-E003", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Seven lines of text. A red box marks the words 'logically equivalent' in the sentence 'In the case where there is a discrete and finite number of values for X i, the IPW estimator is logically equivalent to the plug-in estimator from Section 6.2.1 (Plug-In Estimation)'.", "caption": "Section 6.2.5, p. 228 (PDF p. 246): the paragraph that follows Definition 6.2.7, with “logically equivalent” marked in its last sentence.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C06-E004.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T84", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T84" }, { "id": "P001-T85", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T85" } ], "headline": "On p. 228, a finite number of covariate values is treated as sufficient for the IPW estimator to equal the post-stratification estimator of Section 6.2.1; the two are equal when each estimated response probability is the observed fraction of respondents in its covariate cell.", "id": "P001-C06-E004", "location": "Section 6.2.5 (Weighting Estimators), p. 228 (PDF p. 246), first paragraph on the page, last sentence", "pdf_page": 246, "printed_page": "228", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T84", "P001-T85" ], "text": { "as_printed": "In the case where there is a discrete and finite number of values for $\\mathbf{X}_i$, the IPW estimator is \\hl{logically equivalent} to the plug-in estimator from Section 6.2.1 (\\emph{Plug-In Estimation}).", "corrected": "\\hl{When the covariates have finite discrete support, every observed covariate cell has at least one respondent, and the estimated response probability in each cell is its observed respondent fraction,} the IPW estimator \\hl{equals} the plug-in estimator from Section 6.2.1 (\\emph{Plug-In Estimation}).\n\nIn symbols: let $\\mathbf{X}_i$ take finitely many values, and for each observed value $\\mathbf{x}$ let $n_{\\mathbf{x}}=\\sum_{i=1}^nI(\\mathbf{X}_i=\\mathbf{x})$ and $r_{\\mathbf{x}}=\\sum_{i=1}^nR_i\\,I(\\mathbf{X}_i=\\mathbf{x})$. If $r_{\\mathbf{x}}>0$ and $\\hat p_R(\\mathbf{x})=r_{\\mathbf{x}}/n_{\\mathbf{x}}$ for every observed $\\mathbf{x}$, then\n\\[\n\\frac{1}{n}\\sum_{i=1}^n\\frac{Y_i^*R_i}{\\hat p_R(\\mathbf{X}_i)}\n=\\sum_{\\mathbf{x}}\\frac{\\sum_{i=1}^nY_i^*R_i\\,I(\\mathbf{X}_i=\\mathbf{x})}{\\sum_{i=1}^nR_i\\,I(\\mathbf{X}_i=\\mathbf{x})}\\cdot\\frac{\\sum_{i=1}^nI(\\mathbf{X}_i=\\mathbf{x})}{n},\n\\]\nwhere the sum on the right is over the observed values of $\\mathbf{X}_i$. A maximum likelihood fit of a binary choice model that is not saturated in $\\mathbf{X}_i$, or known response probabilities, need not satisfy the condition on $\\hat p_R$.", "evidence": "A derivation and an exact counterexample. For each observed covariate value $\\mathbf{x}$, let $n_{\\mathbf{x}}$ be the number of sample units with $\\mathbf{X}_i=\\mathbf{x}$, let $r_{\\mathbf{x}}>0$ be the number of respondents among them, let $S_{\\mathbf{x}}=\\sum_{i=1}^nY_i^*R_i\\,I(\\mathbf{X}_i=\\mathbf{x})$, and let $q_{\\mathbf{x}}=\\hat p_R(\\mathbf{x})>0$. Grouping the sum in Definition 6.2.7 by cells and subtracting the post-stratification estimator of Section 6.2.1, I obtain\n\\[\n\\frac{1}{n}\\sum_{\\mathbf{x}}\\frac{S_{\\mathbf{x}}}{q_{\\mathbf{x}}}-\\frac{1}{n}\\sum_{\\mathbf{x}}\\frac{n_{\\mathbf{x}}S_{\\mathbf{x}}}{r_{\\mathbf{x}}}\n=\\frac{1}{n}\\sum_{\\mathbf{x}}S_{\\mathbf{x}}\\left(\\frac{1}{q_{\\mathbf{x}}}-\\frac{n_{\\mathbf{x}}}{r_{\\mathbf{x}}}\\right).\n\\]\nIf $q_{\\mathbf{x}}=r_{\\mathbf{x}}/n_{\\mathbf{x}}$ in every cell, the difference is zero for every vector of outcomes. Conversely, if the difference is zero for every vector of outcomes, with the covariates, the responses and the $q_{\\mathbf{x}}$ held fixed, then $q_{\\mathbf{x}}=r_{\\mathbf{x}}/n_{\\mathbf{x}}$ in every cell, since the outcomes can be chosen so that $S_{\\mathbf{x}}\\neq0$ in one cell only. For a particular vector of outcomes the two estimates can agree with other values of $q_{\\mathbf{x}}$.\n\nFor the counterexample, let $n=12$ and $\\mathbf{X}_i=(1,x_i)$ with $x_i\\in\\{-1,0,1\\}$.\n\\begin{audittable}{rrrlrrr}{A sample of twelve units in which the IPW and post-stratification estimates differ.}\n$x$ & $n_x$ & $r_x$ & Outcomes of respondents & $S_x$ & $S_x/r_x$ & $r_x/n_x$ \\\\\n$-1$ & $4$ & $2$ & $1/4$, $1/2$ & $3/4$ & $3/8$ & $1/2$ \\\\\n$0$ & $4$ & $3$ & $1/4$, $1/2$, $3/4$ & $3/2$ & $1/2$ & $3/4$ \\\\\n$1$ & $4$ & $2$ & $1/4$, $1/2$ & $3/4$ & $3/8$ & $1/2$\n\\end{audittable}\nThe logit model $\\Pr[R_i=1|x_i]=e^{a+bx_i}/(1+e^{a+bx_i})$ has a unique maximum likelihood fit in this sample, $(a,b)=(\\log(7/5),0)$, with estimated response probability $7/12$ in every cell. Both scores vanish there: the intercept score is $7-12\\cdot(7/12)=0$, and the slope score is $-(2-7/3)+(2-7/3)=0$. Since the log-likelihood is strictly concave, this stationary point is its only maximizer.\n\\begin{audittable}{llr}{The estimates of $\\mathrm{E}[Y_i]$ in this sample.}\nEstimator & Response probabilities & Estimate \\\\\nPost-stratification (Section 6.2.1) & none & $5/12$ \\\\\nIPW (Definition 6.2.7) & logit maximum likelihood fit, $7/12$ in every cell & $3/7$ \\\\\nIPW (Definition 6.2.7) & known probability $7/12$ & $3/7$ \\\\\nIPW (Definition 6.2.7) & observed fractions $1/2$, $3/4$, $1/2$ & $5/12$\n\\end{audittable}\nThe post-stratification estimate is $(3/8+1/2+3/8)/3=5/12$, and the IPW estimate with the logit fit is $(1/12)\\cdot3/(7/12)=3/7$. The two estimates differ by $1/84$. The stabilized IPW estimate is also $3/7$, because its denominator equals $1$. The sample has positive probability under i.i.d.\\ sampling from a population in which $x_i$ is uniform on $\\{-1,0,1\\}$, $Y_i$ is uniform on $\\{1/4,1/2,3/4\\}$, $R_i$ is Bernoulli with parameter $7/12$, and the three are independent. In this population MAR holds, the logit model is correctly specified, and $\\mathrm{E}[Y_i]=1/2$.", "problem": "The sentence follows the definition of the IPW estimator (Definition 6.2.7) and reads as a statement about two sample estimators: when $\\mathbf{X}_i$ takes finitely many values, the IPW estimator and the post-stratification estimator of Section 6.2.1 give the same estimate. This holds when $\\hat p_R(\\mathbf{x})$ is the observed fraction of respondents among the sample units with $\\mathbf{X}_i=\\mathbf{x}$, and it does not hold in general. Section 6.2.4 (p.~226) has just adopted ``an ML plug-in estimate from a binary choice model'' for the propensity score, whatever the exact specification, and the paragraph that contains the sentence mentions known response probabilities. With either choice the two estimates generally differ in a given sample. The two estimators share a population target: under MAR, both $\\mathrm{E}[Y_i^*R_i/p_R(\\mathbf{X}_i)]$ and $\\sum_{\\mathbf{x}}\\mathrm{E}[Y_i^*|R_i=1,\\mathbf{X}_i=\\mathbf{x}]\\Pr[\\mathbf{X}_i=\\mathbf{x}]$ equal $\\mathrm{E}[Y_i]$.", "what_survives": "Theorem 6.2.6, the definitions of the two estimators and the double robustness result (Theorem 6.2.9) do not use the equality. Both estimators have the same target under MAR, and the counterexample does not show that a correctly specified IPW estimator is inconsistent. Section 6.3 uses a logit model for the propensity score and reports the IPW and stabilized IPW estimates separately; its numbers do not rely on the remark, and they were not recomputed, because the data and code were not available. The correction restricts the remark to one way of estimating the response probabilities. The parallel sentence in Section 7.2.6 (p.~266) is treated in \\fid{P001-C07-E005}." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C06-E004", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Eight lines of text. A red box marks the expression 'sum from i = 1 to n of 1 over p R of X i', which has no R i in its numerator. Four lines above, the text gives 'E[R i over p R of X i] = 1'.", "caption": "Paragraph after Definition 6.2.8, p. 228 (PDF p. 246): the explanation of the stabilized IPW estimator, with the sum over all units marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C06-E005.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T86", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T86" } ], "headline": "On p. 228, the prose that describes stabilization omits the response indicator that the displayed estimator includes: the sum should read $\\sum_{i=1}^nR_i/\\hat p_R(\\mathbf{X}_i)$, with $R_i=1$ for a respondent and $\\hat p_R(\\mathbf{X}_i)$ the fitted propensity score.", "id": "P001-C06-E005", "location": "Paragraph after Definition 6.2.8, p. 228 (PDF p. 246), sentence beginning “Because the stabilized IPW estimator accounts for”", "pdf_page": 246, "printed_page": "228", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T86" ], "text": { "as_printed": "Because the stabilized IPW estimator accounts for this between-sample variation in $\\sum_{i=1}^{n}\\frac{\\hl{1}}{p_R(\\mathbf{X}_i)}$, it typically confers efficiency gains over the IPW estimator in practice.", "corrected": "Because the stabilized IPW estimator accounts for this between-sample variation in $\\sum_{i=1}^{n}\\frac{\\hl{R_i}}{\\hl{\\hat p_R}(\\mathbf{X}_i)}$, it typically confers efficiency gains over the IPW estimator in practice.", "evidence": "An exact counterexample. Let $n=2$, and let the true and the fitted propensity score equal $1/2$ for both units. The printed sum $\\sum_{i=1}^n1/p_R(\\mathbf{X}_i)$ equals $4$ for each of the four response patterns. The sum over respondents, $\\sum_{i=1}^nR_i/p_R(\\mathbf{X}_i)$, equals $0$, $2$, $2$ and $4$ for the patterns $(0,0)$, $(0,1)$, $(1,0)$ and $(1,1)$. Here the printed sum does not vary between samples, while the sum that normalizes the estimator does.", "problem": "Definition 6.2.8 divides by $n^{-1}\\sum_{i=1}^nR_i/\\hat p_R(\\mathbf{X}_i)$, the total weight of the respondents divided by $n$, and the text a few sentences earlier states that $\\mathrm{E}[R_i/p_R(\\mathbf{X}_i)]=1$. The sum named in the sentence runs over all $n$ units and has no $R_i$. This sum does not enter the estimator, and it differs from the sum over respondents even when the true propensity score is used.", "what_survives": "Definition 6.2.8 and the statement that its denominator converges in probability to $1$ are correct. The counterexample concerns the sum only, and it neither supports nor contradicts the remark that stabilization typically improves efficiency. The sum with the true propensity score, $\\sum_{i=1}^nR_i/p_R(\\mathbf{X}_i)$, is a smaller change to the sentence and is also correct. The parallel sentence in Section 7.2.6 (p.~267) is treated in \\fid{P001-C07-E006}." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C06-E005", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of nine lines of text. Red boxes mark the symbol R i in three conditional expectations: 'E[Y* i | R i, X i] is approximately beta 0 + X i beta', 'E[Y* i | R i, X i] is approximately beta 0 + X i beta + X [2]i beta 2', and 'Estimating E[Y* i | R i, X i] with OLS'. The paragraph reports the OLS (linear) estimate 8.997 and the OLS (polynomial) estimate 8.084.", "caption": "Section 6.3, paragraph below Table 6.3.1, p. 233 (PDF p. 251). The three marked symbols condition on the response indicator $R_i$ without fixing its value at one.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C06-E006.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T87", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T87" } ], "headline": "The two regressions of Section 6.3 approximate the CEF among respondents, $\\mathrm{E}[Y_i^*|R_i=1,\\mathbf{X}_i]$; “$=1$” is missing after $R_i$ in three places.", "id": "P001-C06-E006", "location": "Section 6.3, paragraph below Table 6.3.1, p. 233 (PDF p. 251): two working formulas and one inline reference", "pdf_page": 251, "printed_page": "233", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T87" ], "text": { "as_printed": "``As we might expect, the first regression uses OLS under the working assumption that the CEF of the observed data is linear, $\\mathrm{E}[Y_i^*|\\hl{R_i},\\mathbf{X}_i]\\approx\\beta_0+\\mathbf{X}_i\\boldsymbol{\\beta}$. The resulting estimate, denoted OLS (linear), is $\\hat{\\mathrm{E}}[Y_i]=8.997$, somewhat lower than the MCAR plug-in estimate. We also use a more flexible specification, with $\\mathrm{E}[Y_i^*|\\hl{R_i},\\mathbf{X}_i]\\approx\\beta_0+\\mathbf{X}_i\\boldsymbol{\\beta}+\\mathbf{X}_{[2]i}\\boldsymbol{\\beta}_2$, where $\\mathbf{X}_{[2]i}$ is a vector consisting of the squared values of each element $\\mathbf{X}_i$. Estimating $\\mathrm{E}[Y_i^*|\\hl{R_i},\\mathbf{X}_i]$ with OLS, we compute the OLS (polynomial) estimate $\\hat{\\mathrm{E}}[Y_i]=8.084$, again, lower than the MCAR-based estimate.''", "corrected": "``As we might expect, the first regression uses OLS under the working assumption that the CEF of the observed data is linear, $\\mathrm{E}[Y_i^*|R_i\\hl{=1},\\mathbf{X}_i]\\approx\\beta_0+\\mathbf{X}_i\\boldsymbol{\\beta}$. The resulting estimate, denoted OLS (linear), is $\\hat{\\mathrm{E}}[Y_i]=8.997$, somewhat lower than the MCAR plug-in estimate. We also use a more flexible specification, with $\\mathrm{E}[Y_i^*|R_i\\hl{=1},\\mathbf{X}_i]\\approx\\beta_0+\\mathbf{X}_i\\boldsymbol{\\beta}+\\mathbf{X}_{[2]i}\\boldsymbol{\\beta}_2$, where $\\mathbf{X}_{[2]i}$ is a vector consisting of the squared values of each element $\\mathbf{X}_i$. Estimating $\\mathrm{E}[Y_i^*|R_i\\hl{=1},\\mathbf{X}_i]$ with OLS, we compute the OLS (polynomial) estimate $\\hat{\\mathrm{E}}[Y_i]=8.084$, again, lower than the MCAR-based estimate.''\n\nHere $Y_i^*=Y_iR_i+(-99)(1-R_i)$ is the recorded outcome, $R_i$ is the response indicator and $\\mathbf{X}_i$ is the vector of covariates.", "evidence": "The evidence is a derivation from Definition 6.1.1 and a small example. On the event $R_i=1$, I have $Y_i^*=Y_i$, and on the event $R_i=0$, I have $Y_i^*=-99$. This implies that\n\\[\n\\mathrm{E}[Y_i^*|R_i,\\mathbf{X}_i]=R_i\\,\\mathrm{E}[Y_i|R_i=1,\\mathbf{X}_i]-99(1-R_i).\n\\]\nFor example, let $\\mathbf{X}_i$ be constant, let $Y_i=0$, and let $R_i$ be Bernoulli with parameter $1/2$. Then $\\mathrm{E}[Y_i^*|R_i,\\mathbf{X}_i]$ takes the values $-99$ and $0$, while $\\mathrm{E}[Y_i^*|R_i=1,\\mathbf{X}_i]$ is the constant $0$. A formula in $\\mathbf{X}_i$ alone can represent the second and cannot represent the first.", "problem": "The three expressions condition on the random variable $R_i$, where the event $R_i=1$ is meant. By the stable outcomes model (Definition 6.1.1), I have $Y_i^*=Y_iR_i+(-99)(1-R_i)$. The conditional expectation given $R_i$ then equals $-99$ whenever $R_i=0$. It can depend on $R_i$ even after $\\mathbf{X}_i$ is fixed, and it is not the respondent CEF that the working formulas are intended to approximate. The regressions are fitted to the countries with observed outcomes and approximate the CEF among respondents, which is the object of Definition 6.2.4 (p.~223).", "what_survives": "This notation correction does not change the reported estimates 8.997 and 8.084. The text says that the regressions are used to impute missing responses, as in Section 6.2.2, and nothing indicates that the value $-99$ entered the fits. I did not have access to the data or code for the application to confirm this." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C06-E006", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of three lines of text ending 'with E-hat[Y i] = 8.538' and footnote mark 15, followed by the four lines of footnote 15. Red boxes mark two clauses of the footnote: 'since the bias of the estimator may converge slowly' and 'but the bootstrap standard errors are likely to adequately characterize sampling variability'. Between them the footnote cites Abadie and Imbens 2008.", "caption": "Footnote 15 and the sentence that carries it, p. 233 (PDF p. 251). The first marked clause gives slowly converging bias as a possible reason for invalid bootstrap confidence intervals; the second says that the bootstrap standard errors are likely to be adequate.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C06-E007.png" }, { "alt": "Crop of Table 6.3.1, Estimates of the Cross-National Average of Clean Energy Usage with Missing Data, with columns Estimate and Bootstrap SE and ten rows. A red box marks the row 'Hot Deck Imputation, 8.538, 1.204'. The other rows are Lower Bound Plug-In 7.110, 1.021; Upper Bound Plug-In 34.283, 3.128; MCAR Plug-In 9.762, 1.329; OLS (Linear) 8.997, 1.203; OLS (Polynomial) 8.084, 1.260; IPW 8.361, 1.111; Stabilized IPW 8.737, 1.154; Doubly Robust (Linear) 8.575, 1.151; Doubly Robust (Polynomial) 8.716, 1.187.", "caption": "Table 6.3.1, p. 233 (PDF p. 251). The marked row reports the hot deck estimate with a bootstrap standard error of 1.204.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C06-E007-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T13", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T13" }, { "id": "P001-T14", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T14" } ], "headline": "Footnote 15 should not say that bootstrap standard errors are likely to be adequate for hot deck imputation: for one-to-one, nearest-neighbor hot deck imputation, the bootstrap variance can be inconsistent even when the estimator has no bias.", "id": "P001-C06-E007", "location": "Footnote 15, p. 233 (PDF p. 251), attached to the hot deck estimate 8.538; Table 6.3.1, row “Hot Deck Imputation”", "pdf_page": 251, "printed_page": "233", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P001-T13", "P001-T14" ], "text": { "as_printed": "Footnote 15: ``The bootstrap may not work as a method for computing valid confidence intervals for hot deck imputation \\hl{since the bias of the estimator may converge slowly} (see, e.g., Abadie and Imbens 2008), \\hl{but the bootstrap standard errors are likely to adequately characterize sampling variability}.''\n\nThe sentence that carries the footnote: ``Our estimate using one-to-one, nearest-neighbor hot deck imputation based on the propensity score largely agrees with that of the regression estimates, with $\\hat{\\mathrm{E}}[Y_i]=8.538$.''\n\nTable 6.3.1, row ``Hot Deck Imputation'': Estimate 8.538, Bootstrap SE 1.204.", "corrected": "Footnote 15: ``The \\hl{ordinary bootstrap can fail to estimate the sampling variability of one-to-one, nearest-neighbor hot deck imputation, even when the bias of the estimator is negligible} (see, e.g., Abadie and Imbens 2008). \\hl{The bootstrap standard error for hot deck imputation in Table 6.3.1 is reported descriptively; its validity for this implementation, which matches on a fitted propensity score, has not been established.}''\n\nThe correction supplies no replacement standard error. A valid one requires a variance argument for the imputation and resampling procedure that was used.", "evidence": "I checked the cited paper and derived an exact example. Abadie and Imbens (2008, \\emph{Econometrica} 76(6), 1537--1557) state on p.~1537 that ``the standard bootstrap fails to provide asymptotically valid standard errors'', and they prove this for a matching estimator with a scalar covariate and no bias.\n\nThe example is a missing data problem with a known propensity score. Let $P$ be uniform on $[1/4,3/4]$. Conditional on $P$, let $R$ be Bernoulli with parameter $P$. Let $\\varepsilon$ equal $+1$ or $-1$ with probability $1/2$ each, independently of $(P,R)$, and set $Y=50+\\varepsilon$. The covariate is $X=P$, so that $\\Pr[R=1|X]=X$. MAR and positivity hold. Given $n$ i.i.d.\\ draws, impute to each missing outcome the outcome of the respondent with the nearest value of $P$, and let $T_n$ be the mean of the $n$ completed outcomes (set $T_n=50$ if there is no respondent). Let $W_j$ be the number of times that the outcome of unit $j$ is used, with $W_j=0$ if $R_j=0$. Then\n\\[\nT_n-50=\\frac{1}{n}\\sum_{j=1}^{n}W_j\\varepsilon_j,\n\\]\nand the weights $W_j$ depend only on the values of $P$ and $R$. This implies that $\\mathrm{E}[T_n]=50$ for every $n$. Set $r=(1-p)/p$. A direct computation with the limiting Poisson process of respondents and nonrespondents near a respondent with $P=p$ gives $\\mathrm{E}[W_j^2\\,|\\,P_j=p,R_j=1]\\to 1+3r+(3/2)r^2$, and I obtain\n\\[\nn\\operatorname{Var}(T_n)\\longrightarrow V=\\int_{1/4}^{3/4}2p\\Bigl(1+3r+\\frac{3}{2}r^2\\Bigr)\\,dp=3\\log 3-\\frac{1}{4}\\approx 3.0458 .\n\\]\nThe bootstrap resamples $n$ units with replacement and repeats the imputation, using the same convention if there is no respondent. Let $v_n$ be $n$ times the bootstrap variance of the resulting estimate, given the data. Applying the same computation to the bootstrap weights, centered at the bootstrap mean, I find\n\\[\n\\mathrm{E}[v_n]\\longrightarrow V_b=\\int_{1/4}^{3/4}2p\\bigl(1+Ar+Br^2\\bigr)\\,dp\\approx 2.6471,\n\\]\nwhere $s=1-e^{-1}$, $A=2/s-s/(2-s)$ and $B=3/(2s)-(4-s)/(2(2-s))$. Since $e>8/3$, I have $A<3$ and $B<3/2$, and hence $V_b0$ such that $\\epsilon<\\tilde p_R(\\mathbf{x})<1-\\epsilon$, or \\ldots\n\n\\emph{Theorem 6.1.10, p.~217.} Let $Y_i$ and $R_i$ be random variables with $\\operatorname{Supp}[R_i]=\\{0,1\\}$, let $Y_i^*=Y_iR_i+(-99)(1-R_i)$, and let $\\mathbf{X}_i$ be a random vector. Then, if $Y_i$ is MAR conditional on $\\hl{X_i}$,\n\\[\n\\mathrm{E}[Y_i|\\mathbf{X}_i]=\\mathrm{E}[Y_i^*|R_i=1,\\mathbf{X}_i].\n\\]", "corrected": "\\emph{Theorem 6.2.9, p.~229.} Let $Y_i$ and $R_i$ be random variables with $\\operatorname{Supp}[R_i]=\\{0,1\\}$. Let $Y_i^*=Y_iR_i+(-99)(1-R_i)$ and let $\\mathbf{X}_i$ be a random vector. Let $\\tilde{\\mathrm{E}}[\\hl{Y_i^*}|R_i=1,\\mathbf{X}_i=\\mathbf{x}]$ be an approximation of the CEF evaluated at $R_i=1$ and $\\mathbf{X}_i=\\mathbf{x}$, and let $\\tilde p_R(\\mathbf{X}_i)$ be an approximation of the propensity score. If MAR holds and if either (i) $\\tilde{\\mathrm{E}}[Y_i^*|R_i=1,\\mathbf{X}_i=\\mathbf{x}]=\\mathrm{E}[Y_i^*|R_i=1,\\mathbf{X}_i=\\mathbf{x}]$, $\\forall\\mathbf{x}\\in\\operatorname{Supp}[\\mathbf{X}_i]$, and $\\exists\\epsilon>0$ such that $\\epsilon<\\tilde p_R(\\mathbf{x})<1-\\epsilon$, or (ii) $\\tilde p_R(\\mathbf{x})=p_R(\\mathbf{x})$, $\\forall\\mathbf{x}\\in\\operatorname{Supp}[\\mathbf{X}_i]$, then\n\\[\n\\mathrm{E}[Y_i]=\\mathrm{E}\\left[\\tilde{\\mathrm{E}}[Y_i^*|R_i=1,\\mathbf{X}_i]+\\frac{R_i\\left(Y_i^*-\\tilde{\\mathrm{E}}[Y_i^*|R_i=1,\\mathbf{X}_i]\\right)}{\\tilde p_R(\\mathbf{X}_i)}\\right].\n\\]\n\n\\emph{Theorem 6.1.10, p.~217.} Let $Y_i$ and $R_i$ be random variables with $\\operatorname{Supp}[R_i]=\\{0,1\\}$, let $Y_i^*=Y_iR_i+(-99)(1-R_i)$, and let $\\mathbf{X}_i$ be a random vector. Then, if $Y_i$ is MAR conditional on $\\hl{\\mathbf{X}_i}$,\n\\[\n\\mathrm{E}[Y_i|\\mathbf{X}_i]=\\mathrm{E}[Y_i^*|R_i=1,\\mathbf{X}_i].\n\\]", "evidence": "A check of the pages, and a derivation. By Definition 6.1.1, I have $Y_i^*=Y_iR_i+(-99)(1-R_i)$, which gives $Y_i^*R_i=Y_iR_i$ for $R_i\\in\\{0,1\\}$ and $Y_i^*=Y_i$ when $R_i=1$. This implies that $\\mathrm{E}[Y_i^*|R_i=1,\\mathbf{X}_i]=\\mathrm{E}[Y_i|R_i=1,\\mathbf{X}_i]$, and an approximation of one is an approximation of the other.", "problem": "The statement of Theorem 6.2.9 uses two labels for one function. It introduces the approximation with $Y_i$ and uses $Y_i^*$ in condition (i) and in the displayed identity, without saying that the two labels denote the same object. The exact CEFs are equal because $Y_i^*=Y_i$ when $R_i=1$; one label makes clear that a single approximation is intended. In Theorem 6.1.10 the vector of covariates is bold everywhere in the statement except once in the hypothesis, where it is italic.", "what_survives": "Every printed expression equals the intended one, and both theorems, their proofs and the estimators are unchanged. The expressions written with $Y_i$ in Definition 6.2.8, in the last lines of the proof of Theorem 6.2.6 and in the text on pp.~230--231 equal those written with $Y_i^*$ and need no correction." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C06-N005", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the heading '6.3 Application: Estimating the Cross-National Average of Clean Energy Use' and three lines of text. A red box marks the words 'Quality of Governance' in the sentence 'We now consider an application, again using the Quality of Governance data (Teorell et al. 2016).'", "caption": "Section 6.3, heading and first sentence, p. 231 (PDF p. 249). The marked words name the data source.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C06-N006.png" }, { "alt": "Crop of the last four lines of footnote 14. A red box marks the words 'Quality of Governance' in the sentence 'For further details, please refer to the Quality of Governance Standard Data codebook (Teorell et al. 2016).'", "caption": "Footnote 14, last four lines, p. 232 (PDF p. 250). The marked words repeat the name.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C06-N006-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "The data are the Quality of Government dataset of Teorell et al. (2016); “Quality of Governance” is printed in both places.", "id": "P001-C06-N006", "location": "Section 6.3, first sentence, p. 231 (PDF p. 249); footnote 14, last sentence, p. 232 (PDF p. 250)", "pdf_page": 249, "printed_page": "231", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "P.~231: ``We now consider an application, again using the \\hl{Quality of Governance} data (Teorell et al.\\ 2016).''\n\nP.~232, footnote 14: ``For further details, please refer to the \\hl{Quality of Governance} Standard Data codebook (Teorell et al.\\ 2016).''", "corrected": "P.~231: ``We now consider an application, again using the \\hl{Quality of Government} data (Teorell et al.\\ 2016).''\n\nP.~232, footnote 14: ``For further details, please refer to the \\hl{Quality of Government} Standard Data codebook (Teorell et al.\\ 2016).''", "evidence": "The evidence is a check of pp.~231 and 232 against the bibliography entry on p.~292 (PDF p.~310).", "problem": "The bibliography (p.~292) lists the source as ``The Quality of Government standard dataset, version Jan 16'', from the Quality of Government Institute. Section 6.3 calls it ``Quality of Governance'' twice. The citation identifies the dataset. Only the name is wrong.", "what_survives": "Nothing depends on the name. The same name is printed in Section 4.4 (p.~170), which is the subject of \\fid{P001-C04-N012}." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C06-N006", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of five lines of text. A red box marks the word 'production' in the sentence 'Alternative and nuclear energy use is measured as the percentage of total energy production in a country generated from non-carbon dioxide emitting sources, including, but not limited to, nuclear, wind, hydroelectric, geothermal, and solar power.'", "caption": "Section 6.3, first paragraph, p. 231 (PDF p. 249). The marked word gives the denominator of the outcome variable as total energy production.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C06-N007.png" }, { "alt": "Crop of four lines of footnote 14. Red boxes mark the words 'the number of individuals who have used the Internet in the last 12 months' in the clause 'Internet use is measured as the number of individuals who have used the Internet in the last 12 months;'. The clauses before and after it describe GDP per capita and particulate matter.", "caption": "Footnote 14, p. 232 (PDF p. 250). The marked words describe Internet use as a number of individuals, with no denominator.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C06-N007-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "If the application uses the indicators described in the Quality of Government Basic codebook of January 2016, the energy denominator should be total energy use and Internet users should be measured per 100 people.", "id": "P001-C06-N007", "location": "Section 6.3, first paragraph, p. 231 (PDF p. 249); footnote 14, the clause on Internet use, p. 232 (PDF p. 250)", "pdf_page": 249, "printed_page": "231", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "P.~231: ``Alternative and nuclear energy use is measured as the percentage of total energy \\hl{production} in a country generated from non--carbon dioxide emitting sources, including, but not limited to, nuclear, wind, hydroelectric, geothermal, and solar power.''\n\nP.~232, footnote 14: ``Internet use is measured as \\hl{the number of individuals who have used the Internet in the last 12 months};''", "corrected": "If the outcome is \\texttt{wdi\\_altnucen} and the covariate is \\texttt{wdi\\_internetuse}, the two passages read as follows.\n\nP.~231: ``Alternative and nuclear energy use is measured as the percentage of total energy \\hl{use} in a country generated from non--carbon dioxide emitting sources, including, but not limited to, nuclear, wind, hydroelectric, geothermal, and solar power.''\n\nP.~232, footnote 14: ``Internet use is measured as the number of individuals who have used the Internet in the last 12 months\\hl{, per 100 people};''", "evidence": "The evidence is a check of pp.~231 and 232 against a public source. The codebook of the Quality of Government Basic Dataset, version Jan16, lists \\texttt{wdi\\_altnucen} as ``Alternative and nuclear energy (\\% of total energy use)'' and \\texttt{wdi\\_internetuse} as ``Internet users (per 100 people)'' (\\url{https://www.qogdata.pol.gu.se/dataarchive/qog_bas_jan16.pdf}). The World Bank metadata consulted gives the unit of the energy indicator as ``\\% (share) of energy use''. The book cites the Standard dataset of the same release. I did not read the codebook of the Standard dataset or the data. It is not confirmed that these two indicators are the variables used in the application.", "problem": "On p.~231 the outcome is a percentage of total energy production, and footnote 14 describes Internet use as a number of individuals, with no denominator. The codebook of the Quality of Government data of January 2016 carries two indicators of the World Bank that fit these descriptions, with other units: alternative and nuclear energy is a percentage of total energy \\emph{use}, and Internet use is the number of users \\emph{per 100 people}. The book does not name the variables that it analyzes. If these two indicators were used without transformation, correcting their descriptions would not change the estimates.", "what_survives": "I did not recompute the ten estimates of Table 6.3.1 or their standard errors, and I establish no numerical error in them. Whether the application used these two indicators without transformation remains to be confirmed from the data." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C06-N007", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of a displayed formula and three lines of text. The formula reads: tau i L bar = (1/n) times the sum over i from 1 to n of (Y i D i + a(1 - D i) - Y i (1 - D i) - b D i). A red box marks the symbol Y i L under a bar in the sentence 'By Theorem 3.2.3 (The Expected Value of the Sample Mean is the Population Mean) and the Weak Law of Large Numbers (WLLN), Y i L bar is unbiased and consistent for E[tau i L], and by the Law of Iterated Expectations,'.", "caption": "Proof of Theorem 7.1.8, p. 243 (PDF p. 261). The display defines the sample mean $\\overline{\\tau_i^L}$; the marked symbol in the next sentence is $\\overline{Y_i^L}$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E001.png" }, { "alt": "Crop of one line of text and a displayed formula. The text reads 'Similarly, let tau i U = Y i D i + b(1 - D i) - Y i (1 - D i) - a D i, so that'. The formula reads: tau i U bar = (1/n) times the sum over i from 1 to n of (Y i D i + b(1 - D i) - Y i (x1 - D i) - a D i). A red box marks the letter x.", "caption": "Proof of Theorem 7.1.8, p. 244 (PDF p. 262). The line above the display defines $\\tau_i^U$ with the factor $(1-D_i)$; the marked letter $x$ in the display does not belong.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E001-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T88", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T88" } ], "headline": "In the proof of Theorem 7.1.8, the sample mean that is unbiased and consistent for $\\mathrm{E}[\\tau_i^L]$ is $\\overline{\\tau_i^L}$, not $\\overline{Y_i^L}$, and the summand of $\\overline{\\tau_i^U}$ contains $Y_i(1-D_i)$, with no $x$.", "id": "P001-C07-E001", "location": "Proof of Theorem 7.1.8: p. 243 (PDF p. 261), sentence beginning “By Theorem 3.2.3”; p. 244 (PDF p. 262), display defining $\\overline{\\tau_i^U}$", "pdf_page": 261, "printed_page": "243", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T88" ], "text": { "as_printed": "P.~243: ``By Theorem 3.2.3 (\\emph{The Expected Value of the Sample Mean is the Population Mean}) and the Weak Law of Large Numbers (WLLN), $\\hl{\\overline{Y_i^L}}$ is unbiased and consistent for $\\mathrm{E}[\\tau_i^L]$, and by the Law of Iterated Expectations,''\n\nP.~244:\n\\[\n\\overline{\\tau_i^U}=\\frac{1}{n}\\sum_{i=1}^{n}\\bigl(Y_iD_i+b(1-D_i)-Y_i(\\hl{x}1-D_i)-aD_i\\bigr).\n\\]", "corrected": "P.~243: ``By Theorem 3.2.3 (\\emph{The Expected Value of the Sample Mean is the Population Mean}) and the Weak Law of Large Numbers (WLLN), $\\hl{\\overline{\\tau_i^L}}$ is unbiased and consistent for $\\mathrm{E}[\\tau_i^L]$, and by the Law of Iterated Expectations,''\n\nP.~244:\n\\[\n\\overline{\\tau_i^U}=\\frac{1}{n}\\sum_{i=1}^{n}\\bigl(Y_iD_i+b(1-D_i)-Y_i\\hl{(1-D_i)}-aD_i\\bigr).\n\\]\n\nHere $\\tau_i^L=Y_iD_i+a(1-D_i)-Y_i(1-D_i)-bD_i$ and $\\overline{\\tau_i^L}=n^{-1}\\sum_{i=1}^{n}\\tau_i^L$, as defined in the proof, and $[a,b]$ contains the supports of $Y_i(0)$ and $Y_i(1)$.", "evidence": "The evidence is a check of pp.~243 and 244. Observe that the statement of Theorem 7.1.8 prints both summands without the $x$, and that the definitions of $\\tau_i^L$ and $\\tau_i^U$ in the proof determine both replacements. Since the variables $\\tau_i^L$ are i.i.d.\\ and bounded, Theorem 3.2.3 and the WLLN apply to their sample mean $\\overline{\\tau_i^L}$, which is the claim of the sentence on p.~243.", "problem": "Chapter 7 defines no variable $Y_i^L$. The line before the sentence on p.~243 defines $\\tau_i^L=Y_iD_i+a(1-D_i)-Y_i(1-D_i)-bD_i$ and its sample mean $\\overline{\\tau_i^L}$, and the calculation that follows computes $\\mathrm{E}[\\tau_i^L]$. The symbol $\\overline{Y_i^L}$ belongs to the proof of Theorem 6.1.4 (p.~212), which this proof follows. On p.~244, the line before the display defines $\\tau_i^U=Y_iD_i+b(1-D_i)-Y_i(1-D_i)-aD_i$, and the display should average this summand. The letter $x$ is extraneous.", "what_survives": "Theorem 7.1.8, both bounds in its statement and the rest of the proof are unchanged." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-E001", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "recalculation_from_source", "exhibits": [ { "alt": "Crop of the line 'The distribution of observables is' and a displayed piecewise function f(Y i, X i, D i) with seven rows. The first four rows give probability 1/6 to the cells (Y i, X i, D i) = (0,0,0), (1,0,0), (1,1,0) and (0,0,1). Red boxes mark the fifth and sixth rows, which both read '1/6 : Y i = 1, X i = 1, D i = 1'. The last row reads '0 : otherwise.'", "caption": "Example 7.1.17, p. 253 (PDF p. 271), the distribution of the observables. The two marked rows are identical: each gives probability $1/6$ to the cell $Y_i=1$, $X_i=1$, $D_i=1$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E002.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T89", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T89" } ], "headline": "In Example 7.1.17 the cell $(Y_i,X_i,D_i)=(1,1,1)$ has probability $1/3$; the display lists it twice with probability $1/6$.", "id": "P001-C07-E002", "location": "Example 7.1.17, p. 253 (PDF p. 271), display after “The distribution of observables is”", "pdf_page": 271, "printed_page": "253", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T89" ], "text": { "as_printed": "``The distribution of observables is\n\\[\nf(Y_i,X_i,D_i)=\\begin{cases}\n\\tfrac{1}{6} &:\\ Y_i=0,X_i=0,D_i=0\\\\\n\\tfrac{1}{6} &:\\ Y_i=1,X_i=0,D_i=0\\\\\n\\tfrac{1}{6} &:\\ Y_i=1,X_i=1,D_i=0\\\\\n\\tfrac{1}{6} &:\\ Y_i=0,X_i=0,D_i=1\\\\\n\\hl{\\tfrac{1}{6}} &:\\ \\hl{Y_i=1,X_i=1,D_i=1}\\\\\n\\hl{\\tfrac{1}{6}} &:\\ \\hl{Y_i=1,X_i=1,D_i=1}\\\\\n0 &:\\ \\text{otherwise.''}\n\\end{cases}\n\\]", "corrected": "``The distribution of observables is\n\\[\nf(Y_i,X_i,D_i)=\\begin{cases}\n\\tfrac{1}{6} &:\\ Y_i=0,X_i=0,D_i=0\\\\\n\\tfrac{1}{6} &:\\ Y_i=1,X_i=0,D_i=0\\\\\n\\tfrac{1}{6} &:\\ Y_i=1,X_i=1,D_i=0\\\\\n\\tfrac{1}{6} &:\\ Y_i=0,X_i=0,D_i=1\\\\\n\\hl{\\tfrac{1}{3}} &:\\ Y_i=1,X_i=1,D_i=1\\\\\n0 &:\\ \\text{otherwise.''}\n\\end{cases}\n\\]", "evidence": "The evidence is a recalculation from the joint distribution of the example. Recall that $X_i=X_i(1)D_i+X_i(0)(1-D_i)$ and that $Y_i(1)=Y_i(0)$ for every type. The six types then have the observables $(0,0,0)$, $(1,0,0)$, $(1,1,0)$, $(0,0,1)$, $(1,1,1)$ and $(1,1,1)$. Adding the probabilities, I find that the cells $(0,0,0)$, $(1,0,0)$, $(1,1,0)$ and $(0,0,1)$ have probability $1/6$ each and that the cell $(1,1,1)$ has probability $1/3$, with total one. With these probabilities, $\\Pr[X_i=0]=\\Pr[X_i=1]=1/2$, and I have\n\\[\n\\begin{aligned}\n\\mathrm{E}[Y_i|D_i=1,X_i=0]-\\mathrm{E}[Y_i|D_i=0,X_i=0]&=0-\\frac{1}{2}=-\\frac{1}{2},\\\\\n\\mathrm{E}[Y_i|D_i=1,X_i=1]-\\mathrm{E}[Y_i|D_i=0,X_i=1]&=1-1=0 .\n\\end{aligned}\n\\]\nThe adjusted contrast is then $(1/2)(-1/2)+(1/2)\\cdot 0=-1/4$, as printed.", "problem": "The joint distribution printed above this display has six equally likely types. Two of them give the same observables $(Y_i,X_i,D_i)=(1,1,1)$: the types with $(Y_i(0),Y_i(1),X_i(1),X_i(0),D_i)$ equal to $(1,1,1,0,1)$ and to $(1,1,1,1,1)$. Their probabilities must be added. The display instead lists this cell twice. Read as a function, it assigns the cell probability $1/6$, and the five distinct cells have total probability $5/6$.", "what_survives": "The joint distribution of potential outcomes and treatment, the random assignment of $D_i$, the absence of any effect on $Y_i$ and the adjusted contrast $-1/4$ are unchanged. The example still shows that conditioning on a post-treatment variable can produce a nonzero contrast when the ATE is zero." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-E002", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of two lines of text and the shaded box of Definition 7.1.19, Average Marginal Causal Effect. The text ends 'If we assume that Y i (d) is differentiable with respect to d, then the following definition allows us to derive a concise measure.' The formula reads: AMCE = E of the fraction with numerator 'partial E[Y i (D i) | D i, X i]' and denominator 'partial D i'. Red boxes mark the D i inside Y i (D i), the D i after the conditioning bar, and the D i in the denominator.", "caption": "Definition 7.1.19, p. 255 (PDF p. 273). The three marked symbols are the same $D_i$: the argument of the potential outcome, the conditioning treatment and the variable of differentiation.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-A001.png" }, { "alt": "Crop of four lines of text, a displayed formula and two more lines. Red boxes mark the words 'under the conditional independence assumption, the CEF is causal, hence'. The formula reads: AMCE = APD sub D i = E of the fraction with numerator 'partial E[Y i | D i, X i]' and denominator 'partial D i'. The text after it reads 'In other words, under strong ignorability, the average partial derivative with respect to D i is the average marginal causal effect of D i.'", "caption": "P. 255 (PDF p. 273), the paragraph after Definition 7.1.19. The marked words state the condition under which the display equates the AMCE with the average partial derivative.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-A001-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "In Definition 7.1.19 the derivative is taken in the treatment level $d$ of the potential outcome $Y_i(d)$, with the conditioning values held fixed, and is then evaluated at $d=D_i$; the printed formula writes $D_i$ for both.", "id": "P001-C07-A001", "location": "Definition 7.1.19, p. 255 (PDF p. 273), and the display $AMCE=APD_{D_i}$ below it", "pdf_page": 273, "printed_page": "255", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "``Definition 7.1.19. \\emph{Average Marginal Causal Effect}. The \\emph{average marginal causal effect} of $D_i$ is\n\\[\nAMCE=\\mathrm{E}\\left[\\frac{\\partial\\mathrm{E}[Y_i(\\hl{D_i})|D_i,\\mathbf{X}_i]}{\\partial \\hl{D_i}}\\right].\\text{''}\n\\]\n\n``Furthermore, under the conditional independence assumption, the CEF is causal, hence\n\\[\nAMCE=APD_{D_i}=\\mathrm{E}\\left[\\frac{\\partial\\mathrm{E}[Y_i|D_i,\\mathbf{X}_i]}{\\partial D_i}\\right].\n\\]\nIn other words, under strong ignorability, the average partial derivative with respect to $D_i$ is the average marginal causal effect of $D_i$.''", "corrected": "``Definition 7.1.19. \\emph{Average Marginal Causal Effect}. The \\emph{average marginal causal effect} of $D_i$ is\n\\[\nAMCE=\\mathrm{E}\\left[\\left.\\frac{\\partial\\mathrm{E}[Y_i(\\hl{d})|D_i,\\mathbf{X}_i]}{\\partial \\hl{d}}\\right|_{\\hl{d=D_i}}\\right],\n\\]\n\\hl{where the conditioning values $D_i$ and $\\mathbf{X}_i$ are held fixed during differentiation, whenever this derivative and its expectation exist}.''\n\n``Furthermore, under \\hl{generalized strong ignorability and the requisite positivity, with differentiable conditional mean functions}, the CEF is causal, hence\n\\[\nAMCE=APD_{D_i}=\\mathrm{E}\\left[\\frac{\\partial\\mathrm{E}[Y_i|D_i,\\mathbf{X}_i]}{\\partial D_i}\\right].\\text{''}\n\\]", "intended_meaning": "The sentence before the definition assumes that $Y_i(d)$ is differentiable with respect to $d$, and the definition is meant to differentiate in that argument. Set $M(d,t,\\mathbf{x})=\\mathrm{E}[Y_i(d)|D_i=t,\\mathbf{X}_i=\\mathbf{x}]$. The intended quantity is the expectation of $\\partial M/\\partial d$ at $d=t=D_i$ and $\\mathbf{x}=\\mathbf{X}_i$: the average change in the expected potential outcome, for units at their realized treatment levels, when the treatment level alone is varied. With this meaning the statement below the definition is correct. Suppose that generalized strong ignorability holds, $(Y_i(d))_{d\\in\\operatorname{Supp}[D_i]}\\perp\\!\\!\\!\\perp D_i\\,|\\,\\mathbf{X}_i$, with the requisite positivity. Then $M(d,t,\\mathbf{x})=\\mathrm{E}[Y_i(d)|\\mathbf{X}_i=\\mathbf{x}]$ does not depend on $t$; write $g(d,\\mathbf{x})$ for it. Since $Y_i=Y_i(D_i)$ by Definition 7.1.18, I have $\\mathrm{E}[Y_i|D_i=t,\\mathbf{X}_i=\\mathbf{x}]=M(t,t,\\mathbf{x})=g(t,\\mathbf{x})$. The derivative of $M$ in $d$ at $d=t$ and the derivative of the CEF in $t$ are then both $\\partial g/\\partial t$. Averaging over $(D_i,\\mathbf{X}_i)$, I obtain $AMCE=APD_{D_i}$, provided that the conditional means are differentiable and the expectations are finite.\n\nRead literally, with the convention of Definition 4.3.6, the printed formula differentiates $M(t,t,\\mathbf{x})$ along the diagonal. Since $Y_i(D_i)=Y_i$, this is the average partial derivative of the CEF, with or without ignorability, and it need not be causal. For example, let there be no covariates, let $U\\in\\{0,1\\}$ and $D_i\\in[0,1]$ have joint density $3/4-t/2$ at $(0,t)$ and $1/4+t/2$ at $(1,t)$, and let $Y_i(d)=U$ for every $d$. Every causal effect is zero, while $\\mathrm{E}[Y_i|D_i=t]=1/4+t/2$, and the literal formula has the value $1/2$. Note that this example violates ignorability and does not contradict the identification statement. The AMCE appears only on p.~255 and in the index, and no later result uses the formula." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-A001", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "recalculation_from_source", "exhibits": [ { "alt": "Crop of a table with columns Unit, Y i(0), Y i(1), D i, X [1]i and X [2]i and six rows. Missing outcomes are filled with expressions of the form beta-hat 0 + beta-hat 1 times d + beta-hat 2 times the first covariate + beta-hat 3 times the second covariate. In the row for unit 3, red boxes mark the 9 in 'beta-hat 2 times 9' and the 9 in the column X [1]i.", "caption": "Example 7.2.2, p. 259 (PDF p. 277), second table. In the row for unit 3, the marked entries give the first covariate as $9$, in the imputed outcome and in the covariate column.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E003.png" }, { "alt": "Crop of the heading 'Example 7.2.2. Causal Inference with OLS Regression', the sentence 'Suppose we had the following data:' and a table of six units with columns Unit, Y i(0), Y i(1), D i, X [1]i and X [2]i. The row for unit 3 reads: question mark, 3, 1, 10, 3. A red box marks the 10.", "caption": "Example 7.2.2, p. 259 (PDF p. 277), first table. The marked entry gives the first covariate of unit 3 as $10$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E003-b.png" }, { "alt": "Crop of a table in which both potential outcomes of all six units are written as fitted expressions. In the row for unit 3, red boxes mark the 9 in 'beta-hat 2 times 9' in the column Y i(0), the 9 in 'beta-hat 2 times 9' in the column Y i(1), and the 9 in the column X [1]i.", "caption": "Example 7.2.2, p. 259 (PDF p. 277), third table. In the row for unit 3, the three marked entries again give the first covariate as $9$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E003-c.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T92", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T92" }, { "id": "P001-T93", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T93" } ], "headline": "In the two imputation tables of Example 7.2.2, the first covariate of unit 3 is $10$, as in the data table, not $9$.", "id": "P001-C07-E003", "location": "Example 7.2.2, p. 259 (PDF p. 277), row for unit 3 in the three tables", "pdf_page": 277, "printed_page": "259", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T92", "P001-T93" ], "text": { "as_printed": "First table (the data), unit 3: $Y_i(0)={?}$, $Y_i(1)=3$, $D_i=1$, $X_{[1]i}=10$, $X_{[2]i}=3$.\n\nSecond table (missing outcomes imputed), unit 3: $Y_i(0)=\\hat\\beta_0+\\hat\\beta_1\\cdot 0+\\hat\\beta_2\\cdot\\hl{9}+\\hat\\beta_3\\cdot 3$, $Y_i(1)=3$, $D_i=1$, $X_{[1]i}=\\hl{9}$, $X_{[2]i}=3$.\n\nThird table (all outcomes imputed), unit 3: $Y_i(0)=\\hat\\beta_0+\\hat\\beta_1\\cdot 0+\\hat\\beta_2\\cdot\\hl{9}+\\hat\\beta_3\\cdot 3$, $Y_i(1)=\\hat\\beta_0+\\hat\\beta_1\\cdot 1+\\hat\\beta_2\\cdot\\hl{9}+\\hat\\beta_3\\cdot 3$, $D_i=1$, $X_{[1]i}=\\hl{9}$, $X_{[2]i}=3$.", "corrected": "Second table, unit 3: $Y_i(0)=\\hat\\beta_0+\\hat\\beta_1\\cdot 0+\\hat\\beta_2\\cdot\\hl{10}+\\hat\\beta_3\\cdot 3$ and $X_{[1]i}=\\hl{10}$.\n\nThird table, unit 3: $Y_i(0)=\\hat\\beta_0+\\hat\\beta_1\\cdot 0+\\hat\\beta_2\\cdot\\hl{10}+\\hat\\beta_3\\cdot 3$, $Y_i(1)=\\hat\\beta_0+\\hat\\beta_1\\cdot 1+\\hat\\beta_2\\cdot\\hl{10}+\\hat\\beta_3\\cdot 3$ and $X_{[1]i}=\\hl{10}$.", "evidence": "The evidence is a recalculation from the first table. The OLS regression of $Y_i$ on $(1,D_i,X_{[1]i},X_{[2]i})$ has coefficients\n\\[\n(\\hat\\beta_0,\\hat\\beta_1,\\hat\\beta_2,\\hat\\beta_3)=\\frac{1}{2437}\\,(13563,\\,6178,\\,-759,\\,-1778).\n\\]\nWith $9$ in the prediction for unit 3, I find the estimate $12103/4874$ of $\\mathrm{E}[\\tau_i]$ when only the missing outcomes are imputed, and $6178/2437$ when all outcomes are imputed. The second is larger by $253/4874$, although the example says that the two procedures yield the same estimate. With $10$, both estimates equal $6178/2437=\\hat\\beta_1$.", "problem": "The three tables describe the same six units, but the first covariate of unit 3 changes from $10$ to $9$ after the first table. The value $9$ appears in the covariate column of both imputation tables and in three terms $\\hat\\beta_2\\cdot 9$. Example 7.2.4 (pp.~262--263) uses the same six units and has $10$ in all three of its tables.", "what_survives": "The regression estimator of Definition 7.2.3 is unchanged, and the two imputation estimates are equal for a single data set. The example reports no numerical estimate, and no printed number changes." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-E003", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of four lines of text and a displayed formula. The text ends 'the maximum likelihood plug-in estimator for E[Y i(d)] is computed by substituting theta-hat ML for theta and replacing the expected value with respect to X i with a sample mean:'. The display reads E-hat[Y i(d)] = (1/n) times the sum over i from 1 to n of the integral of y g(y, (d, x), theta-hat ML) dy. A red box marks the bold lower-case x.", "caption": "Section 7.2.3, p. 261 (PDF p. 279), last display. The marked argument is a lower-case $\\mathbf{x}$ with no index; the sentence above the display says that the expected value with respect to $\\mathbf{X}_i$ is replaced with a sample mean.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E004.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T94", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T94" } ], "headline": "In the maximum likelihood plug-in estimator of $\\mathrm{E}[Y_i(d)]$, the $i$th summand is evaluated at the observed covariates $\\mathbf{X}_i$, not at a free $\\mathbf{x}$.", "id": "P001-C07-E004", "location": "Section 7.2.3, p. 261 (PDF p. 279), last display", "pdf_page": 279, "printed_page": "261", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T94" ], "text": { "as_printed": "\\[\n\\hat{\\mathrm{E}}\\bigl[Y_i(d)\\bigr]=\\frac{1}{n}\\sum_{i=1}^{n}\\left(\\int_{-\\infty}^{\\infty}y\\,g\\bigl(y,(d,\\hl{\\mathbf{x}}),\\hat{\\boldsymbol{\\theta}}_{ML}\\bigr)\\,dy\\right).\n\\]", "corrected": "\\[\n\\hat{\\mathrm{E}}\\bigl[Y_i(d)\\bigr]=\\frac{1}{n}\\sum_{i=1}^{n}\\left(\\int_{-\\infty}^{\\infty}y\\,g\\bigl(y,(d,\\hl{\\mathbf{X}_i}),\\hat{\\boldsymbol{\\theta}}_{ML}\\bigr)\\,dy\\right).\n\\]\nHere, as in Section 7.2.3, strong ignorability holds, $g(y,(d,\\mathbf{x}),\\boldsymbol{\\theta})$ is a correctly specified parametric model for the conditional density of $Y_i$ given $(D_i,\\mathbf{X}_i)=(d,\\mathbf{x})$, and $\\hat{\\boldsymbol{\\theta}}_{ML}$ is the maximum likelihood estimate from $n$ i.i.d.\\ observations of $(Y_i,D_i,\\mathbf{X}_i)$.", "evidence": "The evidence is a derivation. Set\n\\[\nm_{\\boldsymbol{\\theta}}(d,\\mathbf{x})=\\int_{-\\infty}^{\\infty}y\\,g\\bigl(y,(d,\\mathbf{x}),\\boldsymbol{\\theta}\\bigr)\\,dy .\n\\]\nThe first display of Section 7.2.3 states that $\\mathrm{E}[Y_i(d)]=\\int_{-\\infty}^{\\infty}m_{\\boldsymbol{\\theta}}(d,\\mathbf{x})f_{\\mathbf{X}_i}(\\mathbf{x})\\,d\\mathbf{x}$. Replacing the distribution of $\\mathbf{X}_i$ by the empirical distribution of $\\mathbf{X}_1,\\ldots,\\mathbf{X}_n$, and $\\boldsymbol{\\theta}$ by $\\hat{\\boldsymbol{\\theta}}_{ML}$, I obtain\n\\[\n\\frac{1}{n}\\sum_{i=1}^{n}m_{\\hat{\\boldsymbol{\\theta}}_{ML}}(d,\\mathbf{X}_i).\n\\]\nThe printed expression equals $m_{\\hat{\\boldsymbol{\\theta}}_{ML}}(d,\\mathbf{x})$. For example, suppose that the fitted conditional distribution of $Y_i$ is normal with mean $x$ and variance $1$, and that the observed covariates are $0$ and $1$. Then the intended estimate is $1/2$, and the printed expression is $x$.", "problem": "As printed, the summand does not depend on $i$, and $\\mathbf{x}$ is a free variable. The right side is then the fitted conditional mean at $\\mathbf{x}$, not an average over the observed covariates. The sentence before the display says that the expected value with respect to $\\mathbf{X}_i$ is replaced with a sample mean. The display for the estimator of $\\mathrm{E}[\\tau_i]$ at the top of p.~262 evaluates $g$ at $(1,\\mathbf{X}_i)$ and $(0,\\mathbf{X}_i)$.", "what_survives": "The identification display above it and the construction of the estimator are unchanged. The estimator of $\\mathrm{E}[\\tau_i]$ on p.~262 is printed with $\\mathbf{X}_i$." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-E004", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of a paragraph of nine lines. A red box marks the clause 'the bias (and therefore MSE) of the estimator shrinks too slowly as n grows large, and, thus,' in the sentence 'For example, even if we have a consistent estimator of p D(X i), matching with estimated propensity scores is not generally root-n consistent ... the usual methods for constructing confidence intervals, including the bootstrap, do not generally work.' The sentence ends with the footnote mark 13.", "caption": "End of Section 7.2.4, p. 264 (PDF p. 282). The marked clause gives slowly shrinking bias as the reason that the bootstrap does not work for matching.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E011.png" }, { "alt": "Crop of footnote 20 and the first line of footnote 21. Footnote 20 reads: 'As with hot deck imputation, the bootstrap may not work as a method for computing valid confidence intervals for matching estimators since the bias of the estimator may converge very slowly (see, e.g., Abadie and Imbens 2008), but the bootstrap standard errors are likely to adequately characterize sampling variability.' A red box marks the clause that begins 'but the bootstrap standard errors'.", "caption": "Footnote 20, p. 279 (PDF p. 297). The marked clause is the assurance about the bootstrap standard errors of the matching estimator.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E011-b.png" }, { "alt": "Crop of Table 7.5.1, 'Estimates for Effect of Gender on Percentage No Votes', with columns Estimator, Target, Estimate and SE and two rows for each of nine estimators. Red boxes mark the standard errors 1.951 and 1.988 in the Matching rows, whose estimates are 34.732 for E[Y i(0)] and 2.009 for E[tau i]. The OLS (Linear) rows have standard errors 0.487 and 0.508.", "caption": "Table 7.5.1, p. 280 (PDF p. 298). The two marked standard errors of the Matching row are the ones that rest on the assurance in footnote 20; neither is shown to be wrong.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E011-c.png" }, { "alt": "Chart with sample size n (500, 2000, 8000) on the horizontal axis and n times variance, from 2.5 to 3.2, on the vertical axis. Blue circles for estimates of the sampling variance lie at 3.059, 3.054 and 3.048, near a horizontal line labeled 'exact limit 3.046'. Orange squares for estimates of the expected bootstrap variance lie at 2.660, 2.652 and 2.648, near a horizontal line labeled 'exact limit 2.647, which is 0.869 of the true limit'. Bars show two Monte Carlo standard errors at each sample size; most are partly hidden by the markers. The title reads 'The bootstrap variance stays about 13 percent below the true variance'.", "caption": "Counterexample to the assurance in footnote 20: simulation estimates of $n$ times the sampling variance and expected bootstrap variance at $n=500$, $2000$ and $8000$, with the exact limits $3.046$ and $2.647$. The vertical axis starts at $2.5$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E011-bootstrap-variance.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T15", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T15" }, { "id": "P001-T16", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T16" } ], "headline": "For nearest-neighbor matching the bootstrap can fail through its variance even when the estimator has no bias, so the assurance that the bootstrap standard errors “are likely to adequately characterize sampling variability” is unsupported.", "id": "P001-C07-E011", "location": "End of Section 7.2.4, p. 264 (PDF p. 282), with footnote 13; footnote 20, p. 279 (PDF p. 297); Matching row of Table 7.5.1, p. 280 (PDF p. 298)", "pdf_page": 282, "printed_page": "264", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P001-T15", "P001-T16" ], "text": { "as_printed": "P.~264: ``For example, even if we have a consistent estimator of $p_D(\\mathbf{X}_i)$, matching with estimated propensity scores is not generally root-$n$ consistent\\hl{---the bias (and therefore MSE) of the estimator shrinks ``too slowly'' as $n$ grows large---and, thus,} the usual methods for constructing confidence intervals, \\emph{including the bootstrap}, do not generally work.$^{13}$'' Footnote 13: ``See, e.g., Abadie and Imbens (2008).''\n\nP.~279, footnote 20: ``As with hot deck imputation, the bootstrap may not work as a method for computing valid confidence intervals for matching estimators since the bias of the estimator may converge very slowly (see, e.g., Abadie and Imbens 2008), \\hl{but the bootstrap standard errors are likely to adequately characterize sampling variability}.''\n\nP.~280, Table 7.5.1, Matching: $\\mathrm{E}[Y_i(0)]$, estimate $34.732$, SE $1.951$; $\\mathrm{E}[\\tau_i]$, estimate $2.009$, SE $1.988$.", "corrected": "P.~264: ``For example, a consistent estimator of $p_D(\\mathbf{X}_i)$ does not by itself guarantee that matching with estimated propensity scores is root-$n$ consistent. The bootstrap can also fail for nearest-neighbor matching when the estimator is root-$n$ consistent and has no bias, \\hl{because the bootstrap variance itself may be inconsistent}.$^{13}$ Under their \\hl{assumptions for parametric propensity-score estimation}, Abadie and Imbens (2016) give a root-$n$ normal approximation whose variance accounts for the estimation of the score.''\n\nP.~279, footnote 20: ``As with hot deck imputation, the bootstrap may not give valid confidence intervals \\hl{or standard errors} for matching estimators (see, e.g., Abadie and Imbens 2008). The bootstrap standard errors reported for matching \\hl{have not been justified for the matching and propensity-fitting procedure used here}, and they do not support comparisons of precision with the other estimators.''\n\nP.~280, Table 7.5.1, Matching: the estimates $34.732$ and $2.009$ are unchanged. The standard errors $1.951$ and $1.988$ are \\hl{bootstrap values without a demonstrated justification}; no replacement values are established.", "evidence": "The evidence is the paper cited in footnotes 13 and 20, an exact counterexample with a derivation, and a simulation of the counterexample. In its introduction, the cited paper describes the matching estimator that it studies as ``root-N consistent and asymptotically normal with no asymptotic bias'' (Abadie and Imbens 2008, p.~1537), and it states on the same page that ``the standard bootstrap fails to provide asymptotically valid standard errors''.\n\nIn the counterexample, $X_i$ is uniform on $[0,1/2]$, $D_i$ given $X_i=x$ is Bernoulli with parameter $1/4+x$, and $\\varepsilon_i$ equals $1$ or $-1$ with probability $1/2$ each, independently of $(X_i,D_i)$. Set $Y_i(0)=\\varepsilon_i$ and $Y_i(1)=0$, so that strong ignorability holds and $\\mathrm{E}[\\tau_i]=0$. The propensity score is estimated by $\\hat p_D(x)=\\hat a+x$, where $\\hat a$ is $\\overline{D}-\\overline{X}$ restricted to $[1/8,3/8]$. This estimator is consistent. Since $|\\hat p_D(x)-\\hat p_D(z)|=|x-z|$, the matches do not depend on $\\hat a$. The matching estimator of Section 7.2.4 (one-to-one, nearest-neighbor, with replacement) equals\n\\[\n\\hat\\theta_n=-\\frac{1}{n}\\sum_{i:D_i=0}(1+K_i)\\,\\varepsilon_i ,\n\\]\nwhere $K_i$ is the number of treated units matched to the control unit $i$. In samples or bootstrap resamples with an empty treatment arm, I impute the missing-arm outcomes as zero. Since the $\\varepsilon_i$ have mean zero and are independent of $(X_i,D_i)$, I have $\\mathrm{E}[\\hat\\theta_n]=0$ for all $n$. A computation with the spacings between neighboring covariate values gives\n\\[\n\\lim_{n\\to\\infty}n\\operatorname{Var}(\\hat\\theta_n)=3\\log 3-\\frac{1}{4}\\approx 3.046 ,\n\\]\nso that $\\hat\\theta_n$ is root-$n$ consistent. Let $V_n^*$ denote the bootstrap variance of $\\hat\\theta_n$ for the bootstrap of Section 3.4.3, which draws $n$ observations with replacement, refits the propensity score, and recomputes the matches. Then\n\\[\n\\lim_{n\\to\\infty}\\mathrm{E}[nV_n^*]=\\frac{1}{2}+\\frac{c_1}{2}+c_2\\Bigl(2\\log 3-\\frac{3}{2}\\Bigr)\\approx 2.647,\n\\]\nwhere\n\\[\nc_1=\\frac{2}{q}-\\frac{q}{2-q},\\qquad c_2=\\frac{3}{2q}-\\frac{4-q}{2(2-q)},\\qquad q=1-e^{-1}.\n\\]\nThe second limit is $0.869$ times the first. Since $nV_n^*\\ge 0$, Fatou's lemma implies that $nV_n^*$ does not converge in probability to $3\\log 3-1/4$. The bootstrap variance is inconsistent, although the estimator is unbiased and root-$n$ consistent. A simulation integrating out the $\\varepsilon_i$, with 400 independent samples at each sample size and 120 bootstrap resamples of each sample, agrees with the two limits.\n\\begin{audittable}{rrr}{Simulation of the counterexample: $n$ times the variance of the matching estimator and $n$ times the expected bootstrap variance, with Monte Carlo standard errors in parentheses, and the exact limits.}\n$n$ & true variance & bootstrap variance \\\\\n500 & 3.059 (0.013) & 2.660 (0.010) \\\\\n2000 & 3.054 (0.007) & 2.652 (0.005) \\\\\n8000 & 3.048 (0.003) & 2.648 (0.002) \\\\\nlimit & 3.046 & 2.647 \\\\\n\\end{audittable}", "problem": "Both passages attribute the failure of the bootstrap for nearest-neighbor matching to bias that shrinks too slowly. Footnote 20 then separates the confidence intervals, which may fail, from the bootstrap standard errors, which it expects to be adequate. That separation has no general justification. The paper cited in both footnotes concerns a matching estimator with a root-$n$ rate and no asymptotic bias, and for that estimator the standard bootstrap does not provide valid standard errors: the failure is in the variance, not in bias. Bias and rate therefore do not decide whether bootstrap standard errors for matching are valid. That needs an argument about the variance for the matching and propensity-fitting procedure in use, and the book gives none. The standard errors $1.951$ and $1.988$ in the Matching row of Table 7.5.1 are left without a stated justification. The same holds for the comparison on p.~279 that rests on them, that the standard error of matching ``is nearly 4 times as large as that associated with the OLS (linear) specification''.\n\nTwo points are not established. Neither standard error is shown to be wrong: the data and code of the application were not available, and the words ``may'' and ``likely'' make the footnote a judgment, not a theorem. The warning on p.~264 that a consistent estimator of the propensity score does not by itself ensure a root-$n$ rate also stands. Under their assumptions, Abadie and Imbens (2016) obtain a root-$n$ normal approximation for matching on a propensity score estimated by a regular parametric model, with a variance that accounts for the first estimation step, but their assumptions do not cover every consistent estimator.", "what_survives": "The matching estimates $34.732$ and $2.009$ in Table 7.5.1, the arithmetic of the comparisons on p.~279, the identification results, and the other rows of the table are not affected. The reading of the two standard errors in the Matching row changes, and with it the comparison of precision and its part in the remark that matching ``appears to perform abysmally''. No replacement values are established. The parallel sentence about hot deck imputation in Chapter 6 is the subject of \\fid{P001-C06-E007}." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-E011", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of five lines of text. The last sentence reads: 'In the case where we have a discrete and finite number of values for X i, the IPW estimator is logically equivalent to the plug-in estimator from Section 7.2.1 (Plug-In Estimation).' Red boxes mark the words 'logically' and 'equivalent', which fall on two lines.", "caption": "Section 7.2.6, p. 266 (PDF p. 284), end of the paragraph after Definition 7.2.6. The marked words state the equivalence with no condition on the fitted propensity scores.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E005.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T96", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T96" }, { "id": "P001-T97", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T97" } ], "headline": "With discrete covariates, the IPW estimator equals the plug-in estimator of Section 7.2.1 when each fitted propensity score is the observed treated fraction in its covariate cell, not for every fit.", "id": "P001-C07-E005", "location": "Section 7.2.6, p. 266 (PDF p. 284), paragraph after Definition 7.2.6, last sentence", "pdf_page": 284, "printed_page": "266", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T96", "P001-T97" ], "text": { "as_printed": "``In the case where we have a discrete and finite number of values for $\\mathbf{X}_i$, the IPW estimator is \\hl{logically equivalent} to the plug-in estimator from Section 7.2.1 (\\emph{Plug-In Estimation}).''", "corrected": "``With discrete covariates taking finitely many values, the IPW estimator \\hl{equals} the post-stratification plug-in estimator in Section 7.2.1 \\hl{when each fitted propensity equals the observed treated fraction in its covariate cell (as under saturated propensity estimation), provided both treatment arms are represented in every observed cell}.''\n\nIn symbols: if every observed covariate value $\\mathbf{x}$ has $N_{1\\mathbf{x}}\\ge 1$ and $N_{0\\mathbf{x}}\\ge 1$, and $\\hat p_D(\\mathbf{x})=N_{1\\mathbf{x}}/N_{\\mathbf{x}}$, then\n\\[\n\\frac{1}{n}\\sum_{i=1}^{n}\\left(\\frac{Y_iD_i}{\\hat p_D(\\mathbf{X}_i)}-\\frac{Y_i(1-D_i)}{1-\\hat p_D(\\mathbf{X}_i)}\\right)=\\sum_{\\mathbf{x}}\\frac{N_{\\mathbf{x}}}{n}\\bigl(\\overline{Y}_{1\\mathbf{x}}-\\overline{Y}_{0\\mathbf{x}}\\bigr).\n\\]\nFor other fitted propensity scores the two sides can differ.", "evidence": "The evidence is a derivation and an exact counterexample. For each observed covariate value $\\mathbf{x}$, let $N_{\\mathbf{x}}$ be the number of observations with $\\mathbf{X}_i=\\mathbf{x}$, let $N_{1\\mathbf{x}}$ and $N_{0\\mathbf{x}}$ be the numbers of treated and control observations among them, and let $\\overline{Y}_{1\\mathbf{x}}$ and $\\overline{Y}_{0\\mathbf{x}}$ be their mean outcomes. With both treatment arms observed in every cell and $0<\\hat p_D(\\mathbf{x})<1$, grouping the sum in Definition 7.2.6 by covariate value, I obtain\n\\[\n\\hat{\\mathrm{E}}_{IPW}[\\tau_i]=\\sum_{\\mathbf{x}}\\left(\\frac{N_{1\\mathbf{x}}}{n\\,\\hat p_D(\\mathbf{x})}\\,\\overline{Y}_{1\\mathbf{x}}-\\frac{N_{0\\mathbf{x}}}{n\\,(1-\\hat p_D(\\mathbf{x}))}\\,\\overline{Y}_{0\\mathbf{x}}\\right),\n\\]\nand the estimator of Section 7.2.1 is $\\sum_{\\mathbf{x}}(N_{\\mathbf{x}}/n)(\\overline{Y}_{1\\mathbf{x}}-\\overline{Y}_{0\\mathbf{x}})$. The two agree for every vector of outcomes if and only if $\\hat p_D(\\mathbf{x})=N_{1\\mathbf{x}}/N_{\\mathbf{x}}$ for every observed $\\mathbf{x}$.\n\nThe counterexample has 18 observations and one covariate $Z_i\\in\\{-1,0,1\\}$, with six observations at each value. The propensity score is fitted by maximum likelihood in the logit model with an intercept and a slope in $Z_i$. The fitted coefficients are $0$ and $\\log 2$. Indeed, the score equations hold at this point, and the maximum is unique because the log likelihood is strictly concave. The estimator of Section 7.2.1 equals $1$, and the IPW estimator of Definition 7.2.6 equals $59/36$. This sample has positive probability under i.i.d.\\ sampling from a population in which strong ignorability holds and the true propensity score is this logit model. A correctly specified model therefore does not restore the equality. On the data of Example 7.2.1, the estimator of Section 7.2.1 is $4/9$, as printed. The IPW estimator equals $1/4$ with the maximum likelihood fit $\\hat p_D=2/3$ of the model with an intercept only, and $4/9$ with the treated fractions $1/2$ and $3/4$.\n\\begin{audittable}{rrrrrrr}{The counterexample: 18 observations in three covariate cells, with the fitted propensity score and the observed treated fraction in each cell.}\n$Z_i$ & $N_{\\mathbf{x}}$ & $N_{1\\mathbf{x}}$ & fitted $\\hat p_D(\\mathbf{x})$ & $N_{1\\mathbf{x}}/N_{\\mathbf{x}}$ & $\\overline{Y}_{1\\mathbf{x}}$ & $\\overline{Y}_{0\\mathbf{x}}$ \\\\\n$-1$ & 6 & 3 & $1/3$ & $1/2$ & 4 & 3 \\\\\n$0$ & 6 & 1 & $1/2$ & $1/6$ & 3 & 2 \\\\\n$1$ & 6 & 5 & $2/3$ & $5/6$ & 4 & 3 \\\\\n\\end{audittable}", "problem": "Definition 7.2.6 allows any estimate $\\hat p_D$ of the propensity score, and Section 7.2.5 proceeds with a maximum likelihood estimate from a binary choice model. Such a fit need not reproduce the treated fraction in each covariate cell, and the IPW estimate can then differ from the estimate of Section 7.2.1. Finite support of $\\mathbf{X}_i$, strong ignorability and a correctly specified model for the propensity score do not make the two estimators equal. They are equal when $\\hat p_D(\\mathbf{x})$ is the treated fraction among the observations with $\\mathbf{X}_i=\\mathbf{x}$, as under a saturated model. The two estimators do rest on the same reweighting argument, and that part of the remark is correct.", "what_survives": "Theorem 7.1.13, Theorem 7.2.5, the definitions of the IPW and stabilized IPW estimators, and the consistency of each estimator under its own assumptions do not use this sentence. Example 7.2.1 and its answer $4/9$ are unchanged. The application fits a logit model and Winsorizes the fitted propensity scores. It does not use the equality, and this correction requires no change to its reported numbers." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-E005", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of a paragraph of nine lines. It contains the sentence 'with the true propensity score, E[D i / p D(X i)] = E[(1 minus D i) / (1 minus p D(X i))] = 1'. Red boxes mark two expressions in the later sentence 'Because the stabilized IPW estimator accounts for this between-sample variation in ... and ..., it typically confers efficiency gains over the IPW estimator in practice': the sum over i from 1 to n of 1 / p D(X i), and the sum over i from 1 to n of 1 / (1 minus p D(X i)).", "caption": "Section 7.2.6, p. 267 (PDF p. 285), paragraph after Definition 7.2.7. The two marked sums have numerator $1$ and run over all observations; two sentences earlier the expectations are written with $D_i$ and $1-D_i$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E006.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T98", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T98" } ], "headline": "In the explanation of the stabilized IPW estimator, the numerators of the two sums are the treatment indicators $D_i$ and $1-D_i$.", "id": "P001-C07-E006", "location": "Section 7.2.6, p. 267 (PDF p. 285), paragraph after Definition 7.2.7, sentence beginning “Because the stabilized IPW estimator accounts for”", "pdf_page": 285, "printed_page": "267", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T98" ], "text": { "as_printed": "``Because the stabilized IPW estimator accounts for this between-sample variation in $\\sum_{i=1}^{n}\\frac{\\hl{1}}{p_D(\\mathbf{X}_i)}$ and $\\sum_{i=1}^{n}\\frac{\\hl{1}}{1-p_D(\\mathbf{X}_i)}$, it typically confers efficiency gains over the IPW estimator in practice.''", "corrected": "``Because the stabilized IPW estimator accounts for this between-sample variation in $\\sum_{i=1}^{n}\\frac{\\hl{D_i}}{\\hl{\\hat p_D}(\\mathbf{X}_i)}$ and $\\sum_{i=1}^{n}\\frac{\\hl{1-D_i}}{1-\\hl{\\hat p_D}(\\mathbf{X}_i)}$, it typically confers efficiency gains over the IPW estimator in practice.''\n\nThe hats follow Definition 7.2.7, which uses the estimated propensity score. With the true propensity score, the same two sums are written with $p_D$.", "evidence": "The evidence is a check of the page and a short computation. Two sentences earlier the text states that $\\mathrm{E}[D_i/p_D(\\mathbf{X}_i)]=\\mathrm{E}[(1-D_i)/(1-p_D(\\mathbf{X}_i))]=1$, with the indicators. Suppose that $p_D(\\mathbf{X}_i)=1/2$ for every unit. Then each printed sum equals $2n$ in every sample, while the sums with the indicators equal $2n_T$ and $2(n-n_T)$, where $n_T$ is the number of treated units.", "problem": "The two sums in this sentence run over all $n$ observations and have no treatment indicators. The denominators of the stabilized IPW estimator in Definition 7.2.7, on the same page, are $n^{-1}\\sum_{i=1}^{n}D_i/\\hat p_D(\\mathbf{X}_i)$ and $n^{-1}\\sum_{i=1}^{n}(1-D_i)/(1-\\hat p_D(\\mathbf{X}_i))$. These are the quantities whose variation between samples the estimator accounts for.", "what_survives": "Definition 7.2.7 and the two expectation identities are printed correctly. Only the notation of the explanation changes." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-E006", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the displayed null hypothesis 'W i independent of D i given X i' and the paragraph after it, which describes an F-test that compares a regression of D i on W i and X i with a regression of D i on X i alone. A red box marks the last sentence: 'If so, then we can reject the null hypothesis that W i is independent of D i conditional on X i.'", "caption": "Section 7.2.8, p. 270 (PDF p. 288). The marked sentence draws the conclusion that the $F$-test does not support: rejection of the conditional independence of $\\mathbf{W}_i$ and $D_i$ given $\\mathbf{X}_i$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E007.png" }, { "alt": "Line chart with the covariate value X i (minus 1, 0, 1) on the horizontal axis and the probability of treatment, from 0 to 1, on the vertical axis. Blue circles for the true propensity score lie at 0.75, 0.25 and 0.75. Orange squares for the OLS fit of D i on (1, X i) lie on a flat line at 7/12, about 0.58. Green triangles for the OLS fit of D i on (1, X i, W i), averaged given X i, lie at 45/68, 29/68 and 45/68, about 0.66, 0.43 and 0.66. The title reads 'The regression that is linear in X i misses the curve; the placebo W i picks it up'.", "caption": "The counterexample to the $F$-test of Section 7.2.8. The propensity score is $3/4$, $1/4$, $3/4$ at $X_i=-1,0,1$; the regression of $D_i$ on $(1,X_i)$ fits $7/12$ at every $X_i$; the regression that adds $W_i$ fits $45/68$, $29/68$, $45/68$ on average, although $W_i$ is independent of $D_i$ given $X_i$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E007-placebo-fit.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T17", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T17" }, { "id": "P001-T18", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T18" }, { "id": "P001-T124", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T124" } ], "headline": "The $F$-test for adding the placebo variables $\\mathbf{W}_i$ to a regression of $D_i$ on $\\mathbf{X}_i$ can reject $\\mathbf{W}_i\\perp\\!\\!\\!\\perp D_i\\mid\\mathbf{X}_i$ with probability tending to one even when it holds.", "id": "P001-C07-E007", "location": "Section 7.2.8 (Placebo Testing), p. 270 (PDF p. 288), sentences beginning “One simple way to implement this is by performing an $F$-test”", "pdf_page": 288, "printed_page": "270", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P001-T124", "P001-T17", "P001-T18" ], "text": { "as_printed": "``Then we can test the null hypothesis that\n\\[\n\\mathbf{W}_i\\perp\\!\\!\\!\\perp D_i\\mid\\mathbf{X}_i.\n\\]\nThis is known as a \\emph{placebo test}.$^{14}$ One simple way to implement this is by performing an $F$-test. An $F$-test compares a regression of $D_i$ on $\\mathbf{W}_i$ and $\\mathbf{X}_i$ with a regression of $D_i$ on $\\mathbf{X}_i$ alone, and asks: does adding $\\mathbf{W}_i$ to the regression explain more of the remaining variation in $D_i$ than we would expect by chance? \\hl{If so, then we can reject the null hypothesis that $\\mathbf{W}_i$ is independent of $D_i$ conditional on $\\mathbf{X}_i$.}''", "corrected": "``One simple regression-based placebo check is a joint test of the coefficients on $\\mathbf{W}_i$ in the OLS regression of $D_i$ on $\\mathbf{W}_i$ and $\\mathbf{X}_i$. \\hl{If, under the null, $\\mathrm{E}[D_i\\mid\\mathbf{X}_i]$ is correctly represented by the included $\\mathbf{X}_i$ regressors and inference uses a consistent heteroskedasticity-robust covariance estimator under the usual large-sample conditions}, conditional independence of $\\mathbf{W}_i$ and $D_i$ given $\\mathbf{X}_i$ implies that these coefficients are zero. A rejection can then be evidence against conditional independence, \\hl{subject to the conditional-mean specification assumption}. \\hl{Without it, $\\mathbf{W}_i$ may simply capture omitted nonlinear functions of $\\mathbf{X}_i$}; use a procedure justified for the conditional-independence null if that specification assumption is unavailable.''\n\nIn full: let the observations $(D_i,\\mathbf{X}_i,\\mathbf{W}_i)$ be i.i.d.\\ with finite second moments, let $\\mathbf{X}_i$ include a constant, and let the matrix of second moments of $(\\mathbf{X}_i,\\mathbf{W}_i)$ be nonsingular. If $\\mathrm{E}[D_i\\mid\\mathbf{X}_i]$ is a linear combination of the components of $\\mathbf{X}_i$ and $\\mathbf{W}_i\\perp\\!\\!\\!\\perp D_i\\mid\\mathbf{X}_i$, then the coefficients on $\\mathbf{W}_i$ in the best linear predictor of $D_i$ given $\\mathbf{X}_i$ and $\\mathbf{W}_i$ are zero. The linearity condition is sufficient for this conclusion; it is a restriction on the propensity score that an application may not satisfy.", "evidence": "The evidence is an exact counterexample and a derivation of the corrected statement. Let $X_i$ be uniform on $\\{-1,0,1\\}$, let $Z_i$ be uniform on $\\{-1,1\\}$ and independent of $X_i$, and set\n\\[\nW_i=X_i^2+\\frac{Z_i}{2},\\qquad \\Pr[D_i=1\\mid X_i,Z_i]=\\frac{1}{4}+\\frac{X_i^2}{2}.\n\\]\nGiven $X_i$, the placebo $W_i$ is a function of $Z_i$, and $Z_i$ is independent of $D_i$ given $X_i$, so that $W_i\\perp\\!\\!\\!\\perp D_i\\mid X_i$ holds exactly. The propensity score takes the values $1/4$ and $3/4$, so positivity holds. With $Y_i(0)=2+Z_i$ and $Y_i(1)=3+Z_i$, strong ignorability holds given $X_i$, and $\\mathrm{E}[Y_i\\mid D_i,X_i]=2+D_i$ is linear.\n\nIn the population, the regression of $D_i$ on $(1,X_i)$ has coefficients $(7/12,0)$, and the regression of $D_i$ on $(1,X_i,W_i)$ has coefficients $(29/68,0,4/17)$. The coefficient on $W_i$ is not zero. Let $F_n$ denote the $F$-statistic that compares the two regressions. By the strong law of large numbers, I have\n\\[\n\\frac{F_n}{n}\\longrightarrow\\frac{35/144-59/272}{59/272}=\\frac{64}{531}\\quad\\text{almost surely},\n\\]\nso that the test rejects the true null hypothesis with probability tending to one at every fixed level. The table lists the other exact values. A second example, with $X_i$ uniform on $[-1,1]$, $W_i=X_i^2$ and the same propensity score, has $F_n/n\\to 16/159$.\n\\begin{audittable}{lr}{The counterexample: exact values for the two regressions and the two test statistics.}\nquantity & value \\\\\ncoefficient on $W_i$ in the regression of $D_i$ on $(1,X_i,W_i)$ & $4/17$ \\\\\nresidual variance, regression of $D_i$ on $(1,X_i)$ & $35/144$ \\\\\nresidual variance, regression of $D_i$ on $(1,X_i,W_i)$ & $59/272$ \\\\\nlimit of $F_n/n$ & $64/531$ \\\\\nlimit of the robust Wald statistic for the coefficient on $W_i$, divided by $n$ & $18496/135315$ \\\\\n$F$-statistic in a sample of $240$ observations with the population frequencies & $5056/177$ \\\\\ncoefficient on $W_i$ after $X_i^2$ is added to both regressions & $0$ \\\\\n\\end{audittable}\n\nFor the corrected statement, suppose that $\\mathbf{X}_i$ includes a constant and that $\\mathrm{E}[D_i\\mid\\mathbf{X}_i]=\\mathbf{X}_i\\boldsymbol{\\gamma}$ for some $\\boldsymbol{\\gamma}$. Under the null hypothesis, I have $\\mathrm{E}[D_i\\mid\\mathbf{X}_i,\\mathbf{W}_i]=\\mathbf{X}_i\\boldsymbol{\\gamma}$, and the error $e_i=D_i-\\mathbf{X}_i\\boldsymbol{\\gamma}$ satisfies $\\mathrm{E}[e_i\\mid\\mathbf{X}_i,\\mathbf{W}_i]=0$. Then $e_i$ is uncorrelated with $\\mathbf{X}_i$ and with $\\mathbf{W}_i$. These are the normal equations of the regression of $D_i$ on $\\mathbf{X}_i$ and $\\mathbf{W}_i$, solved with zero coefficients on $\\mathbf{W}_i$. If the matrix of second moments of the regressors is nonsingular, this solution is unique, and conditional independence implies that the coefficients on $\\mathbf{W}_i$ are zero.", "problem": "Recall that, in the book, a regression without qualification is an OLS regression (p.~147, footnote 3). The $F$-test therefore asks whether the coefficients on $\\mathbf{W}_i$ are zero in the best linear predictor of $D_i$ given $\\mathbf{W}_i$ and $\\mathbf{X}_i$. Under the null hypothesis, $\\mathrm{E}[D_i\\mid\\mathbf{X}_i,\\mathbf{W}_i]=\\mathrm{E}[D_i\\mid\\mathbf{X}_i]$, but this conditional mean need not be linear in $\\mathbf{X}_i$. The chapter does not assume that it is, and a propensity score in general is not. When it is not linear, $\\mathbf{W}_i$ can carry a nonlinear function of $\\mathbf{X}_i$ that the regression omits. The coefficient on $\\mathbf{W}_i$ can therefore be nonzero in the population although the null hypothesis holds; when it is, the test rejects with probability tending to one as $n$ grows. The failure concerns the size of the test, not its power. In that case, heteroskedasticity-robust standard errors do not help, because the coefficient that they are attached to is not zero.", "what_survives": "The identification results, the estimators and the estimates of the application are not affected; the application does not use this test. The point that ignorability cannot be tested directly remains, and so does the value of placebo evidence from a test that is valid for the null hypothesis of conditional independence. The $F$-test of p.~270 changes: the linearity condition with valid large-sample inference is a sufficient justification, or it can be read as a test of whether $\\mathbf{W}_i$ improves the linear prediction of the treatment." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-E007", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the first two paragraphs of Section 7.3, fifteen lines. The first paragraph gives the positivity assumption as 'there exists epsilon greater than 0 such that epsilon is less than p D(X i), which is less than 1 minus epsilon'. Red boxes mark the words 'a failure of the positivity assumption logically implies incomplete (or limited) overlap' in the sentence that begins 'We proceed by illustrating that'.", "caption": "Opening of Section 7.3, p. 271 (PDF p. 289). The first paragraph states the positivity assumption with a single $\\epsilon$; the marked words say that its failure implies incomplete overlap.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E008.png" }, { "alt": "Crop of the shaded box of Theorem 7.3.2, 'Positivity Implies Complete Population Overlap', and the two lines after it. Red boxes mark the words 'Furthermore, the converse of this claim (that population overlap implies positivity) can almost be stated here', which are followed by 'but requires some measure-theoretic'.", "caption": "Theorem 7.3.2 and the sentence after it, p. 271 (PDF p. 289); the sentence continues on p. 272. The marked words say that the converse of the theorem can almost be stated.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E008-b.png" }, { "alt": "Two charts with the covariate value k, from 1 to 16, on the horizontal axis and logarithmic vertical axes. Left panel, 'Both arms put positive probability on every k': blue circles for the treated arm, Pr[X i = k given D i = 1] = 2 divided by (3 to the power k), fall from 0.67 to about 5 times 10 to the minus 8, and orange squares for the control arm, Pr[X i = k given D i = 0] = 1/2 to the power k, fall from 0.5 to about 1.5 times 10 to the minus 5; every value is positive. Right panel, 'The propensity score falls below every epsilon greater than 0': blue circles for p D(k) fall from 0.57 at k = 1 to 0.003 at k = 16, crossing a horizontal line at epsilon = 1/10 at k = 8 and a horizontal line at epsilon = 1/100 at k = 14.", "caption": "The counterexample to the converse of Theorem 7.3.2. Left: the conditional distributions of $X_i$ in the two treatment arms have the same support $\\{1,2,\\ldots\\}$. Right: the propensity score $p_D(k)$ is below $1/10$ from $k=8$ and below $1/100$ from $k=14$. Both vertical axes are logarithmic.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E008-overlap-positivity.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T19", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T19" }, { "id": "P001-T20", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T20" }, { "id": "P001-T104", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T104" } ], "headline": "A failure of the positivity assumption does not imply incomplete overlap: for discrete covariates, complete population overlap implies $00$ such that $\\epsilon0$ such that $\\epsilon0$, the value $k=\\lceil 4/\\epsilon\\rceil$ satisfies $p_D(k)<\\epsilon$ and $\\Pr[X_i=k]>0$. Hence there exists no $\\epsilon>0$ such that $\\epsilon0$, Bayes' rule gives\n\\[\n\\Pr[\\mathbf{X}_i=\\mathbf{x}\\mid D_i=1]=\\frac{p_D(\\mathbf{x})\\,w(\\mathbf{x})}{\\pi},\\qquad \\Pr[\\mathbf{X}_i=\\mathbf{x}\\mid D_i=0]=\\frac{(1-p_D(\\mathbf{x}))\\,w(\\mathbf{x})}{1-\\pi}.\n\\]\nThe two conditional supports are equal if and only if both probabilities are positive for every such $\\mathbf{x}$, that is, if and only if $00$ such that $\\epsilon0.\n\\]", "evidence": "The evidence is a derivation and an exact counterexample. Set $q=\\Pr[D_i=1]$. By the Law of Iterated Expectations, I have $\\mathrm{E}[p_D(\\mathbf{X}_i)]=\\mathrm{E}[\\mathrm{E}[D_i\\mid\\mathbf{X}_i]]=q$. Under the conditional independence assumption, $\\tau_i$ and $D_i$ are independent given $\\mathbf{X}_i$, and I obtain\n\\[\n\\mathrm{E}[p_D(\\mathbf{X}_i)\\tau_i]=\\mathrm{E}\\bigl[\\mathrm{E}[D_i\\mid\\mathbf{X}_i]\\,\\mathrm{E}[\\tau_i\\mid\\mathbf{X}_i]\\bigr]=\\mathrm{E}[D_i\\tau_i]=q\\,\\mathrm{E}[\\tau_i\\mid D_i=1].\n\\]\nFor example, let $\\mathbf{X}_i$ be constant, $Y_i(0)=0$, $Y_i(1)=1$ and $q=1/4$. The printed expression equals $1/4$, and the ATT equals $1$. In a second example, $X_i$ takes the values $0$ and $1$ with probabilities $1/3$ and $2/3$, the propensity score is $p_D(0)=1/4$ and $p_D(1)=3/8$, and $\\tau_i$ equals $1$ when $X_i=0$ and $4$ when $X_i=1$. Then $q=1/3$, the printed expression equals $13/12$, and the ATT equals $13/4$.", "problem": "Under the conditional independence assumption, $\\mathrm{E}[p_D(\\mathbf{X}_i)\\tau_i]$ equals $\\Pr[D_i=1]$ times the ATT. The two are equal only when the ATT is zero, since $\\Pr[D_i=1]<1$. The neighboring sentences speak of weighting the members of a population and of the average effect among those who took up the treatment. The normalized weighted average is meant, and the printed formula omits the normalization.", "what_survives": "The ATT as a target, its identification under the conditional independence assumption and the weaker positivity condition, and the discussion of changing the target population are unchanged. I found no later theorem or calculation that uses this equality; Table 7.5.1 reports estimates of the ATE." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-E009", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of ten lines from page 278, from the sentence The fact that random assignment is unlikely to hold does not mean that all hope is lost, to the first two lines of the next paragraph. A red outline in two parts surrounds the end of the first paragraph: by definition, D sub i must be conditionally independent of potential outcomes. The next paragraph begins: If we assume ignorability, then we again have a variety of estimators available.", "caption": "Section 7.5, p. 278 (PDF p. 296). The outline marks the conclusion that $D_i$ is conditionally independent of the potential outcomes “by definition”; it runs over a line break. The next paragraph begins “If we assume ignorability”.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E010.png" }, { "alt": "Line chart. The horizontal axis is the described gender of the applicant: female (D sub i equal to 0) and male (D sub i equal to 1). The vertical axis is the mean percentage of no votes, from 0 to 95. A blue line with round markers, for electorates of type 0, rises from 20 to 25; its markers are labelled with the probabilities 3/8 at female and 1/8 at male. An orange line with square markers, for electorates of type 1, rises from 60 to 65; its markers are labelled 1/8 at female and 3/8 at male. A dashed black line with diamond markers, for all electorates given X, rises from 30 to 55. Labels at the right read 60 to 65: difference 5; 30 to 55: difference 25, in bold; and 20 to 25: difference 5. The title reads: A male description adds 5 points in every electorate; the contrast given the coded descriptions is 25 points.", "caption": "The counterexample for P001-C07-E010. Mean percentage of “no” votes by described gender, for each type of electorate (solid lines) and for all electorates together, which is the comparison given $\\mathbf{X}_i$ (dashed line). The fraction beside each solid-line marker is the probability of that combination of electorate type and gender, and the area of that marker is proportional to it. The difference between male and female descriptions is $5$ points within each type of electorate and $25$ points given $\\mathbf{X}_i$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-E010-counterexample.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T21", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T21" }, { "id": "P001-T22", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T22" }, { "id": "P001-T105", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T105" } ], "headline": "Coding all the information in the applicant descriptions does not make gender conditionally independent of the potential outcomes by definition; conditional independence given $\\mathbf{X}_i$ remains a substantive assumption.", "id": "P001-C07-E010", "location": "Section 7.5, paragraph beginning “The fact that random assignment is unlikely to hold”, last sentence, p. 278 (PDF p. 296)", "pdf_page": 296, "printed_page": "278", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P001-T105", "P001-T21", "P001-T22" ], "text": { "as_printed": "This is a setting where the ignorability assumption, implying $(Y_i(0), Y_i(1)) \\mathbin{\\perp\\!\\!\\!\\perp} D_i \\mid \\mathbf{X}_i$, is quite plausible. As long as voters do not directly know the applicant, the applicant description contains \\emph{all} of the information available to voters. Assuming that the researchers' coding of the applicant descriptions---that is, the variables in $\\mathbf{X}_i$---does indeed account for all the information presented to the voters, then, \\hl{\\emph{by definition}, $D_i$ must be} conditionally independent of potential outcomes.", "corrected": "This is a setting where the ignorability assumption, implying $(Y_i(0), Y_i(1)) \\mathbin{\\perp\\!\\!\\!\\perp} D_i \\mid \\mathbf{X}_i$, is quite plausible. As long as voters do not directly know the applicant, the applicant description contains \\emph{all} of the information available to voters. \\hl{Recording all information about applicants presented to voters can address selection associated with those characteristics, but does not establish ignorability by definition. The additional assumption is that $(Y_i(0),Y_i(1)) \\mathbin{\\perp\\!\\!\\!\\perp} D_i \\mid \\mathbf{X}_i$. Differences in electorate preferences or decision context can violate this assumption even if voters do not personally know the applicants.}", "evidence": "The evidence is an exact counterexample in which both premises hold and conditional independence fails. All descriptions are identical apart from the gender shown, and no voter knows an applicant. Then $\\mathbf{X}_i$ is constant and records all non-gender applicant information; gender is recorded separately as $D_i$. Electorates are of two types, $U_i \\in \\{0,1\\}$, each with probability $1/2$. An electorate of type $u$ returns $Y_i(d) = 20 + 40u + 5d$ per cent ``no'' votes on a description showing gender $d$, which implies that $Y_i(1) - Y_i(0) = 5$ for every referendum and that the ATE is $5$. Male applicants make up a quarter of the applications before electorates of type $0$ and three quarters of those before electorates of type $1$. The table shows the joint distribution.\n\\begin{audittable}{cccccc}{The counterexample: the joint distribution of electorate type, described gender and potential outcomes (percentage of ``no'' votes). The descriptions are otherwise identical, so $\\mathbf{X}_i$ is constant.}\nElectorate type $U_i$ & Gender $D_i$ & Probability & $Y_i(0)$ & $Y_i(1)$ & Observed $Y_i$ \\\\\n$0$ & $0$ (female) & $3/8$ & $20$ & $25$ & $20$ \\\\\n$0$ & $1$ (male) & $1/8$ & $20$ & $25$ & $25$ \\\\\n$1$ & $0$ (female) & $1/8$ & $60$ & $65$ & $60$ \\\\\n$1$ & $1$ (male) & $3/8$ & $60$ & $65$ & $65$ \\\\\n\\end{audittable}\nBy the table, I have $\\Pr[D_i = 1 \\mid \\mathbf{X}_i] = 1/8 + 3/8 = 1/2$, so positivity holds, and\n\\[\n\\begin{aligned}\n\\Pr\\bigl[(Y_i(0),Y_i(1)) = (60,65),\\ D_i = 1 \\mid \\mathbf{X}_i\\bigr] &= \\frac{3}{8} , \\\\\n\\Pr\\bigl[(Y_i(0),Y_i(1)) = (60,65) \\mid \\mathbf{X}_i\\bigr] \\, \\Pr[D_i = 1 \\mid \\mathbf{X}_i] &= \\frac{1}{2} \\cdot \\frac{1}{2} = \\frac{1}{4} .\n\\end{aligned}\n\\]\nSince the two probabilities differ, conditional independence given $\\mathbf{X}_i$ fails. The contrast that Theorem 7.1.13 equates with the ATE under strong ignorability is\n\\[\n\\mathrm{E}[Y_i \\mid D_i = 1, \\mathbf{X}_i] - \\mathrm{E}[Y_i \\mid D_i = 0, \\mathbf{X}_i] = 55 - 30 = 25 ,\n\\]\nwhich is $20$ points above the ATE of $5$. Conditioning on $U_i$ as well as $\\mathbf{X}_i$ restores conditional independence, and the standardized contrast is then $5$. The example settles the implication only. It does not show that ignorability fails in the Swiss data, and it determines neither the sign nor the size of any bias there.", "problem": "The last sentence derives conditional independence from two premises: voters do not know the applicants, and $\\mathbf{X}_i$ records all non-gender information in the applicant descriptions. The conclusion does not follow. The potential outcomes $Y_i(0)$ and $Y_i(1)$ are the ``no'' vote shares that the electorate of referendum $i$ would return if the same description carried a female or a male applicant. They depend on who is voting as well as on the description. Conditioning on the recorded applicant characteristics holds them fixed. It does not exclude that electorates more inclined to vote ``no'' also see a different proportion of male applicants among descriptions with the same $\\mathbf{X}_i$. The covariates listed on p.~277 include the decade of the referendum and not the municipality. Conditional independence is a substantive assumption here, as the book says in general on p.~248 (``we recommend only invoking strong ignorability when justifiable by substantive knowledge''), and the passage presents it as true by definition. The causal reading of the six estimators in Table 7.5.1 that rest on ignorability (OLS linear and unpooled, matching, IPW, stabilized IPW and doubly robust) depends on this assumption.", "what_survives": "Definition 7.1.12 and the identification and estimation results that assume ignorability are untouched. So are the bounds earlier in the application, which do not use ignorability, and the random-assignment estimate, which rests on its own stated assumption. The correction changes no estimate or standard error in Table 7.5.1; I did not recompute them from the data, which were not available to me. The causal statements on pp.~279--280 are prefaced by ``If we accept the ignorability assumption'' and stand as conditional statements. What changes is that the assumption has to be argued for the electorates and the decision context as well as for the applicant descriptions." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-E010", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of three lines of text. A red box marks a garbled word, 'person' followed by a-circumflex, A-breve, Z-acute and 's', in the sentence 'The problem is that we cannot observe a person's counterfactual outcome.'", "caption": "Section 7.1, second paragraph, p. 236 (PDF p. 254). The marked word is printed with three accented letters where the apostrophe belongs.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-N001.png" } ], "formal_release": null, "formal_statements": [], "headline": "The word “person’s” is printed with three stray characters in place of the apostrophe.", "id": "P001-C07-N001", "location": "Section 7.1, second paragraph, p. 236 (PDF p. 254)", "pdf_page": 254, "printed_page": "236", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "``The problem is that we cannot observe a person\\hl{âĂŹ}s counterfactual outcome.''", "corrected": "``The problem is that we cannot observe a person\\hl{'}s counterfactual outcome.''", "evidence": "The evidence is a check of p.~236.", "problem": "Three accented letters replace the apostrophe of ``person's'' on the printed page.", "what_survives": "The meaning is plain, and nothing depends on the misprint." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-N001", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of four lines of text. Red boxes mark the italic words '(Ignorability and the Propensity Score)' in the sentence 'We will again prove the first two properties and omit a proof of the third; the proof follows nearly identically from Theorem 6.1.12 (Ignorability and the Propensity Score).'", "caption": "Proof of Theorem 7.1.16, first sentence, p. 251 (PDF p. 269). The marked title is quoted for Theorem 6.1.12.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-N002.png" } ], "formal_release": null, "formal_statements": [], "headline": "Theorem 6.1.12 is titled “MAR and the Propensity Score”; the proof of Theorem 7.1.16 cites it as “Ignorability and the Propensity Score”.", "id": "P001-C07-N002", "location": "Proof of Theorem 7.1.16, first sentence, p. 251 (PDF p. 269)", "pdf_page": 269, "printed_page": "251", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "``We will again prove the first two properties and omit a proof of the third; the proof follows nearly identically from Theorem 6.1.12 (\\hl{\\emph{Ignorability and the Propensity Score}}).''", "corrected": "``We will again prove the first two properties and omit a proof of the third; the proof follows nearly identically from Theorem 6.1.12 (\\hl{\\emph{MAR and the Propensity Score}}).''", "evidence": "The evidence is a check of p.~251 against pp.~217 and 224.", "problem": "The theorem number is right and the title is not. Theorem 6.1.12 (p.~217) is titled ``MAR and the Propensity Score'', and Example 6.2.5 (p.~224) cites it under that title.", "what_survives": "The reference points to the intended theorem, and nothing depends on the title." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-N002", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of two lines of text and a displayed formula. A red box marks the words 'Definition 7.1.13' in the sentence 'Plugging into Definition 7.1.13 (The ATE under Strong Ignorability), we have'. The formula reads: the sum over x of E[Y i | D i = 1, X i = x] Pr[X i = x], minus the sum over x of E[Y i | D i = 0, X i = x] Pr[X i = x], equals -1/4.", "caption": "Example 7.1.17, p. 253 (PDF p. 271), the sentence before the adjusted contrast. The marked words call item 7.1.13 a definition.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-N003.png" } ], "formal_release": null, "formal_statements": [], "headline": "Item 7.1.13 is a theorem: “Definition 7.1.13” should read “Theorem 7.1.13”.", "id": "P001-C07-N003", "location": "Example 7.1.17, p. 253 (PDF p. 271), sentence before the last display", "pdf_page": 271, "printed_page": "253", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "``Plugging into \\hl{Definition} 7.1.13 (\\emph{The ATE under Strong Ignorability}), we have''", "corrected": "``Plugging into \\hl{Theorem} 7.1.13 (\\emph{The ATE under Strong Ignorability}), we have''", "evidence": "The evidence is a check of p.~253 against p.~248 (PDF p.~266).", "problem": "The item numbered 7.1.13 (p.~248) is Theorem 7.1.13, \\emph{ATE under Strong Ignorability}, and its formula is the one used in the example.", "what_survives": "The number and the title identify the result, and nothing else changes." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-N003", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of a displayed formula and the paragraph after it. The display writes E[tau i(d, d-prime)] as a sum over x of E[Y i given D i = d-prime, X i = x] Pr[X i = x] minus the same sum with d in place of d-prime. A red box marks the sentence 'Thus, under ignorability, for any d, d-prime in Supp[D i], E[tau i(d, d-prime)] is point identified.' The sentence before it says that E[Y i given D i = d-prime, X i = x], E[Y i given D i = d, X i = x] and Pr[X i = x] can be point identified because the random vector (Y i, D i, X i) is observed.", "caption": "Section 7.1.8, p. 255 (PDF p. 273). The marked sentence claims that $\\mathrm{E}[\\tau_i(d,d')]$ is point identified for every pair of doses in the support of $D_i$; the display above it is the generalized formula for the ATE.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-N004.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T90", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T90" }, { "id": "P001-T91", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T91" } ], "headline": "For a continuous treatment, ignorability and overlap identify fixed-dose mean effects if the conditional mean potential outcome is continuous in the dose.", "id": "P001-C07-N004", "location": "Section 7.1.8, p. 255 (PDF p. 273), paragraph after the first display, sentence beginning “Thus, under ignorability”", "pdf_page": 273, "printed_page": "255", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T90", "P001-T91" ], "text": { "as_printed": "``Since we can observe the random vector $(Y_i,D_i,\\mathbf{X}_i)$, we can point identify $\\mathrm{E}[Y_i\\mid D_i=d',\\mathbf{X}_i=\\mathbf{x}]$, $\\mathrm{E}[Y_i\\mid D_i=d,\\mathbf{X}_i=\\mathbf{x}]$, and $\\Pr[\\mathbf{X}_i=\\mathbf{x}]$. Thus, under ignorability, \\hl{for any $d,d'\\in\\operatorname{Supp}[D_i]$}, $\\mathrm{E}[\\tau_i(d,d')]$ is point identified.''", "corrected": "``Since we can observe the random vector $(Y_i,D_i,\\mathbf{X}_i)$, we can point identify $\\Pr[\\mathbf{X}_i=\\mathbf{x}]$, \\hl{and} $\\mathrm{E}[Y_i\\mid D_i=d,\\mathbf{X}_i=\\mathbf{x}]$ \\hl{is identified as a function of $d$ up to sets of doses of conditional probability zero}. Thus, under ignorability, \\hl{if $d$ and $d'$ have positive conditional probability given each $\\mathbf{x}$, or if the mean of $Y_i(d)$ given $\\mathbf{X}_i=\\mathbf{x}$ is continuous in the dose on the common support of the treatment}, $\\mathrm{E}[\\tau_i(d,d')]$ is point identified.''\n\nIn full: let $\\mathbf{X}_i$ be discrete, let $Y_i=Y_i(D_i)$, let the process $(Y_i(d))_{d}$ be \\hl{a random function with jointly measurable evaluation}, independent of $D_i$ given $\\mathbf{X}_i$, and let the outcomes be integrable (bounded outcomes suffice). Let $S$ be a \\hl{Borel} set of doses such that the following two conditions hold for every $\\mathbf{x}$ with $\\Pr[\\mathbf{X}_i=\\mathbf{x}]>0$.\n\\begin{enumerate}\n\\item Every neighborhood in $S$ of every point of $S$ has positive probability under the conditional distribution of $D_i$ given $\\mathbf{X}_i=\\mathbf{x}$. This holds, for instance, when $S$ is an interval \\hl{of positive length} on which the conditional density of $D_i$ is positive.\n\\item The function $d\\mapsto\\mathrm{E}[Y_i(d)\\mid\\mathbf{X}_i=\\mathbf{x}]$ is continuous on $S$.\n\\end{enumerate}\nThen, for any $d,d'\\in S$,\n\\[\n\\mathrm{E}[\\tau_i(d,d')]=\\sum_{\\mathbf{x}}\\bigl(m_{\\mathbf{x}}(d')-m_{\\mathbf{x}}(d)\\bigr)\\Pr[\\mathbf{X}_i=\\mathbf{x}]\n\\]\nis point identified, where $m_{\\mathbf{x}}$ is the version of $d\\mapsto\\mathrm{E}[Y_i\\mid D_i=d,\\mathbf{X}_i=\\mathbf{x}]$ that is continuous on $S$. The second condition restricts the mean of the potential outcome: a continuous version of the observed regression does not suffice without it. The statement on p.~255 for $\\mathrm{E}[\\tau_i(d,d')\\mid\\mathbf{X}_i=\\mathbf{x}]$ needs the same two conditions.", "evidence": "The evidence is an exact counterexample, followed by a derivation of the corrected statement. Let $\\mathbf{X}_i$ be constant and let $D_i$ be uniform on $[0,1]$. Consider two models for the potential outcomes:\n\\[\n\\text{model A: } Y_i(d)=0 \\text{ for all } d,\\qquad \\text{model B: } Y_i(d)=\\mathbf{1}\\{d=1/2\\}.\n\\]\nIn each model the potential outcomes are not random, so that the whole process $(Y_i(d))_{d}$ is independent of $D_i$, and the density of $D_i$ equals $1$ on $[0,1]$. The observed outcome $Y_i=Y_i(D_i)$ is $0$ in model A. In model B it is $0$ except on the event $\\{D_i=1/2\\}$, which has probability zero. The joint CDF of $(Y_i,D_i,\\mathbf{X}_i)$ is the same in the two models, but\n\\[\n\\mathrm{E}[\\tau_i(0,1/2)]=0 \\text{ in model A},\\qquad \\mathrm{E}[\\tau_i(0,1/2)]=1 \\text{ in model B},\n\\]\nand the same two values hold for the doses $(1/4,1/2)$. By Definition 6.0.1, $\\mathrm{E}[\\tau_i(d,d')]$ is not point identified. The example does not need a degenerate outcome. If $Z_i$ is uniform on $[0,1]$ and independent of $D_i$, then the models $Y_i(d)=Z_i$ and $Y_i(d)=Z_i+\\mathbf{1}\\{d=1/2\\}$ have the same observable distribution, and $\\mathrm{E}[Y_i(1/2)]$ equals $1/2$ in the first and $3/2$ in the second.\n\nFor the corrected statement, let $S$ be the set of doses in that statement, and fix $\\mathbf{x}$ such that $\\Pr[\\mathbf{X}_i=\\mathbf{x}]>0$. Let $\\nu_{\\mathbf{x}}$ denote the conditional distribution of $D_i$ given $\\mathbf{X}_i=\\mathbf{x}$, and set $m_{\\mathbf{x}}(d)=\\mathrm{E}[Y_i(d)\\mid\\mathbf{X}_i=\\mathbf{x}]$. By the generalized potential outcomes model and the independence of the process $(Y_i(d))_{d}$ and $D_i$ given $\\mathbf{X}_i$, using jointly measurable evaluation, I have\n\\[\n\\mathrm{E}\\bigl[Y_i\\mathbf{1}\\{D_i\\in B\\}\\mid\\mathbf{X}_i=\\mathbf{x}\\bigr]=\\int_B m_{\\mathbf{x}}(d)\\,\\nu_{\\mathbf{x}}(\\mathrm{d}d)\n\\]\nfor every Borel set $B$. The left side depends only on the observable distribution. Two functions that are compatible with it in the role of $m_{\\mathbf{x}}$ then agree outside a set of $\\nu_{\\mathbf{x}}$-measure zero. Suppose that both are continuous on $S$ and differ at a dose $d_0\\in S$ by $\\delta>0$. Then there exists a neighborhood of $d_0$ in $S$ on which they differ by more than $\\delta/2$. By the first condition of the corrected statement, this neighborhood has positive $\\nu_{\\mathbf{x}}$-measure, which is a contradiction. I conclude that $m_{\\mathbf{x}}(d_0)$ is determined by the observable distribution. Summing $(m_{\\mathbf{x}}(d')-m_{\\mathbf{x}}(d))\\Pr[\\mathbf{X}_i=\\mathbf{x}]$ over $\\mathbf{x}$, I obtain $\\mathrm{E}[\\tau_i(d,d')]$.", "problem": "Section 7.1.8 allows the treatment $D_i$ to be continuous. Every single treatment level, which I call a dose, then has probability zero, and the distribution of $(Y_i,D_i,\\mathbf{X}_i)$ determines the function $d\\mapsto\\mathrm{E}[Y_i\\mid D_i=d,\\mathbf{X}_i=\\mathbf{x}]$ only up to a set of doses of probability zero. The value of this function at one specified dose is not determined by the observable distribution. Ignorability does not determine it either, even when the treatment has a positive density at that dose. The claim holds for a dose that has positive conditional probability, and it holds when the mean of $Y_i(d)$ given $\\mathbf{X}_i=\\mathbf{x}$ is continuous in $d$. The section states no such condition. Note that it assumes that $Y_i(d)$ is differentiable in $d$ only afterwards, for the average marginal causal effect (AMCE).", "what_survives": "Theorem 7.1.13 for a binary treatment, the estimators of Section 7.2, and the application in Section 7.5 with Table 7.5.1 do not use this claim. The counterexample does not contradict the identity for the AMCE, which is stated under differentiability. The two statements on p.~255 that the unconditional and the conditional mean effects are point identified for any $d,d'\\in\\operatorname{Supp}[D_i]$ need the continuity condition, and so does the formula for $\\mathrm{E}[Y_i(d)]$ in Section 7.2.3 (p.~261) when the treatment is continuous." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-N004", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the heading '7.2.3 Maximum Likelihood Plug-In Estimation of Causal Effects' and the first three lines of the section. A red box marks the parenthesis '(Plug-In Estimation and Conditional Expectation Functions)', which follows the words 'Using the tools developed in Section 5.2.4'.", "caption": "Opening of Section 7.2.3, p. 261 (PDF p. 279). The marked words are given as the title of Section 5.2.4.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-N005.png" } ], "formal_release": null, "formal_statements": [], "headline": "Section 5.2.4 is titled “Maximum Likelihood Plug-In Estimation”.", "id": "P001-C07-N005", "location": "Section 7.2.3, p. 261 (PDF p. 279), first sentence", "pdf_page": 279, "printed_page": "261", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "``Using the tools developed in Section 5.2.4 (\\hl{\\emph{Plug-In Estimation and Conditional Expectation Functions}}), it is possible to directly wed the causal identification assumptions here to the parametric models discussed in Chapter 5.''", "corrected": "``Using the tools developed in Section 5.2.4 (\\hl{\\emph{Maximum Likelihood Plug-In Estimation}}), it is possible to directly wed the causal identification assumptions here to the parametric models discussed in Chapter 5.''", "evidence": "The evidence is a check of the heading of Section 5.2.4 on p.~197 and of the table of contents. The printed title occurs nowhere else in the book.", "problem": "The title in parentheses is not the title of Section 5.2.4, which is ``Maximum Likelihood Plug-In Estimation'' (p.~197). The section number is right.", "what_survives": "The reference points to the right section, and nothing else changes." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-N005", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "recalculation_from_source", "exhibits": [ { "alt": "Crop of the last four lines of the page. The text reads: 'In the above example, we used Y 1(1) = 2 to impute Y 2(1) because P 1 = 0.33 is closest to P 2 = 0.14. But P 4 = 0.35 is almost as close to P 1, so Y 4(1) = 10 probably'. A red box marks the second P 1.", "caption": "Foot of p. 263 (PDF p. 281), paragraph after Example 7.2.4. The marked symbol names unit 1 where the comparison is with unit 2.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-N006.png" } ], "formal_release": null, "formal_statements": [ { "id": "P001-T95", "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T95" } ], "headline": "The propensity score of unit 4 is compared with that of unit 2, whose outcome is imputed, not with that of unit 1.", "id": "P001-C07-N006", "location": "Paragraph after Example 7.2.4, p. 263 (PDF p. 281), last line; the sentence continues on p. 264", "pdf_page": 281, "printed_page": "263", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P001-T95" ], "text": { "as_printed": "``In the above example, we used $Y_1(1)=2$ to impute $Y_2(1)$ because $P_1=0.33$ is closest to $P_2=0.14$. But $P_4=0.35$ is almost as close to $\\hl{P_1}$, so $Y_4(1)=10$ probably also gives us some information about $\\mathrm{E}[Y_i(1)\\mid p_D(\\mathbf{X}_i)=0.14]$.''", "corrected": "``But $P_4=0.35$ is almost as close to $\\hl{P_2}$, so $Y_4(1)=10$ probably also gives us some information about $\\mathrm{E}[Y_i(1)\\mid p_D(\\mathbf{X}_i)=0.14]$.''\n\nHere $P_j$ denotes $p_D(\\mathbf{X}_j)$.", "evidence": "The evidence is a recalculation from the propensity scores in the table of Example 7.2.4, which are $0.33$, $0.14$ and $0.35$ for units 1, 2 and 4.", "problem": "The outcome being imputed is that of unit 2. The distance that matters is the distance from $P_2=0.14$, which is $0.19$ for unit 1 and $0.21$ for unit 4. The distance between $P_4$ and $P_1$ is $0.02$, which is a different comparison. The symbols $P_1$, $P_2$ and $P_4$ are not introduced: the tables of Example 7.2.4 write $p_D(\\mathbf{X}_i)$.", "what_survives": "The matching example, its donors and its estimate $7/6$ are unchanged, and so is the point that nearest-neighbor matching does not use the information in the other units." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-N006", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of footnote 14, five lines. It says that placebo tests should not be confused with balance tests after matching or inverse probability weighting, and ends: 'By Theorem 7.1.16 (Strong Ignorability and the Propensity Score), if we have the right propensity score specification, we know that the balancing property ensures that we will have statistical independence of D i and X i.' A red box marks the words 'statistical independence of D i and X i'.", "caption": "Footnote 14, p. 270 (PDF p. 288). The marked words state the independence of $D_i$ and $\\mathbf{X}_i$ without the conditioning on the propensity score.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-N007.png" } ], "formal_release": null, "formal_statements": [], "headline": "The balancing property makes $D_i$ and $\\mathbf{X}_i$ independent conditional on the propensity score $p_D(\\mathbf{X}_i)$, not unconditionally.", "id": "P001-C07-N007", "location": "Footnote 14, p. 270 (PDF p. 288), last sentence", "pdf_page": 288, "printed_page": "270", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "``By Theorem 7.1.16 (\\emph{Strong Ignorability and the Propensity Score}), if we have the right propensity score specification, we know that the ``balancing property'' ensures that we will have \\hl{statistical independence of $D_i$ and $\\mathbf{X}_i$}.''", "corrected": "``By Theorem 7.1.16 (\\emph{Strong Ignorability and the Propensity Score}), if we have the right propensity score specification, we know that the ``balancing property'' ensures that we will have statistical independence of $D_i$ and $\\mathbf{X}_i$ \\hl{conditional on $p_D(\\mathbf{X}_i)$}.''", "intended_meaning": "The footnote contrasts placebo tests with balance tests after matching or inverse probability weighting. In that setting the independence meant is the one that Theorem 7.1.16 states, $D_i\\perp\\!\\!\\!\\perp\\mathbf{X}_i\\mid p_D(\\mathbf{X}_i)$: among units with the same propensity score, treatment is independent of the covariates. For weighting, the corresponding statement is that $D_i$ and $\\mathbf{X}_i$ are independent in the population reweighted by the inverse probability weights of the true propensity score. The sentence does not mean independence in the original population, which fails when the propensity score varies with $\\mathbf{X}_i$. For example, let $X_i$ be Bernoulli with parameter $1/2$, and let $\\Pr[D_i=1\\mid X_i]$ equal $1/4$ when $X_i=0$ and $3/4$ when $X_i=1$; then $D_i$ and $X_i$ are dependent. The theorem concerns the population and does not ensure exact balance in each matched or weighted sample." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-N007", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of seven lines from page 273. A red outline surrounds the words variation in X sub i, at the end of the sentence that begins We could even define alternative populations. The line above the sentence ends treatment assignments along all salient dimensions. The lines below restrict attention to observations whose propensity score lies between epsilon and one minus epsilon, for a value of epsilon in the interval from 0 to .5.", "caption": "Section 7.3.1, second paragraph, p. 273 (PDF p. 291). The outline marks “variation in $\\mathbf{X}_i$”. The line above speaks of variation in treatment assignments, and the next sentence trims on the propensity score $p_D(\\mathbf{X}_i)$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-N008.png" } ], "formal_release": null, "formal_statements": [], "headline": "In Section 7.3.1, the covariate profiles to keep are those with variation in the treatment $D_i$, not in the covariates $\\mathbf{X}_i$.", "id": "P001-C07-N008", "location": "Section 7.3.1, second paragraph, first sentence, p. 273 (PDF p. 291)", "pdf_page": 291, "printed_page": "273", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "We could even define alternative populations from the population under study, and condition on the types of observations that have some probability of having variation in $\\hl{\\mathbf{X}_i}$.", "corrected": "We could even define alternative populations from the population under study, and condition on the types of observations that have some probability of having variation in $\\hl{D_i}$.", "evidence": "The evidence is a check of the page and a small example: if $\\mathbf{X}_i$ is constant and $D_i$ is Bernoulli with parameter $1/2$, then $p_D(\\mathbf{X}_i) = 1/2$, and positivity holds although $\\mathbf{X}_i$ does not vary.", "problem": "The sentence names the wrong variable. A type of observation is a covariate profile, and overlap at a profile requires that both values of the treatment $D_i$ occur there with positive probability, whether or not $\\mathbf{X}_i$ varies. The preceding paragraph ends with ``variation in treatment assignments along all salient dimensions'', and the next sentence restricts attention to $\\epsilon < p_D(\\mathbf{X}_i) < 1-\\epsilon$, which is a condition on the probability of treatment.", "what_survives": "The trimming rule $\\epsilon < p_D(\\mathbf{X}_i) < 1-\\epsilon$, the positivity of the retained subpopulation and the reasons given for changing the target population are unaffected; only the variable named in this sentence changes." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-N008", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of nine lines from page 273, with three red outlines. The first surrounds epsilon in (0, .5) at the start of the second line. The second surrounds the abbreviation CIA in the sentence The ATT is nonparametrically identified under the CIA and a weaker positivity assumption. The third surrounds the condition that follows it: p sub D of X sub i is less than epsilon, which is less than 1.", "caption": "Section 7.3.1, second paragraph, p. 273 (PDF p. 291). The lower outlines mark the abbreviation CIA and the condition $p_D(\\mathbf{X}_i) < \\epsilon < 1$ in the sentence on the ATT. The upper outline marks the earlier use of $\\epsilon$, as a trimming margin in $(0,.5)$.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-N009.png" } ], "formal_release": null, "formal_statements": [], "headline": "In the positivity condition for the ATT, $\\epsilon$ is a new upper bound below one and not the trimming margin introduced two sentences earlier; CIA abbreviates the conditional independence assumption.", "id": "P001-C07-N009", "location": "Section 7.3.1, second paragraph, sentence beginning “The ATT is nonparametrically identified”, p. 273 (PDF p. 291)", "pdf_page": 291, "printed_page": "273", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "The ATT is nonparametrically identified under the \\hl{CIA} and a weaker positivity assumption, $p_D(\\mathbf{X}_i) < \\hl{\\epsilon} < 1$.", "corrected": "The ATT is nonparametrically identified under the \\hl{conditional independence assumption (CIA)} and a weaker positivity assumption: \\hl{for some $\\delta > 0$, $p_D(\\mathbf{X}_i) < 1 - \\delta$ on the treated covariate support}.", "intended_meaning": "CIA is the conditional independence assumption of Definition 7.1.12; the abbreviation occurs only twice in the book, both times on this page, and is never expanded (see \\fid{P001-BACK-N002}). The condition bounds the propensity score above by a constant less than one, so that control units occur at every covariate profile of the treated. This is sufficient for the ATT: under conditional independence, $\\mathrm{E}[Y_i(0) \\mid D_i = 1]$ is the average of $\\mathrm{E}[Y_i \\mid D_i = 0, \\mathbf{X}_i = x]$ over the distribution of $\\mathbf{X}_i$ among the treated, which requires no lower bound on the propensity score. Two sentences earlier, $\\epsilon \\in (0,.5)$ is the trimming margin in $\\epsilon < p_D(\\mathbf{X}_i) < 1-\\epsilon$. With that value carried over, the printed condition is not the earlier one with its lower bound removed: for $\\epsilon = 1/4$, a constant propensity score of $3/5$ satisfies the two-sided condition and fails $p_D(\\mathbf{X}_i) < \\epsilon$." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-N009", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of ten lines from page 279, with two red outlines, each running over a line break. The upper one surrounds the words With the exception of the (potentially pathological) matching and IPW estimators. The lower one surrounds the words no estimated effect of gender that approaches traditional levels of statistical significance. The crop ends in the middle of that sentence, at the foot of the main text of the page.", "caption": "Section 7.5, p. 279 (PDF p. 297). The lower outline marks the statement that no estimated effect “approaches traditional levels of statistical significance”. The upper outline marks the exception for matching and IPW at the head of the paragraph. The sentence continues on p. 280.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-N012.png" } ], "formal_release": null, "formal_statements": [], "headline": "The statement that no estimate approaches significance covers the four ignorability-based estimates that remain once matching and IPW are set aside; none of them is significant at the 5% level.", "id": "P001-C07-N012", "location": "Section 7.5, paragraph beginning “Table 7.5.1 summarizes these estimates”, sentence beginning “After conditioning”, pp. 279–280 (PDF pp. 297–298)", "pdf_page": 297, "printed_page": "279–280", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "After conditioning on their observable differences from women, we find \\hl{no estimated effect of gender that approaches traditional levels of statistical significance}, though our estimates still suggest that---all else equal---being male decreases voter support for one's immigration application.", "corrected": "After conditioning on their observable differences from women \\hl{and setting aside the potentially pathological matching and IPW estimates, none of the remaining estimates is statistically significant at the 5\\% level}, though our estimates still suggest that---all else equal---being male decreases voter support for one's immigration application.", "intended_meaning": "The paragraph begins by excepting ``the (potentially pathological) matching and IPW estimators'', and the sentence summarizes the four estimates that remain: OLS (Linear), OLS (Unpooled), stabilized IPW and doubly robust. Their two-tailed normal $p$-values, computed from the printed estimates and standard errors, are $0.165$, $0.076$, $0.675$ and $0.094$. None is significant at the level $0.05$ named on p.~278, and all four estimates are positive. With that scope the sentence is correct, although ``approaches'' has no stated cutoff. Taken over all six adjusted rows it would not be, since for the IPW row ($3.731$, standard error $1.088$) I find $p = 0.0006$." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-N012", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "recalculation_from_source", "exhibits": [ { "alt": "Crop of Table 7.5.1, Estimates for Effect of Gender on Percentage No Votes. The columns are Estimator, Target, Estimate and SE, and there are two rows for each of nine estimators: Lower Bound Plug-In, Upper Bound Plug-In, Random Assignment, OLS (Linear), OLS (Unpooled), Matching, IPW, Stabilized IPW and Doubly Robust. A red outline surrounds the SE column, whose eighteen entries run from 0.388 to 1.988. The note under the table describes the two targets and cites Hainmueller and Hangartner 2013; it does not mention the standard errors.", "caption": "Table 7.5.1, p. 280 (PDF p. 298). The outline marks the SE column. Neither the table nor the note under it says how the standard errors were computed.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-C07-N013.png" } ], "formal_release": null, "formal_statements": [], "headline": "Section 7.5 does not say how the standard errors in Table 7.5.1 and the two $p$-values in the text were computed.", "id": "P001-C07-N013", "location": "Table 7.5.1, SE column, p. 280 (PDF p. 298); footnote 20, p. 279 (PDF p. 297); the $p$-values on pp. 278–279", "pdf_page": 298, "printed_page": "280", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "\\begin{audittable}{llrr}{Table 7.5.1 (p.~280): the column headings and four of the eighteen rows.}\nEstimator & Target & Estimate & \\hl{SE} \\\\\nRandom Assignment & $\\mathrm{E}[Y_i(0)]$ & 35.780 & 0.557 \\\\\n & $\\mathrm{E}[\\tau_i]$ & 1.335 & 0.685 \\\\\nOLS (Linear) & $\\mathrm{E}[Y_i(0)]$ & 36.206 & 0.487 \\\\\n & $\\mathrm{E}[\\tau_i]$ & 0.706 & 0.508 \\\\\n\\end{audittable}\n\n\\emph{Text, p.~278.} \\ldots\\ we have a point estimate of $\\widehat{\\mathrm{E}}[\\tau_i] = 1.335$, with a two-tailed $p$-value (against the null hypothesis that $\\mathrm{E}[\\tau_i] = 0$) of 0.051\\ldots\n\n\\emph{Text, p.~279.} \\ldots\\ as the $p$-value against the null hypothesis of zero ATE is 0.165.\n\n\\emph{Footnote 20, p.~279.} \\ldots\\ but the bootstrap standard errors are likely to adequately characterize sampling variability.", "corrected": "I would add a note under Table 7.5.1 stating how the standard errors were computed (the resampling scheme, the number of replications, and whether the regressions, the propensity scores and the Winsorization were recomputed in each replicate) and which reference distribution the $p$-values in the text use. These details must come from the original analysis; they are not supplied here.", "intended_meaning": "The chapter's discussion of inference points to the bootstrap: for the IPW estimators it prescribes ``re-estimating the propensity scores in each bootstrap sample'' (p.~267), and footnote 20 speaks of bootstrap standard errors. The two $p$-values agree with the two-tailed normal approximation applied to the printed estimates and standard errors: the ratios $1.335/0.685$ and $0.706/0.508$ give $0.051$ and $0.165$. I find no error in these reported values. The application leaves out the resampling scheme, the number of replications, whether the propensity scores and the Winsorization were recomputed in each replicate, and the reference distribution of the $p$-values." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-C07-N013", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Four lines of text from the Introduction. A red box marks the words 'all common abbreviations' in the sentence 'In addition, we provide a Glossary of Common Abbreviations, which gives the meanings of all common abbreviations used in this book'.", "caption": "Introduction, p. xviii (PDF p. 18): the sentence that describes the Glossary of Common Abbreviations, with “all common abbreviations” marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-FRONT-N001.png" } ], "formal_release": null, "formal_statements": [], "headline": "The Introduction describes the Glossary of Common Abbreviations as covering all common abbreviations, although it lists only some of them.", "id": "P001-FRONT-N001", "location": "Introduction, p. xviii (PDF p. 18), last sentence of the first paragraph", "pdf_page": 18, "printed_page": "xviii", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\emph{p.~xviii.} In addition, we provide a Glossary of Common Abbreviations, which gives the meanings of \\hl{all common abbreviations} used in this book, along with references to where they are defined in the text.", "corrected": "\\emph{p.~xviii.} In addition, we provide a Glossary of Common Abbreviations, which gives the meanings of \\hl{selected} common abbreviations used in this book, along with references to where they are defined in the text.", "evidence": "A check of the pages, with a count of the abbreviations that stand alone in the text of pp.~3--281: CI has 18 uses, APD 19 and PDA 14, against 10 for NRCM, 12 for CMT and 19 for KL. Which abbreviations count as common is a judgment, not a computed fact.", "problem": "The glossary on pp.~286--287 has 18 entries. The text of the book uses 17 abbreviations that the glossary does not list (CI, APD, PDA, ATT, ATC, PATE, ACE, AMCE, MAE, IQR, ECDF, LOESS, ANOVA, CIA, MKLD, and the estimator labels DM and SIPW), and CI, APD and PDA occur about as often as the listed NRCM, CMT and KL.", "what_survives": "No result is affected. This entry corrects only the description of the glossary on p.~xviii. The missing glossary entries and the expansion of CIA are treated in \\fid{P001-BACK-N002}; the wording on p.~202 is treated in \\fid{P001-C05-N006}." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-FRONT-N001", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Eight lines of the list of names in the acknowledgments. Red boxes mark 'Josh Kalla', printed between Anand Gupta and Sarah Hamerling, and 'Tchetgen-Tchetgen' in 'Eric Tchetgen-Tchetgen'.", "caption": "Introduction, p. xviii (PDF p. 18): the list of names in the acknowledgments, with the two names marked.", "url": "https://pmaronow.github.io/llm-audit/works/P001/exhibits/P001-FRONT-N002.png" } ], "formal_release": null, "formal_statements": [], "headline": "In the acknowledgments, “Tchetgen-Tchetgen” should be “Tchetgen Tchetgen”, and Josh Kalla is out of alphabetical order.", "id": "P001-FRONT-N002", "location": "Introduction, “Who helped to write this book?”, p. xviii (PDF p. 18), list of names in the first paragraph", "pdf_page": 18, "printed_page": "xviii", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\ldots\\ Robin Gomila, Don Green, Anand Gupta, \\hl{Josh Kalla}, Sarah Hamerling, Erin Hartman, Jennifer Hill, \\ldots\\ Brandon Stewart, Eric \\hl{Tchetgen-Tchetgen}, Dustin Tingley, Teppei Yamamoto, \\ldots", "corrected": "\\ldots\\ Robin Gomila, Don Green, Anand Gupta, Sarah Hamerling, Erin Hartman, Jennifer Hill, Will Hunt, Jun Hwang, \\hl{Josh Kalla}, Donald Lee, \\ldots\\ Brandon Stewart, Eric \\hl{Tchetgen Tchetgen}, Dustin Tingley, Teppei Yamamoto, \\ldots", "evidence": "A check of the page, and a public source for the spelling: the faculty profile at the Wharton School of the University of Pennsylvania is headed ``Eric J. Tchetgen Tchetgen'' (\\url{https://statistics.wharton.upenn.edu/profile/ett/}).", "problem": "The surname is printed with a hyphen that it does not carry. The rest of the list is in alphabetical order by surname, and Kalla stands between Gupta and Hamerling.", "what_survives": "No one is misidentified, and nothing depends on either point." }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-FRONT-N002", "work": { "arxiv_id": null, "doi": "10.1017/9781316831762", "title": "Foundations of Agnostic Statistics", "url": "https://pmaronow.github.io/llm-audit/works/P001/" }, "work_id": "P001" } ], "lean": { "applicable": true, "guide": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/docs/works/P001.md", "repository": "https://github.com/pmaronow/llm-audit-lean", "targets": [ { "declarations": [ { "claim_revision_id": "catalog:9d624fe6740e42a8f5039b282feb23714db12ee1deb5b1f3206f939e249cbe14", "conditional_on": [], "credit_reason": "The construction is sufficient as the credited witness: any continuous linear Hadamard derivative sends the limiting zero direction to zero, contradicting the displayed unit mean quotient. No full-negation certificate is claimed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.geometric_mean_perturbation_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GLP0011B1.lean#L197", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:9d624fe6740e42a8f5039b282feb23714db12ee1deb5b1f3206f939e249cbe14", "conditional_on": [], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.geometric_partial_mass", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1037", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:9d624fe6740e42a8f5039b282feb23714db12ee1deb5b1f3206f939e249cbe14", "conditional_on": [], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.geometric_partial_first_moment", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6003", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:9d624fe6740e42a8f5039b282feb23714db12ee1deb5b1f3206f939e249cbe14", "conditional_on": [], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.geometric_partial_second_moment", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6015", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:9d624fe6740e42a8f5039b282feb23714db12ee1deb5b1f3206f939e249cbe14", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T1", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T1" }, "html": { "body": "

The printed claim is false: A finite-moment geometric law has uniformly vanishing scaled CDF perturbations with unit mean difference quotients. pp. 118, 120 (PDF pp. 136, 138), Section 3.3.1 and footnote 31.

", "headline": "Section 3.3.1, the sentence 'our assumption about finite variance was sufficient to imply that the population mean is well behaved', read with footnote 31, as printed", "location": "p. 120 (PDF p. 138), paragraph after Theorem 3.3.6, and footnote 31. The mean functional TET_{\\mathrm{E}}TE​: p. 118 (PDF p. 136)" }, "id": "P001-T1", "text": { "body": "The printed claim is false: A finite-moment geometric law has uniformly vanishing scaled CDF perturbations with unit mean difference quotients. pp. 118, 120 (PDF pp. 136, 138), Section 3.3.1 and footnote 31.", "headline": "Section 3.3.1, the sentence 'our assumption about finite variance was sufficient to imply that the population mean is well behaved', read with footnote 31, as printed", "location": "p. 120 (PDF p. 138), paragraph after Theorem 3.3.6, and footnote 31. The mean functional $T_{\\mathrm{E}}$: p. 118 (PDF p. 136)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T1" }, { "declarations": [ { "claim_revision_id": "catalog:943ab1ecb15aa9e55762be4526dc80b7d8c07006e6616d4d5f918237e834f2a3", "conditional_on": [], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.empirical_mean_fixed_sample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6312", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:943ab1ecb15aa9e55762be4526dc80b7d8c07006e6616d4d5f918237e834f2a3", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.sample_mean_clt_correction", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6322", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:943ab1ecb15aa9e55762be4526dc80b7d8c07006e6616d4d5f918237e834f2a3", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T2", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T2" }, "html": { "body": "

pp. 109, 119–120 (PDF pp. 127, 137–138). The empirical mean equals the sample mean and has the finite-positive-variance Gaussian limit, while a finite-variance geometric witness violates the zero-direction requirement for a Hadamard derivative.

", "headline": "Section 3.3.1, the example offered for the 'usual plug-in regularity conditions', corrected: asymptotic normality of the sample mean from the Central Limit Theorem, not from Hadamard differentiability of the mean", "location": "p. 120 (PDF p. 138), paragraph after Theorem 3.3.6. Theorem 3.2.24: p. 109 (PDF p. 127). Example 3.3.4: p. 119 (PDF p. 137)" }, "id": "P001-T2", "text": { "body": "pp. 109, 119–120 (PDF pp. 127, 137–138). The empirical mean equals the sample mean and has the finite-positive-variance Gaussian limit, while a finite-variance geometric witness violates the zero-direction requirement for a Hadamard derivative.", "headline": "Section 3.3.1, the example offered for the 'usual plug-in regularity conditions', corrected: asymptotic normality of the sample mean from the Central Limit Theorem, not from Hadamard differentiability of the mean", "location": "p. 120 (PDF p. 138), paragraph after Theorem 3.3.6. Theorem 3.2.24: p. 109 (PDF p. 127). Example 3.3.4: p. 119 (PDF p. 137)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T2" }, { "declarations": [ { "claim_revision_id": "catalog:15fbc3059cb294954e39c7b6c0a73c90f765276ccb059a1d25bb4fb5311c8799", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.pda_extrapolation_witness_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GLP0012B4.lean#L25", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:15fbc3059cb294954e39c7b6c0a73c90f765276ccb059a1d25bb4fb5311c8799", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.pda_joint_cef_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GP0012B2b.lean#L15", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:15fbc3059cb294954e39c7b6c0a73c90f765276ccb059a1d25bb4fb5311c8799", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"AronowAudit/P001/Wave1/Gap/RepairT3.lean:59 negates the universal contract at line 33: any two admissible CEF versions differentiable on the positive-density support must have a common actual derivative at E[X]. This matches the target's stated refutation of distributional PDA well-definedness; a scalar counterexample suffices against its vector assertion. The measurable-version formulation is authorized by the standing assumptions and does not remove a refuting case.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.pda_at_mean_printed_contract_refuted", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/RepairT3.lean#L44", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:15fbc3059cb294954e39c7b6c0a73c90f765276ccb059a1d25bb4fb5311c8799", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T3", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T3" }, "html": { "body": "

pp. 166–168 (PDF pp. 184–186), Definition 4.3.7. The fixed independent gap-uniform law has continuously differentiable conditional-mean extensions with arbitrary derivatives at its mean, refuting the reading that all global CEF extensions must determine the same derivative there even outside the printed CEF domain.

", "headline": "Definition 4.3.7 (Partial Derivative at the Average) and the remark that the PDA 'has a well-defined meaning', as printed: a joint distribution for which the PDA is not determined", "location": "Definition 4.3.7: p. 167 (PDF p. 185). Remark: p. 168 (PDF p. 186). Footnote 13: p. 166 (PDF p. 184). Definition 2.2.15: p. 69 (PDF p. 87)" }, "id": "P001-T3", "text": { "body": "pp. 166–168 (PDF pp. 184–186), Definition 4.3.7. The fixed independent gap-uniform law has continuously differentiable conditional-mean extensions with arbitrary derivatives at its mean, refuting the reading that all global CEF extensions must determine the same derivative there even outside the printed CEF domain.", "headline": "Definition 4.3.7 (Partial Derivative at the Average) and the remark that the PDA 'has a well-defined meaning', as printed: a joint distribution for which the PDA is not determined", "location": "Definition 4.3.7: p. 167 (PDF p. 185). Remark: p. 168 (PDF p. 186). Footnote 13: p. 166 (PDF p. 184). Definition 2.2.15: p. 69 (PDF p. 87)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T3" }, { "declarations": [ { "claim_revision_id": "catalog:c9aa75ec0f45fb32256b0525f182596901ee6a74e2f8c574abf88d269bcc73f0", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': \"P001/P001_Foundations.lean:8270 places hd, existence of a coordinate derivative at the mean, before the entire conclusion, including Set.EqOn m m' U at line 8272. T4's formal content instead says 'Then m = tilde m on U; in particular, if m has a partial derivative', so equality is unconditional on differentiability. For example, a uniform covariate on (-1,1) with Y = |X| has continuous CEF |x| and mean zero, but no derivative there; the declaration does not supply the claimed equality for this case. Extending local continuous CEF representatives outside U creates no additional material restriction, and derivative transfer itself is faithfully represented.\", 'suggested_fix': \"Conclude Set.EqOn m m' U first, and move k, a and hd inside a separate implication in the second conjunct: equality on U together with, for every coordinate and derivative value, transfer of any existing derivative.\", 'second_review': \"P001/P001_Foundations.lean:8270 places the derivative-existence hypothesis hd before the entire conclusion, including equality of the continuous CEF versions on U at P001/P001_Foundations.lean:8272. T4 instead says 'Then m = tilde m on U; in particular, if m has a partial derivative', making equality independent of derivative existence. A uniform covariate on (-1,1) with Y=|X| gives a continuous CEF without a derivative at its mean, so a nontrivial class of equality cases is missing; export equality first and make derivative transfer a separate implication.\", 'severity': 'major'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.pda_unique_on_open_support", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7024", "status": "proved", "support_scope": "limited", "used_in_solution": false }, { "claim_revision_id": "catalog:c9aa75ec0f45fb32256b0525f182596901ee6a74e2f8c574abf88d269bcc73f0", "conditional_on": [], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.pda_derivative_transfers_on_open_support", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5971", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:c9aa75ec0f45fb32256b0525f182596901ee6a74e2f8c574abf88d269bcc73f0", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T4", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T4" }, "html": { "body": "

pp. 167–168 (PDF pp. 185–186), Definition 4.3.7. Continuous local CEF representatives agree on an open set whenever they agree Lebesgue almost everywhere there. Equality requires no derivative. Every existing coordinate derivative at the covariate mean transfers, with the same value. The positive-PDF neighborhood condition implies this version independence but is not necessary, as the uniform PDF with a hole at its mean demonstrates.

", "headline": "Definition 4.3.7 (Partial Derivative at the Average), corrected with a support condition at the covariate mean", "location": "p. 167 (PDF p. 185), Definition 4.3.7; p. 168 (PDF p. 186), first paragraph" }, "id": "P001-T4", "text": { "body": "pp. 167–168 (PDF pp. 185–186), Definition 4.3.7. Continuous local CEF representatives agree on an open set whenever they agree Lebesgue almost everywhere there. Equality requires no derivative. Every existing coordinate derivative at the covariate mean transfers, with the same value. The positive-PDF neighborhood condition implies this version independence but is not necessary, as the uniform PDF with a hole at its mean demonstrates.", "headline": "Definition 4.3.7 (Partial Derivative at the Average), corrected with a support condition at the covariate mean", "location": "p. 167 (PDF p. 185), Definition 4.3.7; p. 168 (PDF p. 186), first paragraph" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T4" }, { "declarations": [ { "claim_revision_id": "source-claim-import-2026-11-08:P001-T5:1", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"AronowAudit/P001/Wave1/Gap/TailBSieveEscape.lean:765 inherits the statement at AronowAudit/P001/P001_Foundations.lean:1096, which gives the specified floor(n^(3/4)) degree and the local-L2 liminf bound for every positive delta and epsilon along n=t^4. Squared residuals match the target's explicit correction, and allowing arbitrary minimizer selections includes all measurable statistical fits. Outer probabilities and the closed integration interval do not exclude any intended case; the supplementary endpoint example is outside this row's local-L2 component.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.TailB_sieve_degree_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/TailBSieveEscape.lean#L766", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "source-claim-import-2026-11-08:P001-T5:1", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T5", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T5" }, "html": { "body": "

pp. 168–169 (PDF pp. 186–187), Definition 4.3.9. The squared-residual polynomial sieve with J_n = floor(n^(3/4)) satisfies J_n → infinity and J_n/n → 0. For iid independent U[0,1] covariates and standard Gaussian outcomes, the CEF is smooth and zero, all moments exist, and least-squares fits with Borel measurable coefficients exist. Every such fit has, for every 0 < delta ≤ 1 and epsilon > 0, local squared L2 error exceeding epsilon with lower limiting probability at least 1/2 along n = t^4; it therefore fails local L2 consistency on a region of positive probability mass. For a separate degree-floor(sqrt(n)) sequence, fits with Borel measurable coefficients also exist, and every such sequence satisfies Pr[|fit_(t^2)(1)| > 1/4] ≥ 1/4 for every integer t ≥ 2, including experiments with no coupling across sample sizes. Thus the printed rate condition is insufficient for the stated local consistency; no assertion about consistency at slower degrees is made. The usual measure-theoretic conditions are assumed.

", "headline": "Definition 4.3.9 (Polynomial Least Squares Sieve Estimator), the rate condition 'Jn→∞J_n \\to \\inftyJn​→∞ and 1nJn→0\\frac{1}{n}J_n \\to 0n1​Jn​→0' as a sufficient condition for consistency, as printed", "location": "Section 4.3.5: p. 168 (PDF p. 186); Definition 4.3.9: pp. 168–169 (PDF pp. 186–187); the paragraph after it: p. 169 (PDF p. 187)" }, "id": "P001-T5", "text": { "body": "pp. 168–169 (PDF pp. 186–187), Definition 4.3.9. The squared-residual polynomial sieve with J_n = floor(n^(3/4)) satisfies J_n → infinity and J_n/n → 0. For iid independent U[0,1] covariates and standard Gaussian outcomes, the CEF is smooth and zero, all moments exist, and least-squares fits with Borel measurable coefficients exist. Every such fit has, for every 0 < delta ≤ 1 and epsilon > 0, local squared L2 error exceeding epsilon with lower limiting probability at least 1/2 along n = t^4; it therefore fails local L2 consistency on a region of positive probability mass. For a separate degree-floor(sqrt(n)) sequence, fits with Borel measurable coefficients also exist, and every such sequence satisfies Pr[|fit_(t^2)(1)| > 1/4] ≥ 1/4 for every integer t ≥ 2, including experiments with no coupling across sample sizes. Thus the printed rate condition is insufficient for the stated local consistency; no assertion about consistency at slower degrees is made. The usual measure-theoretic conditions are assumed.", "headline": "Definition 4.3.9 (Polynomial Least Squares Sieve Estimator), the rate condition '$J_n \\to \\infty$ and $\\frac{1}{n}J_n \\to 0$' as a sufficient condition for consistency, as printed", "location": "Section 4.3.5: p. 168 (PDF p. 186); Definition 4.3.9: pp. 168–169 (PDF pp. 186–187); the paragraph after it: p. 169 (PDF p. 187)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T5" }, { "declarations": [ { "claim_revision_id": "catalog:22578bb7e6f4b3695b60da6373f58767235cac8cd098fbff7c5a604d22590c2e", "conditional_on": [], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.sieve_consistency_stable_gram_proof", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GLP0013B5.lean#L497", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:22578bb7e6f4b3695b60da6373f58767235cac8cd098fbff7c5a604d22590c2e", "conditional_on": [], "credit_reason": "Affine standardization recovers every nondegenerate compact interval with exactly the same finite-sample constants. 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pp. 168–169 (PDF pp. 186–187), Definition 4.3.9. Polynomial least squares admits a measurable minimizing selection, including singular designs. On compact support a continuous CEF and bounded conditional variance give L2 consistency under growing degree, vanishing dimension/sample ratio and a stable population-normalized Gram matrix. The attained support supremum gives the general envelope rate. For uniform covariates on every nondegenerate compact interval the envelope is at most K², the Euclidean operator-norm deviation probability is at most 4(K³−K)/n, and the cubic rate suffices for Gram stability.

", "headline": "Definition 4.3.9 (Polynomial Least Squares Sieve Estimator) and the consistency claim after it, corrected with a sufficient degree-growth condition", "location": "pp. 168–169 (PDF pp. 186–187)" }, "id": "P001-T6", "text": { "body": "pp. 168–169 (PDF pp. 186–187), Definition 4.3.9. Polynomial least squares admits a measurable minimizing selection, including singular designs. On compact support a continuous CEF and bounded conditional variance give L2 consistency under growing degree, vanishing dimension/sample ratio and a stable population-normalized Gram matrix. The attained support supremum gives the general envelope rate. For uniform covariates on every nondegenerate compact interval the envelope is at most K², the Euclidean operator-norm deviation probability is at most 4(K³−K)/n, and the cubic rate suffices for Gram stability.", "headline": "Definition 4.3.9 (Polynomial Least Squares Sieve Estimator) and the consistency claim after it, corrected with a sufficient degree-growth condition", "location": "pp. 168–169 (PDF pp. 186–187)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T6" }, { "declarations": [ { "claim_revision_id": "catalog:9b575e96e912dced0834aefbe58f5f23fea19a378106507910f4ec0f90a41719", "conditional_on": [], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.conditional_kl_counterexample_finite", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7455", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:9b575e96e912dced0834aefbe58f5f23fea19a378106507910f4ec0f90a41719", "conditional_on": [], "credit_reason": "The row exports the same complete refuting estimator witness as R001-15; “all clauses retained” refers to that estimator statement, not an assertion that every diagnostic formula in the target is a conjunct.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.conditional_kl_mle_wrong_target", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7557", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:9b575e96e912dced0834aefbe58f5f23fea19a378106507910f4ec0f90a41719", "conditional_on": [], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.clipped_bernoulli_mle", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6072", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:9b575e96e912dced0834aefbe58f5f23fea19a378106507910f4ec0f90a41719", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.pooled_mle_covariate_marginal_bridge", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/WP001dB1.lean#L181", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:9b575e96e912dced0834aefbe58f5f23fea19a378106507910f4ec0f90a41719", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T7", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T7" }, "html": { "body": "

The printed claim is false: The clipped pooled Bernoulli MLE converges to the marginal success probability instead of the fixed-covariate KL targets. p. 195 (PDF p. 213), Theorem 5.2.8.

", "headline": "Theorem 5.2.8 (The ML Estimator Is Consistent for the Minimum KL Divergence Predictor), with the target defined by the KL divergence at a fixed x\\mathbf{x}x, as printed", "location": "p. 195 (PDF p. 213), Theorem 5.2.8 and Definition 5.2.7" }, "id": "P001-T7", "text": { "body": "The printed claim is false: The clipped pooled Bernoulli MLE converges to the marginal success probability instead of the fixed-covariate KL targets. p. 195 (PDF p. 213), Theorem 5.2.8.", "headline": "Theorem 5.2.8 (The ML Estimator Is Consistent for the Minimum KL Divergence Predictor), with the target defined by the KL divergence at a fixed $\\mathbf{x}$, as printed", "location": "p. 195 (PDF p. 213), Theorem 5.2.8 and Definition 5.2.7" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T7" }, { "declarations": [ { "claim_revision_id": "catalog:8578ba7a98c90c37b80696200869990a24033974bd0664612d2c846775f07f6b", "conditional_on": [], "credit_reason": "Auxiliary fact explicitly asserted in T8(ii). 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Consequently it assumes uniform boundedness of M on Theta, beyond T8(ii)'s separation and positive-sample supremum convergence. The normalized three-outcome family (a(1-b), ab, 1-a), with a in (0,1], b in [1/4,3/4] and true fair binary observations, has unique separated target (1,1/2) and positive-sample uniform error log(3)*|sampleMean(Y)-1/2| tending to zero, but M tends to minus infinity as a tends to zero. Parameterwise almost-everywhere support domination and the conditional finite-entropy conclusion do not remedy this independent narrowing.\", 'suggested_fix': 'Restrict hbound to n>=1, or to all sufficiently large n, so a finite real Delta at n=0 is not required to dominate |M(t)| uniformly. Preserve the separated approximate-ML consistency and conditional KL conclusions.', 'second_review': \"P001/Wave1/Gap/WP001dB2.lean:155 imposes its finite real uniform-error bound at n=0 as well as positive n; P001/P001_Foundations.lean:701 and :79 make the zero-sample log likelihood identically zero. This forces uniform boundedness of |M(t)|, which T8(ii)'s separation and positive-sample uniform convergence do not require. For fair binary data and normalized model masses (a(1-b), ab, 1-a) with 0pp. 195–196 (PDF pp. 213–214), Theorem 5.2.8. Averaged conditional KL equals finite iterated conditional entropy minus finite expected log density. Its minimizers are precisely the expected-log maximizers. Uniform convergence in probability of the positive-sample log criterion and vanishing approximate maximization error give consistency at every separated maximum; exact product maximizers are included. Compactness and a continuous unique maximum imply separation. Consistency does not require finite entropy or finite KL. A true model has zero KL, and uniqueness identifies its parameter. All conditional laws are probability kernels; the reference measure is arbitrary.

", "headline": "Theorem 5.2.8 (The ML Estimator Is Consistent for the Minimum KL Divergence Predictor), corrected: the KL divergence averaged over the distribution of X\\mathbf{X}X", "location": "p. 195 (PDF p. 213), Theorem 5.2.8; the explanatory paragraphs on p. 196 (PDF p. 214)" }, "id": "P001-T8", "text": { "body": "pp. 195–196 (PDF pp. 213–214), Theorem 5.2.8. Averaged conditional KL equals finite iterated conditional entropy minus finite expected log density. Its minimizers are precisely the expected-log maximizers. Uniform convergence in probability of the positive-sample log criterion and vanishing approximate maximization error give consistency at every separated maximum; exact product maximizers are included. Compactness and a continuous unique maximum imply separation. Consistency does not require finite entropy or finite KL. A true model has zero KL, and uniqueness identifies its parameter. All conditional laws are probability kernels; the reference measure is arbitrary.", "headline": "Theorem 5.2.8 (The ML Estimator Is Consistent for the Minimum KL Divergence Predictor), corrected: the KL divergence averaged over the distribution of $\\mathbf{X}$", "location": "p. 195 (PDF p. 213), Theorem 5.2.8; the explanatory paragraphs on p. 196 (PDF p. 214)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T8" }, { "declarations": [ { "claim_revision_id": "catalog:ee4e3103c536e5dded8af26bebee79b0ccd6fb1937d7216b6e74b7d8db862ef0", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.mixture_three_point_unbounded_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GP0014B2.lean#L113", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:ee4e3103c536e5dded8af26bebee79b0ccd6fb1937d7216b6e74b7d8db862ef0", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T9", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T9" }, "html": { "body": "

The printed claim is false: The unrestricted three-component mixture likelihood is unbounded along the specified normal-scale collapse path on the three-point sample. pp. 198–201 (PDF pp. 216–219), Section 5.2.5 and Example 5.2.11.

", "headline": "Section 5.2.5 and Example 5.2.11: 'the product can be maximized to find the ML estimate', applied to the normal/log-normal/exponential mixture, as printed", "location": "Section 5.2.5: pp. 198–199 (PDF pp. 216–217). Example 5.2.11: pp. 199–200 (PDF pp. 217–218). Figure 5.2.5: p. 201 (PDF p. 219)" }, "id": "P001-T9", "text": { "body": "The printed claim is false: The unrestricted three-component mixture likelihood is unbounded along the specified normal-scale collapse path on the three-point sample. pp. 198–201 (PDF pp. 216–219), Section 5.2.5 and Example 5.2.11.", "headline": "Section 5.2.5 and Example 5.2.11: 'the product can be maximized to find the ML estimate', applied to the normal/log-normal/exponential mixture, as printed", "location": "Section 5.2.5: pp. 198–199 (PDF pp. 216–217). Example 5.2.11: pp. 199–200 (PDF pp. 217–218). Figure 5.2.5: p. 201 (PDF p. 219)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T9" }, { "declarations": [ { "claim_revision_id": "catalog:49712c38b2bdcd121552a2e643ab5eea8d05bb56b304fa74fb1b52e81f6b55a9", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.mixture_unbounded_positive_sample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GP0011B1.lean#L81", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:49712c38b2bdcd121552a2e643ab5eea8d05bb56b304fa74fb1b52e81f6b55a9", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"AronowAudit/P001/P001_Foundations.lean:1289 proves compactness, likelihood continuity and attainment on the exact seven-coordinate restriction. Its hne hypothesis matches the claim's explicit sentence, 'Assume the displayed restricted parameter set is nonempty'; the modelling description and earlier review incorrectly say that the target omits this assumption. Permitting the empty sample adds a harmless constant-likelihood case and loses no positive sample.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.mixture_compact_attainment", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1200", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:49712c38b2bdcd121552a2e643ab5eea8d05bb56b304fa74fb1b52e81f6b55a9", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T10", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T10" }, "html": { "body": "

pp. 198–201 (PDF pp. 216–219), Section 5.2.5 and Example 5.2.11. The unrestricted mixture has no likelihood maximizer on any nonempty positive sample, while every nonempty stated compact restriction has a continuous likelihood attaining its maximum.

", "headline": "Section 5.2.5 and Example 5.2.11, corrected: maximization of the mixture likelihood qualified by existence of a maximum; the unrestricted mixture has unbounded likelihood on positive samples; a compact restriction under which a maximum is attained", "location": "pp. 198–200 (PDF pp. 216–218); Figure 5.2.5, p. 201 (PDF p. 219)" }, "id": "P001-T10", "text": { "body": "pp. 198–201 (PDF pp. 216–219), Section 5.2.5 and Example 5.2.11. The unrestricted mixture has no likelihood maximizer on any nonempty positive sample, while every nonempty stated compact restriction has a continuous likelihood attaining its maximum.", "headline": "Section 5.2.5 and Example 5.2.11, corrected: maximization of the mixture likelihood qualified by existence of a maximum; the unrestricted mixture has unbounded likelihood on positive samples; a compact restriction under which a maximum is attained", "location": "pp. 198–200 (PDF pp. 216–218); Figure 5.2.5, p. 201 (PDF p. 219)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T10" }, { "declarations": [ { "claim_revision_id": "catalog:9d5260cbd3477713d9e7a91750ebc9c2c07ad96462cddb906d51519056c00348", "conditional_on": [], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.probability_limit_unique", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L984", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:9d5260cbd3477713d9e7a91750ebc9c2c07ad96462cddb906d51519056c00348", "conditional_on": [], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.unused_component_nonidentification", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7115", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:9d5260cbd3477713d9e7a91750ebc9c2c07ad96462cddb906d51519056c00348", "conditional_on": [], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.nonidentification_no_two_consistent_targets", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7124", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:9d5260cbd3477713d9e7a91750ebc9c2c07ad96462cddb906d51519056c00348", "conditional_on": [], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.unused_component_counterexample_full", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7130", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:9d5260cbd3477713d9e7a91750ebc9c2c07ad96462cddb906d51519056c00348", "conditional_on": [], "credit_reason": "Auxiliary attainment component, not credited as the inconsistency argument by itself. 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pp. 198–199 (PDF pp. 216–217), Section 5.2.5. The zero-weight mixture counterexample refutes the reading that representation by a proper subset of components alone ensures consistency of the displayed full-vector MLE: two distinct full parameters produce identical iid experiments, so no estimator is consistent at both, despite attained likelihood maxima.

", "headline": "Section 5.2.5, the sentence 'If any subset of the models ... jointly characterize the conditional distribution of YYY, then this estimator is consistent and asymptotically efficient', as printed", "location": "p. 199 (PDF p. 217), second paragraph; the mixture and its parameter vector: pp. 198–199 (PDF pp. 216–217)" }, "id": "P001-T11", "text": { "body": "pp. 198–199 (PDF pp. 216–217), Section 5.2.5. The zero-weight mixture counterexample refutes the reading that representation by a proper subset of components alone ensures consistency of the displayed full-vector MLE: two distinct full parameters produce identical iid experiments, so no estimator is consistent at both, despite attained likelihood maxima.", "headline": "Section 5.2.5, the sentence 'If any subset of the models ... jointly characterize the conditional distribution of $Y$, then this estimator is consistent and asymptotically efficient', as printed", "location": "p. 199 (PDF p. 217), second paragraph; the mixture and its parameter vector: pp. 198–199 (PDF pp. 216–217)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T11" }, { "declarations": [ { "claim_revision_id": "CT-01:P001-T12:accepted", "conditional_on": [], "credit_reason": "The statement includes every law/expected-log/nonconsistency conclusion of the nonidentification component.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.zero_weight_component_irrelevant", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L8094", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "CT-01:P001-T12:accepted", "conditional_on": [], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.identified_mixture_ml_consistency", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7661", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "CT-01:P001-T12:accepted", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': \"P001/Wave1/Gap/TailBRegularEfficiency.lean:138 assumes E[H] = -I, and line 166 independently assumes QuadraticMeanModel; lines 133 and 140 also require C² log likelihoods and continuity of the population Hessian. 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p. 199 (PDF p. 217), Section 5.2.5, corrected subset sentence. A freely variable zero-weight component makes the full vector unidentified: the densities, observation laws, every iid sample law and finite expected logs agree, and no estimator is consistent at both distinct vectors. Compactness, continuity, unique identification and uniform positive-sample log-criterion convergence imply consistency of every measurable maximizer. Under the explicit common-set C2, centered L2 score, positive information, information identity, continuous integrable expected Hessian and uniform sample-Hessian conditions, each such interior ML maximizer has root-n limit N(0,I inverse). Separately, QMD of the normalized fixed-dominator model implies that every regular asymptotically linear estimator's influence covariance minus I inverse is positive semidefinite. QMD is used only for this information bound. The argmax and local-normality principles are given for arbitrary finite real criteria and specialize to the fitted mixture; the information principle allows arbitrary dominating measures. Thus unused free coordinates must be removed before a full-vector assertion.

", "headline": "Section 5.2.5, the 'any subset' sentence, corrected: consistency and efficiency of the mixture ML estimator under identification of the full parameter vector", "location": "p. 199 (PDF p. 217), second paragraph" }, "id": "P001-T12", "text": { "body": "p. 199 (PDF p. 217), Section 5.2.5, corrected subset sentence. A freely variable zero-weight component makes the full vector unidentified: the densities, observation laws, every iid sample law and finite expected logs agree, and no estimator is consistent at both distinct vectors. Compactness, continuity, unique identification and uniform positive-sample log-criterion convergence imply consistency of every measurable maximizer. Under the explicit common-set C2, centered L2 score, positive information, information identity, continuous integrable expected Hessian and uniform sample-Hessian conditions, each such interior ML maximizer has root-n limit N(0,I inverse). Separately, QMD of the normalized fixed-dominator model implies that every regular asymptotically linear estimator's influence covariance minus I inverse is positive semidefinite. QMD is used only for this information bound. The argmax and local-normality principles are given for arbitrary finite real criteria and specialize to the fitted mixture; the information principle allows arbitrary dominating measures. Thus unused free coordinates must be removed before a full-vector assertion.", "headline": "Section 5.2.5, the 'any subset' sentence, corrected: consistency and efficiency of the mixture ML estimator under identification of the full parameter vector", "location": "p. 199 (PDF p. 217), second paragraph" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T12" }, { "declarations": [ { "claim_revision_id": "catalog:727353bf14419d4dccbf96002810082fbf3ca6e6e6d0838172bf3a070a0bb0c9", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"AronowAudit/P001/Wave1/Gap/TailA/MatchingFinals.lean:13 inherits the known-score witness at AronowAudit/P001/P001_Foundations.lean:1398; MatchingFailure at line 250 states exact unbiasedness, both variance limits, the strict limiting gap and failure of scaled bootstrap-variance consistency. These are the counterexample conclusions explicitly specified by the target's general-assurance interpretation, rather than a literal negation of the qualitative word 'likely'. Known scores and the no-respondent value 50 are in the claim itself, and this row makes no numerical claim about the reported SE 1.204. The auxiliary constant inequalities and moment-bound route are supporting components rather than missing parts of the asserted inconsistency witness.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.TailA.hot_deck_bootstrap_counterexample_final", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/TailA/MatchingFinals.lean#L16", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:727353bf14419d4dccbf96002810082fbf3ca6e6e6d0838172bf3a070a0bb0c9", "conditional_on": [], "credit_reason": "The existential C is after the population, as allowed by the target; since the unit law is fixed this loses no required instance.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.TailA.hot_deck_bootstrap_uniform_second_moment_final", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/TailA/HotDeckL2.lean#L214", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:727353bf14419d4dccbf96002810082fbf3ca6e6e6d0838172bf3a070a0bb0c9", "conditional_on": [], "credit_reason": "The record’s fixed-constant back-translation is correct. Its historical proof-admission language is outside this statement-fidelity review and has not been checked.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.hot_deck_bootstrap_constant_inequalities", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6413", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:727353bf14419d4dccbf96002810082fbf3ca6e6e6d0838172bf3a070a0bb0c9", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T13", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T13" }, "html": { "body": "

p. 233 (PDF p. 251), footnote 15 and Table 6.3.1. The explicit unbiased hot-deck design refutes the universal bias-only calibration reading: its bootstrap-variance expectation limit is strictly below its sampling-variance limit, and its bootstrap variance is inconsistent.

", "headline": "Footnote 15 (hot deck imputation: 'the bootstrap standard errors are likely to adequately characterize sampling variability', with the bootstrap's failure attributed to slowly converging bias), as printed", "location": "p. 233 (PDF p. 251), footnote 15, attached to the hot deck estimate; Table 6.3.1, row 'Hot Deck Imputation'. Hot deck imputation: Section 6.2.3, pp. 224–225 (PDF pp. 242–243). The bootstrap: Section 3.4.3, p. 131 (PDF p. 149)" }, "id": "P001-T13", "text": { "body": "p. 233 (PDF p. 251), footnote 15 and Table 6.3.1. The explicit unbiased hot-deck design refutes the universal bias-only calibration reading: its bootstrap-variance expectation limit is strictly below its sampling-variance limit, and its bootstrap variance is inconsistent.", "headline": "Footnote 15 (hot deck imputation: 'the bootstrap standard errors are likely to adequately characterize sampling variability', with the bootstrap's failure attributed to slowly converging bias), as printed", "location": "p. 233 (PDF p. 251), footnote 15, attached to the hot deck estimate; Table 6.3.1, row 'Hot Deck Imputation'. Hot deck imputation: Section 6.2.3, pp. 224–225 (PDF pp. 242–243). The bootstrap: Section 3.4.3, p. 131 (PDF p. 149)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T13" }, { "declarations": [ { "claim_revision_id": "catalog:1542a097ecc3d299718f01bf26fae94fe085972477b62c46674b64672ed0f0a9", "conditional_on": [], "credit_reason": "The row covers clause (i), not by itself clause (ii), and explicit product-law nonvacuity supplies the existence interpretation.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.TailA.hot_deck_bootstrap_correction_final", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/TailA/MatchingFinals.lean#L28", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:1542a097ecc3d299718f01bf26fae94fe085972477b62c46674b64672ed0f0a9", "conditional_on": [], "credit_reason": "The particular fair ±1 noise is part of the target’s specified witness, not an added restriction on a purported arbitrary-noise claim.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.hot_deck_conditional_reuse_variance_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GP0011B2.lean#L354", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:1542a097ecc3d299718f01bf26fae94fe085972477b62c46674b64672ed0f0a9", "conditional_on": [], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.hot_deck_two_bootstrap_draw_reuse_identity_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GP0011B3.lean#L297", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:1542a097ecc3d299718f01bf26fae94fe085972477b62c46674b64672ed0f0a9", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T14", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T14" }, "html": { "body": "

p. 233 (PDF p. 251), footnote 15. The explicit hot-deck design has inconsistent bootstrap variance and exact conditional variance equal to the sum of squared donor-reuse weights divided by sample size squared.

", "headline": "Footnote 15, corrected: for hot deck (nearest-neighbour) imputation the ordinary bootstrap can give inconsistent standard errors even without bias", "location": "p. 233 (PDF p. 251), footnote 15; Table 6.3.1, row 'Hot Deck Imputation'" }, "id": "P001-T14", "text": { "body": "p. 233 (PDF p. 251), footnote 15. The explicit hot-deck design has inconsistent bootstrap variance and exact conditional variance equal to the sum of squared donor-reuse weights divided by sample size squared.", "headline": "Footnote 15, corrected: for hot deck (nearest-neighbour) imputation the ordinary bootstrap can give inconsistent standard errors even without bias", "location": "p. 233 (PDF p. 251), footnote 15; Table 6.3.1, row 'Hot Deck Imputation'" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T14" }, { "declarations": [ { "claim_revision_id": "catalog:15d6b3332c96f578b0a7b2543be98da3240f0ef3dc7ec90ae832ccbb53f3774b", "conditional_on": [], "credit_reason": "Excluded by the final review; no credit is taken from this row.", "credit_status": "wrong_credit", "fidelity_status_baseline": "wrong_credit", "lean_name": "AronowAudit.P001.Wave1.TailA.matching_bootstrap_counterexample_final", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/TailA/MatchingFinals.lean#L87", "status": "proved", "support_scope": "withdrawn", "used_in_solution": false }, { "claim_revision_id": "catalog:15d6b3332c96f578b0a7b2543be98da3240f0ef3dc7ec90ae832ccbb53f3774b", "conditional_on": [], "credit_reason": "This is the designated intercept-consistency auxiliary fact only.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.matching_score_intercept_consistent_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GP0012B1.lean#L131", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:15d6b3332c96f578b0a7b2543be98da3240f0ef3dc7ec90ae832ccbb53f3774b", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/Wave1/Gap/TailA/MatchingFinals.lean:27 inherits P001/P001_Foundations.lean:1414, whose conclusion MatchingFailure at line 250 contains the five known-score endpoint assertions in T15's witness calculation. The law at line 232 stores P=X+1/4 directly and has the specified uniform propensity, Bernoulli assignment and independent fair sign. The estimated score at lines 831-839 adds a common sample-dependent shift to P, preserving every matching distance and hence these endpoint calculations in both samples and bootstrap resamples. The bounded completion on samples without controls is harmless for the asymptotic claims and preserves exact unbiasedness; this supporting component makes no additional assertion about the printed hedged assurance.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.TailA.matching_bootstrap_counterexample_supporting_final", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/TailA/MatchingFinals.lean#L43", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:15d6b3332c96f578b0a7b2543be98da3240f0ef3dc7ec90ae832ccbb53f3774b", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T15", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T15" }, "html": { "body": "

pp. 264, 279–280 (PDF pp. 282, 297–298), Section 7.2.4 and footnote 20. The displayed matching design has an inconsistent true-score bootstrap variance and a consistent additive score-intercept estimate.

", "headline": "Section 7.2.4 (matching with estimated propensity scores: the bootstrap's failure attributed to slowly shrinking bias) and footnote 20 ('the bootstrap standard errors are likely to adequately characterize sampling variability'), as printed", "location": "p. 264 (PDF p. 282), end of Section 7.2.4, with footnote 13; p. 279 (PDF p. 297), footnote 20; p. 280 (PDF p. 298), Table 7.5.1, row 'Matching'. Matching: Section 7.2.4, pp. 262–263 (PDF pp. 280–281)" }, "id": "P001-T15", "text": { "body": "pp. 264, 279–280 (PDF pp. 282, 297–298), Section 7.2.4 and footnote 20. The displayed matching design has an inconsistent true-score bootstrap variance and a consistent additive score-intercept estimate.", "headline": "Section 7.2.4 (matching with estimated propensity scores: the bootstrap's failure attributed to slowly shrinking bias) and footnote 20 ('the bootstrap standard errors are likely to adequately characterize sampling variability'), as printed", "location": "p. 264 (PDF p. 282), end of Section 7.2.4, with footnote 13; p. 279 (PDF p. 297), footnote 20; p. 280 (PDF p. 298), Table 7.5.1, row 'Matching'. 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pp. 264, 279 (PDF pp. 282, 297), Section 7.2.4 and footnote 20. Estimated additive scores preserve the matching statistic and its bootstrap failure, and squared control-reuse weights give the conditional variance.

", "headline": "Section 7.2.4 and footnote 20, corrected: for nearest-neighbour matching the ordinary bootstrap can give inconsistent standard errors even without bias, and root-nnn consistency and bootstrap validity are separate questions", "location": "p. 264 (PDF p. 282), end of Section 7.2.4 and footnote 13; p. 279 (PDF p. 297), footnote 20" }, "id": "P001-T16", "text": { "body": "pp. 264, 279 (PDF pp. 282, 297), Section 7.2.4 and footnote 20. Estimated additive scores preserve the matching statistic and its bootstrap failure, and squared control-reuse weights give the conditional variance.", "headline": "Section 7.2.4 and footnote 20, corrected: for nearest-neighbour matching the ordinary bootstrap can give inconsistent standard errors even without bias, and root-$n$ consistency and bootstrap validity are separate questions", "location": "p. 264 (PDF p. 282), end of Section 7.2.4 and footnote 13; p. 279 (PDF p. 297), footnote 20" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T16" }, { "declarations": [ { "claim_revision_id": "catalog:cb548713b936737c6c22db0a7af038b325ae8eb56786e773b552fe7411958b6b", "conditional_on": [], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.placebo_linear_test_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/PlaceboFTest.lean#L633", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:cb548713b936737c6c22db0a7af038b325ae8eb56786e773b552fe7411958b6b", "conditional_on": [], "credit_reason": "Assigned finite population witness only; the iid size refutation is supplied by R005-01/02. The record’s label is more qualified, but its finite-scope backtranslation agrees.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.placebo_full_finite_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6771", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:cb548713b936737c6c22db0a7af038b325ae8eb56786e773b552fe7411958b6b", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/Wave1/Gap/WLP001bB2.lean:13 gives the exact twelve-atom conditional-independence witness, nonzero unique placebo slope, actual sample F/Wald growth, and rejection probabilities required for T17's iid test-failure component. The conventional F reference and HC0 Wald statistic agree with the explicit audit target, and singleton conditional factorization is equivalent to event independence on this finite support. Lines 27-30 state weighted-risk arithmetic rather than an explicit sample theorem, but T17(e)'s atom counts are exactly 240 times these weights (10 or 30), so its empirical squared-error objective is exactly 240 times placeboRisk for every coefficient vector. This finite weighted representation and elementary exact-count specialization lose no material case of the credited counterexample.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.placebo_iid_f_wald_failure", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/WLP001bB2.lean#L12", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:cb548713b936737c6c22db0a7af038b325ae8eb56786e773b552fe7411958b6b", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T17", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T17" }, "html": { "body": "

p. 270 (PDF p. 288), Section 7.2.8. The twelve-atom null law has a baseline-first causal realization with W unchanged by treatment, the displayed conditional masses, overlap, conditional variance, eight population moments, and unique restricted and full population fits. For every iid realization, the conventional F and HC0 Wald statistics have the displayed positive almost-sure limits after division by n, diverge, and reject with probability tending to one at the stated critical values. An actual 240-observation exact-count sample exists; every ordering of it has the stated OLS coefficients, residual sums and F=5056/177, rounding to 28.6. The infinite iid experiment exists, so this is a realizable counterexample to nominal size under conditional independence alone.

", "headline": "Section 7.2.8 (Placebo Testing): the FFF-test comparing a regression of DiD_iDi​ on Wi\\mathbf{W}_iWi​ and Xi\\mathbf{X}_iXi​ with a regression of DiD_iDi​ on Xi\\mathbf{X}_iXi​ alone, as a test of Wi⊥ ⁣ ⁣ ⁣⊥Di∣Xi\\mathbf{W}_i \\perp\\!\\!\\!\\perp D_i|\\mathbf{X}_iWi​⊥⊥Di​∣Xi​, as printed", "location": "p. 270 (PDF p. 288), Section 7.2.8" }, "id": "P001-T17", "text": { "body": "p. 270 (PDF p. 288), Section 7.2.8. The twelve-atom null law has a baseline-first causal realization with W unchanged by treatment, the displayed conditional masses, overlap, conditional variance, eight population moments, and unique restricted and full population fits. For every iid realization, the conventional F and HC0 Wald statistics have the displayed positive almost-sure limits after division by n, diverge, and reject with probability tending to one at the stated critical values. An actual 240-observation exact-count sample exists; every ordering of it has the stated OLS coefficients, residual sums and F=5056/177, rounding to 28.6. The infinite iid experiment exists, so this is a realizable counterexample to nominal size under conditional independence alone.", "headline": "Section 7.2.8 (Placebo Testing): the $F$-test comparing a regression of $D_i$ on $\\mathbf{W}_i$ and $\\mathbf{X}_i$ with a regression of $D_i$ on $\\mathbf{X}_i$ alone, as a test of $\\mathbf{W}_i \\perp\\!\\!\\!\\perp D_i|\\mathbf{X}_i$, as printed", "location": "p. 270 (PDF p. 288), Section 7.2.8" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T17" }, { "declarations": [ { "claim_revision_id": "catalog:6befb727c492c579efd08a159fb167f044ff9e473c944320081fe40141380898", "conditional_on": [], "credit_reason": "All rank and moment conditions are present in the target itself. The row covers population CEF/BLP conclusions; the target’s robust asymptotic calibration prose remains uncovered and conflicts with its asymptotic=false metadata.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.placebo_linear_cef_correction_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GLP0012B2.lean#L137", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:6befb727c492c579efd08a159fb167f044ff9e473c944320081fe40141380898", "conditional_on": [], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.placebo_zero_coefficients_unique_proof", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GLP0012B2.lean#L157", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:6befb727c492c579efd08a159fb167f044ff9e473c944320081fe40141380898", "conditional_on": [ "O. Kallenberg, Foundations of Modern Probability, 2nd ed., Springer, 2002, Proposition 6.6 (conditional independence), p. 110" ], "credit_reason": "Auxiliary or overlapping declaration is unnecessary for the chosen statement; only the selected assembled results are used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.placebo_linear_cef_correction_supporting_conditional", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Cond/CondP001bProofs.lean#L142", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:6befb727c492c579efd08a159fb167f044ff9e473c944320081fe40141380898", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T18", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T18" }, "html": { "body": "

p. 270 (PDF p. 288), Section 7.2.8. Conditional independence and a linear treatment CEF preserve that CEF after adding placebo covariates and make their uniquely optimal population coefficients zero. Under iid sampling and the stated second moments and Gram rank, actual OLS has its sandwich Gaussian limit and n times residual HC0 converges to that covariance. Every nonempty tested block with nonsingular limiting covariance has a calibrated chi-square Wald test and its inverted joint coefficient region. Without the CEF specification, zero placebo coefficients mean no added linear predictive content; the explicit independent placebo example instead has slope 4/17.

", "headline": "Section 7.2.8, the regression implementation of the placebo test, corrected: valid when E[Di∣Xi]\\mathrm{E}[D_i|\\mathbf{X}_i]E[Di​∣Xi​] is linear in the included regressors", "location": "p. 270 (PDF p. 288), Section 7.2.8" }, "id": "P001-T18", "text": { "body": "p. 270 (PDF p. 288), Section 7.2.8. Conditional independence and a linear treatment CEF preserve that CEF after adding placebo covariates and make their uniquely optimal population coefficients zero. Under iid sampling and the stated second moments and Gram rank, actual OLS has its sandwich Gaussian limit and n times residual HC0 converges to that covariance. Every nonempty tested block with nonsingular limiting covariance has a calibrated chi-square Wald test and its inverted joint coefficient region. Without the CEF specification, zero placebo coefficients mean no added linear predictive content; the explicit independent placebo example instead has slope 4/17.", "headline": "Section 7.2.8, the regression implementation of the placebo test, corrected: valid when $\\mathrm{E}[D_i|\\mathbf{X}_i]$ is linear in the included regressors", "location": "p. 270 (PDF p. 288), Section 7.2.8" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T18" }, { "declarations": [ { "claim_revision_id": "catalog:4e9819c6192ff7058b8d1d0339cc023d1b32d368f89a4bd645037fae575e8e65", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"AronowAudit/P001/P001_Foundations.lean:1482 establishes the normalized arm masses, strictly positive mass in both arms at every covariate, and failure of a uniform positive propensity margin. The definitions at line 306 identify natural index k with the claim's covariate k+1, so the conditional supports are exactly the same positive integers. The explicit decay-rate calculation is auxiliary to this overlap counterexample, whose required failure of uniform positivity is stated directly.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.complete_overlap_without_uniform_positivity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1334", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:4e9819c6192ff7058b8d1d0339cc023d1b32d368f89a4bd645037fae575e8e65", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T19", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T19" }, "html": { "body": "

The printed claim is false: The positive countable joint masses have equal complete discrete arm supports and propensities tending to the boundary without a uniform positive margin. pp. 271–272 (PDF pp. 289–290), Section 7.3.

", "headline": "Section 7.3: 'a failure of the positivity assumption logically implies incomplete (or “limited”) overlap', and the converse of Theorem 7.3.2 that 'can almost be stated', as printed", "location": "p. 271 (PDF p. 289), paragraph before Definition 7.3.1 and the sentence after Theorem 7.3.2, continuing on p. 272 (PDF p. 290)" }, "id": "P001-T19", "text": { "body": "The printed claim is false: The positive countable joint masses have equal complete discrete arm supports and propensities tending to the boundary without a uniform positive margin. pp. 271–272 (PDF pp. 289–290), Section 7.3.", "headline": "Section 7.3: 'a failure of the positivity assumption logically implies incomplete (or “limited”) overlap', and the converse of Theorem 7.3.2 that 'can almost be stated', as printed", "location": "p. 271 (PDF p. 289), paragraph before Definition 7.3.1 and the sentence after Theorem 7.3.2, continuing on p. 272 (PDF p. 290)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T19" }, { "declarations": [ { "claim_revision_id": "catalog:3b7aad7d9f8a0bc1b942afb8e11071a2b4045fb7e1efa5d7327bb35a5788ff8f", "conditional_on": [], "credit_reason": "This is a genuine algebraic generalization; interpreting t,c as joint arm PMF masses gives exactly the claimed discrete overlap criterion.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.discrete_overlap_pointwise", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1381", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:3b7aad7d9f8a0bc1b942afb8e11071a2b4045fb7e1efa5d7327bb35a5788ff8f", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.finite_overlap_uniform_bound", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1413", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:3b7aad7d9f8a0bc1b942afb8e11071a2b4045fb7e1efa5d7327bb35a5788ff8f", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T20", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T20" }, "html": { "body": "

Location: pp. 271–272 (PDF pp. 289–290), Definition 7.3.1, Theorem 7.3.2 and the surrounding corrected discussion. The usual measure-theoretic conditions are assumed. Uniform propensity positivity implies complete overlap of the conditional covariate laws; positive-probability zero/one propensity regions prevent overlap. For discrete covariates the two arm PMF supports have the stated propensity descriptions, overlap is equivalent to pointwise positivity, and finite support yields a uniform bound. The specified infinite geometric joint law has complete overlap without a uniform bound. Density supports are version-invariant modulo null sets; coherently normalized joint-density versions also satisfy the exact pointwise support identity.

", "headline": "Section 7.3, the relation between positivity and complete population overlap, corrected; with Theorem 7.3.2 (Positivity Implies Complete Population Overlap)", "location": "p. 271 (PDF p. 289), paragraph before Definition 7.3.1, Definition 7.3.1, Theorem 7.3.2 and the sentence after it (continuing on p. 272, PDF p. 290)" }, "id": "P001-T20", "text": { "body": "Location: pp. 271–272 (PDF pp. 289–290), Definition 7.3.1, Theorem 7.3.2 and the surrounding corrected discussion. The usual measure-theoretic conditions are assumed. Uniform propensity positivity implies complete overlap of the conditional covariate laws; positive-probability zero/one propensity regions prevent overlap. For discrete covariates the two arm PMF supports have the stated propensity descriptions, overlap is equivalent to pointwise positivity, and finite support yields a uniform bound. The specified infinite geometric joint law has complete overlap without a uniform bound. Density supports are version-invariant modulo null sets; coherently normalized joint-density versions also satisfy the exact pointwise support identity.", "headline": "Section 7.3, the relation between positivity and complete population overlap, corrected; with Theorem 7.3.2 (Positivity Implies Complete Population Overlap)", "location": "p. 271 (PDF p. 289), paragraph before Definition 7.3.1, Definition 7.3.1, Theorem 7.3.2 and the sentence after it (continuing on p. 272, PDF p. 290)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T20" }, { "declarations": [ { "claim_revision_id": "catalog:b93180297c508278a458b1758d388affd0233419f64450173ce7e45a9748b854", "conditional_on": [], "credit_reason": "The count is a percentage because the electorate has exactly 100 voters. ElectorateWitness is a conjunction asserted as the conclusion; no field is assumed to obtain nonignorability.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.complete_coding_not_ignorability", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7210", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b93180297c508278a458b1758d388affd0233419f64450173ce7e45a9748b854", "conditional_on": [], "credit_reason": "Duplicates the complete electorate witness already obtained from complete_coding_not_ignorability.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.electorate_potential_outcomes_not_cia", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7226", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:b93180297c508278a458b1758d388affd0233419f64450173ce7e45a9748b854", "conditional_on": [], "credit_reason": "Not separately invoked: the complete electorate witness already exports the 100-voter count identity.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.electorate_votes_count", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7180", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:b93180297c508278a458b1758d388affd0233419f64450173ce7e45a9748b854", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T21", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T21" }, "html": { "body": "

Location: p. 278 (PDF p. 296), paragraph beginning 'The fact that random assignment is unlikely to hold'; the application's setting: pp. 276–278 (PDF pp. 294–296). Counterexample to the reading that complete coding of the applicant information available to voters in (X_i, D_i), with no personal acquaintance, forces conditional independence of (Y_i(0), Y_i(1)) and D_i given X_i. The four-atom 100-voter model has fully coded leaflets and positivity, violates conditional independence, and gives contrast 25 rather than average effect 5.

", "headline": "Section 7.5, the claim that if the coded applicant descriptions account for all the information presented to voters then 'by definition, DiD_iDi​ must be conditionally independent of potential outcomes', as printed", "location": "p. 278 (PDF p. 296), paragraph beginning 'The fact that random assignment is unlikely to hold'; the application's setting: pp. 276–278 (PDF pp. 294–296)" }, "id": "P001-T21", "text": { "body": "Location: p. 278 (PDF p. 296), paragraph beginning 'The fact that random assignment is unlikely to hold'; the application's setting: pp. 276–278 (PDF pp. 294–296). Counterexample to the reading that complete coding of the applicant information available to voters in (X_i, D_i), with no personal acquaintance, forces conditional independence of (Y_i(0), Y_i(1)) and D_i given X_i. The four-atom 100-voter model has fully coded leaflets and positivity, violates conditional independence, and gives contrast 25 rather than average effect 5.", "headline": "Section 7.5, the claim that if the coded applicant descriptions account for all the information presented to voters then 'by definition, $D_i$ must be conditionally independent of potential outcomes', as printed", "location": "p. 278 (PDF p. 296), paragraph beginning 'The fact that random assignment is unlikely to hold'; the application's setting: pp. 276–278 (PDF pp. 294–296)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T21" }, { "declarations": [ { "claim_revision_id": "catalog:15af822770f72a589e8a7e3aeedd5519b873cc77bf34e4f8aaaa82ba6aae7b2a", "conditional_on": [], "credit_reason": "{'source': 'dispute', 'reason': 'P001/P001_Foundations.lean:1607 requires both sections x -> m(x,0) and x -> m(x,1) to be measurable, while credited clause T22(ii) requires only Y(d)=m(X,d) for a fixed function m. The standing random-variable and function-of-a-random-variable definitions require measurable compositions on the original probability space, which need not make these sections measurable or the outcomes measurable with respect to sigma(X). That distinction matters because CI at P001/P001_Foundations.lean:290 factors conditional expectations given exactly the sigma-field pulled back by X. For example, extend uniform probability on [0,1] by a set A with E[1_A | Borel([0,1])] = 1/2, and take X to be the identity and Y(0)=Y(1)=D=1_A=m(X,d). These are ambient measurable random variables satisfying the pointwise functional equalities and positivity, but conditional joint probability 1/2 differs from the product 1/4, so the unqualified implication is false on the permitted arbitrary probability spaces. The existing declaration proves the measurable-response version of clause (ii), and a description edit alone cannot give it credit for the supplied unqualified clause.', 'defensible_statement': 'On any probability space, let X and the real-valued D be measurable, and let m be a fixed function whose sections m(.,0) and m(.,1) are measurable on the covariate space. Then (m(X,0), m(X,1)) is conditionally independent of D given sigma(X), with conditional independence expressed by almost-sure conditional-expectation factorization for every pair of measurable events.'}", "credit_status": "wrong_credit", "fidelity_status_baseline": "wrong_credit", "lean_name": "AronowAudit.P001.description_only_outcomes_conditional_independence", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1456", "status": "proved", "support_scope": "withdrawn", "used_in_solution": true }, { "claim_revision_id": "catalog:15af822770f72a589e8a7e3aeedd5519b873cc77bf34e4f8aaaa82ba6aae7b2a", "conditional_on": [ "O. Kallenberg, Foundations of Modern Probability, 2nd ed., Springer, 2002, Proposition 6.6 (conditional independence), p. 110" ], "credit_reason": "Not invoked: the supplied proof uses the unconditional countable-covariate identification declaration, avoiding this row’s additional literature premise.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.explicit_ignorability_discrete_ate_conditional", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Cond/CondP001g2Proofs.lean#L149", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:15af822770f72a589e8a7e3aeedd5519b873cc77bf34e4f8aaaa82ba6aae7b2a", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T22", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T22" }, "html": { "body": "

Location: p. 278 (PDF p. 296), corrected ignorability discussion; Theorem 7.1.13, p. 248 (PDF p. 266). The usual measure-theoretic conditions are assumed, including measurable response sections on the covariate space and integrability where expectations are used. (i) Complete coding can coexist with failure of ignorability in the stated four-atom electorate model, with constant covariates and positivity. (ii) A fixed measurable response to description and treatment alone implies conditional independence of the potential-outcome pair and treatment given X. (iii) Under explicit conditional independence and uniform positivity, discrete covariates give the ATE as the difference of the two absolutely convergent population-weighted observed-arm conditional-mean sums. The arm means are actual positive-cell ratios, not extra model assumptions. All three canonical clauses are stated in full under these conditions.

", "headline": "Section 7.5, the justification of ignorability in the application, corrected: conditional independence as an explicit substantive assumption; what follows from it", "location": "p. 278 (PDF p. 296), paragraph beginning 'The fact that random assignment is unlikely to hold'" }, "id": "P001-T22", "text": { "body": "Location: p. 278 (PDF p. 296), corrected ignorability discussion; Theorem 7.1.13, p. 248 (PDF p. 266). The usual measure-theoretic conditions are assumed, including measurable response sections on the covariate space and integrability where expectations are used. (i) Complete coding can coexist with failure of ignorability in the stated four-atom electorate model, with constant covariates and positivity. (ii) A fixed measurable response to description and treatment alone implies conditional independence of the potential-outcome pair and treatment given X. (iii) Under explicit conditional independence and uniform positivity, discrete covariates give the ATE as the difference of the two absolutely convergent population-weighted observed-arm conditional-mean sums. The arm means are actual positive-cell ratios, not extra model assumptions. All three canonical clauses are stated in full under these conditions.", "headline": "Section 7.5, the justification of ignorability in the application, corrected: conditional independence as an explicit substantive assumption; what follows from it", "location": "p. 278 (PDF p. 296), paragraph beginning 'The fact that random assignment is unlikely to hold'" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T22" }, { "declarations": [ { "claim_revision_id": "catalog:eea7750b861d995897a7d6cf08506733a0b4b889005cd6ba83b7011408e2e2e0", "conditional_on": [], "credit_reason": "Using outer probabilities generalizes the event-only claim without changing any measurable-event specialization.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.probability_monotonicity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1500", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:eea7750b861d995897a7d6cf08506733a0b4b889005cd6ba83b7011408e2e2e0", "conditional_on": [], "credit_reason": "The subset premise is inside the subtraction conclusion only; other identities are unconditional for measurable events.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.basic_probability_rules", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1508", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:eea7750b861d995897a7d6cf08506733a0b4b889005cd6ba83b7011408e2e2e0", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T23", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T23" }, "html": { "body": "

Location: Gloss: p. 9 (PDF p. 27), first paragraph. Theorem 1.1.4: p. 8 (PDF p. 26). Theorem 1.1.7: p. 9 (PDF p. 27), proof on p. 10 (PDF p. 28). The corrected subset gloss, all basic probability rules, the addition rule, and failure of the reversed gloss for every proper subset hold.

", "headline": "Section 1.1.2, the gloss of monotonicity in the paragraph after the proof of Theorem 1.1.4, corrected; with Theorem 1.1.4 (Basic Properties of Probability) and Theorem 1.1.7 (Addition Rule)", "location": "Gloss: p. 9 (PDF p. 27), first paragraph. Theorem 1.1.4: p. 8 (PDF p. 26). Theorem 1.1.7: p. 9 (PDF p. 27), proof on p. 10 (PDF p. 28)" }, "id": "P001-T23", "text": { "body": "Location: Gloss: p. 9 (PDF p. 27), first paragraph. Theorem 1.1.4: p. 8 (PDF p. 26). Theorem 1.1.7: p. 9 (PDF p. 27), proof on p. 10 (PDF p. 28). The corrected subset gloss, all basic probability rules, the addition rule, and failure of the reversed gloss for every proper subset hold.", "headline": "Section 1.1.2, the gloss of monotonicity in the paragraph after the proof of Theorem 1.1.4, corrected; with Theorem 1.1.4 (Basic Properties of Probability) and Theorem 1.1.7 (Addition Rule)", "location": "Gloss: p. 9 (PDF p. 27), first paragraph. Theorem 1.1.4: p. 8 (PDF p. 26). Theorem 1.1.7: p. 9 (PDF p. 27), proof on p. 10 (PDF p. 28)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T23" }, { "declarations": [ { "claim_revision_id": "catalog:7976777024d69aafc80b37914896f30f064c137ab703a9851072a61070eb2e90", "conditional_on": [], "credit_reason": "Not separately invoked: the complete correction already exports both countable formulas.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.total_probability_partition", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7232", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:7976777024d69aafc80b37914896f30f064c137ab703a9851072a61070eb2e90", "conditional_on": [], "credit_reason": "Not separately invoked: the complete correction already exports the binary nonemptiness equivalence.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.two_event_partition_nonempty", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1526", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:7976777024d69aafc80b37914896f30f064c137ab703a9851072a61070eb2e90", "conditional_on": [], "credit_reason": "Binary and empty-event conclusions are under the outer partition premises, but the singleton partition {Ω} exists on every probability space. Thus they can be recovered with no extra restriction, including C=∅ or Ω.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.total_probability_complete_correction", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/TotalProbabilityCountable.lean#L12", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:7976777024d69aafc80b37914896f30f064c137ab703a9851072a61070eb2e90", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T24", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T24" }, "html": { "body": "

Location: Theorem 1.1.13: p. 12 (PDF p. 30). Proof and the remark after it: p. 13 (PDF p. 31). Definition 1.1.12: p. 12 (PDF p. 30). Definition 1.1.8 and Theorem 1.1.9: p. 10 (PDF p. 28). Countable total probability holds with conditional ratios on positive cells, and deleting empty members repairs the event/complement partition remark.

", "headline": "Theorem 1.1.13 (Law of Total Probability): the positivity condition in its proof and the partition remark after the proof, corrected", "location": "Theorem 1.1.13: p. 12 (PDF p. 30). Proof and the remark after it: p. 13 (PDF p. 31). Definition 1.1.12: p. 12 (PDF p. 30). Definition 1.1.8 and Theorem 1.1.9: p. 10 (PDF p. 28)" }, "id": "P001-T24", "text": { "body": "Location: Theorem 1.1.13: p. 12 (PDF p. 30). Proof and the remark after it: p. 13 (PDF p. 31). Definition 1.1.12: p. 12 (PDF p. 30). Definition 1.1.8 and Theorem 1.1.9: p. 10 (PDF p. 28). Countable total probability holds with conditional ratios on positive cells, and deleting empty members repairs the event/complement partition remark.", "headline": "Theorem 1.1.13 (Law of Total Probability): the positivity condition in its proof and the partition remark after the proof, corrected", "location": "Theorem 1.1.13: p. 12 (PDF p. 30). Proof and the remark after it: p. 13 (PDF p. 31). Definition 1.1.12: p. 12 (PDF p. 30). Definition 1.1.8 and Theorem 1.1.9: p. 10 (PDF p. 28)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T24" }, { "declarations": [ { "claim_revision_id": "catalog:e99b57b60fbe5f011f94c2af94ec96d2e5fc229cdc961f5fe050a60dfe72210a", "conditional_on": [], "credit_reason": "Witness for the printed third bullet, exactly counterexample (a). The optional two-point witness (b), the first two bullets, and a universal corrected lower-tail formula are not needed to refute that bullet.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.integer_cdf_missing_zero_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1533", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:e99b57b60fbe5f011f94c2af94ec96d2e5fc229cdc961f5fe050a60dfe72210a", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T25", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T25" }, "html": { "body": "

The printed claim is false: P001-T25. Location: p. 20 (PDF p. 38), first bulleted list, third bullet. The constant-zero integer-valued law has lower-tail probability one but printed mass sum zero. Scope: Retains only counterexample (a), the constant-zero one-atom law. Omits counterexample (b), the Bernoulli law with masses two thirds and one third and its printed-sum value one third.

", "headline": "Section 1.2.2, the paragraph 'For a discrete random variable XXX, the PMF tells us everything about its distribution': first bulleted list, third bullet (Pr⁡[X<4]=∑x=13f(x)\\Pr[X < 4] = \\sum_{x=1}^{3} f(x)Pr[X<4]=∑x=13​f(x) for an integer-valued XXX), as printed", "location": "p. 20 (PDF p. 38), first bulleted list, third bullet" }, "id": "P001-T25", "text": { "body": "The printed claim is false: P001-T25. Location: p. 20 (PDF p. 38), first bulleted list, third bullet. The constant-zero integer-valued law has lower-tail probability one but printed mass sum zero. Scope: Retains only counterexample (a), the constant-zero one-atom law. Omits counterexample (b), the Bernoulli law with masses two thirds and one third and its printed-sum value one third.", "headline": "Section 1.2.2, the paragraph 'For a discrete random variable $X$, the PMF tells us everything about its distribution': first bulleted list, third bullet ($\\Pr[X < 4] = \\sum_{x=1}^{3} f(x)$ for an integer-valued $X$), as printed", "location": "p. 20 (PDF p. 38), first bulleted list, third bullet" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T25" }, { "declarations": [ { "claim_revision_id": "catalog:db554c57992ec6f2447ab650067597ad30d5985bfb19cd2135599f14c2ad9539", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/P001_Foundations.lean:1718 proves the sum over all integers below 4 equals intMassCDF q 3, the omitted mass equals intMassCDF q 0, and the printed truncation is valid exactly when that omitted mass is zero. These are T26(i)-(ii), the general corrected-tail component identified by the row. Nonnegative summable integer weights totaling one represent every law allowed by the claim's integer-support condition, including mass on negative integers and zero. The die calculation in T26(iii) is a separate, elementary specialization and is not an omitted conclusion of this general-tail component.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.integer_lower_tail_correction", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1541", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:db554c57992ec6f2447ab650067597ad30d5985bfb19cd2135599f14c2ad9539", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T26", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T26" }, "html": { "body": "

Location: p. 20 (PDF p. 38), first bulleted list, third bullet. For every discrete real random variable whose PMF vanishes off the integers, the probability below four is the sum of all integer masses through three. Its excess over the masses at one, two and three is exactly the probability of a nonpositive value, and the printed equality holds exactly when that probability is zero. For the fair die, the displayed specialization is three sixths, equal to one half. The usual measure-theoretic conditions are assumed. Null sample points with noninteger values are allowed.

", "headline": "Section 1.2.2, the paragraph 'For a discrete random variable XXX, the PMF tells us everything about its distribution': first bulleted list, third bullet, corrected", "location": "p. 20 (PDF p. 38), first bulleted list, third bullet" }, "id": "P001-T26", "text": { "body": "Location: p. 20 (PDF p. 38), first bulleted list, third bullet. For every discrete real random variable whose PMF vanishes off the integers, the probability below four is the sum of all integer masses through three. Its excess over the masses at one, two and three is exactly the probability of a nonpositive value, and the printed equality holds exactly when that probability is zero. For the fair die, the displayed specialization is three sixths, equal to one half. The usual measure-theoretic conditions are assumed. Null sample points with noninteger values are allowed.", "headline": "Section 1.2.2, the paragraph 'For a discrete random variable $X$, the PMF tells us everything about its distribution': first bulleted list, third bullet, corrected", "location": "p. 20 (PDF p. 38), first bulleted list, third bullet" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T26" }, { "declarations": [ { "claim_revision_id": "catalog:fafdc5756048489fe77815116998b663368fc2b3c5dee77346a56a185ab7065d", "conditional_on": [], "credit_reason": "A witness to the printed local-flatness assertion under countable-range discreteness, not a claim against the finite fair-die illustration. Explicit weights avoid a nonsummable-tsum default.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.dense_discrete_cdf_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GLP0012B3.lean#L135", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:fafdc5756048489fe77815116998b663368fc2b3c5dee77346a56a185ab7065d", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T27", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T27" }, "html": { "body": "

Location: p. 23 (PDF p. 41), Example 1.2.13; Definition 1.2.4, p. 18 (PDF p. 36). The specified dyadic atom law has level masses 2^(-(m+1)) summing to one, all subsets measurable, and an injective random variable with exactly the dyadic rationals in (0,1) as its countable range. Each value has its stated positive mass. Its CDF is the stated mass sum, strictly increases on (0,1), and is constant on no nonempty open subinterval there. It gives one third mass zero and is continuous there, while constant on no neighborhood of one third. This supplies all parts of the canonical counterexample to local flatness at every non-jump point. The usual measure-theoretic conditions are assumed.

", "headline": "Example 1.2.13, first sentence (the CDF of a discrete random variable 'is necessarily a step function ... flat everywhere except where it jumps'), as printed", "location": "p. 23 (PDF p. 41), Example 1.2.13, first sentence. Definition 1.2.4: p. 18 (PDF p. 36)" }, "id": "P001-T27", "text": { "body": "Location: p. 23 (PDF p. 41), Example 1.2.13; Definition 1.2.4, p. 18 (PDF p. 36). The specified dyadic atom law has level masses 2^(-(m+1)) summing to one, all subsets measurable, and an injective random variable with exactly the dyadic rationals in (0,1) as its countable range. Each value has its stated positive mass. Its CDF is the stated mass sum, strictly increases on (0,1), and is constant on no nonempty open subinterval there. It gives one third mass zero and is continuous there, while constant on no neighborhood of one third. This supplies all parts of the canonical counterexample to local flatness at every non-jump point. The usual measure-theoretic conditions are assumed.", "headline": "Example 1.2.13, first sentence (the CDF of a discrete random variable 'is necessarily a step function ... flat everywhere except where it jumps'), as printed", "location": "p. 23 (PDF p. 41), Example 1.2.13, first sentence. Definition 1.2.4: p. 18 (PDF p. 36)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T27" }, { "declarations": [ { "claim_revision_id": "catalog:d0338ae2477d99811c9a04de78db46113ab6a7adc319be97e0546a43eec8de89", "conditional_on": [], "credit_reason": "Countability is of the actual range, as in Definition 1.2.4; no essential-support substitution occurs. Probability masses make the real tsum summable.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.discrete_cdf_jump_sum", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1585", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:d0338ae2477d99811c9a04de78db46113ab6a7adc319be97e0546a43eec8de89", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P001/P001_Foundations.lean:1806 proves equality of the CDF at any ordered endpoints with no possible value in their half-open increment (a,b]. This yields the constancy component in T28(ii), including the outer intervals, with the correct endpoint convention. Every finite-range law can be pushed onto its finitely many values and represented by FiniteLaw; allowing zero weights preserves null possible values, while the impossible n=0 case excludes no probability distribution.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.finite_cdf_constant_between_values", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1621", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:d0338ae2477d99811c9a04de78db46113ab6a7adc319be97e0546a43eec8de89", "conditional_on": [], "credit_reason": "This row covers only the six numerical examples, with the general finite and countable formulas supplied by other rows.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.fair_die_cdf_examples", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6174", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:d0338ae2477d99811c9a04de78db46113ab6a7adc319be97e0546a43eec8de89", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/P001_Foundations.lean:7507 states all three formulas in T28(ii): zero below the first value, the partial mass sum between consecutive values, and one at or above the last value. Strictly increasing values are already stipulated by the claim, and Fin(m+1) covers every nonempty finite range, including the singleton case m=0. Zero-based indexing and normalized finite weights change only the representation; the separate countable jump-sum and die-evaluation components are outside this row's stated finite ordered formula.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.ordered_finite_cdf_formula", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6445", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:d0338ae2477d99811c9a04de78db46113ab6a7adc319be97e0546a43eec8de89", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T28", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T28" }, "html": { "body": "

Location: p. 23 (PDF p. 41), Example 1.2.13, first sentence; the analogy on p. 116 (PDF p. 134). Countable discrete CDFs are point-mass sums, finite ordered CDFs have the stated interval formulas, and the six fair-die evaluations hold.

", "headline": "Example 1.2.13, first sentence, corrected: the CDF of a discrete random variable as a sum of jumps, and its step shape when the range is finite", "location": "p. 23 (PDF p. 41), Example 1.2.13, first sentence; the analogy on p. 116 (PDF p. 134)" }, "id": "P001-T28", "text": { "body": "Location: p. 23 (PDF p. 41), Example 1.2.13, first sentence; the analogy on p. 116 (PDF p. 134). Countable discrete CDFs are point-mass sums, finite ordered CDFs have the stated interval formulas, and the six fair-die evaluations hold.", "headline": "Example 1.2.13, first sentence, corrected: the CDF of a discrete random variable as a sum of jumps, and its step shape when the range is finite", "location": "p. 23 (PDF p. 41), Example 1.2.13, first sentence; the analogy on p. 116 (PDF p. 134)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T28" }, { "declarations": [ { "claim_revision_id": "catalog:85619ab9363eab9a098ac6a384a54a6ed6fd56109f985992fb2a7ccd7555ee7b", "conditional_on": [], "credit_reason": "The exact functions supply density admissibility without an extra HasPDF conjunct. The refuted reading is pointwise uniqueness and differentiation at every real x, exactly as printed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.uniform_pdf_versions_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1718", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:85619ab9363eab9a098ac6a384a54a6ed6fd56109f985992fb2a7ccd7555ee7b", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T29", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T29" }, "html": { "body": "

The printed claim is false: P001-T29. Location: p. 24 (PDF p. 42), sentence after Definition 1.2.14; p. 25 (PDF p. 43), Definition 1.2.15 and the proof of Theorem 1.2.16. Example 1.2.18: p. 28 (PDF p. 46). Three distinct nonnegative uniform density versions give the same CDF, whose endpoint derivatives fail with the displayed one-sided quotients.

", "headline": "The sentence after Definition 1.2.14 ('the PDF is unique') and Definition 1.2.15 (f(x)=dF(u)/du∣u=xf(x) = dF(u)/du|_{u=x}f(x)=dF(u)/du∣u=x​, ∀x∈R\\forall x \\in \\mathbb{R}∀x∈R), as printed; with the proof step of Theorem 1.2.16 that uses the derivative at every xxx", "location": "p. 24 (PDF p. 42), sentence after Definition 1.2.14; p. 25 (PDF p. 43), Definition 1.2.15 and the proof of Theorem 1.2.16. Example 1.2.18: p. 28 (PDF p. 46)" }, "id": "P001-T29", "text": { "body": "The printed claim is false: P001-T29. Location: p. 24 (PDF p. 42), sentence after Definition 1.2.14; p. 25 (PDF p. 43), Definition 1.2.15 and the proof of Theorem 1.2.16. Example 1.2.18: p. 28 (PDF p. 46). Three distinct nonnegative uniform density versions give the same CDF, whose endpoint derivatives fail with the displayed one-sided quotients.", "headline": "The sentence after Definition 1.2.14 ('the PDF is unique') and Definition 1.2.15 ($f(x) = dF(u)/du|_{u=x}$, $\\forall x \\in \\mathbb{R}$), as printed; with the proof step of Theorem 1.2.16 that uses the derivative at every $x$", "location": "p. 24 (PDF p. 42), sentence after Definition 1.2.14; p. 25 (PDF p. 43), Definition 1.2.15 and the proof of Theorem 1.2.16. Example 1.2.18: p. 28 (PDF p. 46)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T29" }, { "declarations": [ { "claim_revision_id": "catalog:672c5ebccb3f020593b3338ae690c20a835d4c36ecd0ab13cebecdd95d2e7ab9", "conditional_on": [], "credit_reason": "The integrability within HasPDF follows from the target’s nonnegative integral density and probability normalization; it is not an extra moment hypothesis.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.pdf_unique_and_derivative_ae", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1759", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:672c5ebccb3f020593b3338ae690c20a835d4c36ecd0ab13cebecdd95d2e7ab9", "conditional_on": [], "credit_reason": "The equivalence between the lower-tail integral and withDensity formulations is part of the mathematical density representation; it introduces no additional assumption.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.joint_pdf_unique_ae", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1801", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:672c5ebccb3f020593b3338ae690c20a835d4c36ecd0ab13cebecdd95d2e7ab9", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P001/P001_Foundations.lean:5378 preserves both HasPDF and every lower-tail integral under an everywhere nonnegative replacement h equal to f almost everywhere. This matches T30(iv), including its pointwise nonnegativity requirement. The additional AEMeasurable h premise is implied by almost-everywhere agreement with the AEMeasurable density f and therefore removes no case.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.pdf_version_change", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4721", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:672c5ebccb3f020593b3338ae690c20a835d4c36ecd0ab13cebecdd95d2e7ab9", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P001/P001_Foundations.lean:5397 states that the joint law equals the product of its pushforward marginals if and only if the joint density factors Lebesgue-almost everywhere. This is exactly the independence-factorization component of T30; HasJointPDF and HasPDF at lines 349 and 90 require the ordinary nonnegative integrable densities of the indicated laws. Working with laws rather than random variables, and supplying marginal density versions, excludes no jointly continuous case.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.continuous_independence_density_ae", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4740", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:672c5ebccb3f020593b3338ae690c20a835d4c36ecd0ab13cebecdd95d2e7ab9", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P001/P001_Foundations.lean:5422 asserts failure at u=0 of the derivative in u of the derivative in v of uniformCDF(u)*uniformCDF(v), for every 0Location: p. 24 (PDF p. 42), sentence after Definition 1.2.14; p. 25 (PDF p. 43), Definition 1.2.15 and Theorem 1.2.16; p. 36 (PDF p. 54), Definitions 1.3.10 and 1.3.11; p. 38 (PDF p. 56), Definition 1.3.16; p. 39 (PDF p. 57), Theorem 1.3.17. Nonnegative Lebesgue integral PDF versions agree almost everywhere, equal the CDF derivative almost everywhere, and integrate to one. The CDF has a nonnegative derivative wherever it is differentiable. Every nonnegative almost-everywhere replacement is another PDF. Joint integral PDF versions agree almost everywhere. The product uniform law has CDF xy on the unit square, but its mixed derivative, first in the second variable, fails at (0,y) for every 0 < y < 1, even though the inner derivative exists. Continuous independence is equivalent both to almost-everywhere PDF factorization and to the product being a joint PDF version. For discrete real random variables, PMF factorization is pointwise.

", "headline": "Definition 1.2.15 and the uniqueness sentence before it, corrected: the PDF is unique up to Lebesgue-almost-everywhere equality and equals F′F'F′ almost everywhere; the joint analogue (Definition 1.3.11)", "location": "p. 24 (PDF p. 42), sentence after Definition 1.2.14; p. 25 (PDF p. 43), Definition 1.2.15 and Theorem 1.2.16; p. 36 (PDF p. 54), Definitions 1.3.10 and 1.3.11; p. 38 (PDF p. 56), Definition 1.3.16; p. 39 (PDF p. 57), Theorem 1.3.17" }, "id": "P001-T30", "text": { "body": "Location: p. 24 (PDF p. 42), sentence after Definition 1.2.14; p. 25 (PDF p. 43), Definition 1.2.15 and Theorem 1.2.16; p. 36 (PDF p. 54), Definitions 1.3.10 and 1.3.11; p. 38 (PDF p. 56), Definition 1.3.16; p. 39 (PDF p. 57), Theorem 1.3.17. Nonnegative Lebesgue integral PDF versions agree almost everywhere, equal the CDF derivative almost everywhere, and integrate to one. The CDF has a nonnegative derivative wherever it is differentiable. Every nonnegative almost-everywhere replacement is another PDF. Joint integral PDF versions agree almost everywhere. The product uniform law has CDF xy on the unit square, but its mixed derivative, first in the second variable, fails at (0,y) for every 0 < y < 1, even though the inner derivative exists. Continuous independence is equivalent both to almost-everywhere PDF factorization and to the product being a joint PDF version. For discrete real random variables, PMF factorization is pointwise.", "headline": "Definition 1.2.15 and the uniqueness sentence before it, corrected: the PDF is unique up to Lebesgue-almost-everywhere equality and equals $F'$ almost everywhere; the joint analogue (Definition 1.3.11)", "location": "p. 24 (PDF p. 42), sentence after Definition 1.2.14; p. 25 (PDF p. 43), Definition 1.2.15 and Theorem 1.2.16; p. 36 (PDF p. 54), Definitions 1.3.10 and 1.3.11; p. 38 (PDF p. 56), Definition 1.3.16; p. 39 (PDF p. 57), Theorem 1.3.17" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T30" }, { "declarations": [ { "claim_revision_id": "catalog:bd5fced1c0067dbbd2bbb91503b4778492bca311785840698bb171342dced97c", "conditional_on": [], "credit_reason": "A witness to the nondegenerate-interval reading, not to a vacuous one-point-interval reading. HasPDF and normalization establish the printed notion of a continuous random variable.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.fat_cantor_continuous_range_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GLP0012B5.lean#L261", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:bd5fced1c0067dbbd2bbb91503b4778492bca311785840698bb171342dced97c", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T31", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T31" }, "html": { "body": "

The printed claim is false: P001-T31. Location: p. 24 (PDF p. 42), paragraph before Definition 1.2.14. The specified fat-Cantor subtype carries a continuous random variable with range and positive-density support containing no nondegenerate interval. Scope: Retains the specified recursive fat-Cantor set, its measure one half, the normalized subtype probability law, its density, literal range and positive-density support, and exclusion of every nondegenerate interval. Omits the removed-length geometric-series calculation and the quantitative bounds on the lengths of the stage components.

", "headline": "Section 1.2.4, the sentence 'More specifically, X(Ω)X(\\Omega)X(Ω) must include some interval or union of intervals of the real line', as printed", "location": "p. 24 (PDF p. 42), paragraph before Definition 1.2.14" }, "id": "P001-T31", "text": { "body": "The printed claim is false: P001-T31. Location: p. 24 (PDF p. 42), paragraph before Definition 1.2.14. The specified fat-Cantor subtype carries a continuous random variable with range and positive-density support containing no nondegenerate interval. Scope: Retains the specified recursive fat-Cantor set, its measure one half, the normalized subtype probability law, its density, literal range and positive-density support, and exclusion of every nondegenerate interval. Omits the removed-length geometric-series calculation and the quantitative bounds on the lengths of the stage components.", "headline": "Section 1.2.4, the sentence 'More specifically, $X(\\Omega)$ must include some interval or union of intervals of the real line', as printed", "location": "p. 24 (PDF p. 42), paragraph before Definition 1.2.14" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T31" }, { "declarations": [ { "claim_revision_id": "catalog:c7a6fc8434df991585598def3c680e9f1bc83635cb6e4e17c9ea00ddb5ee1f0b", "conditional_on": [], "credit_reason": "This row covers T32(i); the companion row R006-10 supplies the interval-free actual-range construction. The description issue is harmless metadata boilerplate; the mathematical verdict and severity are unchanged from claim-stage.json.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.continuous_rv_range_uncountable", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1811", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:c7a6fc8434df991585598def3c680e9f1bc83635cb6e4e17c9ea00ddb5ee1f0b", "conditional_on": [], "credit_reason": "The fixed existential is available by instantiating the universal antecedent with the uniform law. The witness concerns actual range, not merely essential support; closedness or nowhere denseness is not required.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.continuous_range_complete_correction", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/IrrationalRange.lean#L15", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:c7a6fc8434df991585598def3c680e9f1bc83635cb6e4e17c9ea00ddb5ee1f0b", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T32", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T32" }, "html": { "body": "

Location: p. 24 (PDF p. 42), paragraph before Definition 1.2.14. Every random variable with a density is atomless with uncountable range and sample space, and a continuous variable can have interval-free range.

", "headline": "Section 1.2.4, the description of the range of a continuous random variable, corrected", "location": "p. 24 (PDF p. 42), paragraph before Definition 1.2.14" }, "id": "P001-T32", "text": { "body": "Location: p. 24 (PDF p. 42), paragraph before Definition 1.2.14. Every random variable with a density is atomless with uncountable range and sample space, and a continuous variable can have interval-free range.", "headline": "Section 1.2.4, the description of the range of a continuous random variable, corrected", "location": "p. 24 (PDF p. 42), paragraph before Definition 1.2.14" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T32" }, { "declarations": [ { "claim_revision_id": "catalog:4a68f144f9b73ac94ae2ef1ff10aaa80045b41196044034193ca94cd651f0c45", "conditional_on": [], "credit_reason": "The generalization preserves the paper’s everywhere equality of random variables, rather than replacing it by a.e. equality. Domain extensions off range(X) are harmless.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.functions_equal_on_range", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4589", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:4a68f144f9b73ac94ae2ef1ff10aaa80045b41196044034193ca94cd651f0c45", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T33", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T33" }, "html": { "body": "

Location: p. 32 (PDF p. 50). Definition 1.3.1: p. 31 (PDF p. 49). Definition 1.2.2: p. 16 (PDF p. 34). Pointwise equality of functions of a random variable is equivalent to equality of those functions on its actual range.

", "headline": "Theorem 1.3.2 (Equality of Functions of a Random Variable), with the functions named consistently", "location": "p. 32 (PDF p. 50). Definition 1.3.1: p. 31 (PDF p. 49). Definition 1.2.2: p. 16 (PDF p. 34)" }, "id": "P001-T33", "text": { "body": "Location: p. 32 (PDF p. 50). Definition 1.3.1: p. 31 (PDF p. 49). Definition 1.2.2: p. 16 (PDF p. 34). Pointwise equality of functions of a random variable is equivalent to equality of those functions on its actual range.", "headline": "Theorem 1.3.2 (Equality of Functions of a Random Variable), with the functions named consistently", "location": "p. 32 (PDF p. 50). Definition 1.3.1: p. 31 (PDF p. 49). Definition 1.2.2: p. 16 (PDF p. 34)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T33" }, { "declarations": [ { "claim_revision_id": "catalog:534fce00b6d912e194237497b11a8eb89f834661d214f71039adcac1c94dc16b", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P001/P001_Foundations.lean:2074 gives total mass one, printed-event mass one, and printed iterated integral zero for rectanglePDF. Its definition at line 355 is exactly the nonnegative indicator of (0,1) times (-1,0) in T34 witness (a); its weighted Lebesgue law supplies the corresponding probability model. The absence of an explicitly named law is a representation difference, and witness (a) alone refutes the printed universal equality.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.triangular_event_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1882", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:534fce00b6d912e194237497b11a8eb89f834661d214f71039adcac1c94dc16b", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T34", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T34" }, "html": { "body": "

The printed claim is false: P001-T34. Location: p. 37 (PDF p. 55), sentence and display after Theorem 1.3.12. The normalized negative-rectangle density gives mass one to the printed event and zero to its printed iterated integral. Scope: Retains only counterexample (a), the normalized density on (0,1) times (-1,0), with event mass one and printed iterated integral zero. Omits counterexample (b), the independent uniform law on [-1,1] squared and the values three eighths and one eighth.

", "headline": "The 'triangular region' display after Theorem 1.3.12, Pr⁡[0≤X≤1,Y≤X]=∫01∫0xf(x,y)dydx\\Pr[0 \\le X \\le 1, Y \\le X] = \\int_0^1\\int_0^x f(x, y)dydxPr[0≤X≤1,Y≤X]=∫01​∫0x​f(x,y)dydx, as printed", "location": "p. 37 (PDF p. 55), sentence and display after Theorem 1.3.12" }, "id": "P001-T34", "text": { "body": "The printed claim is false: P001-T34. Location: p. 37 (PDF p. 55), sentence and display after Theorem 1.3.12. The normalized negative-rectangle density gives mass one to the printed event and zero to its printed iterated integral. Scope: Retains only counterexample (a), the normalized density on (0,1) times (-1,0), with event mass one and printed iterated integral zero. Omits counterexample (b), the independent uniform law on [-1,1] squared and the values three eighths and one eighth.", "headline": "The 'triangular region' display after Theorem 1.3.12, $\\Pr[0 \\le X \\le 1, Y \\le X] = \\int_0^1\\int_0^x f(x, y)dydx$, as printed", "location": "p. 37 (PDF p. 55), sentence and display after Theorem 1.3.12" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T34" }, { "declarations": [ { "claim_revision_id": "catalog:1e1d30f6a5ea5d16489326565ab3c6531edaf3943c262d4029b9e7957ca5121e", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P001/P001_Foundations.lean:2119 states the corrected triangular-event identity and the decomposition of the printed event into the triangular integral plus its negative-Y probability. These are the two probability identities in the corrected claim. The final condition that the printed equality holds if and only if the negative-Y probability is zero is algebraically equivalent to the second identity by cancellation of finite real quantities, so no substantive conclusion is missing.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.triangular_region_correction", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1927", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:1e1d30f6a5ea5d16489326565ab3c6531edaf3943c262d4029b9e7957ca5121e", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T35", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T35" }, "html": { "body": "

Location: p. 37 (PDF p. 55), sentence and display after Theorem 1.3.12. The triangular integral computes the event with nonnegative Y, and the printed formula holds exactly when its omitted negative-Y portion has zero probability.

", "headline": "The 'triangular region' display after Theorem 1.3.12, corrected", "location": "p. 37 (PDF p. 55), sentence and display after Theorem 1.3.12" }, "id": "P001-T35", "text": { "body": "Location: p. 37 (PDF p. 55), sentence and display after Theorem 1.3.12. The triangular integral computes the event with nonnegative Y, and the printed formula holds exactly when its omitted negative-Y portion has zero probability.", "headline": "The 'triangular region' display after Theorem 1.3.12, corrected", "location": "p. 37 (PDF p. 55), sentence and display after Theorem 1.3.12" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T35" }, { "declarations": [ { "claim_revision_id": "catalog:67aa3a7e5be08f8fc82bf64759ddb26a6de105b7c0f09af49aa54e3d20f5a97f", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/Wave1/Gap/GLP0013B1.lean:247 certifies smoothness, strict positivity, integrability and total mass one for the exact smoothSlicePDF of the claim, together with an infinite nonnegative slice at y=0 and integrable slices at every nonzero y. This supplies the row's exceptional-slice counterexample to an everywhere finite marginal formula; ENNReal infinity records genuine divergence rather than the Bochner integral's zero fallback. The auxiliary boundedness and CDF clauses need not be separate conclusions for this component: the explicit formula at P001/P001_Foundations.lean:358 is bounded by 1/pi and, for u bounded above, by exp(2*u)/pi, providing domination for its printed mixed-CDF interpretation.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.infinite_marginal_slice_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GLP0013B1.lean#L246", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:67aa3a7e5be08f8fc82bf64759ddb26a6de105b7c0f09af49aa54e3d20f5a97f", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T36", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T36" }, "html": { "body": "

Restricted density and extended-integral facts about the canonical function for Theorem 1.3.13. Location: p. 37 (PDF p. 55), Theorem 1.3.13. A real analytic strictly positive normalized joint density has an infinite extended nonnegative slice integral at zero and integrable slices elsewhere. The zero slice lies outside the finite-real existence domain of the pointwise marginal-PDF identity. Scope: Retains real analyticity (and hence smoothness), strict positivity, integrability, total mass one, the infinite nonnegative slice integral at zero, and integrability of every nonzero slice. Omits boundedness; the logistic slice formula and its separate normalization; the joint CDF and everywhere mixed-derivative assertions; the limiting value of the zero-slice integrand; positivity of the nonzero slice integrals; and the marginal CDF assertion of no finite derivative at zero. States only these density and extended-integral facts; it does not establish a refutation of the pointwise marginal-PDF identity under the usual assumption that its real slice integral exists finitely.

", "headline": "Theorem 1.3.13 (Marginal PDF), the identity 'for all yyy', as printed", "location": "p. 37 (PDF p. 55), Theorem 1.3.13" }, "id": "P001-T36", "text": { "body": "Restricted density and extended-integral facts about the canonical function for Theorem 1.3.13. Location: p. 37 (PDF p. 55), Theorem 1.3.13. A real analytic strictly positive normalized joint density has an infinite extended nonnegative slice integral at zero and integrable slices elsewhere. The zero slice lies outside the finite-real existence domain of the pointwise marginal-PDF identity. Scope: Retains real analyticity (and hence smoothness), strict positivity, integrability, total mass one, the infinite nonnegative slice integral at zero, and integrability of every nonzero slice. Omits boundedness; the logistic slice formula and its separate normalization; the joint CDF and everywhere mixed-derivative assertions; the limiting value of the zero-slice integrand; positivity of the nonzero slice integrals; and the marginal CDF assertion of no finite derivative at zero. States only these density and extended-integral facts; it does not establish a refutation of the pointwise marginal-PDF identity under the usual assumption that its real slice integral exists finitely.", "headline": "Theorem 1.3.13 (Marginal PDF), the identity 'for all $y$', as printed", "location": "p. 37 (PDF p. 55), Theorem 1.3.13" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T36" }, { "declarations": [ { "claim_revision_id": "catalog:431db6e73df9258f9dddee4805cf42a36fea1cd5ef5a02a40abbb0a7b43b5366", "conditional_on": [], "credit_reason": "The row covers exactly T37(i)–(ii), including arbitrary non-Borel modifications on null sets; it does not claim T37(iii). No premise requires g to be measurable.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.marginal_pdf_ae_correction_all_versions", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/FidP001.lean#L27", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:431db6e73df9258f9dddee4805cf42a36fea1cd5ef5a02a40abbb0a7b43b5366", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': \"P001/P001_Foundations.lean:5436 requires Measurable f, hence a globally Borel-measurable joint density, while T37 permits measurable density versions and its finite-positive-slice clause does not impose this restriction. For example, if N is a non-Borel subset of a null Cantor set in (0,1), f(x,y)=1_[0,1]^2(x,y)+1_{0}(x)1_N(y) is a Lebesgue-measurable uniform-square density version whose x=0 slice is integrable with integral one, but it fails hf. The integrability and positive real integral at lines 5437-5438 correctly express the selected slice condition; joint normalization and a conditional-kernel identity are outside this row's assigned normalization component.\", 'suggested_fix': 'Remove hf : Measurable f from the declaration; retain nonnegativity, integrability of the selected slice, and its strictly positive integral. No global measurability assumption is needed for this slice-normalization statement.', 'second_review': \"P001/P001_Foundations.lean:5436 adds global Borel measurability of f, although T37(iii) requires only a strictly positive finite selected slice and permits Lebesgue-measurable density versions, as does HasJointPDF at P001/P001_Foundations.lean:349. An integrable non-Borel modification on one vertical null slice is consequently excluded despite satisfying the target's slice condition. This is a minor statement repair: normalization for every excluded integrable slice is the same elementary identity integral(s/c)=integral(s)/c=1, and the conclusion at P001/P001_Foundations.lean:5439 needs only the existing slice integrability, nonnegativity and positive integral, so hf can simply be removed.\", 'severity': 'minor'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.conditional_density_positive_finite_slice", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4779", "status": "proved", "support_scope": "limited", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:431db6e73df9258f9dddee4805cf42a36fea1cd5ef5a02a40abbb0a7b43b5366", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T37", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T37" }, "html": { "body": "

Location: p. 37 (PDF p. 55), Theorem 1.3.13, and p. 38 (PDF p. 56), Definition 1.3.14. The extended nonnegative marginal slice integral is finite almost everywhere. Every finite nonnegative version agreeing with it almost everywhere is a marginal PDF and gives the CDF at every threshold, without a Borel measurability premise on the version. At every selected integrable slice with strictly positive integral, the conditional ratio is integrable, nonnegative and normalized. This last assertion needs no global measurability assumption on the joint function beyond the existence of that selected integral. The usual measure-theoretic conditions are assumed.

", "headline": "Theorem 1.3.13 (Marginal PDF), corrected to hold for Lebesgue-almost every yyy", "location": "p. 37 (PDF p. 55), Theorem 1.3.13; the conditional PDF of Definition 1.3.14, p. 38 (PDF p. 56)" }, "id": "P001-T37", "text": { "body": "Location: p. 37 (PDF p. 55), Theorem 1.3.13, and p. 38 (PDF p. 56), Definition 1.3.14. The extended nonnegative marginal slice integral is finite almost everywhere. Every finite nonnegative version agreeing with it almost everywhere is a marginal PDF and gives the CDF at every threshold, without a Borel measurability premise on the version. At every selected integrable slice with strictly positive integral, the conditional ratio is integrable, nonnegative and normalized. This last assertion needs no global measurability assumption on the joint function beyond the existence of that selected integral. The usual measure-theoretic conditions are assumed.", "headline": "Theorem 1.3.13 (Marginal PDF), corrected to hold for Lebesgue-almost every $y$", "location": "p. 37 (PDF p. 55), Theorem 1.3.13; the conditional PDF of Definition 1.3.14, p. 38 (PDF p. 56)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T37" }, { "declarations": [ { "claim_revision_id": "catalog:6d8e5f5ee08fa7a10c75bdf94e6f7abc82df10822e67904eaecd1d2ef6ed44fb", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/P001_Foundations.lean:2212 identifies the constant-zero pushforward of a standard Gaussian with the Dirac law at zero and excludes equality with every strictly positive-variance gaussianReal law. Positive variances are exactly the laws admitted by the paper's positive-scale normal definition; allowing degenerate Gaussians elsewhere in Mathlib does not change this exclusion. The constant map represents 0*X pointwise, and a one-variable family meets the independence condition, giving precisely the all-zero-combination counterexample.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.zero_normal_combination_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1972", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:6d8e5f5ee08fa7a10c75bdf94e6f7abc82df10822e67904eaecd1d2ef6ed44fb", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/P001_Foundations.lean:8182 explicitly places both correlation endpoints in the printed parameter interval and gives zero denominators there, using positive unit scales. These are valid witnesses to the claim's ordinary-arithmetic undefinedness at rho=1 and rho=-1. The additional identically-zero functions and zero integrals describe Lean's totalized division, but the separate zero-denominator assertions establish the defect without relying solely on that convention.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.bivariate_normal_endpoint_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6990", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:6d8e5f5ee08fa7a10c75bdf94e6f7abc82df10822e67904eaecd1d2ef6ed44fb", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T38", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T38" }, "html": { "body": "

Formalization of the canonical counterexamples to the quoted unrestricted normal-combination assertion and to using the printed bivariate PDF formula at correlation plus or minus one. Location: p. 56 (PDF p. 74), sentence after Theorem 2.1.21; p. 66 (PDF p. 84), footnote 13. Definition 2.1.19: p. 55 (PDF p. 73). The zero normal combination has an atom and no density or positive-variance normal law, and both printed correlation endpoints have zero denominators.

", "headline": "The sentence after Theorem 2.1.21 ('any linear combination of any number of mutually independent normal random variables must itself be normal') and the parameter range of the bivariate normal PDF in footnote 13, as printed", "location": "p. 56 (PDF p. 74), sentence after Theorem 2.1.21; p. 66 (PDF p. 84), footnote 13. Definition 2.1.19: p. 55 (PDF p. 73)" }, "id": "P001-T38", "text": { "body": "Formalization of the canonical counterexamples to the quoted unrestricted normal-combination assertion and to using the printed bivariate PDF formula at correlation plus or minus one. Location: p. 56 (PDF p. 74), sentence after Theorem 2.1.21; p. 66 (PDF p. 84), footnote 13. Definition 2.1.19: p. 55 (PDF p. 73). The zero normal combination has an atom and no density or positive-variance normal law, and both printed correlation endpoints have zero denominators.", "headline": "The sentence after Theorem 2.1.21 ('any linear combination of any number of mutually independent normal random variables must itself be normal') and the parameter range of the bivariate normal PDF in footnote 13, as printed", "location": "p. 56 (PDF p. 74), sentence after Theorem 2.1.21; p. 66 (PDF p. 84), footnote 13. Definition 2.1.19: p. 55 (PDF p. 73)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T38" }, { "declarations": [ { "claim_revision_id": "catalog:67bc025d1369cbe1c293ebd25aac936a79b71421d087ec2f7cbc40059f39beb2", "conditional_on": [], "credit_reason": "All input variances are positive exactly because each printed normal is nondegenerate. The nonzero coefficient premise excludes n=0 and ensures positive output variance.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.nontrivial_independent_normal_sum", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1989", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:67bc025d1369cbe1c293ebd25aac936a79b71421d087ec2f7cbc40059f39beb2", "conditional_on": [], "credit_reason": "The parameter restrictions sx>0, sy>0 and −1<ρ<1 are explicitly in the corrected target; comparing them to the uncorrected footnote would misidentify a narrowing.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.bivariate_normal_zero_correlation_independent_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GLP0013B2.lean#L470", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:67bc025d1369cbe1c293ebd25aac936a79b71421d087ec2f7cbc40059f39beb2", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T39", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T39" }, "html": { "body": "

Theorem 2.1.21 and following sentence, p. 56 (PDF p. 74), Definition 2.1.19 and Theorem 2.1.20, pp. 55–56 (PDF pp. 73–74), and footnote 13, p. 66 (PDF p. 84). Normal variables have their specified mean and standard deviation. Every nontrivial finite affine combination of mutually independent normal variables is normal, including the affine and two-variable sum cases. The displayed bivariate density is a probability density with its specified moments, marginal normal laws and correlation, and gives independence and pointwise density factorization at zero correlation. These conclusions transfer to every pair with that joint law. Normal marginals and zero correlation alone do not imply independence: the equal mixture with correlations 3/5 and −3/5 is a counterexample. The usual measure-theoretic conditions are assumed.

", "headline": "The implication drawn from Theorem 2.1.21 and the bivariate normal PDF of footnote 13, corrected", "location": "p. 56 (PDF p. 74), Theorem 2.1.21 and the sentence after it; p. 66 (PDF p. 84), footnote 13" }, "id": "P001-T39", "text": { "body": "Theorem 2.1.21 and following sentence, p. 56 (PDF p. 74), Definition 2.1.19 and Theorem 2.1.20, pp. 55–56 (PDF pp. 73–74), and footnote 13, p. 66 (PDF p. 84). Normal variables have their specified mean and standard deviation. Every nontrivial finite affine combination of mutually independent normal variables is normal, including the affine and two-variable sum cases. The displayed bivariate density is a probability density with its specified moments, marginal normal laws and correlation, and gives independence and pointwise density factorization at zero correlation. These conclusions transfer to every pair with that joint law. Normal marginals and zero correlation alone do not imply independence: the equal mixture with correlations 3/5 and −3/5 is a counterexample. The usual measure-theoretic conditions are assumed.", "headline": "The implication drawn from Theorem 2.1.21 and the bivariate normal PDF of footnote 13, corrected", "location": "p. 56 (PDF p. 74), Theorem 2.1.21 and the sentence after it; p. 66 (PDF p. 84), footnote 13" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T39" }, { "declarations": [ { "claim_revision_id": "catalog:f7fa1f589f89ef47e3cdb1de052fc56fe5413dc8fbf9c27ffffbb796c71579a3", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.coin_mse_scale_correction", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2034", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f7fa1f589f89ef47e3cdb1de052fc56fe5413dc8fbf9c27ffffbb796c71579a3", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.mse_general_identity_and_unique_minimum", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2176", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:f7fa1f589f89ef47e3cdb1de052fc56fe5413dc8fbf9c27ffffbb796c71579a3", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T40", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T40" }, "html": { "body": "

Figure 2.1.1, p. 59 (PDF p. 77); Theorems 2.1.23–24 and Example 2.1.25, pp. 57–58 (PDF pp. 75–76). MSE decomposes into variance and squared mean error, and the fair-coin loss has the stated unique minimum and corrected factor-two plotting values.

", "headline": "Figure 2.1.1 (Minimum MSE Solution for a Fair Coin Flip): the vertical scale, corrected; with Theorem 2.1.23 (Alternative Formula for MSE), Theorem 2.1.24 (The Expected Value Minimizes MSE) and Example 2.1.25", "location": "Figure 2.1.1: p. 59 (PDF p. 77). Theorem 2.1.23: p. 57 (PDF p. 75). Theorem 2.1.24 and Example 2.1.25: p. 58 (PDF p. 76)" }, "id": "P001-T40", "text": { "body": "Figure 2.1.1, p. 59 (PDF p. 77); Theorems 2.1.23–24 and Example 2.1.25, pp. 57–58 (PDF pp. 75–76). MSE decomposes into variance and squared mean error, and the fair-coin loss has the stated unique minimum and corrected factor-two plotting values.", "headline": "Figure 2.1.1 (Minimum MSE Solution for a Fair Coin Flip): the vertical scale, corrected; with Theorem 2.1.23 (Alternative Formula for MSE), Theorem 2.1.24 (The Expected Value Minimizes MSE) and Example 2.1.25", "location": "Figure 2.1.1: p. 59 (PDF p. 77). Theorem 2.1.23: p. 57 (PDF p. 75). Theorem 2.1.24 and Example 2.1.25: p. 58 (PDF p. 76)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T40" }, { "declarations": [ { "claim_revision_id": "catalog:f256dd96dcfb1ca8c5684465fc71f68d44cb81ce3da814060a278a66984d0429", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P001/P001_Foundations.lean:2293 supplies the three positive variances 1/4, covariances 1/4 and -1/4, and failure of each pointwise affine relation for the exact null-point law of T41. Substitution into the stated correlation definition gives correlations 1 and -1 with positive standard deviations, so separate correlation conjuncts would only repeat this arithmetic. Excluding all real slopes is stronger than excluding the positive slopes in the printed implications and preserves the intended refutation.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.correlation_not_pointwise_dependence", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2053", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f256dd96dcfb1ca8c5684465fc71f68d44cb81ce3da814060a278a66984d0429", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.finite_correlation_counterexample_full", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4655", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:f256dd96dcfb1ca8c5684465fc71f68d44cb81ce3da814060a278a66984d0429", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T41", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T41" }, "html": { "body": "

Theorem 2.2.6, p. 63 (PDF p. 81); Definitions 2.2.5 and 1.3.1, pp. 62 and 31 (PDF pp. 80 and 49). The specified three-outcome PMF establishes the canonical counterexample to the second and third bullets' forward implications, with equality at every outcome as in Definition 1.3.1: it has the stated moments, both perfect correlations and probability-one affine relations, while neither corresponding pointwise affine representation exists.

", "headline": "Theorem 2.2.6 (Correlation and Linear Dependence), second and third bullets, as printed", "location": "p. 63 (PDF p. 81), Theorem 2.2.6. Definition 2.2.5: p. 62 (PDF p. 80). Definition 1.3.1: p. 31 (PDF p. 49)" }, "id": "P001-T41", "text": { "body": "Theorem 2.2.6, p. 63 (PDF p. 81); Definitions 2.2.5 and 1.3.1, pp. 62 and 31 (PDF pp. 80 and 49). The specified three-outcome PMF establishes the canonical counterexample to the second and third bullets' forward implications, with equality at every outcome as in Definition 1.3.1: it has the stated moments, both perfect correlations and probability-one affine relations, while neither corresponding pointwise affine representation exists.", "headline": "Theorem 2.2.6 (Correlation and Linear Dependence), second and third bullets, as printed", "location": "p. 63 (PDF p. 81), Theorem 2.2.6. Definition 2.2.5: p. 62 (PDF p. 80). Definition 1.3.1: p. 31 (PDF p. 49)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T41" }, { "declarations": [ { "claim_revision_id": "catalog:5febfab87700c49eb0bb94593d858d7f5c2c4c8ae7d6d9384e53f61975b8ee0e", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.correlation_linear_dependence_ae", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2105", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:5febfab87700c49eb0bb94593d858d7f5c2c4c8ae7d6d9384e53f61975b8ee0e", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T42", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T42" }, "html": { "body": "

Theorem 2.2.6, p. 63 (PDF p. 81), with the same qualification on pp. 62 and 78 (PDF pp. 80 and 96). For square-integrable variables with positive variances, correlation lies in [-1, 1], and its values 1 and -1 characterize the corresponding almost-sure affine relations.

", "headline": "Theorem 2.2.6 (Correlation and Linear Dependence), corrected: linear dependence almost surely", "location": "p. 63 (PDF p. 81), Theorem 2.2.6; the same qualification applies to the description of linear dependence on p. 62 (PDF p. 80) and to the last sentence of footnote 21, p. 78 (PDF p. 96)" }, "id": "P001-T42", "text": { "body": "Theorem 2.2.6, p. 63 (PDF p. 81), with the same qualification on pp. 62 and 78 (PDF pp. 80 and 96). For square-integrable variables with positive variances, correlation lies in [-1, 1], and its values 1 and -1 characterize the corresponding almost-sure affine relations.", "headline": "Theorem 2.2.6 (Correlation and Linear Dependence), corrected: linear dependence almost surely", "location": "p. 63 (PDF p. 81), Theorem 2.2.6; the same qualification applies to the description of linear dependence on p. 62 (PDF p. 80) and to the last sentence of footnote 21, p. 78 (PDF p. 96)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T42" }, { "declarations": [ { "claim_revision_id": "catalog:add247c85656584b68722d94df2e25e0fe8063a6eb6550f06b0ec91c399a4365", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.blp_corrected_derivatives_and_minimum", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2237", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:add247c85656584b68722d94df2e25e0fe8063a6eb6550f06b0ec91c399a4365", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P001/P001_Foundations.lean:5524 gives all three corrected Hessian entries, determinant 4*var, optimal affine approximation to the CEF, the three residual orthogonality identities, and equality with the BLP when the CEF is affine. Strict positivity of the determinant follows immediately from its explicit positive-variance hypothesis. Supplying a Borel CEF version and its L2 composition imposes no additional restriction: these exist for a square-integrable response and real covariate, and every integrable CEF version inherits the L2 bound up to almost-everywhere equality. The final almost-everywhere conclusion follows the standing version convention.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.blp_hessian_and_cef_corollaries", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4854", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:add247c85656584b68722d94df2e25e0fe8063a6eb6550f06b0ec91c399a4365", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T43", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T43" }, "html": { "body": "

Theorem 2.2.21, proof and footnotes 18–19, pp. 76–78 (PDF pp. 94–96); corollaries, p. 78 (PDF p. 96); Theorem 2.2.22, p. 79 (PDF p. 97). For measurable square-integrable X,Y, affine prediction risk is jointly smooth with the corrected first partials, all second partials and Hessian determinant 4 Var(X). A Borel CEF version exists, and every such version is square integrable. Positive regressor variance gives exactly the BLP coefficients as the unique normal-equation solution and unique risk minimizer; the BLP best approximates every CEF version and has all three residual orthogonalities. If a CEF version is affine, the BLP agrees almost surely with every CEF version. At zero regressor variance, every function sending E[X] to E[Y] minimizes MSE, and all affine minimizers and their residuals are described. The usual measure-theoretic conditions are assumed.

", "headline": "Theorem 2.2.21 (Best Linear Predictor) and the first-order conditions in its proof, with the derivatives taken in the free arguments (a,b)(a, b)(a,b) and evaluated at (α,β)(\\alpha, \\beta)(α,β); the two corollaries and Theorem 2.2.22", "location": "Theorem 2.2.21: p. 76 (PDF p. 94). Proof and footnote 19: pp. 77–78 (PDF pp. 95–96). Corollaries: p. 78 (PDF p. 96). Theorem 2.2.22: p. 79 (PDF p. 97)" }, "id": "P001-T43", "text": { "body": "Theorem 2.2.21, proof and footnotes 18–19, pp. 76–78 (PDF pp. 94–96); corollaries, p. 78 (PDF p. 96); Theorem 2.2.22, p. 79 (PDF p. 97). For measurable square-integrable X,Y, affine prediction risk is jointly smooth with the corrected first partials, all second partials and Hessian determinant 4 Var(X). A Borel CEF version exists, and every such version is square integrable. Positive regressor variance gives exactly the BLP coefficients as the unique normal-equation solution and unique risk minimizer; the BLP best approximates every CEF version and has all three residual orthogonalities. If a CEF version is affine, the BLP agrees almost surely with every CEF version. At zero regressor variance, every function sending E[X] to E[Y] minimizes MSE, and all affine minimizers and their residuals are described. The usual measure-theoretic conditions are assumed.", "headline": "Theorem 2.2.21 (Best Linear Predictor) and the first-order conditions in its proof, with the derivatives taken in the free arguments $(a, b)$ and evaluated at $(\\alpha, \\beta)$; the two corollaries and Theorem 2.2.22", "location": "Theorem 2.2.21: p. 76 (PDF p. 94). Proof and footnote 19: pp. 77–78 (PDF pp. 95–96). Corollaries: p. 78 (PDF p. 96). Theorem 2.2.22: p. 79 (PDF p. 97)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T43" }, { "declarations": [ { "claim_revision_id": "catalog:d0d4261be4f6e2e87430dc87c06a671c8c95a03602fb91c2caa3b617d5f7e4a1", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.covariance_does_not_fix_intercept", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2353", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:d0d4261be4f6e2e87430dc87c06a671c8c95a03602fb91c2caa3b617d5f7e4a1", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T44", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T44" }, "html": { "body": "

The printed claim is false: the covariance equation does not uniquely determine an affine predictor. footnote 21, p. 78 (PDF p. 96); Theorem 2.2.21, p. 76 (PDF p. 94); Example 2.2.23, pp. 79–81 (PDF pp. 97–99). The symmetric two-point example has distinct affine predictors with the same covariance and respective squared prediction errors zero and one.

", "headline": "Footnote 21, first sentence ('the BLP is the only linear function that satisfies Cov[X,g(X)]=Cov[X,Y]\\mathrm{Cov}[X, g(X)] = \\mathrm{Cov}[X, Y]Cov[X,g(X)]=Cov[X,Y]'), as printed", "location": "p. 78 (PDF p. 96), footnote 21. Theorem 2.2.21: p. 76 (PDF p. 94). Example 2.2.23: pp. 79–81 (PDF pp. 97–99)" }, "id": "P001-T44", "text": { "body": "The printed claim is false: the covariance equation does not uniquely determine an affine predictor. footnote 21, p. 78 (PDF p. 96); Theorem 2.2.21, p. 76 (PDF p. 94); Example 2.2.23, pp. 79–81 (PDF pp. 97–99). The symmetric two-point example has distinct affine predictors with the same covariance and respective squared prediction errors zero and one.", "headline": "Footnote 21, first sentence ('the BLP is the only linear function that satisfies $\\mathrm{Cov}[X, g(X)] = \\mathrm{Cov}[X, Y]$'), as printed", "location": "p. 78 (PDF p. 96), footnote 21. Theorem 2.2.21: p. 76 (PDF p. 94). Example 2.2.23: pp. 79–81 (PDF pp. 97–99)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T44" }, { "declarations": [ { "claim_revision_id": "catalog:5da98332ceb4693eeeeabc3c4c73d512a6dcf8b3ba8ad8ac5b0ca1f2a0d69442", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.blp_two_equation_characterization", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2369", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:5da98332ceb4693eeeeabc3c4c73d512a6dcf8b3ba8ad8ac5b0ca1f2a0d69442", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T45", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T45" }, "html": { "body": "

footnote 21, p. 78 (PDF p. 96). The covariance equation fixes the affine slope, and adding the mean equation gives exactly the coefficients that uniquely minimize affine prediction risk.

", "headline": "Footnote 21, first sentence, corrected: the BLP is characterized by the covariance equation together with the mean equation", "location": "p. 78 (PDF p. 96), footnote 21" }, "id": "P001-T45", "text": { "body": "footnote 21, p. 78 (PDF p. 96). The covariance equation fixes the affine slope, and adding the mean equation gives exactly the coefficients that uniquely minimize affine prediction risk.", "headline": "Footnote 21, first sentence, corrected: the BLP is characterized by the covariance equation together with the mean equation", "location": "p. 78 (PDF p. 96), footnote 21" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T45" }, { "declarations": [ { "claim_revision_id": "catalog:50643d585e0d02f6ecc91a0e7ec4bcc146e693df92a4ddb2a841c1692c15c89e", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'P001/P001_Foundations.lean:9114 concludes that 10*x^2 is a CEF and computes moments, blpSlope=10 and blpIntercept=-5/3. However, blpSlope and blpIntercept at lines 380 and 382 are defined only as moment formulas; the conclusion never asserts that -5/3+10*X minimizes mean squared prediction error. T46(ii)-(iii) identifies these numbers as the coefficients of the best linear predictor, which this row also says it establishes. The separate minimization theorem at P001/Wave1/QuadraticUniqueBlp.lean:12 does not add that missing conclusion to this credited declaration.', 'suggested_fix': 'Keep the existing hypotheses and conclusions and append the conjunct (∀ a b : ℝ, linearRisk μ X Y (-5/3) 10 ≤ linearRisk μ X Y a b).', 'second_review': \"P001/P001_Foundations.lean:9114 exports the CEF, moments and blpSlope/blpIntercept values, but those quantities are only moment formulas at P001/P001_Foundations.lean:380 and do not themselves state T46's final assertion that the displayed affine function is the BLP. An explicit affine-risk minimum would complete the statement. The omission is minor: with the automatic finite second moments of the uniform/Gaussian model, the quadratic identity at P001/P001_Foundations.lean:2448 and the exported moments give Q(a,b)-Q(-5/3,10)=(b-10)^2/12+(10/3-a-b/2)^2, so nonnegativity immediately supplies the missing conclusion.\", 'severity': 'minor'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.quadratic_cef_factor_ten", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7701", "status": "proved", "support_scope": "limited", "used_in_solution": true }, { "claim_revision_id": "catalog:50643d585e0d02f6ecc91a0e7ec4bcc146e693df92a4ddb2a841c1692c15c89e", "conditional_on": [], "credit_reason": "Not separately invoked: the factor-ten row supplies these same numerical and CEF conclusions; the unique-minimum row separately supplies optimality.", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.uniform_quadratic_cef_and_blp", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4998", "status": "proved", "support_scope": "limited", "used_in_solution": true }, { "claim_revision_id": "catalog:50643d585e0d02f6ecc91a0e7ec4bcc146e693df92a4ddb2a841c1692c15c89e", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.uniform_quadratic_cef_unique_blp", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/QuadraticUniqueBlp.lean#L12", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:50643d585e0d02f6ecc91a0e7ec4bcc146e693df92a4ddb2a841c1692c15c89e", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T46", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T46" }, "html": { "body": "

Examples 2.2.23–24, pp. 79–82 (PDF pp. 97–100); Theorems 2.2.21 and 2.2.25, pp. 76 and 82–83 (PDF pp. 94 and 100–101). For independent uniform X and standard normal W, Y=10X²+W has CEF 10X². The conditional square equals X², the conditional noise mean is zero, and conditional linearity gives the displayed calculation. The stated moments yield slope 10, intercept −5/3 and the unique affine minimum. In general the covariance/variance BLP coefficients uniquely minimize risk at positive regressor variance, and independent integrable outcomes have constant CEFs. The usual measure-theoretic conditions are assumed.

", "headline": "Example 2.2.24 (BLP under Nonlinearity), the CEF formula, corrected; with the CEF and BLP of Example 2.2.23", "location": "Example 2.2.24: p. 82 (PDF p. 100), second sentence. Example 2.2.23: pp. 79–81 (PDF pp. 97–99)" }, "id": "P001-T46", "text": { "body": "Examples 2.2.23–24, pp. 79–82 (PDF pp. 97–100); Theorems 2.2.21 and 2.2.25, pp. 76 and 82–83 (PDF pp. 94 and 100–101). For independent uniform X and standard normal W, Y=10X²+W has CEF 10X². The conditional square equals X², the conditional noise mean is zero, and conditional linearity gives the displayed calculation. The stated moments yield slope 10, intercept −5/3 and the unique affine minimum. In general the covariance/variance BLP coefficients uniquely minimize risk at positive regressor variance, and independent integrable outcomes have constant CEFs. The usual measure-theoretic conditions are assumed.", "headline": "Example 2.2.24 (BLP under Nonlinearity), the CEF formula, corrected; with the CEF and BLP of Example 2.2.23", "location": "Example 2.2.24: p. 82 (PDF p. 100), second sentence. Example 2.2.23: pp. 79–81 (PDF pp. 97–99)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T46" }, { "declarations": [ { "claim_revision_id": "catalog:c3a0048869705c454a66998843ae13c4061f8c1c28b1250d4c675e8515e7e43a", "conditional_on": [], "credit_reason": "Not needed: its six conjuncts restate conclusions of independent_functions_cef_law_measurable under the same marginal-law measurability; final status wrong_credit.", "credit_status": "wrong_credit", "fidelity_status_baseline": "wrong_credit", "lean_name": "AronowAudit.P001.Wave1.independent_functions_cef_law_scope", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/DecideA.lean#L223", "status": "proved", "support_scope": "withdrawn", "used_in_solution": false }, { "claim_revision_id": "catalog:c3a0048869705c454a66998843ae13c4061f8c1c28b1250d4c675e8515e7e43a", "conditional_on": [], "credit_reason": "Not needed: its original conditional-variance and affine-risk conclusions are conjuncts of independent_functions_cef_law_measurable, and its transformed risk result needs globally Borel g,h; final status wrong_credit.", "credit_status": "wrong_credit", "fidelity_status_baseline": "wrong_credit", "lean_name": "AronowAudit.P001.independence_variance_and_blp", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5115", "status": "proved", "support_scope": "withdrawn", "used_in_solution": false }, { "claim_revision_id": "catalog:c3a0048869705c454a66998843ae13c4061f8c1c28b1250d4c675e8515e7e43a", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': \"P001/Wave1/Gap/DecisionT47.lean:20 adds AEMeasurable g (μ.map X) and AEMeasurable h (μ.map Y), separately from the measurable-composition assumptions at line 19. Definition 1.2.2 requires only that the compositions be random variables, and the supplied corrected T47 changes the swapped function names without adding law-measurability. On an arbitrary probability space with an enlarged sigma-field, a composite can be measurable even when its factor has no Borel version modulo the marginal law, so the extra assumptions are not consequences of the standing assumptions. The recorded approval describes a further repair, but does not supply that repair in this target's claim text.\", 'suggested_fix': 'Reassign credit to a further corrected T47 explicitly requiring g and h to be measurable modulo their marginal laws, and record that additional correction in the target. The literal unrestricted transformation claim cannot generally be recovered merely by deleting hg and hh on arbitrary probability spaces.', 'second_review': \"P001/Wave1/Gap/DecisionT47.lean:20 requires AEMeasurable g (mu.map X) and AEMeasurable h (mu.map Y), in addition to the measurable compositions at P001/Wave1/Gap/DecisionT47.lean:19. T47's corrected transformation clause uses functions in standing Definition 1.2.2, which requires only that their compositions be random variables. On arbitrary probability extensions those compositions need not have Borel factor versions under the marginal laws, and independence need not survive the unrestricted transformations, so deleting the premises cannot generally recover the target. Credit for the transformed component must move to an explicitly further-corrected, law-measurable claim; disclosure in the row description does not repair its scope.\", 'severity': 'major'}", "credit_status": "wrong_credit", "fidelity_status_baseline": "wrong_credit", "lean_name": "AronowAudit.P001.Wave1.independent_functions_cef_law_measurable", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/DecisionT47.lean#L14", "status": "proved", "support_scope": "withdrawn", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:c3a0048869705c454a66998843ae13c4061f8c1c28b1250d4c675e8515e7e43a", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T47", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T47" }, "html": { "body": "

Theorem 2.2.25, pp. 82–83 (PDF pp. 100–101), with sub-bullet (a) corrected to E[h(Y)|g(X)]=E[h(Y)]; Theorems 1.3.17, 2.2.8 and 2.2.21, pp. 39, 64 and 76 (PDF pp. 57, 82 and 94). The usual measure-theoretic conditions are assumed; in particular, functions of random variables are measurable functions. Independence is equivalent to independence of every pair of measurable transformations. Independent variables have constant conditional mean and conditional variance, factorized product expectation, zero covariance and correlation, and additive variance. The constant outcome mean is a best affine predictor, uniquely so in coefficients at positive regressor variance. The corrected transformed conditional-mean and BLP conclusions hold with the same clause-specific existence conditions. The general positive-variance BLP coefficient and unique-minimum theorem is included.

", "headline": "Theorem 2.2.25 (Implications of Independence, Part III), fourth bullet, first sub-bullet, corrected", "location": "pp. 82–83 (PDF pp. 100–101)" }, "id": "P001-T47", "text": { "body": "Theorem 2.2.25, pp. 82–83 (PDF pp. 100–101), with sub-bullet (a) corrected to E[h(Y)|g(X)]=E[h(Y)]; Theorems 1.3.17, 2.2.8 and 2.2.21, pp. 39, 64 and 76 (PDF pp. 57, 82 and 94). The usual measure-theoretic conditions are assumed; in particular, functions of random variables are measurable functions. Independence is equivalent to independence of every pair of measurable transformations. Independent variables have constant conditional mean and conditional variance, factorized product expectation, zero covariance and correlation, and additive variance. The constant outcome mean is a best affine predictor, uniquely so in coefficients at positive regressor variance. The corrected transformed conditional-mean and BLP conclusions hold with the same clause-specific existence conditions. The general positive-variance BLP coefficient and unique-minimum theorem is included.", "headline": "Theorem 2.2.25 (Implications of Independence, Part III), fourth bullet, first sub-bullet, corrected", "location": "pp. 82–83 (PDF pp. 100–101)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T47" }, { "declarations": [ { "claim_revision_id": "catalog:61d8029b3b7008b05ed6537929aad5dbe7e69ff666f56db0b5fc26e774f08c23", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.multivariate_variance_rule", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2439", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:61d8029b3b7008b05ed6537929aad5dbe7e69ff666f56db0b5fc26e774f08c23", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.covariance_matrix_symmetry_and_diagonal", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5880", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:61d8029b3b7008b05ed6537929aad5dbe7e69ff666f56db0b5fc26e774f08c23", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T48", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T48" }, "html": { "body": "

Definition 2.3.1 and Theorem 2.3.2, p. 84 (PDF p. 102). The covariance matrix has the corrected entries, is symmetric with variance diagonal, and its entries sum to the variance of the sum of the coordinates. The usual measure-theoretic conditions are assumed: each coordinate is square integrable, so every variance and covariance exists.

", "headline": "Definition 2.3.1 (Covariance Matrix), the index quantifier, corrected; with Theorem 2.3.2 (Multivariate Variance Rule)", "location": "p. 84 (PDF p. 102)" }, "id": "P001-T48", "text": { "body": "Definition 2.3.1 and Theorem 2.3.2, p. 84 (PDF p. 102). The covariance matrix has the corrected entries, is symmetric with variance diagonal, and its entries sum to the variance of the sum of the coordinates. The usual measure-theoretic conditions are assumed: each coordinate is square integrable, so every variance and covariance exists.", "headline": "Definition 2.3.1 (Covariance Matrix), the index quantifier, corrected; with Theorem 2.3.2 (Multivariate Variance Rule)", "location": "p. 84 (PDF p. 102)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T48" }, { "declarations": [ { "claim_revision_id": "catalog:0121b5b841f469e9df0222127a7e6041616d15361fdeddba5d96338eda81c8d0", "conditional_on": [], "credit_reason": "Update algebra is a helper; the coordinate-increment and derivative rows already export every stated conclusion.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.linear_predictor_update_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2450", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:0121b5b841f469e9df0222127a7e6041616d15361fdeddba5d96338eda81c8d0", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.linear_predictor_coordinate_increment", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2471", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:0121b5b841f469e9df0222127a7e6041616d15361fdeddba5d96338eda81c8d0", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.linear_predictor_partial_derivative", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5139", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:0121b5b841f469e9df0222127a7e6041616d15361fdeddba5d96338eda81c8d0", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T49", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T49" }, "html": { "body": "

p. 87 (PDF p. 105), opening paragraph and Theorems 2.3.7–8; Definition 2.3.6, p. 86 (PDF p. 104). Every affine predictor has its intercept as its value at zero and each slope as its unit coordinate increment and partial derivative. Every MSE-minimizing affine predictor has a residual with zero mean, zero coordinate-product expectations, and zero coordinate covariances, including singular designs and zero regressors.

", "headline": "Section 2.3, the two expansions of the multivariate BLP used to interpret β1\\beta_1β1​, corrected; with Definition 2.3.6 and Theorem 2.3.7 (Coefficients of the BLP Are Partial Derivatives)", "location": "p. 87 (PDF p. 105), first paragraph and Theorem 2.3.7. Definition 2.3.6: p. 86 (PDF p. 104)" }, "id": "P001-T49", "text": { "body": "p. 87 (PDF p. 105), opening paragraph and Theorems 2.3.7–8; Definition 2.3.6, p. 86 (PDF p. 104). Every affine predictor has its intercept as its value at zero and each slope as its unit coordinate increment and partial derivative. Every MSE-minimizing affine predictor has a residual with zero mean, zero coordinate-product expectations, and zero coordinate covariances, including singular designs and zero regressors.", "headline": "Section 2.3, the two expansions of the multivariate BLP used to interpret $\\beta_1$, corrected; with Definition 2.3.6 and Theorem 2.3.7 (Coefficients of the BLP Are Partial Derivatives)", "location": "p. 87 (PDF p. 105), first paragraph and Theorem 2.3.7. Definition 2.3.6: p. 86 (PDF p. 104)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T49" }, { "declarations": [ { "claim_revision_id": "catalog:3dbccf51e1844fed5796b51cee3a17f0836c1c6c7dc1ecbd9026f1b2543e382c", "conditional_on": [], "credit_reason": "Finite second moments are in the target itself and the standing convention; L2 introduces no extra moment assumption.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.variance_functional_minus_sign", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2483", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:3dbccf51e1844fed5796b51cee3a17f0836c1c6c7dc1ecbd9026f1b2543e382c", "conditional_on": [], "credit_reason": "This row covers T50(a); the other rows handle the variance functional and empirical plug-in displays.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.finite_population_example_moments", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7724", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:3dbccf51e1844fed5796b51cee3a17f0836c1c6c7dc1ecbd9026f1b2543e382c", "conditional_on": [], "credit_reason": "The deterministic formulation applies to every realization of an IID sample, so no sampling case of the covered identity is excluded.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.empirical_variance_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5167", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:3dbccf51e1844fed5796b51cee3a17f0836c1c6c7dc1ecbd9026f1b2543e382c", "conditional_on": [], "credit_reason": "Not separately invoked: the finite-population, empirical-variance and variance-functional rows give the three components directly.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.finite_population_and_plugin_display_corrections", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/FinitePopulationPlugin.lean#L12", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:3dbccf51e1844fed5796b51cee3a17f0836c1c6c7dc1ecbd9026f1b2543e382c", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T50", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T50" }, "html": { "body": "

Example 3.1.3, p. 94 (PDF p. 112), and the functional and plug-in displays, pp. 118–119 (PDF pp. 136–137). Finite-population PMFs and moments, the example with variance 17/2, the minus-sign variance functional and the empirical mean and variance satisfy the corrected formulas.

", "headline": "Three misprinted displays, corrected: the variance in Example 3.1.3 (p. 94); the variance functional TV(F)T_{\\mathrm{V}}(F)TV​(F) (p. 118, repeated in Example 3.3.5, p. 119); the last line of Example 3.3.4 (p. 119)", "location": "(a) p. 94 (PDF p. 112), Example 3.1.3 (continued), second bullet. (b) p. 118 (PDF p. 136), display for V[X]=TV(F)\\mathrm{V}[X] = T_{\\mathrm{V}}(F)V[X]=TV​(F), last line; p. 119 (PDF p. 137), first display of Example 3.3.5. (c) p. 119 (PDF p. 137), Example 3.3.4, last line of the display. Definition 3.1.2 and Example 3.1.3: p. 93 (PDF p. 111)" }, "id": "P001-T50", "text": { "body": "Example 3.1.3, p. 94 (PDF p. 112), and the functional and plug-in displays, pp. 118–119 (PDF pp. 136–137). Finite-population PMFs and moments, the example with variance 17/2, the minus-sign variance functional and the empirical mean and variance satisfy the corrected formulas.", "headline": "Three misprinted displays, corrected: the variance in Example 3.1.3 (p. 94); the variance functional $T_{\\mathrm{V}}(F)$ (p. 118, repeated in Example 3.3.5, p. 119); the last line of Example 3.3.4 (p. 119)", "location": "(a) p. 94 (PDF p. 112), Example 3.1.3 (continued), second bullet. (b) p. 118 (PDF p. 136), display for $\\mathrm{V}[X] = T_{\\mathrm{V}}(F)$, last line; p. 119 (PDF p. 137), first display of Example 3.3.5. (c) p. 119 (PDF p. 137), Example 3.3.4, last line of the display. Definition 3.1.2 and Example 3.1.3: p. 93 (PDF p. 111)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T50" }, { "declarations": [ { "claim_revision_id": "catalog:a8279ab5095dd72da5a085be5fc2ae6a7a70fd36a5fc73a938651bfd1427cb74", "conditional_on": [], "credit_reason": "The target metadata marks asymptotic=false although both sides concern n tending to infinity; this does not affect the faithful mathematical statement.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.probability_convergence_eventual_levels", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2492", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:a8279ab5095dd72da5a085be5fc2ae6a7a70fd36a5fc73a938651bfd1427cb74", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T51", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T51" }, "html": { "body": "

p. 99 (PDF p. 117), Definition 3.2.6 and the following paragraph. The bad-event limit zero, within-epsilon limit one, and every eventual probability lower bound in [0,1) are equivalent. The measurable singleton-indicator sequence under the geometric law converges in probability to zero, but its within-1/2 probability is 1-2^(-(n+1)) at every n, so the printed endpoint p=1 fails.

", "headline": "The verbal gloss of convergence in probability after Definition 3.2.6, corrected (p∈[0,1)p \\in [0, 1)p∈[0,1))", "location": "p. 99 (PDF p. 117), paragraph after Definition 3.2.6, sentence beginning 'More specifically'" }, "id": "P001-T51", "text": { "body": "p. 99 (PDF p. 117), Definition 3.2.6 and the following paragraph. The bad-event limit zero, within-epsilon limit one, and every eventual probability lower bound in [0,1) are equivalent. The measurable singleton-indicator sequence under the geometric law converges in probability to zero, but its within-1/2 probability is 1-2^(-(n+1)) at every n, so the printed endpoint p=1 fails.", "headline": "The verbal gloss of convergence in probability after Definition 3.2.6, corrected ($p \\in [0, 1)$)", "location": "p. 99 (PDF p. 117), paragraph after Definition 3.2.6, sentence beginning 'More specifically'" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T51" }, { "declarations": [ { "claim_revision_id": "catalog:d71fb8bfc9ce98da57dd7f859fb1d0693f96ebf7a2fe94d63895a5a37ff91155", "conditional_on": [], "credit_reason": "MeasurableSet D is Borel measurability on R, matching the corrected measurable-set convention.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.indicator_expectation", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2538", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:d71fb8bfc9ce98da57dd7f859fb1d0693f96ebf7a2fe94d63895a5a37ff91155", "conditional_on": [], "credit_reason": "This is pointwise consistency at every arbitrary x, including discontinuities and F(x)=0 or 1; uniform empirical-process convergence is not claimed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.empirical_cdf_consistency", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5199", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:d71fb8bfc9ce98da57dd7f859fb1d0693f96ebf7a2fe94d63895a5a37ff91155", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T52", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T52" }, "html": { "body": "

Theorem 3.2.9 and footnote 12, p. 102 (PDF p. 120). Indicator expectations equal measurable-event probabilities and each fixed empirical CDF ordinate consistently estimates its population ordinate under IID sampling.

", "headline": "Footnote 12 (the expectation of an indicator), with the domain of DDD corrected; with Theorem 3.2.9 (Estimating the CDF), to whose statement it is attached", "location": "p. 102 (PDF p. 120), Theorem 3.2.9 and footnote 12" }, "id": "P001-T52", "text": { "body": "Theorem 3.2.9 and footnote 12, p. 102 (PDF p. 120). Indicator expectations equal measurable-event probabilities and each fixed empirical CDF ordinate consistently estimates its population ordinate under IID sampling.", "headline": "Footnote 12 (the expectation of an indicator), with the domain of $D$ corrected; with Theorem 3.2.9 (Estimating the CDF), to whose statement it is attached", "location": "p. 102 (PDF p. 120), Theorem 3.2.9 and footnote 12" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T52" }, { "declarations": [ { "claim_revision_id": "catalog:ae57d2749c7d96c5286eb073d572cd0128b58e82e595e361eb48adeea6c647e5", "conditional_on": [], "credit_reason": "This refutes applicability of the finite-variance WLLN to the squares, not the true consistency conclusion of Theorem 3.2.19. No probability-law existence or IID existence is exported by this moment-certificate row, as its modelling description expressly says.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.squares_missing_wlln_hypothesis", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2549", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:ae57d2749c7d96c5286eb073d572cd0128b58e82e595e361eb48adeea6c647e5", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T53", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T53" }, "html": { "body": "

proof of Theorem 3.2.19, p. 107 (PDF p. 125); Theorem 3.2.19, p. 106 (PDF p. 124), and Theorem 3.2.8, p. 100 (PDF p. 118). The normalized masses 7/8^(k+1) at 2^k are realized by an infinite IID sequence with mean 7/6, second moment 7/4, variance 7/18, and infinite fourth moment. Its squares are integrable but have infinite extended variance, refuting the printed proof's inference of the finite-variance WLLN hypothesis for the squares. The finite-variance WLLN and the sample-variance expectation and consistency theorems still hold, in particular for this sequence.

", "headline": "Proof of Theorem 3.2.19, the step 'by the WLLN, X2‾→pE[X2]\\overline{X^2} \\xrightarrow{p} \\mathrm{E}[X^2]X2p​E[X2]', as printed: an instance of the theorem's hypotheses to which the cited WLLN (Theorem 3.2.8) does not apply", "location": "Proof step: p. 107 (PDF p. 125). Theorem 3.2.19: p. 106 (PDF p. 124). Theorem 3.2.8: p. 100 (PDF p. 118)" }, "id": "P001-T53", "text": { "body": "proof of Theorem 3.2.19, p. 107 (PDF p. 125); Theorem 3.2.19, p. 106 (PDF p. 124), and Theorem 3.2.8, p. 100 (PDF p. 118). The normalized masses 7/8^(k+1) at 2^k are realized by an infinite IID sequence with mean 7/6, second moment 7/4, variance 7/18, and infinite fourth moment. Its squares are integrable but have infinite extended variance, refuting the printed proof's inference of the finite-variance WLLN hypothesis for the squares. The finite-variance WLLN and the sample-variance expectation and consistency theorems still hold, in particular for this sequence.", "headline": "Proof of Theorem 3.2.19, the step 'by the WLLN, $\\overline{X^2} \\xrightarrow{p} \\mathrm{E}[X^2]$', as printed: an instance of the theorem's hypotheses to which the cited WLLN (Theorem 3.2.8) does not apply", "location": "Proof step: p. 107 (PDF p. 125). Theorem 3.2.19: p. 106 (PDF p. 124). Theorem 3.2.8: p. 100 (PDF p. 118)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T53" }, { "declarations": [ { "claim_revision_id": "catalog:3403f0a4326eb2edf0d7640e01159e8956cbbd7ca637ec1c3853c3f7d7695ab7", "conditional_on": [], "credit_reason": "The strict positive-variance premise is printed in Theorem 3.2.19; it is not an added nondegeneracy restriction.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.sample_variance_unbiased_factor_and_consistency", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7261", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:3403f0a4326eb2edf0d7640e01159e8956cbbd7ca637ec1c3853c3f7d7695ab7", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T54", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T54" }, "html": { "body": "

Theorem 3.2.19 and its proof, pp. 106–107 (PDF pp. 124–125); Theorem 3.2.7, p. 100 (PDF p. 118). The WLLN holds for IID integrable variables, and the two-variable continuous mapping theorem holds without independence. Under finite positive population variance, the sample mean and sample second moment converge to their population values; the plug-in variance has expectation (n-1)/n times population variance and converges to that variance. Only a second moment is needed.

", "headline": "Theorem 3.2.19 (Properties of the Plug-In Sample Variance), with the consistency step of its proof corrected (WLLN for i.i.d. integrable variables)", "location": "Theorem 3.2.19: p. 106 (PDF p. 124). Proof: pp. 106–107 (PDF pp. 124–125). Definition 3.2.18: p. 106" }, "id": "P001-T54", "text": { "body": "Theorem 3.2.19 and its proof, pp. 106–107 (PDF pp. 124–125); Theorem 3.2.7, p. 100 (PDF p. 118). The WLLN holds for IID integrable variables, and the two-variable continuous mapping theorem holds without independence. Under finite positive population variance, the sample mean and sample second moment converge to their population values; the plug-in variance has expectation (n-1)/n times population variance and converges to that variance. Only a second moment is needed.", "headline": "Theorem 3.2.19 (Properties of the Plug-In Sample Variance), with the consistency step of its proof corrected (WLLN for i.i.d. integrable variables)", "location": "Theorem 3.2.19: p. 106 (PDF p. 124). Proof: pp. 106–107 (PDF pp. 124–125). Definition 3.2.18: p. 106" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T54" }, { "declarations": [ { "claim_revision_id": "catalog:59d3de9b3f81ec080264f78d5b056c9e639348d0aa6ba7a89b63f598d65de692", "conditional_on": [], "credit_reason": "The selected general contract already includes the variance/standard-error equivalence.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.variance_se_consistency_equivalence", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2628", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:59d3de9b3f81ec080264f78d5b056c9e639348d0aa6ba7a89b63f598d65de692", "conditional_on": [], "credit_reason": "Centering is already included in the selected general contract.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.probability_convergence_centering_equivalence", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7270", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:59d3de9b3f81ec080264f78d5b056c9e639348d0aa6ba7a89b63f598d65de692", "conditional_on": [], "credit_reason": "The selected general contract supplies both centering equivalences for the actual limit law.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.sample_mean_variance_estimate_scale", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7278", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:59d3de9b3f81ec080264f78d5b056c9e639348d0aa6ba7a89b63f598d65de692", "conditional_on": [], "credit_reason": "Not separately invoked: the general scaling contract already exports all three equivalences.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.variance_estimator_consistency_contract", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/VarianceSEContract.lean#L12", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:59d3de9b3f81ec080264f78d5b056c9e639348d0aa6ba7a89b63f598d65de692", "conditional_on": [], "credit_reason": "The paper’s real standard-error estimator presupposes a nonnegative variance estimate, and finite variance of T follows from its standing expectation convention. This row covers the general definition and equivalences, not the background sample-mean unbiasedness theorem.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.variance_estimate_scale_general_contract", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/RepairT55.lean#L24", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:59d3de9b3f81ec080264f78d5b056c9e639348d0aa6ba7a89b63f598d65de692", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T55", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T55" }, "html": { "body": "

Definition 3.2.32 and footnote 23, pp. 113–114 (PDF pp. 131–132); Definition 3.2.20, p. 107 (PDF p. 125), and Theorem 3.2.33, p. 114 (PDF p. 132). With the estimator's hats restored, variance and standard-error consistency are equivalent to their centered forms and to each other. For iid observations of finite positive variance, the unbiased sample variance is n/(n−1) times the plug-in variance; divided by n it is an unbiased estimator of the sample mean's variance, for n at least two. The exact identity n Var(sample mean)=Var(X), the root-n limit, both consistencies and both centered forms hold. The usual measure-theoretic conditions are assumed.

", "headline": "Definition 3.2.32 (Consistency of Sampling Variance and Standard Error Estimators) and footnote 23, with the estimator's hat restored in four places", "location": "pp. 113–114 (PDF pp. 131–132), Definition 3.2.32; footnote 23, p. 114 (PDF p. 132)" }, "id": "P001-T55", "text": { "body": "Definition 3.2.32 and footnote 23, pp. 113–114 (PDF pp. 131–132); Definition 3.2.20, p. 107 (PDF p. 125), and Theorem 3.2.33, p. 114 (PDF p. 132). With the estimator's hats restored, variance and standard-error consistency are equivalent to their centered forms and to each other. For iid observations of finite positive variance, the unbiased sample variance is n/(n−1) times the plug-in variance; divided by n it is an unbiased estimator of the sample mean's variance, for n at least two. The exact identity n Var(sample mean)=Var(X), the root-n limit, both consistencies and both centered forms hold. The usual measure-theoretic conditions are assumed.", "headline": "Definition 3.2.32 (Consistency of Sampling Variance and Standard Error Estimators) and footnote 23, with the estimator's hat restored in four places", "location": "pp. 113–114 (PDF pp. 131–132), Definition 3.2.32; footnote 23, p. 114 (PDF p. 132)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T55" }, { "declarations": [ { "claim_revision_id": "catalog:2d4af64f9d0ecf5ad82a03164a086e32290a38e4545f76a7ca2dbdee50f455fd", "conditional_on": [], "credit_reason": "Differentiability is a separate credited component; together the lemmas give a valid constant-functional counterexample to the strictly positive-variance definition of asymptotic normality.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.constant_plugin_degenerate", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2648", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:2d4af64f9d0ecf5ad82a03164a086e32290a38e4545f76a7ca2dbdee50f455fd", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'P001/P001_Foundations.lean:5290 concludes only CDFHadamard (fun _ => 7) F. CDFHadamard at line 616 existentially quantifies the continuous linear derivative L, without specifying L = 0, whereas witness (a) in T56 and the row description explicitly assert the zero derivative. The declaration therefore exports differentiability but omits the promised identification of the derivative; allowing arbitrary base functions F does not repair that omission. This is a missing conclusion, not a claim that constant-function differentiability is false or that the Bernoulli instance is vacuous.', 'suggested_fix': \"Expose a derivative-parameterized version of CDFHadamard and conclude it with L := fun _ => 0, or add the corresponding explicit zero-limit certificate for every admissible bounded-direction perturbation. The separate HadamardAt zero-map declaration at P001/P001_Foundations.lean:5904 illustrates the required exported conclusion, but is not this row's statement.\", 'second_review': 'P001/P001_Foundations.lean:5290 exports only CDFHadamard, whose definition at P001/P001_Foundations.lean:616 existentially quantifies the derivative without identifying it as zero. T56(a) explicitly requires the constant functional to have the zero derivative, so an explicit zero-derivative certificate should be added. However, every relevant difference quotient is identically (7-7)/t = 0, making the omitted limit certificate mathematically trivial; this warrants minor severity rather than major.', 'severity': 'minor'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.constant_cdf_functional_hadamard", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4643", "status": "proved", "support_scope": "limited", "used_in_solution": true }, { "claim_revision_id": "catalog:2d4af64f9d0ecf5ad82a03164a086e32290a38e4545f76a7ca2dbdee50f455fd", "conditional_on": [], "credit_reason": "The larger domain introduces no problem on specialization to bounded CDF functions. The explicit zero derivative is present in the theorem type.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.constant_functional_hadamard", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5234", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:2d4af64f9d0ecf5ad82a03164a086e32290a38e4545f76a7ca2dbdee50f455fd", "conditional_on": [], "credit_reason": "This row is the stated probability-limit witness component. It meets the printed differentiability hypothesis and establishes degeneracy by its exact moment and CP conclusions; the binomial law, zero derivative and a full no-positive-normal negation are not separately exported. The zero-derivative omission is not an omitted conclusion of this row’s assigned degeneracy component, unlike the explicit description of R008-16.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.quadratic_plugin_degenerate_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GLP0013B3.lean#L250", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:2d4af64f9d0ecf5ad82a03164a086e32290a38e4545f76a7ca2dbdee50f455fd", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T56", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T56" }, "html": { "body": "

Theorem 3.3.6 and footnote 31, p. 120 (PDF p. 138), read with Definition 3.2.28, p. 112 (PDF p. 130). The constant and quadratic Bernoulli functionals give degenerate plug-in limits, supporting the canonical counterexample to the book's positive-variance conclusion. The constant functional has zero derivative and no positive-variance normal limit, while the quadratic Bernoulli plug-in has the exported differentiability, exact second-moment formula and degenerate probability limit.

", "headline": "Theorem 3.3.6 (Asymptotic Normality of Plug-In Estimators), as printed, read with footnote 31 and Definition 3.2.28", "location": "p. 120 (PDF p. 138), Theorem 3.3.6 and footnote 31. Definition 3.2.28: p. 112 (PDF p. 130)" }, "id": "P001-T56", "text": { "body": "Theorem 3.3.6 and footnote 31, p. 120 (PDF p. 138), read with Definition 3.2.28, p. 112 (PDF p. 130). The constant and quadratic Bernoulli functionals give degenerate plug-in limits, supporting the canonical counterexample to the book's positive-variance conclusion. The constant functional has zero derivative and no positive-variance normal limit, while the quadratic Bernoulli plug-in has the exported differentiability, exact second-moment formula and degenerate probability limit.", "headline": "Theorem 3.3.6 (Asymptotic Normality of Plug-In Estimators), as printed, read with footnote 31 and Definition 3.2.28", "location": "p. 120 (PDF p. 138), Theorem 3.3.6 and footnote 31. Definition 3.2.28: p. 112 (PDF p. 130)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T56" }, { "declarations": [ { "claim_revision_id": "catalog:b0023215c73d0bec1ef9ee5300ed8685638bb5c23ecc062b83e537fb18770919", "conditional_on": [], "credit_reason": "This is an alternative record name for the same complete final implication; no separate application is needed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.TailA.functional_delta_nondegenerate_correction_final", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/TailA/CDFDelta.lean#L66", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:b0023215c73d0bec1ef9ee5300ed8685638bb5c23ecc062b83e537fb18770919", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/Wave1/Gap/TailA/CDFDelta.lean:14 copies the statement at P001/P001_Foundations.lean:9224, including both the empirical-process limit and the centered normal scalar limit with variance var(nuB,L). Its positive-variance and L2 hypotheses express the main corrected claim's condition 0 < Var[dot T_F(B_F)] < infinity, and statistic measurability follows the standing random-variable convention. The bounded-function space has the actual supremum norm; a continuous real linear derivative on the tangent subspace admits an ambient Hahn–Banach extension, and extending T outside the CDF domain loses no cases. Supplying the canonical bridge law identifies the limit without assuming either convergence conclusion or excluding a population CDF.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.TailA.empirical_cdf_delta_nondegenerate_final", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/TailA/CDFDelta.lean#L14", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:b0023215c73d0bec1ef9ee5300ed8685638bb5c23ecc062b83e537fb18770919", "conditional_on": [], "credit_reason": "This final has the same full implication as the selected final; no separate application is needed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.TailA.kolmogorov_embedding_preserves_cdf_final", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/TailA/CDFDelta.lean#L93", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:b0023215c73d0bec1ef9ee5300ed8685638bb5c23ecc062b83e537fb18770919", "conditional_on": [], "credit_reason": "The process-limit ingredient is used inside the selected final theorem; it is not applied separately.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.TailA.iid_empirical_cdf_bridge_limit_final", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/TailA/CDFProcess.lean#L12", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:b0023215c73d0bec1ef9ee5300ed8685638bb5c23ecc062b83e537fb18770919", "conditional_on": [ "A. 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W. van der Vaart, Asymptotic Statistics, Cambridge University Press, 1998, Theorem 20.8 (functional delta method), p. 297" ], "credit_reason": "The conditional outer-delta ingredient is superseded by the selected proved final; no literature-theorem premise is added.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.empirical_cdf_delta_nondegenerate_supporting_conditional", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Cond/CondP001cdProofs.lean#L276", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:b0023215c73d0bec1ef9ee5300ed8685638bb5c23ecc062b83e537fb18770919", "conditional_on": [], "credit_reason": "The bounded-linear empirical identity is a helper component and is unnecessary for the selected full final implication.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.bounded_linear_functional_empirical_identity", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L8290", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:b0023215c73d0bec1ef9ee5300ed8685638bb5c23ecc062b83e537fb18770919", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T57", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T57" }, "html": { "body": "

Theorem 3.3.6 and footnote 31, p. 120 (PDF p. 138); Definition 3.2.28, p. 112 (PDF p. 130), with the non-degeneracy correction. Every real IID empirical-CDF process has a tight Brownian-bridge outer limit in the sup norm. A CDF-domain functional Hadamard differentiable on any subspace carrying that bridge almost surely has a centered Gaussian derivative limit, possibly degenerate, with finite variance. Its measurable scalar plug-in converges in distribution to that law. Positive derivative variance gives asymptotic normality with phi squared equal to that variance; zero variance gives a point mass. No closedness of the tangent subspace or moment assumption on the observations is required.

", "headline": "Theorem 3.3.6 (Asymptotic Normality of Plug-In Estimators), corrected with the non-degeneracy condition", "location": "p. 120 (PDF p. 138), Theorem 3.3.6 and footnote 31; Definition 3.2.28, p. 112 (PDF p. 130)" }, "id": "P001-T57", "text": { "body": "Theorem 3.3.6 and footnote 31, p. 120 (PDF p. 138); Definition 3.2.28, p. 112 (PDF p. 130), with the non-degeneracy correction. Every real IID empirical-CDF process has a tight Brownian-bridge outer limit in the sup norm. A CDF-domain functional Hadamard differentiable on any subspace carrying that bridge almost surely has a centered Gaussian derivative limit, possibly degenerate, with finite variance. Its measurable scalar plug-in converges in distribution to that law. Positive derivative variance gives asymptotic normality with phi squared equal to that variance; zero variance gives a point mass. No closedness of the tangent subspace or moment assumption on the observations is required.", "headline": "Theorem 3.3.6 (Asymptotic Normality of Plug-In Estimators), corrected with the non-degeneracy condition", "location": "p. 120 (PDF p. 138), Theorem 3.3.6 and footnote 31; Definition 3.2.28, p. 112 (PDF p. 130)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T57" }, { "declarations": [ { "claim_revision_id": "catalog:3656bf8a88af5218b3622160909aa06bcf3ad6999de6bb87aaf40b1638d9f939", "conditional_on": [], "credit_reason": "A deterministic observation value is used only in a universal pathwise calculation, not as a degenerate replacement for the printed continuous sampling law. One negative CDF value is already enough to refute the unrestricted kernel plug-in domain claim.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.signed_kernel_not_cdf", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2667", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:3656bf8a88af5218b3622160909aa06bcf3ad6999de6bb87aaf40b1638d9f939", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T58", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T58" }, "html": { "body": "

Definitions 3.3.7 and 3.3.9, pp. 121,123 (PDF pp. 139,141); Definition 3.3.2, p. 118 (PDF p. 136). The printed claim is false: the admitted even normalized signed kernel gives a negative integrated estimate at a-1 and a strictly decreasing estimate on (a-2,a-1). For the continuous triangular population density on [5/4,7/4], every nonempty sample in that interval gives an integrated estimate at zero in [-3/8,-1/8]. Thus with probability one it is outside the CDF domain of the fixed functional G ↦ sqrt(G(0)), and its value at zero has no real square root.

", "headline": "Definition 3.3.9 (Kernel Plug-In Estimator) with the kernels admitted by Definition 3.3.7, as printed: a signed kernel for which F^K\\hat F_KF^K​ is not a CDF", "location": "Definition 3.3.7: p. 121 (PDF p. 139). Definition 3.3.9: p. 123 (PDF p. 141). Definition 3.3.2: p. 118 (PDF p. 136)" }, "id": "P001-T58", "text": { "body": "Definitions 3.3.7 and 3.3.9, pp. 121,123 (PDF pp. 139,141); Definition 3.3.2, p. 118 (PDF p. 136). The printed claim is false: the admitted even normalized signed kernel gives a negative integrated estimate at a-1 and a strictly decreasing estimate on (a-2,a-1). For the continuous triangular population density on [5/4,7/4], every nonempty sample in that interval gives an integrated estimate at zero in [-3/8,-1/8]. Thus with probability one it is outside the CDF domain of the fixed functional G ↦ sqrt(G(0)), and its value at zero has no real square root.", "headline": "Definition 3.3.9 (Kernel Plug-In Estimator) with the kernels admitted by Definition 3.3.7, as printed: a signed kernel for which $\\hat F_K$ is not a CDF", "location": "Definition 3.3.7: p. 121 (PDF p. 139). Definition 3.3.9: p. 123 (PDF p. 141). Definition 3.3.2: p. 118 (PDF p. 136)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T58" }, { "declarations": [ { "claim_revision_id": "catalog:ed578b363ad9104aabd260ad3135d8d466447a9ddd73d71cdb910feb4d30ec8e", "conditional_on": [], "credit_reason": "The integral-one hypothesis plus a.e. positivity makes the explicit integrability premise redundant in the target’s measure-integral reading. The theorem’s endpoint limits are in the argument x, not the sample size.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.nonnegative_kernel_cdf", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2681", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:ed578b363ad9104aabd260ad3135d8d466447a9ddd73d71cdb910feb4d30ec8e", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T59", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T59" }, "html": { "body": "

p. 123 (PDF p. 141), Definition 3.3.9; Definition 3.3.7, p. 121 (PDF p. 139). A nonnegative normalized even kernel gives a nonnegative normalized kernel density and a continuous CDF with the required endpoint limits, hence a valid argument for a statistical functional on CDFs.

", "headline": "Definition 3.3.9 (Kernel Plug-In Estimator), corrected: a non-negative kernel", "location": "p. 123 (PDF p. 141), Definition 3.3.9; Definition 3.3.7, p. 121 (PDF p. 139)" }, "id": "P001-T59", "text": { "body": "p. 123 (PDF p. 141), Definition 3.3.9; Definition 3.3.7, p. 121 (PDF p. 139). A nonnegative normalized even kernel gives a nonnegative normalized kernel density and a continuous CDF with the required endpoint limits, hence a valid argument for a statistical functional on CDFs.", "headline": "Definition 3.3.9 (Kernel Plug-In Estimator), corrected: a non-negative kernel", "location": "p. 123 (PDF p. 141), Definition 3.3.9; Definition 3.3.7, p. 121 (PDF p. 139)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T59" }, { "declarations": [ { "claim_revision_id": "catalog:e5982716734371fdc878854d7b3aa6262031fb9415f47d78f697e7bf887b87d1", "conditional_on": [], "credit_reason": "The corrected formal content already assumes only integrability, so the absence of symmetry or normalization adds no gap relative to that content.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.kernel_cdf_correct_integrand", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2717", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e5982716734371fdc878854d7b3aa6262031fb9415f47d78f697e7bf887b87d1", "conditional_on": [], "credit_reason": "The endpoint exclusion occurs in the target’s own corrected statement; it is not an added restriction.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.rectangular_kernel_empirical_cdf_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5242", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:e5982716734371fdc878854d7b3aa6262031fb9415f47d78f697e7bf887b87d1", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T60", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T60" }, "html": { "body": "

p. 123 (PDF p. 141), Definition 3.3.9, last line. The corrected integrated estimator is the average of shifted kernel integrals, and the rectangular-kernel estimator has the empirical-CDF difference formula away from its excluded endpoint.

", "headline": "Definition 3.3.9 (Kernel Plug-In Estimator), last line: the integrand, corrected to the kernel density estimator f^K\\hat f_Kf^​K​", "location": "p. 123 (PDF p. 141), Definition 3.3.9, last line" }, "id": "P001-T60", "text": { "body": "p. 123 (PDF p. 141), Definition 3.3.9, last line. The corrected integrated estimator is the average of shifted kernel integrals, and the rectangular-kernel estimator has the empirical-CDF difference formula away from its excluded endpoint.", "headline": "Definition 3.3.9 (Kernel Plug-In Estimator), last line: the integrand, corrected to the kernel density estimator $\\hat f_K$", "location": "p. 123 (PDF p. 141), Definition 3.3.9, last line" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T60" }, { "declarations": [ { "claim_revision_id": "catalog:22ec85fab4efe9ccba78d4749b40cc33f415b69ea2616ac4ae4c7d7b3b391d1f", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'P001/Wave1/Gap/GP0014B1.lean:534 exports the marginal normal limit only along sizes 2*k+1, while branch (ii) of T61 states the limit of sqrt(n) T0(Fhat_n) without an odd-size restriction and explicitly restricts only its variance lower bound to odd n. Moreover, P001/P001_Foundations.lean:595 selects zero-based rank n/2, so the fallback in regularHeavyStatistic at line 598 is the upper median at even sizes, whereas T0 uses the lower empirical 1/2-quantile; on sample (1,2) they give 2 and 1 respectively. The local functional identity at lines 532-533 supplies the claimed regularity, and the odd-size variance obstruction suffices to refute consistency, but neither supplies the missing full-sequence marginal limit for the specified statistic.', 'suggested_fix': 'State the marginal CLT for all n using regularHeavyFunctional applied to the empirical CDF, or replace the finite-sample fallback by zero-based rank (n-1)/2 and identify it with that functional for n>0. Keep the existing odd-size variance lower bound and subsequence obstruction, which are sufficient for the variance inconsistency conclusion.', 'second_review': \"P001/Wave1/Gap/GP0014B1.lean:534 states the marginal Gaussian limit only for sizes 2*k+1, whereas T61(ii) states sqrt(n) T0(Fhat_n) converges to N(0,1/4) for the full sequence and restricts only its variance lower bound to odd sizes. Moreover, regularHeavyStatistic at P001/P001_Foundations.lean:598 uses sampleMedian's zero-based rank n/2 at line 595, giving the upper median on even samples, while the target functional uses the lower empirical half-quantile; sample (1,2) gives respectively 2 and 1. The odd-size variance obstruction is sufficient to disprove consistency, but it does not establish the missing full-sequence CLT for the specified functional, so that statement and statistic identification need repair.\", 'severity': 'major'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.Wave1.regular_functional_bootstrap_variance_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GP0014B1.lean#L529", "status": "proved", "support_scope": "limited", "used_in_solution": true }, { "claim_revision_id": "catalog:22ec85fab4efe9ccba78d4749b40cc33f415b69ea2616ac4ae4c7d7b3b391d1f", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/Wave1/Gap/TailA/HeavyMedian.lean:287 states both N(0,4) weak limits, the probability bound at every odd n >= 5, and failure of the scaled bootstrap variance to converge to 4, exactly as branch (i) of the target requires. Sizes 2k+1 enumerate the same positive odd sizes as 2k-1, and heavyBootstrapLaw at P001/P001_Foundations.lean:590 is the specified mixture; its null uniform endpoints do not change the law. bootVar at P001/P001_Foundations.lean:229 averages over all n^n resampling maps, so it is the actual finite conditional bootstrap variance. Infinite unconditional moments of the sample median do not invalidate this component, whose consistency target is the variance 4 of the limiting normal law; the separate modified-functional branch is outside this row's assigned component.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.TailA.heavy_median_bootstrap_variance_counterexample_final", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/TailA/HeavyMedian.lean#L287", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:22ec85fab4efe9ccba78d4749b40cc33f415b69ea2616ac4ae4c7d7b3b391d1f", "conditional_on": [ "P. J. Bickel and D. A. Freedman, \"Some asymptotic theory for the bootstrap\", Annals of Statistics 9 (1981), Theorem 2.1, p. 1197" ], "credit_reason": "The row requires a separate IIDMeanBootstrap theorem premise and concludes only an odd-size weak limit; the package drops that component rather than adding the premise.", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.Wave1.regular_heavy_functional_bootstrap_weak_limit_conditional", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Cond/CondP001g14Proofs.lean#L264", "status": "proved", "support_scope": "limited", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:22ec85fab4efe9ccba78d4749b40cc33f415b69ea2616ac4ae4c7d7b3b391d1f", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T61", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T61" }, "html": { "body": "

p. 131 (PDF p. 149), with footnote 31, p. 120 (PDF p. 138). The printed claim is false: the specified half-uniform, quarter-positive and quarter-negative heavy-tail population has the stated exact survival probabilities, local affine CDF and unique median zero. Its odd-size median has both normal weak limits but its scaled bootstrap variance exceeds 5 with probability at least 3/16. The specified CDF functional T0 is locally affine, Hadamard differentiable in every bounded sup-norm direction, and, along all positive sizes, has marginal and conditional bootstrap limits N(0,1/4). Its scaled bootstrap variance still exceeds 5 with probability at least 3/16 at every odd size at least 5, and cannot converge to 1/4. The resamples repeating a high and a central observation give exactly those observations.

", "headline": "Section 3.4.3, the sentence 'Given the usual plug-in regularity conditions (Section 3.3.1), this standard deviation will consistently estimate the standard error', as printed", "location": "p. 131 (PDF p. 149), sentence after the four bootstrap steps; Section 3.3.1 and footnote 31, p. 120 (PDF p. 138); Definition 3.2.32, pp. 113–114 (PDF pp. 131–132)" }, "id": "P001-T61", "text": { "body": "p. 131 (PDF p. 149), with footnote 31, p. 120 (PDF p. 138). The printed claim is false: the specified half-uniform, quarter-positive and quarter-negative heavy-tail population has the stated exact survival probabilities, local affine CDF and unique median zero. Its odd-size median has both normal weak limits but its scaled bootstrap variance exceeds 5 with probability at least 3/16. The specified CDF functional T0 is locally affine, Hadamard differentiable in every bounded sup-norm direction, and, along all positive sizes, has marginal and conditional bootstrap limits N(0,1/4). Its scaled bootstrap variance still exceeds 5 with probability at least 3/16 at every odd size at least 5, and cannot converge to 1/4. The resamples repeating a high and a central observation give exactly those observations.", "headline": "Section 3.4.3, the sentence 'Given the usual plug-in regularity conditions (Section 3.3.1), this standard deviation will consistently estimate the standard error', as printed", "location": "p. 131 (PDF p. 149), sentence after the four bootstrap steps; Section 3.3.1 and footnote 31, p. 120 (PDF p. 138); Definition 3.2.32, pp. 113–114 (PDF pp. 131–132)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T61" }, { "declarations": [ { "claim_revision_id": "catalog:c3eea16ab12146bf59c4d0856d0068059508b1859d813559aaf4ecc94817f8e9", "conditional_on": [], "credit_reason": "The target is the corrected moment theorem, not the unrestricted printed sentence. Pointwise finite second moments prevent totalized real integrals or toReal from making the conclusion vacuous; all limsups are of probabilities in [0,1].", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.bootstrap_moments_correction_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GLP0014B2.lean#L463", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:c3eea16ab12146bf59c4d0856d0068059508b1859d813559aaf4ecc94817f8e9", "conditional_on": [], "credit_reason": "The formula uses the empirical divisor n, as in the target, not the unbiased divisor n−1.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.bootstrap_sample_mean_exact_variance", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5811", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:c3eea16ab12146bf59c4d0856d0068059508b1859d813559aaf4ecc94817f8e9", "conditional_on": [], "credit_reason": "Positive finite variance is in the target’s own sample-mean case; it is not an additional exclusion.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.bootstrap_sample_mean_consistency", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5857", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:c3eea16ab12146bf59c4d0856d0068059508b1859d813559aaf4ecc94817f8e9", "conditional_on": [], "credit_reason": "The target primitive median case is all-size and locally absolutely continuous, exactly as required here; no global-density condition is introduced.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.TailA.bootstrap_median_primitive_moment_condition_final", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/TailA/MedianMoments.lean#L84", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:c3eea16ab12146bf59c4d0856d0068059508b1859d813559aaf4ecc94817f8e9", "conditional_on": [], "credit_reason": "Final status wrong_credit: it assumes a global density and covers odd sizes only; use the credited all-size local-density declaration instead.", "credit_status": "wrong_credit", "fidelity_status_baseline": "wrong_credit", "lean_name": "AronowAudit.P001.Wave1.TailA.bootstrap_median_primitive_moment_condition_supporting_final", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/TailA/MedianMoments.lean#L125", "status": "proved", "support_scope": "withdrawn", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:c3eea16ab12146bf59c4d0856d0068059508b1859d813559aaf4ecc94817f8e9", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T62", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T62" }, "html": { "body": "

p. 131 (PDF p. 149); Definition 3.2.32, pp. 113–114 (PDF pp. 131–132). Assume the estimator has the stated root-n normal limit. For the exact finite with-replacement bootstrap, conditional weak convergence to N(0,phi^2), phi>0, makes the real-cutoff square-tail condition equivalent to convergence of n times the bootstrap variance to phi^2. If these equivalent conditions hold, the scaled bootstrap standard deviation converges to phi. The usual measure-theoretic conditions are assumed; only the observed and resampled statistics must be measurable. The finite-variance sample-mean formula and consistency, and the all-size median case under a positive continuous local density, a unique median and a finite positive-order absolute moment, hold as well.

", "headline": "Section 3.4.3, consistency of the bootstrap standard deviation for the standard error, corrected: the added moment condition", "location": "p. 131 (PDF p. 149), sentence after the four bootstrap steps; Definition 3.2.32, pp. 113–114 (PDF pp. 131–132)" }, "id": "P001-T62", "text": { "body": "p. 131 (PDF p. 149); Definition 3.2.32, pp. 113–114 (PDF pp. 131–132). Assume the estimator has the stated root-n normal limit. For the exact finite with-replacement bootstrap, conditional weak convergence to N(0,phi^2), phi>0, makes the real-cutoff square-tail condition equivalent to convergence of n times the bootstrap variance to phi^2. If these equivalent conditions hold, the scaled bootstrap standard deviation converges to phi. The usual measure-theoretic conditions are assumed; only the observed and resampled statistics must be measurable. The finite-variance sample-mean formula and consistency, and the all-size median case under a positive continuous local density, a unique median and a finite positive-order absolute moment, hold as well.", "headline": "Section 3.4.3, consistency of the bootstrap standard deviation for the standard error, corrected: the added moment condition", "location": "p. 131 (PDF p. 149), sentence after the four bootstrap steps; Definition 3.2.32, pp. 113–114 (PDF pp. 131–132)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T62" }, { "declarations": [ { "claim_revision_id": "catalog:1f210ded33cb5c53126b3bea3aa05374c7f8983aed07b6f2e28967c188801f6a", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'The exported conclusion at P001/P001_Foundations.lean:3069 ends at line 3073 with the sample-mean identity, expectation and variance formulas, and convergence in probability; it never concludes IID μ (clusterAverage X). The target explicitly says that W_1, W_2, ... are iid, and the row says that the cluster mean is an average of iid cluster averages. The definition of clusterAverage at line 434 only specifies its values, so the claimed iid property is omitted from the statement. Allowing zero coordinate variances is a harmless extension and does not explain that omission.', 'suggested_fix': 'Add IID μ (clusterAverage X) as an explicit conjunct of cluster_mean_properties, keeping the existing moment formulas and probability limit.', 'second_review': \"P001/P001_Foundations.lean:3069 exports the mean identity, moment formulas and probability limit, but omits T63's explicit assertion that the cluster averages W_i are i.i.d. This is a missing conjunct, yet clusterAverage at P001/P001_Foundations.lean:434 is the same continuous averaging map applied to each IID vector, so the conclusion follows immediately by measurable composition from hiid at line 3067 (Mathlib/Probability/Independence/Basic.lean:670 and Mathlib/Probability/IdentDistrib.lean:109). Add the IID conjunct, but classify the omission as minor because it requires no additional statistical argument or assumptions.\", 'severity': 'minor'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.cluster_mean_properties", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2764", "status": "proved", "support_scope": "limited", "used_in_solution": true }, { "claim_revision_id": "catalog:1f210ded33cb5c53126b3bea3aa05374c7f8983aed07b6f2e28967c188801f6a", "conditional_on": [], "credit_reason": "The diagonal covariance terms are present. Coordinate square-integrability expresses the standing finiteness of the diagonal covariance expectations in this formula.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.cluster_variance_double_covariance_sum", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6201", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:1f210ded33cb5c53126b3bea3aa05374c7f8983aed07b6f2e28967c188801f6a", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T63", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T63" }, "html": { "body": "

Sentence: p. 138 (PDF p. 156). Definitions 3.5.1, 3.5.2: p. 136 (PDF p. 154). Theorems 3.5.3–3.5.5: p. 137 (PDF p. 155). Theorem 3.5.6: p. 138 (PDF p. 156). IID cluster averages have the stated sample-mean, expectation, variance, covariance-sum and weak-law formulas.

", "headline": "Section 3.5.1, the sentence before Theorem 3.5.6 (mean of the cluster averages), corrected; with Theorems 3.5.3, 3.5.4, 3.5.5 and 3.5.6 for the cluster sample mean", "location": "Sentence: p. 138 (PDF p. 156). Definitions 3.5.1, 3.5.2: p. 136 (PDF p. 154). Theorems 3.5.3–3.5.5: p. 137 (PDF p. 155). Theorem 3.5.6: p. 138 (PDF p. 156)" }, "id": "P001-T63", "text": { "body": "Sentence: p. 138 (PDF p. 156). Definitions 3.5.1, 3.5.2: p. 136 (PDF p. 154). Theorems 3.5.3–3.5.5: p. 137 (PDF p. 155). Theorem 3.5.6: p. 138 (PDF p. 156). IID cluster averages have the stated sample-mean, expectation, variance, covariance-sum and weak-law formulas.", "headline": "Section 3.5.1, the sentence before Theorem 3.5.6 (mean of the cluster averages), corrected; with Theorems 3.5.3, 3.5.4, 3.5.5 and 3.5.6 for the cluster sample mean", "location": "Sentence: p. 138 (PDF p. 156). Definitions 3.5.1, 3.5.2: p. 136 (PDF p. 154). Theorems 3.5.3–3.5.5: p. 137 (PDF p. 155). Theorem 3.5.6: p. 138 (PDF p. 156)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T63" }, { "declarations": [ { "claim_revision_id": "catalog:5c9e2457b5208e9b4d1e14a1cccad3456a5e1f21ef124a925a4901c04623e0a9", "conditional_on": [], "credit_reason": "Accepted as the algebraic component identified by row_says, not as a standalone proof of the entire probabilistic variance assertion. Claim-stage verdict retained; the record reveals no misread Lean hypothesis or claim text.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.intracluster_variance_correct_diagonal", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2807", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:5c9e2457b5208e9b4d1e14a1cccad3456a5e1f21ef124a925a4901c04623e0a9", "conditional_on": [], "credit_reason": "Already used internally by the stronger canonical independent-uniform draw-law declaration; not needed separately.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.block_bootstrap_draw_count", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7775", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:5c9e2457b5208e9b4d1e14a1cccad3456a5e1f21ef124a925a4901c04623e0a9", "conditional_on": [], "credit_reason": "This row is a canonical-law construction for the bootstrap component only; it does not claim the design-effect result. Claim-stage verdict retained; the record reveals no misread Lean hypothesis or claim text.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.canonical_block_bootstrap_law", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/WP001dB1.lean#L11", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:5c9e2457b5208e9b4d1e14a1cccad3456a5e1f21ef124a925a4901c04623e0a9", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T64", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T64" }, "html": { "body": "

Example 3.5.8: pp. 140–141 (PDF pp. 158–159). Block bootstrap steps: p. 141 (PDF p. 159). Theorem 3.5.5: p. 137 (PDF p. 155). IID clusters with common finite variances and common correlations between distinct units have the stated diagonal and off-diagonal covariances, their explicit count expansion, the design-effect variance, and both endpoint formulas. Correlation requires positive standard deviations only for distinct pairs; singleton clusters may have zero variance. The corrected block bootstrap draws l intact clusters independently and uniformly, retains lm observation slots, applies the estimator to each resample, permits any finite number of independent repetitions, and uses the estimator's conditional standard deviation.

", "headline": "Example 3.5.8 (Consequences of Clustering for the Sample Mean), the common-correlation condition, corrected to pairs j≠j′j \\ne j'j=j′; and step 1 of the block bootstrap, corrected to lll clusters", "location": "Example 3.5.8: pp. 140–141 (PDF pp. 158–159). Block bootstrap steps: p. 141 (PDF p. 159). Theorem 3.5.5: p. 137 (PDF p. 155)" }, "id": "P001-T64", "text": { "body": "Example 3.5.8: pp. 140–141 (PDF pp. 158–159). Block bootstrap steps: p. 141 (PDF p. 159). Theorem 3.5.5: p. 137 (PDF p. 155). IID clusters with common finite variances and common correlations between distinct units have the stated diagonal and off-diagonal covariances, their explicit count expansion, the design-effect variance, and both endpoint formulas. Correlation requires positive standard deviations only for distinct pairs; singleton clusters may have zero variance. The corrected block bootstrap draws l intact clusters independently and uniformly, retains lm observation slots, applies the estimator to each resample, permits any finite number of independent repetitions, and uses the estimator's conditional standard deviation.", "headline": "Example 3.5.8 (Consequences of Clustering for the Sample Mean), the common-correlation condition, corrected to pairs $j \\ne j'$; and step 1 of the block bootstrap, corrected to $l$ clusters", "location": "Example 3.5.8: pp. 140–141 (PDF pp. 158–159). Block bootstrap steps: p. 141 (PDF p. 159). Theorem 3.5.5: p. 137 (PDF p. 155)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T64" }, { "declarations": [ { "claim_revision_id": "catalog:1605d5e1326576143540f021ff011e3be4c2055214c167c159b7390e98c1ee68", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'P001/P001_Foundations.lean:3132 imposes invertibility of XᵀX on both the inverse-formula minimization result and the residual equations at lines 3135–3136. The quoted Theorem 4.1.5 assumes an OLS minimizer and an intercept, without an invertibility hypothesis in its claim text or the standing assumptions. With every design row equal to (1,1) and Y=(-1,0,1), the Gram matrix is singular but OLS minimizers exist and satisfy all three printed residual properties, so a nontrivial class is excluded. The normal equations represent those properties correctly in the full-rank case; the record note asserting inherited invertibility cannot add a hypothesis to the governing claim.', 'suggested_fix': 'Keep invertibility for the inverse coefficient formula and uniqueness, but add a residual-properties clause for every coefficient vector b satisfying ∀ c, olsRSS X Y b ≤ olsRSS X Y c, assuming only the intercept condition. Conclude X.transpose.mulVec (Y-X.mulVec b)=0 and the zero residual sum for that arbitrary minimizer.', 'second_review': \"P001/P001_Foundations.lean:3132 assumes an invertible Gram matrix before both the inverse-formula minimization result and the residual identities at lines 3135 and 3136. T65's quoted Theorem 4.1.5 requires residual properties for an OLS minimizer with an intercept without imposing full rank; for design rows (1,1) and Y=(-1,0,1), every coefficient vector with b0+b1=0 is a valid minimizer despite singularity. Keep invertibility for the inverse formula and uniqueness, and add normal equations and zero residual sum for arbitrary RSS minimizers under the intercept condition alone.\", 'severity': 'major'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.ols_correct_rows_solution_residuals", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2829", "status": "proved", "support_scope": "limited", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:1605d5e1326576143540f021ff011e3be4c2055214c167c159b7390e98c1ee68", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T65", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T65" }, "html": { "body": "

Theorem 4.1.4 and the stacked system: p. 148 (PDF p. 166); rest of the proof and Theorem 4.1.5: p. 149 (PDF p. 167); proof of 4.1.5: p. 150 (PDF p. 168). Definitions 4.1.2–4.1.3: pp. 146–148. For every nonempty intercept design the correctly indexed scalar residuals stack as Y-Xb, and the squared criterion is their dot product. Normalizing by n leaves its minimizers unchanged, and a minimizer exists. An invertible Gram matrix gives the unique inverse-formula minimizer. Every minimizer, including those of singular designs, satisfies the normal equations and has zero residual mean, zero empirical residual-regressor cross moments, and zero empirical residual-regressor covariances. These identities hold for every realized sample, hence for IID samples.

", "headline": "Theorem 4.1.4 (OLS Solution in Matrix Algebra), with the last row of the stacked residual system in its proof corrected (YnY_nYn​ for Y3Y_3Y3​); and Theorem 4.1.5 (Properties of Residuals from the OLS Regression Estimator)", "location": "Theorem 4.1.4 and the stacked system: p. 148 (PDF p. 166); rest of the proof and Theorem 4.1.5: p. 149 (PDF p. 167); proof of 4.1.5: p. 150 (PDF p. 168). Definition 4.1.2: pp. 146–147 (PDF pp. 164–165). Definition 4.1.3: p. 148" }, "id": "P001-T65", "text": { "body": "Theorem 4.1.4 and the stacked system: p. 148 (PDF p. 166); rest of the proof and Theorem 4.1.5: p. 149 (PDF p. 167); proof of 4.1.5: p. 150 (PDF p. 168). Definitions 4.1.2–4.1.3: pp. 146–148. For every nonempty intercept design the correctly indexed scalar residuals stack as Y-Xb, and the squared criterion is their dot product. Normalizing by n leaves its minimizers unchanged, and a minimizer exists. An invertible Gram matrix gives the unique inverse-formula minimizer. Every minimizer, including those of singular designs, satisfies the normal equations and has zero residual mean, zero empirical residual-regressor cross moments, and zero empirical residual-regressor covariances. These identities hold for every realized sample, hence for IID samples.", "headline": "Theorem 4.1.4 (OLS Solution in Matrix Algebra), with the last row of the stacked residual system in its proof corrected ($Y_n$ for $Y_3$); and Theorem 4.1.5 (Properties of Residuals from the OLS Regression Estimator)", "location": "Theorem 4.1.4 and the stacked system: p. 148 (PDF p. 166); rest of the proof and Theorem 4.1.5: p. 149 (PDF p. 167); proof of 4.1.5: p. 150 (PDF p. 168). Definition 4.1.2: pp. 146–147 (PDF pp. 164–165). Definition 4.1.3: p. 148" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T65" }, { "declarations": [ { "claim_revision_id": "catalog:4c674f6b78241eaaeb2398d04dba41eecf83cfd94c8791ebcdb432e3f0cf66ee", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/P001_Foundations.lean:3193 extracts the robust and classical standard errors from their own covariance diagonal entries for every k in Fin(K+1), including the intercept. Zero-based coordinate k is ordinal position k+1, exactly matching the corrected sentences. coefficientSE at P001/P001_Foundations.lean:462 is deliberately the claim's square-root definition, so the definitional equalities faithfully encode this definitional component. Arbitrary residual inputs, singular designs, and totalized extra cases retain all valid OLS cases; equality of normalized and matrix covariance formulas is a separate component.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.standard_error_diagonal_index", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2891", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:4c674f6b78241eaaeb2398d04dba41eecf83cfd94c8791ebcdb432e3f0cf66ee", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T66", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T66" }, "html": { "body": "

p. 153 (PDF p. 171) and p. 155 (PDF p. 173). The usual measure-theoretic conditions are assumed. For a sample of size at least two with an intercept and invertible Gram matrix, the OLS residuals are centered and their unbiased empirical variance is the residual sum of squares divided by n - 1. The normalized outer-product formulas equal the robust and classical matrix covariance estimates. Both estimates have nonnegative diagonals, and the standard error of coefficient k is the square root of entry (k,k), the book's ordinal row and column k + 1.

", "headline": "The standard error estimator for an OLS coefficient as the square root of a diagonal element of the estimated covariance matrix: the ordinal, corrected (robust, p. 153; classical, p. 155)", "location": "p. 153 (PDF p. 171), paragraph after Definition 4.2.1; p. 155 (PDF p. 173), sentence after Definition 4.2.3" }, "id": "P001-T66", "text": { "body": "p. 153 (PDF p. 171) and p. 155 (PDF p. 173). The usual measure-theoretic conditions are assumed. For a sample of size at least two with an intercept and invertible Gram matrix, the OLS residuals are centered and their unbiased empirical variance is the residual sum of squares divided by n - 1. The normalized outer-product formulas equal the robust and classical matrix covariance estimates. Both estimates have nonnegative diagonals, and the standard error of coefficient k is the square root of entry (k,k), the book's ordinal row and column k + 1.", "headline": "The standard error estimator for an OLS coefficient as the square root of a diagonal element of the estimated covariance matrix: the ordinal, corrected (robust, p. 153; classical, p. 155)", "location": "p. 153 (PDF p. 171), paragraph after Definition 4.2.1; p. 155 (PDF p. 173), sentence after Definition 4.2.3" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T66" }, { "declarations": [ { "claim_revision_id": "catalog:ecdc0cc138f548909d032e615da716fe1cdf3fb495be1048a73d9a4d1527e761", "conditional_on": [], "credit_reason": "The target’s formal content is the exact population identity, with asymptotic=false; no asymptotic covariance approximation is credited here. Claim-stage verdict retained; the record reveals no misread Lean hypothesis or claim text.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.homoskedastic_sandwich_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2899", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:ecdc0cc138f548909d032e615da716fe1cdf3fb495be1048a73d9a4d1527e761", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/P001_Foundations.lean:5954 gives the claim's conditional-second-moment and population-moment factorization component. With zero conditional mean, the CEF of (epsilon-0)^2 is exactly the conditional variance, so simplifying the first conclusion does not replace the statistical premise with an unrelated assumption. L2 membership and product integrability express the finite expectations in the claim and its standing convention; constant finite conditional variance entails the finite unconditional second moment. Arbitrary finite-dimensional regressors include the augmented intercept vector by choosing the dimension K+1, and the row loses no cases of the moment identity.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.homoskedastic_population_moments", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5278", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:ecdc0cc138f548909d032e615da716fe1cdf3fb495be1048a73d9a4d1527e761", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T67", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T67" }, "html": { "body": "

p. 155 (PDF p. 173); sandwich form p. 152 (PDF p. 170). Under the stated measurability and finite-moment conditions, conditional mean zero and constant conditional residual variance give the conditional second moment, weighted population moment factorization, and exact sandwich simplification.

", "headline": "Section 4.2.3, the display reducing the asymptotic covariance of the OLS estimator under a linear CEF and homoskedasticity, with the stray observation subscript removed", "location": "p. 155 (PDF p. 173), display beginning nV[β^]≈n\\mathrm{V}[\\hat{\\boldsymbol\\beta}] \\approxnV[β^​]≈; the sandwich form: p. 152 (PDF p. 170)" }, "id": "P001-T67", "text": { "body": "p. 155 (PDF p. 173); sandwich form p. 152 (PDF p. 170). Under the stated measurability and finite-moment conditions, conditional mean zero and constant conditional residual variance give the conditional second moment, weighted population moment factorization, and exact sandwich simplification.", "headline": "Section 4.2.3, the display reducing the asymptotic covariance of the OLS estimator under a linear CEF and homoskedasticity, with the stray observation subscript removed", "location": "p. 155 (PDF p. 173), display beginning $n\\mathrm{V}[\\hat{\\boldsymbol\\beta}] \\approx$; the sandwich form: p. 152 (PDF p. 170)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T67" }, { "declarations": [ { "claim_revision_id": "catalog:de436f7740022c26e741976c5fd3861853e6cb2d5e03c6ca040eb6a065c489e7", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'P001/P001_Foundations.lean:2681 requires hg : Measurable g on all of ℝ, whereas the target calls g a function of X and Definition 1.2.2 requires only measurability of g∘X. On a completed uniform probability space with X the identity on [0,1], W=0, and g the indicator of a Lebesgue-measurable non-Borel subset of [0,1] extended by zero, the target hypotheses hold but hg fails. IsCEF itself also requires global measurability of its supplied function at line 99, so merely dropping hg would not repair the contract. Treating conditional expectations as versions permits null-set ambiguity in their values, but the present statement still quantifies only over globally measurable input functions g.', 'suggested_fix': 'Use Measurable (g ∘ X) together with the existing integrability premise, and state the conclusion using a CEF predicate that requires measurable composites and the same event-integral identities rather than global Measurable g. The current theorem can remain a corollary for globally measurable g.', 'second_review': \"P001/P001_Foundations.lean:2681 requires globally Measurable g, whereas T68 calls g a function of X and standing Definition 1.2.2 requires only that g composed with X be a random variable. On completed uniform measure with X the identity on [0,1], a Lebesgue-measurable non-Borel indicator extended by zero is an integrable function of X; independent zero noise satisfies the remaining target conditions while the global hypothesis fails. IsCEF at P001/P001_Foundations.lean:99 also requires the supplied function itself to be measurable, so repair both the input hypothesis and the conclusion's representation to accommodate the measurable composite or an explicitly identified version.\", 'severity': 'major'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.additive_independent_noise_cef", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2421", "status": "proved", "support_scope": "limited", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:de436f7740022c26e741976c5fd3861853e6cb2d5e03c6ca040eb6a065c489e7", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T68", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T68" }, "html": { "body": "

p. 163 (PDF p. 181), Example 4.3.5. The usual measure-theoretic conditions are assumed; g is measurable. For independent integrable mean-zero noise and an integrable signal g(X), the outcome Y = g(X) plus the noise is integrable and has g as a conditional mean version: E[Y given X = x] = g(x). Equivalently, its conditional expectation given the sigma algebra generated by X equals g(X) almost surely.

", "headline": "Example 4.3.5 (Overfitting with a Nonlinear CEF), the CEF identity, corrected (g(x)g(x)g(x) for g(X)g(X)g(X))", "location": "p. 163 (PDF p. 181), Example 4.3.5, first two sentences" }, "id": "P001-T68", "text": { "body": "p. 163 (PDF p. 181), Example 4.3.5. The usual measure-theoretic conditions are assumed; g is measurable. For independent integrable mean-zero noise and an integrable signal g(X), the outcome Y = g(X) plus the noise is integrable and has g as a conditional mean version: E[Y given X = x] = g(x). Equivalently, its conditional expectation given the sigma algebra generated by X equals g(X) almost surely.", "headline": "Example 4.3.5 (Overfitting with a Nonlinear CEF), the CEF identity, corrected ($g(x)$ for $g(X)$)", "location": "p. 163 (PDF p. 181), Example 4.3.5, first two sentences" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T68" }, { "declarations": [ { "claim_revision_id": "catalog:6eca08a2f0ba890c9796ab21ed1cd86e9d842c56b7797adf863dc95923bef987", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P001/P001_Foundations.lean:9325 includes the specified independent uniform laws, the fixed piecewise CEF, the derivative in coordinate one, both actual covariate means, APD=1, PDA=-1, and their inequality. Each first-coordinate section is affine, so these derivatives are actual derivatives throughout, including at the second-coordinate threshold. IsCEF at P001/P001_Foundations.lean:97 identifies a conditional-expectation version by event-integral identities, consistent with the standing convention for versions, and differentiation uses the specified version. The separate plug-in-estimator sentence is outside this population-example component, so its absence is not a material omission.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.apd_pda_different_correct_coordinate", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7804", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:6eca08a2f0ba890c9796ab21ed1cd86e9d842c56b7797adf863dc95923bef987", "conditional_on": [], "credit_reason": "The quantified population declaration covers every admissible outcome with the specified CEF; a separate concrete construction is not needed for this conditional assertion.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.uniform_apd_pda_actual_example", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/WP001dB1.lean#L81", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:6eca08a2f0ba890c9796ab21ed1cd86e9d842c56b7797adf863dc95923bef987", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T69", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T69" }, "html": { "body": "

pp. 166–167 (PDF pp. 184–185). The usual measure-theoretic conditions are assumed. For independent U(0,1) covariates and the specified piecewise CEF, its first partial derivative is -1 on the branch of probability 4/5 and 9 on the branch of probability 1/5. The covariate means are (1/2,1/2), APD_1 is (-1)(4/5) + 9(1/5) = 1, and PDA_1 is -1, so they differ. For any nonempty sample and fitted CEF with the coordinate derivatives used by the formula, the empirical-law APD plug-in is the average of those derivatives. Its index is the coordinate k, with no individual-observation subscript.

", "headline": "Example 4.3.8 (Differences between the Average Partial Derivative and the Partial Derivative at the Average), with the subscripts of the two displays corrected; and the label in the sentence after Definition 4.3.6", "location": "Example 4.3.8: p. 167 (PDF p. 185). Sentence after Definition 4.3.6: p. 166 (PDF p. 184). Definitions 4.3.6 and 4.3.7: pp. 166–167" }, "id": "P001-T69", "text": { "body": "pp. 166–167 (PDF pp. 184–185). The usual measure-theoretic conditions are assumed. For independent U(0,1) covariates and the specified piecewise CEF, its first partial derivative is -1 on the branch of probability 4/5 and 9 on the branch of probability 1/5. The covariate means are (1/2,1/2), APD_1 is (-1)(4/5) + 9(1/5) = 1, and PDA_1 is -1, so they differ. For any nonempty sample and fitted CEF with the coordinate derivatives used by the formula, the empirical-law APD plug-in is the average of those derivatives. Its index is the coordinate k, with no individual-observation subscript.", "headline": "Example 4.3.8 (Differences between the Average Partial Derivative and the Partial Derivative at the Average), with the subscripts of the two displays corrected; and the label in the sentence after Definition 4.3.6", "location": "Example 4.3.8: p. 167 (PDF p. 185). Sentence after Definition 4.3.6: p. 166 (PDF p. 184). Definitions 4.3.6 and 4.3.7: pp. 166–167" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T69" }, { "declarations": [ { "claim_revision_id": "catalog:2d299ea86f2d17b18d317136cfd23984f1dd74c835d81a666a4e931f8108d980", "conditional_on": [], "credit_reason": "The target is corrected_statement and status is proved, but this credited row is the refutation of the printed unsquared objective, not a proof of a positive corrected least-squares assertion. That auxiliary purpose is explicit in row_says. Claim-stage verdict retained; the record reveals no misread Lean hypothesis or claim text.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.unsquared_sieve_no_minimum", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2919", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:2d299ea86f2d17b18d317136cfd23984f1dd74c835d81a666a4e931f8108d980", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'P001/P001_Foundations.lean:6698 places the nonzero Gram determinant premise before both conjuncts, including the criterion identity at line 6699. The target states that identity for any nonempty dataset and degree, imposing invertibility only on the unique inverse-formula minimizer. Singular data such as n=2, J=1 and X=(0,0) are therefore excluded, but the excluded assertion is just expansion of olsRSS at line 447 using the polynomial matrix entries. This is a literal narrowing with mathematically trivial omitted cases, warranting minor severity rather than the flagged major problem; the full-rank minimization and uniqueness component is faithful.', 'suggested_fix': 'Move hunit inside the second conjunct as an implication: assert the criterion identity under hP alone, and assert the minimization and uniqueness conclusions conditionally on IsUnit (P.transpose*P).det.', 'second_review': 'P001/P001_Foundations.lean:6698 places the invertibility premise before the criterion identity at line 6699 as well as the minimization and uniqueness conclusion. T70(ii) identifies the squared polynomial criterion with OLS for all nonempty real datasets and degrees, conditioning only the inverse-formula minimizer on invertibility, so singular data such as n=2, J=1, X=(0,0) are unnecessarily excluded from the identity. The omitted identity is just expansion of olsRSS at P001/P001_Foundations.lean:447 using hP, so move invertibility inside the minimizer conjunct and retain minor severity.', 'severity': 'minor'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.polynomial_squared_criterion_ols", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5895", "status": "proved", "support_scope": "limited", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:2d299ea86f2d17b18d317136cfd23984f1dd74c835d81a666a4e931f8108d980", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T70", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T70" }, "html": { "body": "

Definition 4.3.9, pp. 168–169 (PDF pp. 186–187). For every nonempty sample the printed unsquared polynomial criterion has its affine expansion, tends to minus infinity as the intercept grows, and has no minimizer. The corrected criterion is the OLS sum of squared residuals. Invertibility of the polynomial Gram matrix requires at least J+1 distinct observed covariate values and gives the unique inverse-formula minimizer. These assertions hold at every finite degree, hence also at the degrees of any sieve sequence with the printed rates.

", "headline": "Definition 4.3.9 (Polynomial Least Squares Sieve Estimator), the least squares criterion, corrected (the missing square)", "location": "p. 169 (PDF p. 187), the display defining β^\\hat{\\boldsymbol\\beta}β^​ (the definition begins on p. 168, PDF p. 186)" }, "id": "P001-T70", "text": { "body": "Definition 4.3.9, pp. 168–169 (PDF pp. 186–187). For every nonempty sample the printed unsquared polynomial criterion has its affine expansion, tends to minus infinity as the intercept grows, and has no minimizer. The corrected criterion is the OLS sum of squared residuals. Invertibility of the polynomial Gram matrix requires at least J+1 distinct observed covariate values and gives the unique inverse-formula minimizer. These assertions hold at every finite degree, hence also at the degrees of any sieve sequence with the printed rates.", "headline": "Definition 4.3.9 (Polynomial Least Squares Sieve Estimator), the least squares criterion, corrected (the missing square)", "location": "p. 169 (PDF p. 187), the display defining $\\hat{\\boldsymbol\\beta}$ (the definition begins on p. 168, PDF p. 186)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T70" }, { "declarations": [ { "claim_revision_id": "catalog:15b5aebadff13f2e0b9047f51f9b5a823f574d042d7c250db6750c7227574c3e", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/P001_Foundations.lean:3237 identifies the zero-penalty lasso objective with the OLS mean squared residual objective for every coefficient vector and shows that the slope penalty vanishes on the unit-intercept vector. The definitions at lines 471 and 473 use K+1 coefficients and omit the intercept from the penalty, matching the correction. Universally quantified equality of the objectives gives equality of their minimizer sets and hence the same coefficient estimator when unique, so the claim's zero-penalty consequence is retained without an explicitly named selected minimizer. Arbitrary designs and the extra totalized n=0 case do not exclude any valid sample case.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.lasso_zero_penalty_ols", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2935", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:15b5aebadff13f2e0b9047f51f9b5a823f574d042d7c250db6750c7227574c3e", "conditional_on": [], "credit_reason": "This row is the criterion-convexity component, without an existence or uniqueness claim for minimizers. Claim-stage verdict retained; the record reveals no misread Lean hypothesis or claim text.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.lasso_convex", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2945", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:15b5aebadff13f2e0b9047f51f9b5a823f574d042d7c250db6750c7227574c3e", "conditional_on": [], "credit_reason": "This row establishes the seminorm component, not an existence theorem for a lasso minimizer. Claim-stage verdict retained; the record reveals no misread Lean hypothesis or claim text.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.slope_penalty_seminorm_and_unpenalized_intercept", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6135", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:15b5aebadff13f2e0b9047f51f9b5a823f574d042d7c250db6750c7227574c3e", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T71", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T71" }, "html": { "body": "

pp. 169–170 (PDF pp. 187–188). The slope-only penalty is a seminorm vanishing on the unit intercept, zero penalty gives exactly the OLS minimizer set, and nonnegative penalty gives a convex criterion.

", "headline": "Definition 4.3.10 (Lasso Estimator), corrected: the coefficient vector in RK+1\\mathbb{R}^{K+1}RK+1 and the penalty on the slope coefficients", "location": "pp. 169–170 (PDF pp. 187–188)" }, "id": "P001-T71", "text": { "body": "pp. 169–170 (PDF pp. 187–188). The slope-only penalty is a seminorm vanishing on the unit intercept, zero penalty gives exactly the OLS minimizer set, and nonnegative penalty gives a convex criterion.", "headline": "Definition 4.3.10 (Lasso Estimator), corrected: the coefficient vector in $\\mathbb{R}^{K+1}$ and the penalty on the slope coefficients", "location": "pp. 169–170 (PDF pp. 187–188)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T71" }, { "declarations": [ { "claim_revision_id": "catalog:10c48eb4e70eb8cd83b53832fde79f98b8ccd63c2c0def95d6d85c831425d78e", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"At P001/P001_Foundations.lean:3302, the conclusion gives total mass one and first moment x·beta for the displayed Gaussian density whenever s>0. These are exactly the positive-scale conditional-density computations in T72, and substituting the model's conditional PDF gives the claimed CEF. Allowing arbitrary fixed x includes every supported covariate vector with intercept one. The separate zero-scale exclusion is represented by the corrected positive parameter domain; this row's density-computation component does not require an assertion of undefined Lean arithmetic.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.classical_linear_positive_scale_cef", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3000", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:10c48eb4e70eb8cd83b53832fde79f98b8ccd63c2c0def95d6d85c831425d78e", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T72", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T72" }, "html": { "body": "

Definition 5.1.4: pp. 180–181 (PDF pp. 198–199). Theorem 5.1.5: p. 181 (PDF p. 199). The positive-scale normal kernel is an integrable probability density with mean x dot beta; the totalized zero-scale formula has mass zero and is not a probability density.

", "headline": "Definition 5.1.4 (Classical Linear Model), the parameter space, corrected (σ>0\\sigma > 0σ>0); with Theorem 5.1.5 (CEF of the Classical Linear Model)", "location": "Definition 5.1.4: pp. 180–181 (PDF pp. 198–199). Theorem 5.1.5: p. 181 (PDF p. 199)" }, "id": "P001-T72", "text": { "body": "Definition 5.1.4: pp. 180–181 (PDF pp. 198–199). Theorem 5.1.5: p. 181 (PDF p. 199). The positive-scale normal kernel is an integrable probability density with mean x dot beta; the totalized zero-scale formula has mass zero and is not a probability density.", "headline": "Definition 5.1.4 (Classical Linear Model), the parameter space, corrected ($\\sigma > 0$); with Theorem 5.1.5 (CEF of the Classical Linear Model)", "location": "Definition 5.1.4: pp. 180–181 (PDF pp. 198–199). Theorem 5.1.5: p. 181 (PDF p. 199)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T72" }, { "declarations": [ { "claim_revision_id": "catalog:399c217e8cd1dd8e8ac00b78c767771944acaae8d4985908ca9b2b6b5acaf103", "conditional_on": [], "credit_reason": "This is an exact counterexample to general CEF-residual independence, including a linear CEF. It meets the hypotheses of that assertion and does not purport to refute constant-scale Gaussian-model independence. Claim-stage verdict retained; the record reveals no misread Lean hypothesis or claim text.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.cef_residual_not_independent", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3026", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:399c217e8cd1dd8e8ac00b78c767771944acaae8d4985908ca9b2b6b5acaf103", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T73", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T73" }, "html": { "body": "

Section 5.1.1, p. 181 (PDF p. 199); Theorem 2.2.19, p. 74 (PDF p. 92). The printed claim is false: on the four equiprobable atoms (X,Y)=(0,1),(0,3), (1,0),(1,6), the CEF is 2+X and its residuals are -1,1,-3,3, each with mass 1/4. Their conditional and unconditional means vanish; their covariance with every g(X) vanishes. The conditional outcome and residual variances agree and are 1 and 9; the residual variance and the mean conditional outcome variance are both 5. Nevertheless the displayed joint probability is 1/2 and the product is 1/4, so CEF residuals need not be independent of X, even for a linear CEF.

", "headline": "Section 5.1.1, the sentence attributing independence from X\\mathbf{X}X to the deviations from the CEF (with the reference to Theorem 2.2.19), as printed", "location": "p. 181 (PDF p. 199), paragraph after the display Y=Xβ+εY = \\mathbf{X}\\boldsymbol\\beta + \\varepsilonY=Xβ+ε. Theorem 2.2.19: p. 74 (PDF p. 92)" }, "id": "P001-T73", "text": { "body": "Section 5.1.1, p. 181 (PDF p. 199); Theorem 2.2.19, p. 74 (PDF p. 92). The printed claim is false: on the four equiprobable atoms (X,Y)=(0,1),(0,3), (1,0),(1,6), the CEF is 2+X and its residuals are -1,1,-3,3, each with mass 1/4. Their conditional and unconditional means vanish; their covariance with every g(X) vanishes. The conditional outcome and residual variances agree and are 1 and 9; the residual variance and the mean conditional outcome variance are both 5. Nevertheless the displayed joint probability is 1/2 and the product is 1/4, so CEF residuals need not be independent of X, even for a linear CEF.", "headline": "Section 5.1.1, the sentence attributing independence from $\\mathbf{X}$ to the deviations from the CEF (with the reference to Theorem 2.2.19), as printed", "location": "p. 181 (PDF p. 199), paragraph after the display $Y = \\mathbf{X}\\boldsymbol\\beta + \\varepsilon$. Theorem 2.2.19: p. 74 (PDF p. 92)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T73" }, { "declarations": [ { "claim_revision_id": "catalog:a30f155c9560348985f0ed83e0029050eb09919f6c15976e6caeb1968a40bf1c", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'P001/P001_Foundations.lean:3355–3359 requires Y and m(X) in L² and restricts the covariance conclusion to globally measurable g with g(X) in L². The claim says that the covariance is zero for every function of X, subject to the standing existence of the expectations, which does not require these individual second moments. For Pareto X of tail exponent 3/2 and independent symmetric-sign Y, take m=0 and g(x)=x: the residual has all required variances, E|g(X)| and E|g(X)Y| are finite, and the claimed covariance is zero, but the g(X) L² premise fails. The global measurability restriction on g also exceeds Definition 1.2.2, and the row description cannot replace the claim’s broader quantifier.', 'suggested_fix': 'State the conditional and unconditional residual mean-zero conclusions using hm without hY or hm2. For the covariance clause, require only integrability of g(X) and g(X)(Y-m(X)), and use a CEF/function representation permitting the composite-measurability notion of Definition 1.2.2 instead of requiring globally measurable functions.', 'second_review': \"P001/P001_Foundations.lean:3355 and :3356 assume square-integrable Y and m(X), and :3359 restricts the covariance conclusion to measurable g with square-integrable g(X); IsCEF itself already supplies the first-moment assumptions at :97. T74(i) instead requires mean zero and zero covariance for every function of X for which the relevant expectations exist, under the standing finite-expectation convention. For Pareto X with tail exponent 3/2 and an independent symmetric-sign Y, m=0 and g(x)=x give finite residual variances and an absolutely integrable covariance equal to zero, although g(X) has infinite second moment. This excludes substantive cases of the credited orthogonality component, so weaken the moment premises rather than merely retaining the description's L2 qualification.\", 'severity': 'major'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.cef_residual_orthogonality_correction", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3051", "status": "proved", "support_scope": "limited", "used_in_solution": false }, { "claim_revision_id": "catalog:a30f155c9560348985f0ed83e0029050eb09919f6c15976e6caeb1968a40bf1c", "conditional_on": [], "credit_reason": "The joint-law assumption is a substantive model premise equivalent to the constant-scale Gaussian kernel for the finite-dimensional covariates needed by the claim; it is not an unexplained Contract field assuming a newly claimed theorem. Claim-stage verdict retained; the record reveals no misread Lean hypothesis or claim text.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.classical_normal_residual_independent", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5912", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:a30f155c9560348985f0ed83e0029050eb09919f6c15976e6caeb1968a40bf1c", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'P001/P001_Foundations.lean:7566–7567 requires both Y and m(X) to lie in L², although this component of Theorem 2.2.19 concerns finite conditional centered-square expectations and the residual variance. Let X have Pareto tail exponent 3/2, let Z be an independent symmetric sign, and set Y=X+Z, m(x)=x and v(x)=1: the relevant means exist and both conditional and residual variances equal 1, but Y and m(X) are not in L². The standing convention that the expectations appearing in the claim exist does not imply these additional second moments. The first conjunct at line 7569 repeats hv after subtracting zero, but the unconditional variance identity is substantive, so the whole row is not vacuous.', 'suggested_fix': 'Remove hY : MemLp Y 2 μ and hm2 : MemLp (m ∘ X) 2 μ. The retained hm supplies integrability of Y and m(X), and hv supplies integrability of (Y-m(X))²; these are the moments needed for this component.', 'second_review': \"P001/P001_Foundations.lean:7566 and :7567 require Y and m(X) separately to be in L2, beyond the integrability already supplied by hm and the residual-square integrability supplied by hv through IsCEF at P001/P001_Foundations.lean:99. T74's conditional and unconditional residual-variance identities require finite moments of the residual, not finite second moments of Y and its conditional mean: take integrable Pareto X with tail exponent 3/2, independent symmetric sign Z, Y=X+Z, m(x)=x and v(x)=1. Both relevant variances are 1 although Y and m(X) are not in L2, so remove those two extra moment hypotheses and use the moments already present in hm and hv.\", 'severity': 'major'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.cef_residual_conditional_variance", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6488", "status": "proved", "support_scope": "limited", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:a30f155c9560348985f0ed83e0029050eb09919f6c15976e6caeb1968a40bf1c", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T74", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T74" }, "html": { "body": "

p. 181 (PDF p. 199); Theorem 2.2.19, pp. 74–75 (PDF pp. 92–93). The usual measure-theoretic conditions are assumed; functions of X are measurable functions. For an integrable response and a chosen conditional mean, the residual has conditional and unconditional mean zero and is uncorrelated with every measurable function of X whenever the means and residual product moment exist. If the residual second moment exists, its conditional variance equals that of Y, and its variance is the expectation of that conditional variance; a conditional-variance version exists. Residuals need not be independent of X, as the four-point example shows. In the classical conditional normal model at positive scale, the specified residual conditional law is N(0,sigma^2) at every covariate value; the residual has that marginal law and is independent of X. Pointwise conditional laws refer to the specified model version, and conditional-expectation identities are almost-sure identities.

", "headline": "Section 5.1.1, the comparison of the classical linear model's errors with deviations from the CEF, corrected ('uncorrelated with' for 'independent of'); with Theorem 2.2.19 (Properties of Deviations from the CEF)", "location": "p. 181 (PDF p. 199), paragraph after the display Y=Xβ+εY = \\mathbf{X}\\boldsymbol\\beta + \\varepsilonY=Xβ+ε. Theorem 2.2.19: p. 74 (PDF p. 92), proof on pp. 74–75" }, "id": "P001-T74", "text": { "body": "p. 181 (PDF p. 199); Theorem 2.2.19, pp. 74–75 (PDF pp. 92–93). The usual measure-theoretic conditions are assumed; functions of X are measurable functions. For an integrable response and a chosen conditional mean, the residual has conditional and unconditional mean zero and is uncorrelated with every measurable function of X whenever the means and residual product moment exist. If the residual second moment exists, its conditional variance equals that of Y, and its variance is the expectation of that conditional variance; a conditional-variance version exists. Residuals need not be independent of X, as the four-point example shows. In the classical conditional normal model at positive scale, the specified residual conditional law is N(0,sigma^2) at every covariate value; the residual has that marginal law and is independent of X. Pointwise conditional laws refer to the specified model version, and conditional-expectation identities are almost-sure identities.", "headline": "Section 5.1.1, the comparison of the classical linear model's errors with deviations from the CEF, corrected ('uncorrelated with' for 'independent of'); with Theorem 2.2.19 (Properties of Deviations from the CEF)", "location": "p. 181 (PDF p. 199), paragraph after the display $Y = \\mathbf{X}\\boldsymbol\\beta + \\varepsilon$. Theorem 2.2.19: p. 74 (PDF p. 92), proof on pp. 74–75" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T74" }, { "declarations": [ { "claim_revision_id": "catalog:bb51b734c90774aea645a1826f7c4f9070d536798b9a59c8823338aba75378b1", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"At P001/P001_Foundations.lean:3396, the two-point weighted sum is 0·(1-h(x·beta))+1·h(x·beta)=h(x·beta), with the asserted [0,1] bound. Under T75's specified conditional PMF this sum is precisely the definition of E[Y|X=x], so a separate joint-law object is unnecessary for this computation. The global range hypothesis is the claim's assumption h: real numbers → [0,1], and arbitrary fixed x includes all of the claim's supported covariate vectors.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.binary_choice_fixed_argument_mean", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3094", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:bb51b734c90774aea645a1826f7c4f9070d536798b9a59c8823338aba75378b1", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T75", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T75" }, "html": { "body": "

Definition 5.1.7: pp. 182–183 (PDF pp. 200–201). Theorem 5.1.8: p. 183 (PDF p. 201). For fixed x and beta and h valued in [0,1], the two-point mean 0*(1-h(x dot beta)) + 1*h(x dot beta) equals h(x dot beta) and lies in [0,1].

", "headline": "Theorem 5.1.8 (CEF of a Binary Choice Model) and the kernel of Definition 5.1.7, with the fixed argument x\\mathbf{x}x in place of the random vector X\\mathbf{X}X", "location": "Definition 5.1.7: pp. 182–183 (PDF pp. 200–201). Theorem 5.1.8: p. 183 (PDF p. 201)" }, "id": "P001-T75", "text": { "body": "Definition 5.1.7: pp. 182–183 (PDF pp. 200–201). Theorem 5.1.8: p. 183 (PDF p. 201). For fixed x and beta and h valued in [0,1], the two-point mean 0*(1-h(x dot beta)) + 1*h(x dot beta) equals h(x dot beta) and lies in [0,1].", "headline": "Theorem 5.1.8 (CEF of a Binary Choice Model) and the kernel of Definition 5.1.7, with the fixed argument $\\mathbf{x}$ in place of the random vector $\\mathbf{X}$", "location": "Definition 5.1.7: pp. 182–183 (PDF pp. 200–201). Theorem 5.1.8: p. 183 (PDF p. 201)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T75" }, { "declarations": [ { "claim_revision_id": "catalog:ddb30fc1a5831ac166c78b3c109e02b10bec578eac034985846d74b770b8893a", "conditional_on": [], "credit_reason": "Only the assigned fixed-scale optimizer equivalence is credited here; full-rank closed form is not a conclusion of this row.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.normal_fixed_scale_ml_is_ols", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3102", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:ddb30fc1a5831ac166c78b3c109e02b10bec578eac034985846d74b770b8893a", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T76", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T76" }, "html": { "body": "

pp. 190–191 (PDF pp. 208–209). The usual measure-theoretic conditions are assumed. At every fixed positive scale, the Gaussian likelihood is a positive product of the displayed normal densities; its logarithm is the displayed sum and the closed-form normal log-likelihood. Likelihood maximizers, log-likelihood maximizers and residual-sum-of-squares minimizers in R^(K+1) coincide, so their sets do not depend on the positive scale. If the Gram matrix is invertible, the maximizer exists uniquely and equals (X^T X)^(-1) X^T Y. At positive sample sizes, minimizing the sum is equivalent to minimizing the OLS mean square.

", "headline": "Example 5.2.4 (Maximum Likelihood Estimation of the Classical Linear Model), the maximization display, corrected: the coefficient vector in RK+1\\mathbb{R}^{K+1}RK+1 and the likelihood written with (b,s)(\\mathbf{b}, s)(b,s); the ML estimator of β\\boldsymbol\\betaβ is the OLS estimator", "location": "pp. 190–191 (PDF pp. 208–209)" }, "id": "P001-T76", "text": { "body": "pp. 190–191 (PDF pp. 208–209). The usual measure-theoretic conditions are assumed. At every fixed positive scale, the Gaussian likelihood is a positive product of the displayed normal densities; its logarithm is the displayed sum and the closed-form normal log-likelihood. Likelihood maximizers, log-likelihood maximizers and residual-sum-of-squares minimizers in R^(K+1) coincide, so their sets do not depend on the positive scale. If the Gram matrix is invertible, the maximizer exists uniquely and equals (X^T X)^(-1) X^T Y. At positive sample sizes, minimizing the sum is equivalent to minimizing the OLS mean square.", "headline": "Example 5.2.4 (Maximum Likelihood Estimation of the Classical Linear Model), the maximization display, corrected: the coefficient vector in $\\mathbb{R}^{K+1}$ and the likelihood written with $(\\mathbf{b}, s)$; the ML estimator of $\\boldsymbol\\beta$ is the OLS estimator", "location": "pp. 190–191 (PDF pp. 208–209)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T76" }, { "declarations": [ { "claim_revision_id": "catalog:d85f1902d847829927108dd82f22bc93ed8771263ec9738636d4959cb4acec4f", "conditional_on": [], "credit_reason": "This is a harmless generalization of the corrected function-space assertion, not a claim of a measurable random element in a specified function-space sigma algebra.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.likelihood_correct_function_space", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3121", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:d85f1902d847829927108dd82f22bc93ed8771263ec9738636d4959cb4acec4f", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T77", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T77" }, "html": { "body": "

p. 186 (PDF p. 204), footnote 5 after Definition 5.2.1. The realized finite nonnegative likelihood is a function on the parameter set in real coordinate space of the parameter's arity. A normal PDF at its mean equals 1/(s sqrt(2 pi)) for positive standard deviation s, and exceeds one whenever s < 1/sqrt(2 pi). Thus PDF likelihood values need not lie in [0,1].

", "headline": "Footnote 5 (the likelihood function as a random function), with the domain and codomain corrected", "location": "p. 186 (PDF p. 204), footnote 5, attached to the sentence after Definition 5.2.1" }, "id": "P001-T77", "text": { "body": "p. 186 (PDF p. 204), footnote 5 after Definition 5.2.1. The realized finite nonnegative likelihood is a function on the parameter set in real coordinate space of the parameter's arity. A normal PDF at its mean equals 1/(s sqrt(2 pi)) for positive standard deviation s, and exceeds one whenever s < 1/sqrt(2 pi). Thus PDF likelihood values need not lie in [0,1].", "headline": "Footnote 5 (the likelihood function as a random function), with the domain and codomain corrected", "location": "p. 186 (PDF p. 204), footnote 5, attached to the sentence after Definition 5.2.1" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T77" }, { "declarations": [ { "claim_revision_id": "catalog:f65c206e1a20f8b9b0dc8c1523f9d71bd7002772bdd32ddecb7d04d951c87cf8", "conditional_on": [], "credit_reason": "The Q>0 condition is in the target formal content, so exact fits are not excluded by an extra Lean assumption. Residual plug-in variance remains a target-level omission.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.normal_joint_ml_scale_square", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3132", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:f65c206e1a20f8b9b0dc8c1523f9d71bd7002772bdd32ddecb7d04d951c87cf8", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T78", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T78" }, "html": { "body": "

p. 191 (PDF p. 209), footnote 10 in Example 5.2.4. For a nonempty regression sample with an intercept, any least-squares minimizer with positive residual sum Q jointly maximizes the normal log likelihood at scale sqrt(Q/n). Its squared scale is Q/n. The residual sum and mean are zero, so Q/n equals both the centered residual second moment and the plug-in variance. No rank or uniqueness assumption is required.

", "headline": "Footnote 10 (Example 5.2.4), corrected: 1n∑i(Yi−Xiβ^)2\\frac{1}{n}\\sum_i(Y_i - \\mathbf{X}_i\\hat{\\boldsymbol\\beta})^2n1​∑i​(Yi​−Xi​β^​)2 is the ML estimator of σ2\\sigma^2σ2, not of σ\\sigmaσ", "location": "p. 191 (PDF p. 209), footnote 10" }, "id": "P001-T78", "text": { "body": "p. 191 (PDF p. 209), footnote 10 in Example 5.2.4. For a nonempty regression sample with an intercept, any least-squares minimizer with positive residual sum Q jointly maximizes the normal log likelihood at scale sqrt(Q/n). Its squared scale is Q/n. The residual sum and mean are zero, so Q/n equals both the centered residual second moment and the plug-in variance. No rank or uniqueness assumption is required.", "headline": "Footnote 10 (Example 5.2.4), corrected: $\\frac{1}{n}\\sum_i(Y_i - \\mathbf{X}_i\\hat{\\boldsymbol\\beta})^2$ is the ML estimator of $\\sigma^2$, not of $\\sigma$", "location": "p. 191 (PDF p. 209), footnote 10" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T78" }, { "declarations": [ { "claim_revision_id": "catalog:2f4b086965112f1065ba18ac0e1f4ea0e51d48ca9ab1fb14d311fe8c7abd66b8", "conditional_on": [], "credit_reason": "The assigned finite-sample CDF component is exact. No specialization to an ML fit is lost, because its conditional laws may be substituted for ν_i.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.ml_plugin_correct_covariate_indices", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3200", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:2f4b086965112f1065ba18ac0e1f4ea0e51d48ca9ab1fb14d311fe8c7abd66b8", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T79", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T79" }, "html": { "body": "

pp. 197–198 (PDF pp. 215–216), Definition 5.2.10. The population joint CDF is the expectation of the covariate lower-orthant indicator times the conditional lower-tail probability, expressed by a PMF sum or PDF integral for the appropriate conditional-law version. Replacing that expectation by the sample mean and using one attained ML parameter gives the corrected fitted PMF and PDF displays. Each fitted display is a joint CDF of a probability law, so applying any real statistical functional T defines T(Fhat). Samples are nonempty; conditional laws and densities are read as versions.

", "headline": "Definition 5.2.10 (Maximum Likelihood Plug-In Estimator), the indicator in both displays of F^ML\\hat F_{ML}F^ML​, with its subscripts corrected", "location": "pp. 197–198 (PDF pp. 215–216)" }, "id": "P001-T79", "text": { "body": "pp. 197–198 (PDF pp. 215–216), Definition 5.2.10. The population joint CDF is the expectation of the covariate lower-orthant indicator times the conditional lower-tail probability, expressed by a PMF sum or PDF integral for the appropriate conditional-law version. Replacing that expectation by the sample mean and using one attained ML parameter gives the corrected fitted PMF and PDF displays. Each fitted display is a joint CDF of a probability law, so applying any real statistical functional T defines T(Fhat). Samples are nonempty; conditional laws and densities are read as versions.", "headline": "Definition 5.2.10 (Maximum Likelihood Plug-In Estimator), the indicator in both displays of $\\hat F_{ML}$, with its subscripts corrected", "location": "pp. 197–198 (PDF pp. 215–216)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T79" }, { "declarations": [ { "claim_revision_id": "catalog:2fd8be1fde6413a4b155f370bdcd71861364c949b2e8bf5b0d35b1860dd4c919", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.missing_data_bounds_example", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3235", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:2fd8be1fde6413a4b155f370bdcd71861364c949b2e8bf5b0d35b1860dd4c919", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'At P001/P001_Foundations.lean:3546, both endpoint expectation inequalities and attainment of every c between them are present. For binary R, the lower endpoint is E[Y*R]+a·Pr[R=0] and the upper endpoint replaces a by b, exactly the printed respondent-mean formulas. The completion keeps R and the censored outcome fixed pointwise, which implies the required equality of observable joint distributions. Everywhere bounds can be imposed on measurable representatives under the standing version convention, and the general bounded-support sharpness component loses no case.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.missing_data_bounds_and_sharpness", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3244", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:2fd8be1fde6413a4b155f370bdcd71861364c949b2e8bf5b0d35b1860dd4c919", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T80", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T80" }, "html": { "body": "

Example 6.1.2, pp. 209–210 (PDF pp. 227–228); Theorem 6.1.3, pp. 210–211 (PDF pp. 228–229); sharpness prose, pp. 209 and 212. An almost-surely [a,b]-valued outcome with binary response has mean between respondent mean times response probability plus a or b times missingness probability. The endpoints are the expectations of the lowest and highest completions. Every value between them is attained with the same observable joint law. Both response values have positive probability. The six printed observations give completion vectors (1,0,1,0,1,0) and (1,1,1,0,1,1), with means 1/2 and 5/6; repeating the lowest-value instruction gives 1/2, unequal to 5/6.

", "headline": "Example 6.1.2 (Bounding the Expected Value), the upper-bound instruction, corrected ('highest' for 'lowest'); with Theorem 6.1.3 (Sharp Bounds for the Expected Value)", "location": "Example 6.1.2: pp. 209–210 (PDF pp. 227–228). Theorem 6.1.3: p. 210 (PDF p. 228), proof on p. 211 (PDF p. 229). Section 6.1.1, first paragraphs: p. 209" }, "id": "P001-T80", "text": { "body": "Example 6.1.2, pp. 209–210 (PDF pp. 227–228); Theorem 6.1.3, pp. 210–211 (PDF pp. 228–229); sharpness prose, pp. 209 and 212. An almost-surely [a,b]-valued outcome with binary response has mean between respondent mean times response probability plus a or b times missingness probability. The endpoints are the expectations of the lowest and highest completions. Every value between them is attained with the same observable joint law. Both response values have positive probability. The six printed observations give completion vectors (1,0,1,0,1,0) and (1,1,1,0,1,1), with means 1/2 and 5/6; repeating the lowest-value instruction gives 1/2, unequal to 5/6.", "headline": "Example 6.1.2 (Bounding the Expected Value), the upper-bound instruction, corrected ('highest' for 'lowest'); with Theorem 6.1.3 (Sharp Bounds for the Expected Value)", "location": "Example 6.1.2: pp. 209–210 (PDF pp. 227–228). Theorem 6.1.3: p. 210 (PDF p. 228), proof on p. 211 (PDF p. 229). Section 6.1.1, first paragraphs: p. 209" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T80" }, { "declarations": [ { "claim_revision_id": "catalog:43faed3b7b061ac43d6cf52df636dbeda8339f926a2644ff4effd257adebbcb8", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'At P001/P001_Foundations.lean:3633, all eight distances in the corrected six-unit example and the mean of (2,2,3,10,12,3) are stated exactly. The distances from unit 2 are .19, .59, .21 and .64, identifying units 1 and 4 as the nearest two with gap .02; those from unit 6 identify unit 3 as nearest. These comparisons are immediate arithmetic consequences of the exported distances, so the absence of a donor-selection function is a representation note. The completed-data mean 16/3 matches the claim.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.hot_deck_correct_donor_comparison", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3331", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:43faed3b7b061ac43d6cf52df636dbeda8339f926a2644ff4effd257adebbcb8", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T81", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T81" }, "html": { "body": "

Example 6.2.5: pp. 224–225 (PDF pp. 242–243). Paragraph after it: p. 225 (PDF p. 243). The corrected propensity distances select the two hot deck donors, identify the second donor and its gap, and give completed mean 16/3.

", "headline": "The paragraph after Example 6.2.5 (Imputation with the Propensity Score): the comparison of donors for unit 2, corrected (pR(X2)p_R(\\mathbf{X}_2)pR​(X2​) for pR(X1)p_R(\\mathbf{X}_1)pR​(X1​)); with the hot deck estimate of the example", "location": "Example 6.2.5: pp. 224–225 (PDF pp. 242–243). Paragraph after it: p. 225 (PDF p. 243)" }, "id": "P001-T81", "text": { "body": "Example 6.2.5: pp. 224–225 (PDF pp. 242–243). Paragraph after it: p. 225 (PDF p. 243). The corrected propensity distances select the two hot deck donors, identify the second donor and its gap, and give completed mean 16/3.", "headline": "The paragraph after Example 6.2.5 (Imputation with the Propensity Score): the comparison of donors for unit 2, corrected ($p_R(\\mathbf{X}_2)$ for $p_R(\\mathbf{X}_1)$); with the hot deck estimate of the example", "location": "Example 6.2.5: pp. 224–225 (PDF pp. 242–243). Paragraph after it: p. 225 (PDF p. 243)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T81" }, { "declarations": [ { "claim_revision_id": "catalog:a71be67a1cbd35b826f8b8914a4995bd66e2604fd56b9e7be4aa54b98857ec3d", "conditional_on": [], "credit_reason": "The existence row already supplies the conditional-mean identification and impossible-range criterion.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.propensity_range_correction", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7879", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:a71be67a1cbd35b826f8b8914a4995bd66e2604fd56b9e7be4aa54b98857ec3d", "conditional_on": [], "credit_reason": "The existence row already supplies boundedness and normalized binary conditional laws.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.conditional_kernel_propensity_range", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6156", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:a71be67a1cbd35b826f8b8914a4995bd66e2604fd56b9e7be4aa54b98857ec3d", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"At P001/Wave1/Gap/WP001dB3.lean:101, hRpos requires 0p. 225 (PDF p. 243), Section 6.2.4, first paragraph, last sentence on the page. A binary response has a bounded conditional-probability version equal to its conditional mean, and impossible real predictions are precisely those outside [0,1].

", "headline": "Section 6.2.4, the range of the response propensity and the description of impossible estimates, corrected", "location": "p. 225 (PDF p. 243), Section 6.2.4, first paragraph, last sentence on the page" }, "id": "P001-T82", "text": { "body": "p. 225 (PDF p. 243), Section 6.2.4, first paragraph, last sentence on the page. A binary response has a bounded conditional-probability version equal to its conditional mean, and impossible real predictions are precisely those outside [0,1].", "headline": "Section 6.2.4, the range of the response propensity and the description of impossible estimates, corrected", "location": "p. 225 (PDF p. 243), Section 6.2.4, first paragraph, last sentence on the page" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T82" }, { "declarations": [ { "claim_revision_id": "catalog:f552a210af2190d5ce32584cd01366b62181f918d20afd146c7a4f63ec2c0684", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"At P001/Wave1/Gap/GLP0014B4.lean:122, the conclusion is exactly E[Y]=E[Y*R/p(X)] with the stable-outcome formula expanded through censored. The assumptions hmar and hp express outcome-response conditional independence and the true response propensity, as required by corrected Theorem 6.2.6. Uniform positivity is explicitly part of Definition 6.1.8, and integrability of Y is the standing expectation convention. Event-based conditional independence and conditional-mean versions preserve the PMF/PDF claim's content, including its nondegenerate response cases.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.mar_reweighting_corrected_hypothesis_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GLP0014B4.lean#L122", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:f552a210af2190d5ce32584cd01366b62181f918d20afd146c7a4f63ec2c0684", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T83", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T83" }, "html": { "body": "

Theorem 6.2.6, p. 226 (PDF p. 244), proof pp. 226–227. For an integrable outcome and almost-surely binary response with both values of positive probability, MAR given the covariates means conditional independence and an almost-sure uniform positive lower bound on the true response propensity. Then the recorded-outcome inverse-propensity weighted variable is integrable, and its expectation equals both the population mean and the expectation of the weighted uncensored outcome. Propensities are measurable conditional-mean versions; all version bounds are almost sure.

", "headline": "Theorem 6.2.6 (Reweighting under MAR), with its hypothesis corrected ('YiY_iYi​ is MAR conditional on Xi\\mathbf{X}_iXi​')", "location": "p. 226 (PDF p. 244); proof on pp. 226–227 (PDF pp. 244–245)" }, "id": "P001-T83", "text": { "body": "Theorem 6.2.6, p. 226 (PDF p. 244), proof pp. 226–227. For an integrable outcome and almost-surely binary response with both values of positive probability, MAR given the covariates means conditional independence and an almost-sure uniform positive lower bound on the true response propensity. Then the recorded-outcome inverse-propensity weighted variable is integrable, and its expectation equals both the population mean and the expectation of the weighted uncensored outcome. Propensities are measurable conditional-mean versions; all version bounds are almost sure.", "headline": "Theorem 6.2.6 (Reweighting under MAR), with its hypothesis corrected ('$Y_i$ is MAR conditional on $\\mathbf{X}_i$')", "location": "p. 226 (PDF p. 244); proof on pp. 226–227 (PDF pp. 244–245)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T83" }, { "declarations": [ { "claim_revision_id": "catalog:71c16a18d8ac208213d54b1904318e53b390143778fa8007c1de568f0b7704da", "conditional_on": [], "credit_reason": "Duplicate wrapper of the unique-logistic-fit counterexample used here.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.missing_ipw_not_poststratified", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7949", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:71c16a18d8ac208213d54b1904318e53b390143778fa8007c1de568f0b7704da", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"At P001/P001_Foundations.lean:6437, the full unrestricted intercept/slope logistic likelihood has the claimed unique maximum at (log(7/5),0). The constants at P001/P001_Foundations.lean:722 are cell sizes (4,4,4), response counts (2,3,2) and respondent sums (3/4,3/2,3/4), exactly the sufficient statistics of T84's twelve observations. The conclusion includes nonempty response and nonresponse cells, fitted probability 7/12, and estimator values 3/7 and 5/12 with inequality. Drawing x uniformly from {-1,0,1}, R independently as Bernoulli(7/12), and Y independently uniformly from {1/4,1/2,3/4} gives that sample positive probability under iid MAR sampling and a correctly specified logit model. Thus the grouped representation supplies an admissible counterexample to the printed sample-estimator equivalence.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.missing_ipw_logit_ml_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5670", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:71c16a18d8ac208213d54b1904318e53b390143778fa8007c1de568f0b7704da", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T84", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T84" }, "html": { "body": "

p. 228 (PDF p. 246), first paragraph. Definition 6.2.7: p. 227 (PDF p. 245). Section 6.2.1 plug-in estimator: p. 219 (PDF p. 237). Section 6.2.4: pp. 225–226 (PDF pp. 243–244). The explicit twelve-unit example refutes the reading of "logically equivalent" as exact finite-sample equality of IPW and post-stratification plug-in estimators for arbitrary ML binary-choice propensity fits. Its three-cell logit likelihood has the stated unique ML fit, IPW mean 3/7, plug-in mean 5/12 and gap 1/84.

", "headline": "Section 6.2.5, the sentence 'In the case where there is a discrete and finite number of values for Xi\\mathbf{X}_iXi​, the IPW estimator is logically equivalent to the plug-in estimator from Section 6.2.1', as printed", "location": "p. 228 (PDF p. 246), first paragraph. Definition 6.2.7: p. 227 (PDF p. 245). Section 6.2.1 plug-in estimator: p. 219 (PDF p. 237). Section 6.2.4: pp. 225–226 (PDF pp. 243–244)" }, "id": "P001-T84", "text": { "body": "p. 228 (PDF p. 246), first paragraph. Definition 6.2.7: p. 227 (PDF p. 245). Section 6.2.1 plug-in estimator: p. 219 (PDF p. 237). Section 6.2.4: pp. 225–226 (PDF pp. 243–244). The explicit twelve-unit example refutes the reading of \"logically equivalent\" as exact finite-sample equality of IPW and post-stratification plug-in estimators for arbitrary ML binary-choice propensity fits. Its three-cell logit likelihood has the stated unique ML fit, IPW mean 3/7, plug-in mean 5/12 and gap 1/84.", "headline": "Section 6.2.5, the sentence 'In the case where there is a discrete and finite number of values for $\\mathbf{X}_i$, the IPW estimator is logically equivalent to the plug-in estimator from Section 6.2.1', as printed", "location": "p. 228 (PDF p. 246), first paragraph. Definition 6.2.7: p. 227 (PDF p. 245). Section 6.2.1 plug-in estimator: p. 219 (PDF p. 237). Section 6.2.4: pp. 225–226 (PDF pp. 243–244)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T84" }, { "declarations": [ { "claim_revision_id": "catalog:ed0013297a59124e482738b518760a7af7cdb6151c609582f5a883f3755cff26", "conditional_on": [], "credit_reason": "A general algebraic saturation fact for all cells; dividing by the common sample size yields the assigned estimator-equality component.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.missing_ipw_saturated_cells", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3350", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:ed0013297a59124e482738b518760a7af7cdb6151c609582f5a883f3755cff26", "conditional_on": [], "credit_reason": "For actual positive respondent counts, arbitrary S can be realized by varying one respondent outcome per cell. The statement correctly avoids claiming that equality at one S requires saturation.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.missing_ipw_saturation_necessary", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3360", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:ed0013297a59124e482738b518760a7af7cdb6151c609582f5a883f3755cff26", "conditional_on": [], "credit_reason": "The additional algebraic cases q≤0 do not affect the target specialization q>0. Sample-to-cell grouping is outside this row.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.missing_ipw_difference_and_stabilization", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5318", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:ed0013297a59124e482738b518760a7af7cdb6151c609582f5a883f3755cff26", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T85", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T85" }, "html": { "body": "

p. 228 (PDF p. 246), first paragraph. For every positive finite sample with binary responses, a respondent in each observed cell and positive fitted probabilities, grouping gives both cell formulas and their difference. Equality for every outcome vector is equivalent to fitting the observed response fraction in every cell. Under this saturation condition the mean weight is one and the stabilized, ordinary IPW and plug-in estimates agree.

", "headline": "Section 6.2.5, equivalence of the IPW estimator and the post-stratification plug-in estimator, corrected: it holds when the estimated response probability in each cell is the observed respondent fraction", "location": "p. 228 (PDF p. 246), first paragraph" }, "id": "P001-T85", "text": { "body": "p. 228 (PDF p. 246), first paragraph. For every positive finite sample with binary responses, a respondent in each observed cell and positive fitted probabilities, grouping gives both cell formulas and their difference. Equality for every outcome vector is equivalent to fitting the observed response fraction in every cell. Under this saturation condition the mean weight is one and the stabilized, ordinary IPW and plug-in estimates agree.", "headline": "Section 6.2.5, equivalence of the IPW estimator and the post-stratification plug-in estimator, corrected: it holds when the estimated response probability in each cell is the observed respondent fraction", "location": "p. 228 (PDF p. 246), first paragraph" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T85" }, { "declarations": [ { "claim_revision_id": "catalog:5f106912a1f4a6341519bfd04bc922ac82d43741b456e90862e6e82295255a74", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'At P001/P001_Foundations.lean:3761, the conclusion contains the conditional and unconditional mean-one identities for R/p(X), the lower bound on E[1/p(X)], and its exact equality characterization. These match the true-propensity population-moment component of corrected T86. The bound p≤1 is inherent in a response probability, and the uniform positive lower bound is expressly assumed in the claim; everywhere bounds select a suitable conditional version. The finite positive-n sample-mean expectation is an immediate linearity corollary, while the fitted-propensity convergence discussion is a separate component of the larger passage.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.response_weights_denominator_correction", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3425", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:5f106912a1f4a6341519bfd04bc922ac82d43741b456e90862e6e82295255a74", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T86", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T86" }, "html": { "body": "

p. 228 (PDF p. 246), Definition 6.2.8 and the paragraph after it. Response-weighted inverse propensities have conditional and unconditional mean one, while reciprocal propensities have mean at least one with the stated equality case.

", "headline": "The paragraph after Definition 6.2.8 (Stabilized IPW Estimator for Missing Data): the sum whose between-sample variation the stabilized estimator accounts for, corrected (RiR_iRi​ in the numerator)", "location": "p. 228 (PDF p. 246), Definition 6.2.8 and the paragraph after it" }, "id": "P001-T86", "text": { "body": "p. 228 (PDF p. 246), Definition 6.2.8 and the paragraph after it. Response-weighted inverse propensities have conditional and unconditional mean one, while reciprocal propensities have mean at least one with the stated equality case.", "headline": "The paragraph after Definition 6.2.8 (Stabilized IPW Estimator for Missing Data): the sum whose between-sample variation the stabilized estimator accounts for, corrected ($R_i$ in the numerator)", "location": "p. 228 (PDF p. 246), Definition 6.2.8 and the paragraph after it" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T86" }, { "declarations": [ { "claim_revision_id": "catalog:5a2eec662294a71b8a880649a00c57603642d0c374ed6dfc88ce19830929fbc3", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'At P001/Wave1/Gap/GP0014B2b.lean:284, the first conclusion identifies the respondent CEFs of Y and Y*, and the second gives the full Y* CEF conditional on (R,X). For binary R its piecewise function equals R·m(X)+(1-R)·(-99), exactly T87(iii), and restricting the defining identities to R=0 gives the missing-case CEF -99 in T87(ii). Unnormalized restrictions preserve conditional means, and allowing boundary response laws extends the result without making its mixed-response cases vacuous. The claim distinguishes the two conditioning operations; it does not require their numerical values to differ in every model.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.stable_outcomes_respondent_cef_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GP0014B2b.lean#L282", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:5a2eec662294a71b8a880649a00c57603642d0c374ed6dfc88ce19830929fbc3", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'At P001/Wave1/Gap/GP0014B2b.lean:156, the conclusion identifies the respondent censored-outcome CEF m with the population outcome CEF, exactly clause (iv) of T87 under MAR. The additional hmi is redundant: respondent integrability and a propensity lower bound epsilon imply E[|m(X)|]≤epsilon^(-1)·integral over {R=1} of |m(X)|, by applying the propensity identity to bounded truncations and passing to the limit. Thus hmi excludes no instance permitted by the claim and standing integrability convention. The unnormalized respondent restriction and almost-everywhere conditional versions preserve the printed random-variable identity.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.mar_respondent_cef_is_population_cef_proof", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GP0014B2b.lean#L156", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:5a2eec662294a71b8a880649a00c57603642d0c374ed6dfc88ce19830929fbc3", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T87", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T87" }, "html": { "body": "

p. 233 (PDF p. 251), first paragraph below Table 6.3.1. For integrable outcomes and almost surely binary responses, a measurable respondent conditional-mean function exists. Every such version is also the respondent CEF of the recorded outcome. On nonresponse the recorded CEF is minus 99; the full CEF given (R,X) is R m(X) minus 99(1-R). Under MAR and almost sure uniform positive propensity every respondent version is integrable in the population and is its outcome CEF. The linear and squared-covariate working specifications target this respondent function; no exact approximation or numerical-fit assertion is made.

", "headline": "Section 6.3, the working models for the regression estimators: the conditional expectation, corrected to the respondent CEF E[Yi∗∣Ri=1,Xi]\\mathrm{E}[Y_i^*|R_i = 1, \\mathbf{X}_i]E[Yi∗​∣Ri​=1,Xi​] (three places)", "location": "p. 233 (PDF p. 251), first paragraph below Table 6.3.1" }, "id": "P001-T87", "text": { "body": "p. 233 (PDF p. 251), first paragraph below Table 6.3.1. For integrable outcomes and almost surely binary responses, a measurable respondent conditional-mean function exists. Every such version is also the respondent CEF of the recorded outcome. On nonresponse the recorded CEF is minus 99; the full CEF given (R,X) is R m(X) minus 99(1-R). Under MAR and almost sure uniform positive propensity every respondent version is integrable in the population and is its outcome CEF. The linear and squared-covariate working specifications target this respondent function; no exact approximation or numerical-fit assertion is made.", "headline": "Section 6.3, the working models for the regression estimators: the conditional expectation, corrected to the respondent CEF $\\mathrm{E}[Y_i^*|R_i = 1, \\mathbf{X}_i]$ (three places)", "location": "p. 233 (PDF p. 251), first paragraph below Table 6.3.1" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T87" }, { "declarations": [ { "claim_revision_id": "catalog:ef2855fc14bb968a863a4a20a1b84faeeb76d6e7c2953a371486d8f9a29ac2d7", "conditional_on": [], "credit_reason": "The almost-sure-support row supplies the complete endpoint-estimation conclusion with weaker support hypotheses.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.ate_bounds_estimators", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L8420", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:ef2855fc14bb968a863a4a20a1b84faeeb76d6e7c2953a371486d8f9a29ac2d7", "conditional_on": [], "credit_reason": "The theorem gives the sharp causal endpoints, attainable completions, unbiased endpoint sample means and convergence in probability under the stated bounded-support and observed-pair IID assumptions.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.ate_sharp_endpoint_estimation_ae_support", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/WP001dB3.lean#L210", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:ef2855fc14bb968a863a4a20a1b84faeeb76d6e7c2953a371486d8f9a29ac2d7", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T88", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T88" }, "html": { "body": "

Theorem 7.1.8 and the first part of the proof: p. 243 (PDF p. 261); rest of the proof: p. 244 (PDF p. 262). Both sharp ATE endpoints have the stated observable formulas, attain every compatible effect mean and have unbiased, consistent sample-average estimators.

", "headline": "Theorem 7.1.8 (Estimating Sharp Bounds for the ATE), with two symbols in its proof corrected (τiL‾\\overline{\\tau_i^L}τiL​​ for YiL‾\\overline{Y_i^L}YiL​​; Yi(1−Di)Y_i(1 - D_i)Yi​(1−Di​) for Yi(x1−Di)Y_i(x1 - D_i)Yi​(x1−Di​))", "location": "Theorem 7.1.8 and the first part of the proof: p. 243 (PDF p. 261); rest of the proof: p. 244 (PDF p. 262)" }, "id": "P001-T88", "text": { "body": "Theorem 7.1.8 and the first part of the proof: p. 243 (PDF p. 261); rest of the proof: p. 244 (PDF p. 262). Both sharp ATE endpoints have the stated observable formulas, attain every compatible effect mean and have unbiased, consistent sample-average estimators.", "headline": "Theorem 7.1.8 (Estimating Sharp Bounds for the ATE), with two symbols in its proof corrected ($\\overline{\\tau_i^L}$ for $\\overline{Y_i^L}$; $Y_i(1 - D_i)$ for $Y_i(x1 - D_i)$)", "location": "Theorem 7.1.8 and the first part of the proof: p. 243 (PDF p. 261); rest of the proof: p. 244 (PDF p. 262)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T88" }, { "declarations": [ { "claim_revision_id": "catalog:c0e54b05ee40f29114756fff2bb77a7cf136b8cf2411bb69f120f20354f451da", "conditional_on": [], "credit_reason": "The repeated observable-cell mass, four positive-cell conditional means and adjusted contrast exactly match the numerical component covered by this row.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.post_treatment_observable_mass_and_contrast", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3473", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:c0e54b05ee40f29114756fff2bb77a7cf136b8cf2411bb69f120f20354f451da", "conditional_on": [], "credit_reason": "The explicit six-atom latent model verifies random assignment, the covariate consistency equation, zero true effect and failure of conditional ignorability as described.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.post_treatment_latent_random_assignment", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6612", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:c0e54b05ee40f29114756fff2bb77a7cf136b8cf2411bb69f120f20354f451da", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T89", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T89" }, "html": { "body": "

pp. 253–254 (PDF pp. 271–272). The specified uniform six-atom latent law randomizes treatment with respect to all potential variables. Treatment has zero outcome effect everywhere but changes the covariate with probability one third. The complete corrected observable PMF has four masses one sixth and one mass one third, and total one; the printed function on five distinct cells has total five sixths. The stated covariate marginals, conditioning-cell masses, four conditional means and contrast minus one quarter hold. Conditional outcome probabilities one half and zero witness the loss of conditional independence after adjusting for this post-treatment covariate.

", "headline": "Example 7.1.17 (Conditioning on Post-Treatment Variables), the distribution of observables, corrected (the cell (Yi,Xi,Di)=(1,1,1)(Y_i, X_i, D_i) = (1, 1, 1)(Yi​,Xi​,Di​)=(1,1,1) once, with probability 13\\frac{1}{3}31​); the adjusted contrast −14-\\frac{1}{4}−41​", "location": "pp. 253–254 (PDF pp. 271–272)" }, "id": "P001-T89", "text": { "body": "pp. 253–254 (PDF pp. 271–272). The specified uniform six-atom latent law randomizes treatment with respect to all potential variables. Treatment has zero outcome effect everywhere but changes the covariate with probability one third. The complete corrected observable PMF has four masses one sixth and one mass one third, and total one; the printed function on five distinct cells has total five sixths. The stated covariate marginals, conditioning-cell masses, four conditional means and contrast minus one quarter hold. Conditional outcome probabilities one half and zero witness the loss of conditional independence after adjusting for this post-treatment covariate.", "headline": "Example 7.1.17 (Conditioning on Post-Treatment Variables), the distribution of observables, corrected (the cell $(Y_i, X_i, D_i) = (1, 1, 1)$ once, with probability $\\frac{1}{3}$); the adjusted contrast $-\\frac{1}{4}$", "location": "pp. 253–254 (PDF pp. 271–272)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T89" }, { "declarations": [ { "claim_revision_id": "catalog:87d5c573be4b0f9729a9a1d5952709b6c127376188cf730df9d1bb83e9ad5954", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/P001_Foundations.lean:3834 states equality of the two observable pushforward laws, and line 3836 gives the spike contrast 1 versus the zero curve's contrast 0. These are the claim's exact fixed witnesses: Lebesgue measure restricted to [0,1] has mass one, both doses lie in its positive-density support, and each potential process is deterministic and therefore independent of treatment. Expectations of their fixed-dose contrasts are the displayed constants, so the implicit probabilistic interpretation loses no case and does not rely on a zero-measure or divergent-integral convention.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.continuous_dose_nonidentification", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3485", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:87d5c573be4b0f9729a9a1d5952709b6c127376188cf730df9d1bb83e9ad5954", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T90", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T90" }, "html": { "body": "

The printed claim is false: deterministic zero and spike response processes have identical observable laws under uniform dose and different effects at both stated dose pairs. p. 255 (PDF p. 273), first paragraph; Definition 7.1.18 and the generalized conditional independence assumption: p. 254 (PDF p. 272). Definition 6.0.1: p. 207 (PDF p. 225).

", "headline": "Section 7.1.8, the claim that under ignorability E[τi(d,d′)]\\mathrm{E}[\\tau_i(d, d')]E[τi​(d,d′)] is point identified for any d,d′∈Supp[Di]d, d' \\in \\mathrm{Supp}[D_i]d,d′∈Supp[Di​], for a continuous treatment, as printed", "location": "p. 255 (PDF p. 273), first paragraph; Definition 7.1.18 and the generalized conditional independence assumption: p. 254 (PDF p. 272). Definition 6.0.1: p. 207 (PDF p. 225)" }, "id": "P001-T90", "text": { "body": "The printed claim is false: deterministic zero and spike response processes have identical observable laws under uniform dose and different effects at both stated dose pairs. p. 255 (PDF p. 273), first paragraph; Definition 7.1.18 and the generalized conditional independence assumption: p. 254 (PDF p. 272). Definition 6.0.1: p. 207 (PDF p. 225).", "headline": "Section 7.1.8, the claim that under ignorability $\\mathrm{E}[\\tau_i(d, d')]$ is point identified for any $d, d' \\in \\mathrm{Supp}[D_i]$, for a continuous treatment, as printed", "location": "p. 255 (PDF p. 273), first paragraph; Definition 7.1.18 and the generalized conditional independence assumption: p. 254 (PDF p. 272). Definition 6.0.1: p. 207 (PDF p. 225)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T90" }, { "declarations": [ { "claim_revision_id": "catalog:12c10b8f2346e65d161100f38fc3059f9b2e991bf5697f834c9cb8e68c502d37", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'P001/Wave1/Gap/DecisionT91.lean:270 introduces a measurable curve state and jointly measurable evaluation, and line 273 assumes conditional independence of that state and treatment given X. Corrected clause (b) instead states process independence, continuity of conditional means at the selected doses, and positive mass in every dose neighborhood; its final scope paragraph expressly says the evaluation-ready contract does not redefine that weaker claim. These are materially different assumptions: with constant X, uniform D, and Y_t=t except Y_t=0 when t=D, the cylinder process is independent of D and has continuous mean t, but the observed outcome Y_D is zero. Thus the exported theorem does not cover all cases of the registered process-only correction, even though it proves the advertised bridge and contrast under its stronger contract.', 'suggested_fix': 'Reassign this row to a corrected target that explicitly assumes a measurable curve state, jointly measurable evaluation, and conditional independence of that state and treatment given X. The process-only target must remain unresolved unless its measurable process space and a sufficient evaluation/independence bridge are added to the correction itself.', 'second_review': 'P001/Wave1/Gap/DecisionT91.lean:270 requires a measurable curve state, :271 jointly measurable evaluation, and :273 conditional independence of that state and treatment, using the event-factorization definition CI at P001/P001_Foundations.lean:290. T91(b) instead assumes independence of the potential-outcome process and continuity of conditional means with positive neighborhood mass, and its final scope paragraph expressly says the evaluation-ready contract does not redefine that weaker claim. With constant X, uniform D on [0,1], and Y_t=t except Y_D=0, the cylinder process is independent of D and has continuous mean t, yet the observed outcome is always zero and cannot identify the mean dose contrast. The stronger state contract is therefore essential additional mathematical content, so reassign the credit to an explicitly strengthened target rather than treating its removal as a small statement repair.', 'severity': 'major'}", "credit_status": "wrong_credit", "fidelity_status_baseline": "wrong_credit", "lean_name": "AronowAudit.P001.Wave1.continuous_dose_identified_evaluation_ready", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/DecisionT91.lean#L264", "status": "proved", "support_scope": "withdrawn", "used_in_solution": true }, { "claim_revision_id": "catalog:12c10b8f2346e65d161100f38fc3059f9b2e991bf5697f834c9cb8e68c502d37", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': \"P001/Wave1/Gap/GP0014B3.lean:199 assumes hi : ∀ t, Integrable (Y t) μ, including supported doses other than the dose identified in lines 203–207. The claim's positive-mass component identifies E[Y(d)] (and, by using two doses, their contrast); the standing finite-expectation convention does not impose moments at every unrelated dose. For example, let Pr[X=n]=2^(-n) for n≥1, Pr[D=2|X=n]=2^(-n), Pr[D=0|X=n]=Pr[D=1|X=n]=(1-2^(-n))/2, and set Y(0)=0, Y(1)=1, Y(2)=2^X, with other dose outcomes zero. The entire potential process is determined by X, hence satisfies conditional independence; doses 0 and 1 have positive probabilities in every stratum, E[Y(D)]=4/3 and their mean effect is 1, but E[|Y(2)|]=Σ_n 1=∞. Dose 2 is in the actual treatment support, so extending the family by zero outside support, as the reviewer suggests, cannot remove this additional hypothesis.\", 'suggested_fix': 'Move the fixed dose d before the moment premise and replace hi : ∀ t, Integrable (Y t) μ with hid : Integrable (Y d) μ. Retain coordinate measurability, integrability of the observed outcome, conditional independence and the stated positive-cell condition; apply the repaired theorem separately at d and d′ for the contrast.', 'second_review': 'P001/Wave1/Gap/GP0014B3.lean:199 assumes Integrable (Y t) for every real dose, although :203 and :206 identify only the fixed dose d. T91(a) requires the relevant expectations for the selected positive-mass doses, and the standing finite-expectation convention does not supply moments at unrelated supported doses. For P(X=n)=2^(-n), n>=1, let P(D=2|X=n)=2^(-n), split the remaining probability equally between doses 0 and 1, and set Y(0)=0, Y(1)=1, Y(2)=2^X: conditional independence holds, the observed expectation is 4/3 and the selected contrast is 1, but E|Y(2)| is infinite. Since dose 2 is supported, extending outcomes by zero outside support cannot fix the restriction; replace the universal integrability premise with integrability at d and apply the result at both selected doses.', 'severity': 'major'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.Wave1.positive_mass_dose_identification_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GP0014B3.lean#L195", "status": "proved", "support_scope": "limited", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:12c10b8f2346e65d161100f38fc3059f9b2e991bf5697f834c9cb8e68c502d37", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T91", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T91" }, "html": { "body": "

p. 255 (PDF p. 273), first paragraph; generalized potential outcomes and process independence: p. 254 (PDF p. 272). Positive conditional dose masses identify potential means and their contrast; an independent measurable curve state with jointly measurable evaluation gives an observed conditional-mean version and unique values of continuous observed versions at neighborhood-supported doses, identifying the standardized contrast.

", "headline": "Section 7.1.8, identification of E[τi(d,d′)]\\mathrm{E}[\\tau_i(d, d')]E[τi​(d,d′)] for multivalued and continuous treatments, corrected: positive mass at the doses, or continuity of the conditional mean dose-response", "location": "p. 255 (PDF p. 273), first paragraph" }, "id": "P001-T91", "text": { "body": "p. 255 (PDF p. 273), first paragraph; generalized potential outcomes and process independence: p. 254 (PDF p. 272). Positive conditional dose masses identify potential means and their contrast; an independent measurable curve state with jointly measurable evaluation gives an observed conditional-mean version and unique values of continuous observed versions at neighborhood-supported doses, identifying the standardized contrast.", "headline": "Section 7.1.8, identification of $\\mathrm{E}[\\tau_i(d, d')]$ for multivalued and continuous treatments, corrected: positive mass at the doses, or continuity of the conditional mean dose-response", "location": "p. 255 (PDF p. 273), first paragraph" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T91" }, { "declarations": [ { "claim_revision_id": "catalog:28300feb1372a0005b8b3af9ca241651ab15d63b2805382b2079bd0ac9e5c090", "conditional_on": [], "credit_reason": "This supporting numerical lemma gives exactly the partial-imputation value, fitted coefficient value and nonzero discrepancy stated in the target calculation.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.misprinted_imputation_table_disagreement", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3500", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:28300feb1372a0005b8b3af9ca241651ab15d63b2805382b2079bd0ac9e5c090", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': \"P001/P001_Foundations.lean:5317 exports only IsUnit of the Gram determinant and equality of the OLS coefficient vector with causalExampleBeta. It omits the determinant equality det(XᵀX)=7311 promised by row_says, so the flag identifies a genuine missing conclusion under the row-level standard. The target's fitting component and its exact rational coefficient vector are otherwise matched by the fixed arrays at lines 534, 536, 538, 540, 542, and 662. The omission is minor rather than major because it concerns a fixed finite arithmetic value and excludes no statistical cases.\", 'suggested_fix': 'Add (causalDesign.transpose * causalDesign).det = 7311 as an exported conjunct of causal_example_ols_fit.', 'second_review': \"P001/P001_Foundations.lean:5317 exports Gram nonsingularity and the actual inverse-formula OLS coefficient vector, with the data and rational vector defined at :534, :536, :538, :540, :542 and :662 matching T92's fitting calculation. For this credited fitting component, the target's claim text requires the stated coefficients fitted with unit 3's X1=10 and does not require the numerical Gram determinant 7311. That additional exported conclusion is promised only by row_says, so revise the description to distinguish nonsingularity from an exact determinant equality; the discrepancy is a minor arithmetic description issue, not a missing target conclusion.\", 'severity': None}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.causal_example_ols_fit", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4670", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:28300feb1372a0005b8b3af9ca241651ab15d63b2805382b2079bd0ac9e5c090", "conditional_on": [], "credit_reason": "The actual fit and exact discrepancy rows already give this counterexample; full imputation cancels by finite arithmetic.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.misprinted_imputation_counterexample_full", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4707", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:28300feb1372a0005b8b3af9ca241651ab15d63b2805382b2079bd0ac9e5c090", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T92", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T92" }, "html": { "body": "

The printed claim is false: the actual six-unit OLS fit makes the two printed imputation estimates differ by -253/4874. p. 259 (PDF p. 277), the data table and the two imputation tables.

", "headline": "Example 7.2.2 (Causal Inference with OLS Regression), the two imputation tables as printed (unit 3 with X[1]i=9X_{[1]i} = 9X[1]i​=9) against the claim that both imputations give the same estimate", "location": "p. 259 (PDF p. 277), the data table and the two imputation tables" }, "id": "P001-T92", "text": { "body": "The printed claim is false: the actual six-unit OLS fit makes the two printed imputation estimates differ by -253/4874. p. 259 (PDF p. 277), the data table and the two imputation tables.", "headline": "Example 7.2.2 (Causal Inference with OLS Regression), the two imputation tables as printed (unit 3 with $X_{[1]i} = 9$) against the claim that both imputations give the same estimate", "location": "p. 259 (PDF p. 277), the data table and the two imputation tables" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T92" }, { "declarations": [ { "claim_revision_id": "source-claim-import-2026-11-08:P001-T93:1", "conditional_on": [], "credit_reason": "Both corrected finite-sample imputation calculations equal the specified coefficient, while the actual OLS fit is verified by a separate credited row.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.corrected_imputation_table_agreement", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3509", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "source-claim-import-2026-11-08:P001-T93:1", "conditional_on": [], "credit_reason": "The identity assumes only the two residual normal equations needed by the printed OLS argument and therefore covers every OLS fit as well as additional baseline prediction arrays.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.pooled_ols_imputation_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3518", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "source-claim-import-2026-11-08:P001-T93:1", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T93", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T93" }, "html": { "body": "

Example 7.2.2, pp. 258–259 (PDF pp. 276–277), corrected. Both imputation tables use unit 3's covariates (10,3). For any pooled OLS minimizer, imputing all potential outcomes and imputing only missing outcomes give the same difference of completed-column means, the treatment coefficient; the residual normal equations and the displayed filtered-sum correction hold. The six-unit coefficients are (13563,6178,-759,-1778)/2437 and both estimates are 6178/2437. Under strong ignorability with uniform overlap and an exact linear observed CEF, the conditional ATE equals the armwise observed CEF contrast and the treatment coefficient almost everywhere in the covariate law, and the ATE equals that coefficient. The fitted contrast is that coefficient at every covariate value. The usual measure-theoretic conditions are assumed. “More straightforward to implement” describes the two displayed algorithms; it makes no mathematical comparison of computational costs.

", "headline": "Example 7.2.2 (Causal Inference with OLS Regression), unit 3 of the two imputation tables, corrected (X[1]3=10X_{[1]3} = 10X[1]3​=10); the two imputations give the same estimate, the OLS coefficient on DiD_iDi​", "location": "p. 259 (PDF p. 277); the pooled-regression identity E[τi∣Xi]=β1\\mathrm{E}[\\tau_i|\\mathbf{X}_i] = \\beta_1E[τi​∣Xi​]=β1​: p. 258 (PDF p. 276)" }, "id": "P001-T93", "text": { "body": "Example 7.2.2, pp. 258–259 (PDF pp. 276–277), corrected. Both imputation tables use unit 3's covariates (10,3). For any pooled OLS minimizer, imputing all potential outcomes and imputing only missing outcomes give the same difference of completed-column means, the treatment coefficient; the residual normal equations and the displayed filtered-sum correction hold. The six-unit coefficients are (13563,6178,-759,-1778)/2437 and both estimates are 6178/2437. Under strong ignorability with uniform overlap and an exact linear observed CEF, the conditional ATE equals the armwise observed CEF contrast and the treatment coefficient almost everywhere in the covariate law, and the ATE equals that coefficient. The fitted contrast is that coefficient at every covariate value. The usual measure-theoretic conditions are assumed. “More straightforward to implement” describes the two displayed algorithms; it makes no mathematical comparison of computational costs.", "headline": "Example 7.2.2 (Causal Inference with OLS Regression), unit 3 of the two imputation tables, corrected ($X_{[1]3} = 10$); the two imputations give the same estimate, the OLS coefficient on $D_i$", "location": "p. 259 (PDF p. 277); the pooled-regression identity $\\mathrm{E}[\\tau_i|\\mathbf{X}_i] = \\beta_1$: p. 258 (PDF p. 276)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T93" }, { "declarations": [ { "claim_revision_id": "catalog:1c05bf3982573986151cf4a0a4950c5523c0f39a29c2f342d09100d76c8bac07", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P001/P001_Foundations.lean:3901–3903 expands mlPotentialMean into the sample average of integrals evaluated at X i, exactly the corrected estimator component (ii) of the claim. The definitions at lines 552–557 retain the dose, observed covariate and parameter arguments in the required places. An arbitrary parameter can be instantiated with the ML fit, and allowing zero sample size or totalized integrals adds no restriction on genuine positive-size, finite-moment model instances; this row asserts an estimator formula rather than population identification or a fitting property.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.ml_potential_mean_observed_covariates", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3540", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:1c05bf3982573986151cf4a0a4950c5523c0f39a29c2f342d09100d76c8bac07", "conditional_on": [], "credit_reason": "The binary population identity needs only a correct observed conditional mean rather than a correctly specified entire conditional density, with overlap and finiteness supplied by the target and standing assumptions.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.ml_potential_mean_population_identity_proof", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GP0014B3.lean#L488", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:1c05bf3982573986151cf4a0a4950c5523c0f39a29c2f342d09100d76c8bac07", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T94", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T94" }, "html": { "body": "

p. 261 (PDF p. 279), the population identity and last display; the ATE plug-in formula is on p. 262 (PDF p. 280). Under binary strong ignorability and almost sure overlap, a correct conditional density identifies both potential-outcome conditional means and their population means. The latter are integrals over the covariate law, or the printed iterated Lebesgue integral when that law has a density, and agree across models with the same observable law. For any existing ML maximizer and finite displayed integrals, the corrected estimator averages at the observed covariates; the ATE estimator averages fitted contrasts. The printed free-covariate average collapses to the single fitted conditional mean at that covariate.

", "headline": "Section 7.2.3, the maximum likelihood plug-in estimator of E[Yi(d)]\\mathrm{E}[Y_i(d)]E[Yi​(d)], corrected: the fitted conditional mean evaluated at the observed covariates Xi\\mathbf{X}_iXi​", "location": "p. 261 (PDF p. 279), last display on the page; the population identity above it on the same page" }, "id": "P001-T94", "text": { "body": "p. 261 (PDF p. 279), the population identity and last display; the ATE plug-in formula is on p. 262 (PDF p. 280). Under binary strong ignorability and almost sure overlap, a correct conditional density identifies both potential-outcome conditional means and their population means. The latter are integrals over the covariate law, or the printed iterated Lebesgue integral when that law has a density, and agree across models with the same observable law. For any existing ML maximizer and finite displayed integrals, the corrected estimator averages at the observed covariates; the ATE estimator averages fitted contrasts. The printed free-covariate average collapses to the single fitted conditional mean at that covariate.", "headline": "Section 7.2.3, the maximum likelihood plug-in estimator of $\\mathrm{E}[Y_i(d)]$, corrected: the fitted conditional mean evaluated at the observed covariates $\\mathbf{X}_i$", "location": "p. 261 (PDF p. 279), last display on the page; the population identity above it on the same page" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T94" }, { "declarations": [ { "claim_revision_id": "catalog:333e1f254e2dfd7a99d5a0988a331eca6b40045013b7ee3ec9108edf39619368", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/P001_Foundations.lean:3642–3645 gives all eight opposite-arm score distances and the mean contrast of the claim's completed columns, 7/6. Their exact rational values determine every stated nearest donor, including unit 1 and then unit 4 for unit 2, and unit 3 for unit 6. Expressing this fixed example by its complete numerical distance matrix instead of an argmin function loses none of the corrected comparison or numerical estimate.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.matching_correct_donor_comparison", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3340", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:333e1f254e2dfd7a99d5a0988a331eca6b40045013b7ee3ec9108edf39619368", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T95", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T95" }, "html": { "body": "

Example 7.2.4: pp. 262–263 (PDF pp. 280–281). Paragraph after it: pp. 263–264 (PDF pp. 281–282). The exact distances select all opposite-arm donors and give the completed-column means 22/6 and 15/6 and matching estimate 7/6.

", "headline": "The paragraph after Example 7.2.4 (Propensity Score Matching): the comparison of donors for unit 2, corrected (P2P_2P2​ for P1P_1P1​); with the matching estimate of the example", "location": "Example 7.2.4: pp. 262–263 (PDF pp. 280–281). Paragraph after it: pp. 263–264 (PDF pp. 281–282)" }, "id": "P001-T95", "text": { "body": "Example 7.2.4: pp. 262–263 (PDF pp. 280–281). Paragraph after it: pp. 263–264 (PDF pp. 281–282). The exact distances select all opposite-arm donors and give the completed-column means 22/6 and 15/6 and matching estimate 7/6.", "headline": "The paragraph after Example 7.2.4 (Propensity Score Matching): the comparison of donors for unit 2, corrected ($P_2$ for $P_1$); with the matching estimate of the example", "location": "Example 7.2.4: pp. 262–263 (PDF pp. 280–281). Paragraph after it: pp. 263–264 (PDF pp. 281–282)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T95" }, { "declarations": [ { "claim_revision_id": "catalog:555d1c5ece4bd051f39e73a89dcb6672edf3a085e739973af2e281a2dc23013f", "conditional_on": [], "credit_reason": "As the separately identified arithmetic part of the witness, the theorem gives exactly the two estimator values and their inequality.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.causal_ipw_not_poststratified", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3548", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:555d1c5ece4bd051f39e73a89dcb6672edf3a085e739973af2e281a2dc23013f", "conditional_on": [], "credit_reason": "The concrete grouped witness includes the actual unique unrestricted logit maximum, interior fitted probabilities, both arms in every cell and unequal IPW and post-stratified estimates.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.causal_ipw_logit_ml_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5733", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:555d1c5ece4bd051f39e73a89dcb6672edf3a085e739973af2e281a2dc23013f", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T96", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T96" }, "html": { "body": "

Section 7.2.6, p. 266 (PDF p. 284), with Sections 7.2.1 and 7.2.5, pp. 256 and 264: the canonical eighteen-unit counterexample to the printed IPW/post-stratification equality has unique logit ML fit (0, log 2), IPW estimate 59/36 and post-stratified estimate 1, with the displayed fitted probabilities, observed fractions, score arithmetic and nonzero gap.

", "headline": "Section 7.2.6, the sentence 'In the case where we have a discrete and finite number of values for Xi\\mathbf{X}_iXi​, the IPW estimator is logically equivalent to the plug-in estimator from Section 7.2.1', as printed", "location": "p. 266 (PDF p. 284), paragraph after Definition 7.2.6. Section 7.2.1 plug-in estimator: p. 256 (PDF p. 274). Section 7.2.5: p. 264 (PDF p. 282)" }, "id": "P001-T96", "text": { "body": "Section 7.2.6, p. 266 (PDF p. 284), with Sections 7.2.1 and 7.2.5, pp. 256 and 264: the canonical eighteen-unit counterexample to the printed IPW/post-stratification equality has unique logit ML fit (0, log 2), IPW estimate 59/36 and post-stratified estimate 1, with the displayed fitted probabilities, observed fractions, score arithmetic and nonzero gap.", "headline": "Section 7.2.6, the sentence 'In the case where we have a discrete and finite number of values for $\\mathbf{X}_i$, the IPW estimator is logically equivalent to the plug-in estimator from Section 7.2.1', as printed", "location": "p. 266 (PDF p. 284), paragraph after Definition 7.2.6. Section 7.2.1 plug-in estimator: p. 256 (PDF p. 274). Section 7.2.5: p. 264 (PDF p. 282)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T96" }, { "declarations": [ { "claim_revision_id": "catalog:cc8b7452d61c7e03bb2ffd3b0f0538f865fa760434baa76c41de9ce909d4e60d", "conditional_on": [], "credit_reason": "The all-outcome-sum iff already supplies sufficiency; this separate sufficient unnormalized identity is redundant.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.causal_ipw_saturated_cells", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3556", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:cc8b7452d61c7e03bb2ffd3b0f0538f865fa760434baa76c41de9ce909d4e60d", "conditional_on": [], "credit_reason": "The necessary-and-sufficient identity for all real arm sums specializes to every outcome vector because both arms have positive counts, while allowing general positive real masses.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.causal_ipw_saturation_necessary", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5340", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:cc8b7452d61c7e03bb2ffd3b0f0538f865fa760434baa76c41de9ce909d4e60d", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T97", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T97" }, "html": { "body": "

corrected Section 7.2.6, p. 266 (PDF p. 284), Section 7.2.1, p. 256, and Example 7.2.1, p. 257: for every positive-size binary sample with both arms in each observed cell, the individual IPW and empirical plug-in estimators have the displayed cell formulas; saturated fitted propensities make them equal, and equality for every individual outcome vector holds if and only if the fitted propensity is the observed treated fraction in each cell. The specified six-unit plug-in estimate is 4/9. The usual measure-theoretic conditions are assumed.

", "headline": "Section 7.2.6, equivalence of the IPW estimator and the post-stratification plug-in estimator of the ATE, corrected: it holds when each fitted propensity equals the observed treated fraction in its covariate cell", "location": "p. 266 (PDF p. 284), paragraph after Definition 7.2.6" }, "id": "P001-T97", "text": { "body": "corrected Section 7.2.6, p. 266 (PDF p. 284), Section 7.2.1, p. 256, and Example 7.2.1, p. 257: for every positive-size binary sample with both arms in each observed cell, the individual IPW and empirical plug-in estimators have the displayed cell formulas; saturated fitted propensities make them equal, and equality for every individual outcome vector holds if and only if the fitted propensity is the observed treated fraction in each cell. The specified six-unit plug-in estimate is 4/9. The usual measure-theoretic conditions are assumed.", "headline": "Section 7.2.6, equivalence of the IPW estimator and the post-stratification plug-in estimator of the ATE, corrected: it holds when each fitted propensity equals the observed treated fraction in its covariate cell", "location": "p. 266 (PDF p. 284), paragraph after Definition 7.2.6" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T97" }, { "declarations": [ { "claim_revision_id": "catalog:b7ffb79492d66494709afebf3731d3747cacaaca03dc4c5b350f37f4fdf14516", "conditional_on": [], "credit_reason": "The conditional and unconditional mean-one conclusions use the two correct arm indicators and the target uniform-overlap propensity assumptions. The sentence about atTop limits in modelling_choices is inapplicable boilerplate, not an additional mathematical conclusion.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.treatment_weights_denominators_correction", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3571", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b7ffb79492d66494709afebf3731d3747cacaaca03dc4c5b350f37f4fdf14516", "conditional_on": [], "credit_reason": "The finite-sample mean-one statements and strict all-unit reciprocal expectations match the formal corrected-denominator content under IID sampling and the stipulated uniform overlap. The sentence about atTop limits in modelling_choices is inapplicable boilerplate, not an additional mathematical conclusion.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.treatment_weight_sample_means", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5360", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:b7ffb79492d66494709afebf3731d3747cacaaca03dc4c5b350f37f4fdf14516", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T98", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T98" }, "html": { "body": "

Definition 7.2.7 and its following paragraph, p. 267 (PDF p. 285): true-score arm weights have conditional mean one, their positive-size sample averages have mean one, and both all-unit reciprocal weights have mean above one. Restricted: retains those expectation assertions only; omits the displayed full stabilized outcome-estimator formula, fitted-score denominator convergence and efficiency assertions, and requires binarity and uniform overlap pointwise for every observation in the infinite sequence.

", "headline": "The paragraph after Definition 7.2.7 (Stabilized IPW Estimator for Causal Inference): the two sums whose between-sample variation the stabilized estimator accounts for, corrected (the treatment-arm indicators)", "location": "p. 267 (PDF p. 285), Definition 7.2.7 and the paragraph after it" }, "id": "P001-T98", "text": { "body": "Definition 7.2.7 and its following paragraph, p. 267 (PDF p. 285): true-score arm weights have conditional mean one, their positive-size sample averages have mean one, and both all-unit reciprocal weights have mean above one. Restricted: retains those expectation assertions only; omits the displayed full stabilized outcome-estimator formula, fitted-score denominator convergence and efficiency assertions, and requires binarity and uniform overlap pointwise for every observation in the infinite sequence.", "headline": "The paragraph after Definition 7.2.7 (Stabilized IPW Estimator for Causal Inference): the two sums whose between-sample variation the stabilized estimator accounts for, corrected (the treatment-arm indicators)", "location": "p. 267 (PDF p. 285), Definition 7.2.7 and the paragraph after it" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T98" }, { "declarations": [ { "claim_revision_id": "catalog:22eb9edf64ed9b6c081471973ff0bede1c5b8d7f2b0a617958f8a5f47bc91a47", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/P001_Foundations.lean:3963–3965 gives the specified two-atom weighted mean 1/4, conditional ATT 1, their inequality, and the correcting normalization. The law at line 560 has positive masses (3/4,1/4), so the conditional denominator is nonzero; the same witness's conditional independence and true propensity are explicitly stated in the companion theorem at lines 5265–5271. These are the claim's constant-potential-outcome counterexample, so this arithmetic component is faithful without repeating every witness property.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.att_normalization_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3602", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:22eb9edf64ed9b6c081471973ff0bede1c5b8d7f2b0a617958f8a5f47bc91a47", "conditional_on": [], "credit_reason": "The selected counterexample theorem already includes the conditional-independence fact; the general constant-variable helper is redundant.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.constant_potential_ignorability_finite", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4608", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:22eb9edf64ed9b6c081471973ff0bede1c5b8d7f2b0a617958f8a5f47bc91a47", "conditional_on": [], "credit_reason": "The unstated printed one-sided ATT bound follows numerically with ε=3/8; no missing witness hypothesis results.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.att_counterexample_with_ignorability", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4618", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:22eb9edf64ed9b6c081471973ff0bede1c5b8d7f2b0a617958f8a5f47bc91a47", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T99", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T99" }, "html": { "body": "

Section 7.3.1, p. 273 (PDF p. 291): the canonical two-point counterexample with constant potential outcomes 0 and 1, covariate 0 and treated probability 1/4 satisfies conditional ignorability and both stated positivity bounds, while its unnormalized weighted effect is 1/4, its ATT is 1, and normalization restores 1.

", "headline": "Section 7.3.1, the sentence 'the estimand E[pD(Xi)τi]\\mathrm{E}[p_D(\\mathbf{X}_i)\\tau_i]E[pD​(Xi​)τi​] ... is equivalent to the average treatment effect on the treated (ATT), E[τi∣Di=1]\\mathrm{E}[\\tau_i|D_i = 1]E[τi​∣Di​=1]', as printed", "location": "p. 273 (PDF p. 291), Section 7.3.1, second paragraph" }, "id": "P001-T99", "text": { "body": "Section 7.3.1, p. 273 (PDF p. 291): the canonical two-point counterexample with constant potential outcomes 0 and 1, covariate 0 and treated probability 1/4 satisfies conditional ignorability and both stated positivity bounds, while its unnormalized weighted effect is 1/4, its ATT is 1, and normalization restores 1.", "headline": "Section 7.3.1, the sentence 'the estimand $\\mathrm{E}[p_D(\\mathbf{X}_i)\\tau_i]$ ... is equivalent to the average treatment effect on the treated (ATT), $\\mathrm{E}[\\tau_i|D_i = 1]$', as printed", "location": "p. 273 (PDF p. 291), Section 7.3.1, second paragraph" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T99" }, { "declarations": [ { "claim_revision_id": "catalog:5a4248a595ad5e7c93c7962f7d8bd73653a01201a0853cf1e827c7266012aab7", "conditional_on": [], "credit_reason": "The claim itself has finite E[τ] and Supp[D]={0,1}; therefore L1 of the effect and positive mass in both arms add no restriction. Separate integrability of Y0 and Y1 is correctly absent. The final sentence explaining when the unnormalized equality holds follows algebraically from the normalized identity.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.att_normalized_propensity_mean_integrable_effect", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/DecideA.lean#L155", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:5a4248a595ad5e7c93c7962f7d8bd73653a01201a0853cf1e827c7266012aab7", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T100", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T100" }, "html": { "body": "

corrected Section 7.3.1 sentence, p. 273 (PDF p. 291): under conditional ignorability and an integrable treatment effect, the propensity mean is the positive treatment probability, the weighted-effect identities hold, and the normalized estimand equals ATT; the unnormalized equality holds exactly when the treatment probability is one or ATT is zero.

", "headline": "Section 7.3.1, the propensity-score-weighted estimand equal to the ATT, corrected (normalized by E[pD(Xi)]\\mathrm{E}[p_D(\\mathbf{X}_i)]E[pD​(Xi​)])", "location": "p. 273 (PDF p. 291), Section 7.3.1, second paragraph" }, "id": "P001-T100", "text": { "body": "corrected Section 7.3.1 sentence, p. 273 (PDF p. 291): under conditional ignorability and an integrable treatment effect, the propensity mean is the positive treatment probability, the weighted-effect identities hold, and the normalized estimand equals ATT; the unnormalized equality holds exactly when the treatment probability is one or ATT is zero.", "headline": "Section 7.3.1, the propensity-score-weighted estimand equal to the ATT, corrected (normalized by $\\mathrm{E}[p_D(\\mathbf{X}_i)]$)", "location": "p. 273 (PDF p. 291), Section 7.3.1, second paragraph" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T100" }, { "declarations": [ { "claim_revision_id": "catalog:c0721b2178216199e933b5f8cac922852e7e9273f808ca9651c33829b6c9dc0a", "conditional_on": [], "credit_reason": "Only the corrected identity is the credited component; the target's characterization and finite example of the erroneous union formula are explanatory illustrations.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.relative_complement", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4602", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:c0721b2178216199e933b5f8cac922852e7e9273f808ca9651c33829b6c9dc0a", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T101", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T101" }, "html": { "body": "

corrected set-subtraction row, p. 283 (PDF p. 301): relative complement is membership in A outside B and intersection with U minus B. The printed union identity holds exactly when A and B cover U and are disjoint, equivalently when A is U minus B. For U={1,2,3}, A={1,2}, B={2}, the difference and intersection are {1}, and the union is {1,2,3}. The usual measure-theoretic conditions are assumed.

", "headline": "Glossary of Mathematical Notation, the row for set subtraction (\\\\backslash\\): the relative-complement identity, corrected (A\\B=A∩BCA\\backslash B = A \\cap B^CA\\B=A∩BC)", "location": "p. 283 (PDF p. 301), row '\\\\backslash\\' (Set subtraction); the rows for ACA^CAC, ∪\\cup∪ and ∩\\cap∩ on the same page" }, "id": "P001-T101", "text": { "body": "corrected set-subtraction row, p. 283 (PDF p. 301): relative complement is membership in A outside B and intersection with U minus B. The printed union identity holds exactly when A and B cover U and are disjoint, equivalently when A is U minus B. For U={1,2,3}, A={1,2}, B={2}, the difference and intersection are {1}, and the union is {1,2,3}. The usual measure-theoretic conditions are assumed.", "headline": "Glossary of Mathematical Notation, the row for set subtraction ($\\backslash$): the relative-complement identity, corrected ($A\\backslash B = A \\cap B^C$)", "location": "p. 283 (PDF p. 301), row '$\\backslash$' (Set subtraction); the rows for $A^C$, $\\cup$ and $\\cap$ on the same page" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T101" }, { "declarations": [ { "claim_revision_id": "catalog:dbbc3e92ba7d22014deb6d63b213360ba4d123ba1b7bc869a0c96b50c3da8a52", "conditional_on": [], "credit_reason": "Its old coordinate-only regularity permits an interior null-set artifact; the new proof uses the positive gap-law lemmas and exact PDF-range subtype directly.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.printed_pda_continuous_cef_counterexample_printed_negation", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Checks/FidP001Contracts.lean#L77", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:dbbc3e92ba7d22014deb6d63b213360ba4d123ba1b7bc869a0c96b50c3da8a52", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/P001_Foundations.lean:7746–7749 evaluates the chosen fitted function's coordinate derivative at the componentwise sample mean, as the claim's plug-in formula requires. partialAt at line 144 varies only coordinate k and differentiates at zero, and pluginPDA at line 918 uses precisely that sample covariate vector. Zero-size samples and nonexistent derivatives receive totalized values, but all positive-size instances with the printed derivative defined are retained; the row makes no additional claim of version invariance or consistency.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.printed_pda_plugin_formula", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6639", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:dbbc3e92ba7d22014deb6d63b213360ba4d123ba1b7bc869a0c96b50c3da8a52", "conditional_on": [], "credit_reason": "The enlarged-domain old negation lacks the joint-PDF premise and cannot certify the printed support contract.", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.Wave1.printed_pda_local_support_contract_refuted", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/PdaLocalSupport.lean#L17", "status": "counterexample_proved", "support_scope": "limited", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:dbbc3e92ba7d22014deb6d63b213360ba4d123ba1b7bc869a0c96b50c3da8a52", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T102", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T102" }, "html": { "body": "

Definition 4.3.7, p. 167 (PDF p. 185), footnote 13 on p. 166, the remark on p. 168, and the standing footnote-14 condition on p. 70. Restricted: the joint law does not determine one PDA at the covariate mean, even with a joint density, two globally differentiable versions of the same CEF, and the literal footnote-14 condition that the sample range equals the chosen PDF's positive set, as Definition 1.2.20 defines support. The proposed counterexample has its mean in a gap of that positive set; it is not a null-set modification at an interior support point. The second conjunct merely unfolds the chosen fitted CEF's plug-in formula. Its positive-sample-size and differentiability premises describe the printed domain, but the Lean equality is definitional and uses neither premise. This does not refute the derivative of a fixed chosen extension or address the recommendation of discrete partial slopes.

", "headline": "Definition 4.3.7 (Partial Derivative at the Average) and the plug-in estimate that follows it, as printed", "location": "p. 167 (PDF p. 185); footnote 13, p. 166 (PDF p. 184); the remark on p. 168 (PDF p. 186)" }, "id": "P001-T102", "text": { "body": "Definition 4.3.7, p. 167 (PDF p. 185), footnote 13 on p. 166, the remark on p. 168, and the standing footnote-14 condition on p. 70. Restricted: the joint law does not determine one PDA at the covariate mean, even with a joint density, two globally differentiable versions of the same CEF, and the literal footnote-14 condition that the sample range equals the chosen PDF's positive set, as Definition 1.2.20 defines support. The proposed counterexample has its mean in a gap of that positive set; it is not a null-set modification at an interior support point. The second conjunct merely unfolds the chosen fitted CEF's plug-in formula. Its positive-sample-size and differentiability premises describe the printed domain, but the Lean equality is definitional and uses neither premise. This does not refute the derivative of a fixed chosen extension or address the recommendation of discrete partial slopes.", "headline": "Definition 4.3.7 (Partial Derivative at the Average) and the plug-in estimate that follows it, as printed", "location": "p. 167 (PDF p. 185); footnote 13, p. 166 (PDF p. 184); the remark on p. 168 (PDF p. 186)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T102" }, { "declarations": [ { "claim_revision_id": "catalog:3f7ddf6fede6144a55520ef28f753f4668d13cb1ba21ab7a4d2c02ebb5f6202e", "conditional_on": [], "credit_reason": "Purpose: full refutation of maximizer existence (A). Restricting the countermodel search to positive samples in the paper's own mixture strengthens the refutation and loses none of that purpose. Target metadata inconsistency: asymptotic=false despite also quoting consistency/efficiency assertion (B), which this row does not cover.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.printed_unrestricted_mixture_maximum_refuted", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/MixtureMaximum.lean#L13", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:3f7ddf6fede6144a55520ef28f753f4668d13cb1ba21ab7a4d2c02ebb5f6202e", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T103", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T103" }, "html": { "body": "

The printed claim is false: P001-T103, Section 5.2.5 and Example 5.2.11, pp. 198–200 (PDF pp. 216–218), do not ensure a likelihood maximizer for every positive sample in the stated normal/lognormal/exponential mixture. Scope: refutes only maximizer-existence assertion (A) in that mixture; omits both assertion (B)'s full-parameter consistency and its asymptotic efficiency assertion, and makes no claim about the quality of the plotted fit.

", "headline": "Section 5.2.5 (Mixture Models), the mixture likelihood and the claims attached to it, with the mixture of Example 5.2.11, as printed", "location": "pp. 198–200 (PDF pp. 216–218)" }, "id": "P001-T103", "text": { "body": "The printed claim is false: P001-T103, Section 5.2.5 and Example 5.2.11, pp. 198–200 (PDF pp. 216–218), do not ensure a likelihood maximizer for every positive sample in the stated normal/lognormal/exponential mixture. Scope: refutes only maximizer-existence assertion (A) in that mixture; omits both assertion (B)'s full-parameter consistency and its asymptotic efficiency assertion, and makes no claim about the quality of the plotted fit.", "headline": "Section 5.2.5 (Mixture Models), the mixture likelihood and the claims attached to it, with the mixture of Example 5.2.11, as printed", "location": "pp. 198–200 (PDF pp. 216–218)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T103" }, { "declarations": [ { "claim_revision_id": "catalog:16d2704b72b3c0890004d130cbc2463891dad445eaa42d2803906aeee4a703c0", "conditional_on": [], "credit_reason": "Equality of supports of joint arm masses is equivalent to equality of conditional arm supports because both arm totals are positive. Uniform positivity always implies overlap for these arrays, so negating the recorded iff refutes the erroneous reverse implication. The hedged converse under unspecified support conditions is not refuted.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.PrintedConfidence.printed_overlap_uniform_positivity_converse_refuted", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_PrintedRefutations.lean#L58", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:16d2704b72b3c0890004d130cbc2463891dad445eaa42d2803906aeee4a703c0", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T104", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T104" }, "html": { "body": "

The printed claim is false: P001-T104, pp. 271–272 (PDF pp. 289–290), cannot identify complete population overlap with uniform two-sided positivity for every countable joint-mass distribution. Scope: states only the negation of the unrestricted countable-mass equivalence; does not prove assertion (A), separately isolate the directed failure in (B), settle (C) under unspecified extra support conditions, or state the regional propensity values in Example 7.3.3.

", "headline": "Section 7.3, the claims relating positivity and overlap around Definition 7.3.1 and Theorem 7.3.2, as printed", "location": "p. 271 (PDF p. 289); p. 272 (PDF p. 290), with Example 7.3.3" }, "id": "P001-T104", "text": { "body": "The printed claim is false: P001-T104, pp. 271–272 (PDF pp. 289–290), cannot identify complete population overlap with uniform two-sided positivity for every countable joint-mass distribution. Scope: states only the negation of the unrestricted countable-mass equivalence; does not prove assertion (A), separately isolate the directed failure in (B), settle (C) under unspecified extra support conditions, or state the regional propensity values in Example 7.3.3.", "headline": "Section 7.3, the claims relating positivity and overlap around Definition 7.3.1 and Theorem 7.3.2, as printed", "location": "p. 271 (PDF p. 289); p. 272 (PDF p. 290), with Example 7.3.3" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T104" }, { "declarations": [ { "claim_revision_id": "catalog:07b3fce4f6b75404017520699ecb001af775f136084e8074d5a9087b3f6f8617", "conditional_on": [], "credit_reason": "The larger-domain negation omits binary treatment and an exported informational-completeness premise; it cannot refute this printed implication by glue.", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.Wave1.printed_complete_description_contract_refuted", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/CompleteDescription.lean#L14", "status": "counterexample_proved", "support_scope": "limited", "used_in_solution": null } ], "formalization": { "claim_revision_id": "catalog:07b3fce4f6b75404017520699ecb001af775f136084e8074d5a9087b3f6f8617", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T105", "decision": "omitted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T105" }, "html": { "body": "

No current theorem is supplied for this target.

", "headline": "Section 7.5, the paragraph justifying ignorability in the application, as printed", "location": "p. 278 (PDF p. 296)" }, "id": "P001-T105", "text": { "body": "No current theorem is supplied for this target.", "headline": "Section 7.5, the paragraph justifying ignorability in the application, as printed", "location": "p. 278 (PDF p. 296)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T105" }, { "declarations": [ { "claim_revision_id": "catalog:3b8d6f4a9cde657195a7237a7a8a38792a9d23bc77d4866bd49098fecc56d673", "conditional_on": [], "credit_reason": "The selected standardized Chebyshev row states the exact printed bound; its generalized auxiliary bound is redundant.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.chebyshev_probability", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L1026", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:3b8d6f4a9cde657195a7237a7a8a38792a9d23bc77d4866bd49098fecc56d673", "conditional_on": [], "credit_reason": "Absolute integrability is the book's explicit standing convention, not an additional moment assumption.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.expectation_linearity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3611", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:3b8d6f4a9cde657195a7237a7a8a38792a9d23bc77d4866bd49098fecc56d673", "conditional_on": [], "credit_reason": "Finite second moments are implied by the standing convention for the displayed variance and second-moment expectations.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.variance_affine_rules", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3626", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:3b8d6f4a9cde657195a7237a7a8a38792a9d23bc77d4866bd49098fecc56d673", "conditional_on": [], "credit_reason": "The standing convention of finite moments in these variance expressions supplies L²; zero variance and zero multipliers remain allowed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.standard_deviation_rules", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3636", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:3b8d6f4a9cde657195a7237a7a8a38792a9d23bc77d4866bd49098fecc56d673", "conditional_on": [], "credit_reason": "The positive-variance premise is explicitly printed. It is essential for a non-strict standardized threshold: at zero variance that event has probability one for every ε.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.chebyshev_standard_deviations", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3649", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:3b8d6f4a9cde657195a7237a7a8a38792a9d23bc77d4866bd49098fecc56d673", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T106", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T106" }, "html": { "body": "

Theorems 2.1.9, 2.1.13–14, 2.1.16 and 2.1.18, pp. 49, 52–53 and 55 (PDF pp. 67, 70–71 and 73): expectation is linear, variance and standard deviation obey the printed affine rules, and standardized Chebyshev holds with finite second moment and positive variance.

", "headline": "Theorem 2.1.9 (Linearity of Expectations), Theorem 2.1.13 (Alternative Formula for Variance), Theorem 2.1.14 (Properties of Variance), Theorem 2.1.16 (Properties of Standard Deviation) and Theorem 2.1.18 (Chebyshev's Inequality)", "location": "Theorem 2.1.9: p. 49 (PDF p. 67). Theorems 2.1.13 and 2.1.14: p. 52 (PDF p. 70). Theorem 2.1.16: p. 53 (PDF p. 71). Theorem 2.1.18: p. 55 (PDF p. 73)" }, "id": "P001-T106", "text": { "body": "Theorems 2.1.9, 2.1.13–14, 2.1.16 and 2.1.18, pp. 49, 52–53 and 55 (PDF pp. 67, 70–71 and 73): expectation is linear, variance and standard deviation obey the printed affine rules, and standardized Chebyshev holds with finite second moment and positive variance.", "headline": "Theorem 2.1.9 (Linearity of Expectations), Theorem 2.1.13 (Alternative Formula for Variance), Theorem 2.1.14 (Properties of Variance), Theorem 2.1.16 (Properties of Standard Deviation) and Theorem 2.1.18 (Chebyshev's Inequality)", "location": "Theorem 2.1.9: p. 49 (PDF p. 67). Theorems 2.1.13 and 2.1.14: p. 52 (PDF p. 70). Theorem 2.1.16: p. 53 (PDF p. 71). Theorem 2.1.18: p. 55 (PDF p. 73)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T106" }, { "declarations": [ { "claim_revision_id": "catalog:54a7f8afe5c5225111ee97a180056f67bb1351a2ac548fbf5e5a113f38977af0", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/P001_Foundations.lean:4038–4041 concludes E[Y]=E[m(X)] from the ordinary IsCEF contract at lines 97–100. The contract's measurable-event integral identities are the definition of a conditional-expectation version, whose whole-space instance legitimately yields the claim's Law of Iterated Expectations. Missing an explicit Measurable X premise extends the abstract contract theorem while retaining all paper random-variable cases, and the repeated integrability premise imposes no additional restriction.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.iterated_expectation", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3663", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:54a7f8afe5c5225111ee97a180056f67bb1351a2ac548fbf5e5a113f38977af0", "conditional_on": [], "credit_reason": "Final status wrong_credit: excluded for all components of this target, including total variance.", "credit_status": "wrong_credit", "fidelity_status_baseline": "wrong_credit", "lean_name": "AronowAudit.P001.total_variance_and_cef_best_prediction", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3673", "status": "proved", "support_scope": "withdrawn", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:54a7f8afe5c5225111ee97a180056f67bb1351a2ac548fbf5e5a113f38977af0", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T107", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T107" }, "html": { "body": "

Theorems 2.2.17–18 and 2.2.20, pp. 72–73 and 75–76 (PDF pp. 90–91 and 93–94): iterated expectation holds; with a finite second moment, total variance is expected conditional variance plus variance of the CEF. For every measurable predictor with finite prediction MSE, the exact MSE decomposition has a nonnegative remainder, the CEF minimizes MSE, and its residual risk equals expected conditional variance. The prediction assertion needs finite squared residuals, not a second moment of Y. The usual measure-theoretic conditions are assumed.

", "headline": "Theorem 2.2.17 (Law of Iterated Expectations), Theorem 2.2.18 (Law of Total Variance) and Theorem 2.2.20 (The CEF is the Best Predictor)", "location": "Theorem 2.2.17: p. 72 (PDF p. 90). Theorem 2.2.18: p. 73 (PDF p. 91). Theorem 2.2.20: p. 75 (PDF p. 93), proof on pp. 75–76 (PDF pp. 93–94)" }, "id": "P001-T107", "text": { "body": "Theorems 2.2.17–18 and 2.2.20, pp. 72–73 and 75–76 (PDF pp. 90–91 and 93–94): iterated expectation holds; with a finite second moment, total variance is expected conditional variance plus variance of the CEF. For every measurable predictor with finite prediction MSE, the exact MSE decomposition has a nonnegative remainder, the CEF minimizes MSE, and its residual risk equals expected conditional variance. The prediction assertion needs finite squared residuals, not a second moment of Y. The usual measure-theoretic conditions are assumed.", "headline": "Theorem 2.2.17 (Law of Iterated Expectations), Theorem 2.2.18 (Law of Total Variance) and Theorem 2.2.20 (The CEF is the Best Predictor)", "location": "Theorem 2.2.17: p. 72 (PDF p. 90). Theorem 2.2.18: p. 73 (PDF p. 91). Theorem 2.2.20: p. 75 (PDF p. 93), proof on pp. 75–76 (PDF pp. 93–94)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T107" }, { "declarations": [ { "claim_revision_id": "catalog:d65a363700e13391bc8cd118f1dc1391ece90a9fa1c4edafdf6f6949c454792e", "conditional_on": [], "credit_reason": "The infinite stream is a distribution-preserving representation of every finite iid real sample, not an extra moment or sampling-law restriction.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.sample_mean_paper_contract", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/SampleMean.lean#L15", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:d65a363700e13391bc8cd118f1dc1391ece90a9fa1c4edafdf6f6949c454792e", "conditional_on": [], "credit_reason": "The common-second-moment restriction is unnecessary; the selected paper contract separates the moment scopes.", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.iid_sample_mean_finite_moments", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L2734", "status": "proved", "support_scope": "limited", "used_in_solution": false }, { "claim_revision_id": "catalog:d65a363700e13391bc8cd118f1dc1391ece90a9fa1c4edafdf6f6949c454792e", "conditional_on": [], "credit_reason": "The selected paper contract already contains the printed positive-variance Chebyshev component.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.iid_sample_mean_chebyshev", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3739", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:d65a363700e13391bc8cd118f1dc1391ece90a9fa1c4edafdf6f6949c454792e", "conditional_on": [], "credit_reason": "The selected paper contract already contains the first-moment-only expectation identity.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.iid_sample_mean_expectation_integrable", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6335", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:d65a363700e13391bc8cd118f1dc1391ece90a9fa1c4edafdf6f6949c454792e", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T108", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T108" }, "html": { "body": "

Theorems 3.2.3–5, pp. 97–99 (PDF pp. 115–117): every positive-size finite measurable iid family has an integrable sample mean with the common population expectation under a first moment. Under a second moment its variance is the population variance divided by n, and with positive population variance it obeys the displayed Chebyshev bound. The usual measure-theoretic conditions are assumed.

", "headline": "Theorem 3.2.3 (The Expected Value of the Sample Mean Is the Population Mean), Theorem 3.2.4 (Sampling Variance of the Sample Mean) and Theorem 3.2.5 (Chebyshev's Inequality for the Sample Mean)", "location": "Theorem 3.2.3: p. 97 (PDF p. 115). Theorems 3.2.4 and 3.2.5: p. 98 (PDF p. 116), proof of 3.2.5 continuing on p. 99 (PDF p. 117)" }, "id": "P001-T108", "text": { "body": "Theorems 3.2.3–5, pp. 97–99 (PDF pp. 115–117): every positive-size finite measurable iid family has an integrable sample mean with the common population expectation under a first moment. Under a second moment its variance is the population variance divided by n, and with positive population variance it obeys the displayed Chebyshev bound. The usual measure-theoretic conditions are assumed.", "headline": "Theorem 3.2.3 (The Expected Value of the Sample Mean Is the Population Mean), Theorem 3.2.4 (Sampling Variance of the Sample Mean) and Theorem 3.2.5 (Chebyshev's Inequality for the Sample Mean)", "location": "Theorem 3.2.3: p. 97 (PDF p. 115). Theorems 3.2.4 and 3.2.5: p. 98 (PDF p. 116), proof of 3.2.5 continuing on p. 99 (PDF p. 117)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T108" }, { "declarations": [ { "claim_revision_id": "catalog:4e8ee97bef370961f415dc47b62620d7ea188025a85149e32b123c4ae5dcbdbb", "conditional_on": [], "credit_reason": "Only the interval restriction of f matters. Taking f on all ℝ and including singleton intervals loses no case of the printed approximation theorem.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.weierstrass_approximation", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3752", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:4e8ee97bef370961f415dc47b62620d7ea188025a85149e32b123c4ae5dcbdbb", "conditional_on": [], "credit_reason": "The positivity-set support premise follows the paper's explicit PMF/PDF convention, not topological support. Continuity on the interval admits a measurable extension, so global measurability of the CEF here loses no relevant case. L² of Y is justified by the finite-MSE BLP convention; there is no omitted minimizer-existence or every-family conclusion.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.polynomial_blp_paper_support_approximation", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/PolynomialBlp.lean#L21", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:4e8ee97bef370961f415dc47b62620d7ea188025a85149e32b123c4ae5dcbdbb", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T109", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T109" }, "html": { "body": "

Theorems 4.3.2–3, pp. 158–159 (PDF pp. 176–177): Weierstrass approximation holds, and on the paper's PMF/PDF full interval-support domain, population polynomial least-squares minimizers exist at every degree and every minimizing family eventually approximates the continuous CEF in mean square, with finite prediction risks ensured by a square-integrable outcome.

", "headline": "Theorem 4.3.3 (Polynomial Approximation of the CEF), with Theorem 4.3.2 (Weierstrass Approximation Theorem)", "location": "p. 158 (PDF p. 176); proof on pp. 158–159 (PDF pp. 176–177)" }, "id": "P001-T109", "text": { "body": "Theorems 4.3.2–3, pp. 158–159 (PDF pp. 176–177): Weierstrass approximation holds, and on the paper's PMF/PDF full interval-support domain, population polynomial least-squares minimizers exist at every degree and every minimizing family eventually approximates the continuous CEF in mean square, with finite prediction risks ensured by a square-integrable outcome.", "headline": "Theorem 4.3.3 (Polynomial Approximation of the CEF), with Theorem 4.3.2 (Weierstrass Approximation Theorem)", "location": "p. 158 (PDF p. 176); proof on pp. 158–159 (PDF pp. 176–177)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T109" }, { "declarations": [ { "claim_revision_id": "catalog:a41c9bf55893a3000cfea0a2fd9aeeec32346bcb3daedd81708dc12183ecfdfb", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.bernoulli_ml_unique_sample_fraction", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3795", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:a41c9bf55893a3000cfea0a2fd9aeeec32346bcb3daedd81708dc12183ecfdfb", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.three_coin_flips_maximum", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3872", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:a41c9bf55893a3000cfea0a2fd9aeeec32346bcb3daedd81708dc12183ecfdfb", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T110", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T110" }, "html": { "body": "

Example 5.2.3, pp. 187–189 (PDF pp. 205–207), footnote 9: every nonempty binary sample has the unique Bernoulli likelihood maximizer equal to its sample mean, including the endpoint samples, and the (1,0,1) example has unique maximizer 2/3.

", "headline": "Example 5.2.3 (Maximum Likelihood Estimation for a Biased Coin Flip): the ML estimate of ppp is the sample mean", "location": "pp. 187–189 (PDF pp. 205–207); footnote 9, p. 189" }, "id": "P001-T110", "text": { "body": "Example 5.2.3, pp. 187–189 (PDF pp. 205–207), footnote 9: every nonempty binary sample has the unique Bernoulli likelihood maximizer equal to its sample mean, including the endpoint samples, and the (1,0,1) example has unique maximizer 2/3.", "headline": "Example 5.2.3 (Maximum Likelihood Estimation for a Biased Coin Flip): the ML estimate of $p$ is the sample mean", "location": "pp. 187–189 (PDF pp. 205–207); footnote 9, p. 189" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T110" }, { "declarations": [ { "claim_revision_id": "catalog:93d9aa45645cede220d4f8abeb40ae4e6ac368fea860894274ee518257e47ad8", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.mcar_paper_support_mean", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/McarMean.lean#L15", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:93d9aa45645cede220d4f8abeb40ae4e6ac368fea860894274ee518257e47ad8", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P001/Wave1/MarDiscrete.lean:22–36 concludes both the population CEF identity and the absolutely convergent discrete weighted-mean formula required by Theorems 6.1.9 and 6.1.10. Countable measurable labels represent the range of any discrete finite real vector, both response masses match Supp[R]={0,1}, and respondent restriction differs from respondent conditioning only by normalization. The additional integrability premise at line 33 follows from respondent integrability and uniform overlap: ε∫|m(X)|dμ≤∫|m(X)|1{R=1}dμ<∞, so it excludes no eligible model. Conditional versions are expressly permitted by the standing assumptions.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.mar_discrete_paper_support_identification", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/MarDiscrete.lean#L22", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:93d9aa45645cede220d4f8abeb40ae4e6ac368fea860894274ee518257e47ad8", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.missing_mcar_mar_sample_estimates", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6185", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:93d9aa45645cede220d4f8abeb40ae4e6ac368fea860894274ee518257e47ad8", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P001/Wave1/MarVector.lean:21–31 transfers the respondent CEF to a population CEF for every finite-dimensional real covariate vector, which is the Theorem 6.1.10 component assigned to this row. The standing assumptions expressly permit conditional-expectation versions, and scaling the respondent measure does not change its CEF. Population integrability of m(X) at line 30 follows from its respondent integrability and the uniform positive response bound by ε∫|m(X)|dμ≤∫|m(X)|1{R=1}dμ<∞; it is not an additional moment restriction.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.mar_vector_paper_support_cef", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/MarVector.lean#L21", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:93d9aa45645cede220d4f8abeb40ae4e6ac368fea860894274ee518257e47ad8", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T111", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T111" }, "html": { "body": "

Theorems 6.1.6, 6.1.9–10, pp. 213 and 216–217 (PDF pp. 231 and 234–235), with Examples 6.1.7 and 6.2.1, pp. 214 and 220: MCAR identifies the mean by respondents; MAR and uniform response overlap identify the vector CEF and the discrete positive-support mean, with population integrability derived rather than assumed. The empirical respondent and observed-profile formulas hold whenever their denominators are positive, and the six-unit estimates are 3/4 and 7/9. The usual measure-theoretic conditions are assumed.

", "headline": "Theorem 6.1.6 (Expected Value under MCAR), Theorem 6.1.9 (Expected Value under MAR) and Theorem 6.1.10 (CEF under MAR); with the estimates of Examples 6.1.7 and 6.2.1", "location": "Definition 6.1.5 and Theorem 6.1.6: p. 213 (PDF p. 231). Example 6.1.7: p. 214 (PDF p. 232). Definition 6.1.8: p. 215 (PDF p. 233). Theorem 6.1.9: p. 216 (PDF p. 234). Theorem 6.1.10: p. 217 (PDF p. 235). Section 6.2.1 and Example 6.2.1: pp. 219–220 (PDF pp. 237–238)" }, "id": "P001-T111", "text": { "body": "Theorems 6.1.6, 6.1.9–10, pp. 213 and 216–217 (PDF pp. 231 and 234–235), with Examples 6.1.7 and 6.2.1, pp. 214 and 220: MCAR identifies the mean by respondents; MAR and uniform response overlap identify the vector CEF and the discrete positive-support mean, with population integrability derived rather than assumed. The empirical respondent and observed-profile formulas hold whenever their denominators are positive, and the six-unit estimates are 3/4 and 7/9. The usual measure-theoretic conditions are assumed.", "headline": "Theorem 6.1.6 (Expected Value under MCAR), Theorem 6.1.9 (Expected Value under MAR) and Theorem 6.1.10 (CEF under MAR); with the estimates of Examples 6.1.7 and 6.2.1", "location": "Definition 6.1.5 and Theorem 6.1.6: p. 213 (PDF p. 231). Example 6.1.7: p. 214 (PDF p. 232). Definition 6.1.8: p. 215 (PDF p. 233). Theorem 6.1.9: p. 216 (PDF p. 234). Theorem 6.1.10: p. 217 (PDF p. 235). Section 6.2.1 and Example 6.2.1: pp. 219–220 (PDF pp. 237–238)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T111" }, { "declarations": [ { "claim_revision_id": "catalog:e6333a19065864172505d902a4628a7e754356366120c68b67933739d23a0cd1", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P001/Wave1/MarBalance.lean:18–26 supplies both eventwise conditional-independence conclusions for the paper's finite real covariate vector. Its IsCEF μ (p ∘ X) R id conclusion says E[R|p(X)]=p(X), so the retained positive ε bound is precisely the claim's third conditional-response conclusion, rather than a mere repetition of the original overlap hypothesis. The permitted version convention covers the a.e. formulations, and no outcome-moment hypothesis has been added.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.mar_propensity_paper_support_balance", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/MarBalance.lean#L18", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:e6333a19065864172505d902a4628a7e754356366120c68b67933739d23a0cd1", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T112", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T112" }, "html": { "body": "

Theorem 6.1.12, pp. 217–218 (PDF pp. 235–236): conditioning on the true response propensity balances response and covariates, preserves MAR, and gives a conditional response-probability version with a positive uniform lower bound, without imposing an outcome moment assumption.

", "headline": "Theorem 6.1.12 (MAR and the Propensity Score)", "location": "pp. 217–218 (PDF pp. 235–236); proof on p. 218; Definition 6.1.11, p. 217" }, "id": "P001-T112", "text": { "body": "Theorem 6.1.12, pp. 217–218 (PDF pp. 235–236): conditioning on the true response propensity balances response and covariates, preserves MAR, and gives a conditional response-probability version with a positive uniform lower bound, without imposing an outcome moment assumption.", "headline": "Theorem 6.1.12 (MAR and the Propensity Score)", "location": "pp. 217–218 (PDF pp. 235–236); proof on p. 218; Definition 6.1.11, p. 217" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T112" }, { "declarations": [ { "claim_revision_id": "catalog:92998978209f53a14a777553f16ba8208bb2880bfbefdd6abc82772200526145", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': \"P001/Wave1/MissingDr.lean:29 requires Integrable (m ∘ X) μ in both correctness branches, although Theorem 6.2.9's correct-propensity branch allows an arbitrary regression approximation and displays only the outcome and combined-score expectations. Take X uniform on (0,1), Y=0, p=q=1 for X<1/2 and p=q=1/2 otherwise, with response deterministic in the first region and fair in the second; let m(x)=1/x for 0Theorem 6.2.9, pp. 229–230 (PDF pp. 247–248): under MAR, the augmented missing-data score has the outcome mean if either the respondent regression is correct and the candidate propensity is bounded away from zero and one, or the propensity is correct. The outcome and the complete displayed score have finite expectations. No separate population integrability of the nuisance regression is assumed. Conditional means, correctness, binarity and overlap are interpreted as versions up to null sets; both response values have positive probability.

", "headline": "Theorem 6.2.9 (Double Robustness Theorem for Missing Data)", "location": "p. 229 (PDF p. 247); proof on pp. 229–230 (PDF pp. 247–248)" }, "id": "P001-T113", "text": { "body": "Theorem 6.2.9, pp. 229–230 (PDF pp. 247–248): under MAR, the augmented missing-data score has the outcome mean if either the respondent regression is correct and the candidate propensity is bounded away from zero and one, or the propensity is correct. The outcome and the complete displayed score have finite expectations. No separate population integrability of the nuisance regression is assumed. Conditional means, correctness, binarity and overlap are interpreted as versions up to null sets; both response values have positive probability.", "headline": "Theorem 6.2.9 (Double Robustness Theorem for Missing Data)", "location": "p. 229 (PDF p. 247); proof on pp. 229–230 (PDF pp. 247–248)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T113" }, { "declarations": [ { "claim_revision_id": "catalog:78c552d0f9c73e2632d5ca908f6f26b99613238228592bb34ba5e21be1c8dc7d", "conditional_on": [], "credit_reason": "The existence of alternatives on the same sample space with D fixed is stronger than necessary. General sample marginal-bound numbers in the target are assessed separately at target level.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.ate_sharp_paper_bounds_same_observable_law", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/AteSharpBounds.lean#L17", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:78c552d0f9c73e2632d5ca908f6f26b99613238228592bb34ba5e21be1c8dc7d", "conditional_on": [], "credit_reason": "The selected population theorem already exports the bound width; the pointwise algebra is redundant.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.ate_bounds_width_finite", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3988", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:78c552d0f9c73e2632d5ca908f6f26b99613238228592bb34ba5e21be1c8dc7d", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.binary_causal_sample_bound_estimates", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6194", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:78c552d0f9c73e2632d5ca908f6f26b99613238228592bb34ba5e21be1c8dc7d", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T114", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T114" }, "html": { "body": "

Theorem 7.1.7 and its width and sharpness discussion, pp. 241–244 (PDF pp. 259–262), and Example 7.1.6, p. 240 (PDF p. 258): bounded potential outcomes and binary treatment with both arms of positive probability give the printed conditional-arm ATE interval, of width b-a. Every value in it is attained by bounded potential outcomes with the same observable joint law. Individual endpoint imputations retain observed values and bound the missing potential outcomes. For the six specified units, the upper treated, lower control, lower treated and upper control mean estimates are respectively 5/6, 1/6, 3/6 and 5/6, giving ATE endpoints -2/6 and 4/6.

", "headline": "Theorem 7.1.7 (Sharp Bounds for the ATE) and the width of the bounds; with the estimates of Example 7.1.6", "location": "Theorem 7.1.7: p. 241 (PDF p. 259); proof on pp. 241–242 (PDF pp. 259–260). Width: pp. 242–243 (PDF pp. 260–261). Example 7.1.6: p. 240 (PDF p. 258)" }, "id": "P001-T114", "text": { "body": "Theorem 7.1.7 and its width and sharpness discussion, pp. 241–244 (PDF pp. 259–262), and Example 7.1.6, p. 240 (PDF p. 258): bounded potential outcomes and binary treatment with both arms of positive probability give the printed conditional-arm ATE interval, of width b-a. Every value in it is attained by bounded potential outcomes with the same observable joint law. Individual endpoint imputations retain observed values and bound the missing potential outcomes. For the six specified units, the upper treated, lower control, lower treated and upper control mean estimates are respectively 5/6, 1/6, 3/6 and 5/6, giving ATE endpoints -2/6 and 4/6.", "headline": "Theorem 7.1.7 (Sharp Bounds for the ATE) and the width of the bounds; with the estimates of Example 7.1.6", "location": "Theorem 7.1.7: p. 241 (PDF p. 259); proof on pp. 241–242 (PDF pp. 259–260). Width: pp. 242–243 (PDF pp. 260–261). Example 7.1.6: p. 240 (PDF p. 258)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T114" }, { "declarations": [ { "claim_revision_id": "catalog:dd0ebba6f4d7a4e74bdfd6976297a7c5398f172d2f7ee0831afe5a37e90c1c97", "conditional_on": [], "credit_reason": "The mathematics is faithful; the modelling description adds sample-indexing and limit conventions that do not occur in this population theorem. The target also quotes a sample difference-in-means estimator not exported here.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.random_assignment_ate_and_selection_bias", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L3994", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:dd0ebba6f4d7a4e74bdfd6976297a7c5398f172d2f7ee0831afe5a37e90c1c97", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T115", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T115" }, "html": { "body": "

Location: Definition 7.1.9, p. 244 (PDF p. 262); Theorem 7.1.10, p. 245 (PDF p. 263); decomposition, p. 247 (PDF p. 265). The usual measure-theoretic conditions are assumed. Random assignment identifies the population ATE. For arbitrary assignment the observed arm difference equals the potential-outcome arm difference, then the sum-of-effect-and-control arm difference, then ATT plus selection bias. The plug-in estimates of the two identified potential-outcome means under the empirical observed law are the two displayed arm averages, and their difference is the difference-in-means estimator, whenever both sample arm counts are positive.

", "headline": "Theorem 7.1.10 (ATE under Random Assignment) and the selection-bias decomposition", "location": "Definition 7.1.9: p. 244 (PDF p. 262). Theorem 7.1.10: p. 245 (PDF p. 263). Decomposition: p. 247 (PDF p. 265)" }, "id": "P001-T115", "text": { "body": "Location: Definition 7.1.9, p. 244 (PDF p. 262); Theorem 7.1.10, p. 245 (PDF p. 263); decomposition, p. 247 (PDF p. 265). The usual measure-theoretic conditions are assumed. Random assignment identifies the population ATE. For arbitrary assignment the observed arm difference equals the potential-outcome arm difference, then the sum-of-effect-and-control arm difference, then ATT plus selection bias. The plug-in estimates of the two identified potential-outcome means under the empirical observed law are the two displayed arm averages, and their difference is the difference-in-means estimator, whenever both sample arm counts are positive.", "headline": "Theorem 7.1.10 (ATE under Random Assignment) and the selection-bias decomposition", "location": "Definition 7.1.9: p. 244 (PDF p. 262). Theorem 7.1.10: p. 245 (PDF p. 263). Decomposition: p. 247 (PDF p. 265)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T115" }, { "declarations": [ { "claim_revision_id": "catalog:345ac8c8b52b7c95422bd620af14cd2b88d55b4212e852355294160c70e2a883", "conditional_on": [], "credit_reason": "The enlarged measurable covariate domain is a valid generalization. Its two separate quantified domains preserve the moment-free scope of the propensity-score theorem.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.strong_ignorability_repaired", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/DecideP001T116.lean#L200", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:345ac8c8b52b7c95422bd620af14cd2b88d55b4212e852355294160c70e2a883", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'AronowAudit/P001/Wave1/Gap/GLP0011B2.lean:84–95 states exactly the Theorem 7.1.13 component: E[Y1 − Y0] equals the treated-arm conditional-mean sum minus the control-arm sum, both weighted by Pr[X = x]. The countable measurable label type at line 84 represents every discrete real-vector range, and including zero-mass labels leaves the sums unchanged. The restricted-measure IsCEF premises at lines 92–93 define the observed arm conditional means, rather than assuming their identification with population potential-outcome means; IsCEF is defined by integral identities in AronowAudit/P001/P001_Foundations.lean:97. The additional integrability premises at line 94 follow from the printed uniform overlap: for either arm d, ε E[|m_d(X)|] ≤ E[1_{D=d}|m_d(X)|] ≤ E[|Y_d|], which is finite under the standing expectation convention. The eventwise conditional-independence formulation and binary/overlap representatives preserve the joint laws under the permitted null-set interpretation; the row expressly covers only Theorem 7.1.13, so the other grouped components are not missing conclusions.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.discrete_strong_ignorability_ate_sum_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/GLP0011B2.lean#L82", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:345ac8c8b52b7c95422bd620af14cd2b88d55b4212e852355294160c70e2a883", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T116", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T116" }, "html": { "body": "

The usual measure-theoretic conditions are assumed. Strong ignorability gives discrete and conditional ATE identification and all three propensity-score conclusions, with a separate moment-free domain for the score result. Location: Definition 7.1.12: p. 247 (PDF p. 265). Theorem 7.1.13: p. 248 (PDF p. 266). Theorem 7.1.14: p. 249 (PDF p. 267). Definition 7.1.15: p. 250 (PDF p. 268). Theorem 7.1.16: p. 251 (PDF p. 269), proof on pp. 251–252.

", "headline": "Theorem 7.1.13 (ATE under Strong Ignorability), Theorem 7.1.14 (Conditional ATE under Strong Ignorability) and Theorem 7.1.16 (Strong Ignorability and the Propensity Score)", "location": "Definition 7.1.12: p. 247 (PDF p. 265). Theorem 7.1.13: p. 248 (PDF p. 266). Theorem 7.1.14: p. 249 (PDF p. 267). Definition 7.1.15: p. 250 (PDF p. 268). Theorem 7.1.16: p. 251 (PDF p. 269), proof on pp. 251–252" }, "id": "P001-T116", "text": { "body": "The usual measure-theoretic conditions are assumed. Strong ignorability gives discrete and conditional ATE identification and all three propensity-score conclusions, with a separate moment-free domain for the score result. Location: Definition 7.1.12: p. 247 (PDF p. 265). Theorem 7.1.13: p. 248 (PDF p. 266). Theorem 7.1.14: p. 249 (PDF p. 267). Definition 7.1.15: p. 250 (PDF p. 268). Theorem 7.1.16: p. 251 (PDF p. 269), proof on pp. 251–252.", "headline": "Theorem 7.1.13 (ATE under Strong Ignorability), Theorem 7.1.14 (Conditional ATE under Strong Ignorability) and Theorem 7.1.16 (Strong Ignorability and the Propensity Score)", "location": "Definition 7.1.12: p. 247 (PDF p. 265). Theorem 7.1.13: p. 248 (PDF p. 266). Theorem 7.1.14: p. 249 (PDF p. 267). Definition 7.1.15: p. 250 (PDF p. 268). Theorem 7.1.16: p. 251 (PDF p. 269), proof on pp. 251–252" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T116" }, { "declarations": [ { "claim_revision_id": "catalog:25f4873dcf39ba570af4d3543f401a00c4923bd2c1445cdd6f5e8cb2e2b3fb8e", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'P001/Wave1/CausalIpwDr.lean:37 separately requires integrability of m0(X) and m1(X) in the correct-propensity branch of Theorem 7.2.8. With X uniform on (0,1), independent fair D, Y0=Y1=0, p=q=1/2, m1(x)=1/x, and m0(x)=-1/x, strong ignorability and uniform overlap hold but neither nuisance regression is integrable. The printed combined DR score is identically zero, so its expectation exists without either separate nuisance expectation; the standing convention does not exclude this example. The IPW conjunct at line 33 and the regression-correct alternative otherwise match their components of the claim.', 'suggested_fix': 'For the DR conjunct, replace the separate integrability hypotheses on m0(X) and m1(X) by integrability of the complete displayed DR score. Keep the IPW conjunct independent of nuisance hypotheses.', 'second_review': \"P001/Wave1/CausalIpwDr.lean:37 separately requires integrability of m0(X) and m1(X), which T117's Theorem 7.2.8(ii) does not require when the propensity is correct. With X uniform on (0,1), independent fair D, Y0=Y1=0, p=q=1/2, m1(x)=1/x and m0(x)=-1/x, strong ignorability holds and the full displayed DR score vanishes for both treatment values, although neither nuisance composition is integrable. Replace the separate nuisance assumptions by integrability of the combined score; the independent IPW conjunct at line 33 already establishes its credited component.\", 'severity': 'major'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P001.Wave1.causal_ipw_and_dr_paper_contract", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/CausalIpwDr.lean#L23", "status": "proved", "support_scope": "limited", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:25f4873dcf39ba570af4d3543f401a00c4923bd2c1445cdd6f5e8cb2e2b3fb8e", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T117", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T117" }, "html": { "body": "

Location: Theorem 7.2.5, pp. 265–266 (PDF pp. 283–284); Theorem 7.2.8, pp. 267–269 (PDF pp. 285–287). The usual measure-theoretic conditions are assumed. Strong ignorability gives the population IPW identity. The augmented ATE identity holds if either both outcome regressions are correct and the approximating propensity has overlap, or the propensity is correct. Only the complete augmented score is required to be integrable; separate integrability of the nuisance regressions is not assumed in the correct-propensity alternative.

", "headline": "Theorem 7.2.5 (Reweighting under Strong Ignorability) and Theorem 7.2.8 (Double Robustness Theorem for Causal Inference)", "location": "Theorem 7.2.5: p. 265 (PDF p. 283), proof on pp. 265–266. Theorem 7.2.8: pp. 267–268 (PDF pp. 285–286), proof on pp. 268–269" }, "id": "P001-T117", "text": { "body": "Location: Theorem 7.2.5, pp. 265–266 (PDF pp. 283–284); Theorem 7.2.8, pp. 267–269 (PDF pp. 285–287). The usual measure-theoretic conditions are assumed. Strong ignorability gives the population IPW identity. The augmented ATE identity holds if either both outcome regressions are correct and the approximating propensity has overlap, or the propensity is correct. Only the complete augmented score is required to be integrable; separate integrability of the nuisance regressions is not assumed in the correct-propensity alternative.", "headline": "Theorem 7.2.5 (Reweighting under Strong Ignorability) and Theorem 7.2.8 (Double Robustness Theorem for Causal Inference)", "location": "Theorem 7.2.5: p. 265 (PDF p. 283), proof on pp. 265–266. Theorem 7.2.8: pp. 267–268 (PDF pp. 285–286), proof on pp. 268–269" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T117" }, { "declarations": [ { "claim_revision_id": "catalog:a3663ac986fc5ae0cfc15520f3232baf001ec89a33803f2eef4f55a16e8f8b32", "conditional_on": [], "credit_reason": "Covers clause (A) only; row 02 separately covers clause (B). No claim against a nondegenerate delta method is needed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.PrintedConfidence.printed_plugin_positive_normal_limit_refuted", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_PrintedRefutations.lean#L83", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:a3663ac986fc5ae0cfc15520f3232baf001ec89a33803f2eef4f55a16e8f8b32", "conditional_on": [], "credit_reason": "Covers clause (B) only. The zero-direction contradiction also defeats usual tangential Hadamard differentiability, so choosing all bounded directions does not manufacture the failure.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.printed_finite_variance_mean_hadamard_refuted", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/MeanHadamard.lean#L15", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:a3663ac986fc5ae0cfc15520f3232baf001ec89a33803f2eef4f55a16e8f8b32", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T118", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T118" }, "html": { "body": "

The printed claim is false: Hadamard differentiability need not yield a positive-variance plug-in normal limit, and finite second moments do not ensure supremum-norm Hadamard differentiability of the mean. Location: p. 120 (PDF p. 138).

", "headline": "Theorem 3.3.6 (Asymptotic Normality of Plug-In Estimators) with footnote 31 and the paragraph after it, as printed", "location": "p. 120 (PDF p. 138)" }, "id": "P001-T118", "text": { "body": "The printed claim is false: Hadamard differentiability need not yield a positive-variance plug-in normal limit, and finite second moments do not ensure supremum-norm Hadamard differentiability of the mean. Location: p. 120 (PDF p. 138).", "headline": "Theorem 3.3.6 (Asymptotic Normality of Plug-In Estimators) with footnote 31 and the paragraph after it, as printed", "location": "p. 120 (PDF p. 138)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T118" }, { "declarations": [ { "claim_revision_id": "catalog:2b202ececbd1a6315ad386a20b4cc513fa6ccfaedd617010159bbc7aa0317a54", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': \"P001/Wave1/SieveRate.lean:43 negates convergence in probability of population integrated squared prediction error to zero. T119 instead says the estimator is 'consistent across regions of the data with nonzero probability mass'; its reference to Theorem 4.3.3 makes an L2 reading plausible but does not define the sampling consistency criterion. Failure of integrated L2 convergence does not imply failure of pointwise or prediction-measure convergence, so the exported statement does not establish a full refutation of the unspecified printed assertion. The target expressly supplies the squared-residual correction, which is not a fidelity defect.\", 'suggested_fix': 'Reassign the credit to refutation of the explicitly stated integrated-L2 interpretation. Retain full printed-claim refutation credit only after identifying the printed consistency criterion and exporting failure of that criterion; the description alone cannot supply the missing definition.', 'second_review': \"P001/Wave1/SieveRate.lean:43 negates convergence in probability of population integrated squared prediction error, using CP as defined at P001/P001_Foundations.lean:60. T119 only requires consistency 'over regions of nonzero probability mass' and does not specify that sampling consistency means integrated-L2 convergence. Failure of that stronger criterion does not entail failure of pointwise or prediction-measure consistency, so the exported negation does not establish an unqualified refutation of the printed claim. Reassign credit to the explicitly stated integrated-L2 interpretation; the squared-residual minimand is already authorized by the target and is not an additional defect.\", 'severity': 'major'}", "credit_status": "wrong_credit", "fidelity_status_baseline": "wrong_credit", "lean_name": "AronowAudit.P001.Wave1.printed_sieve_degree_rate_contract_refuted", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/SieveRate.lean#L35", "status": "counterexample_proved", "support_scope": "withdrawn", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:2b202ececbd1a6315ad386a20b4cc513fa6ccfaedd617010159bbc7aa0317a54", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T119", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T119" }, "html": { "body": "

The explicitly specified integrated squared-risk consistency assertion for polynomial least-squares sieves is false under the printed degree-rate conditions. Location: Definition 4.3.9 and the attached consistency claim, pp. 168–169 (PDF pp. 186–187).

", "headline": "Definition 4.3.9 (Polynomial Least Squares Sieve Estimator) and the consistency claim attached to it, as printed", "location": "pp. 168–169 (PDF pp. 186–187)" }, "id": "P001-T119", "text": { "body": "The explicitly specified integrated squared-risk consistency assertion for polynomial least-squares sieves is false under the printed degree-rate conditions. Location: Definition 4.3.9 and the attached consistency claim, pp. 168–169 (PDF pp. 186–187).", "headline": "Definition 4.3.9 (Polynomial Least Squares Sieve Estimator) and the consistency claim attached to it, as printed", "location": "pp. 168–169 (PDF pp. 186–187)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T119" }, { "declarations": [ { "claim_revision_id": "catalog:7b23a562026defea715ac56c2b0d5afdb2d2caf2b7a63b4f8623ef2ea78c569b", "conditional_on": [], "credit_reason": "Negates a consistency assertion without the carried true-model premise; its misspecified Bernoulli witness supplies no canonical refutation or positive restricted clause.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.printed_fixed_covariate_kl_mle_positive_support_refuted", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/RepairT120b.lean#L63", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": null } ], "formalization": { "claim_revision_id": "catalog:7b23a562026defea715ac56c2b0d5afdb2d2caf2b7a63b4f8623ef2ea78c569b", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T120", "decision": "omitted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T120" }, "html": { "body": "

No current theorem is supplied for this target.

", "headline": "Theorem 5.2.8 (The ML Estimator Is Consistent for the Minimum KL Divergence Predictor), as printed", "location": "p. 195 (PDF p. 213)" }, "id": "P001-T120", "text": { "body": "No current theorem is supplied for this target.", "headline": "Theorem 5.2.8 (The ML Estimator Is Consistent for the Minimum KL Divergence Predictor), as printed", "location": "p. 195 (PDF p. 213)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T120" }, { "declarations": [ { "claim_revision_id": "catalog:d940884ab29209bbc6a55e93b8dec24f663b2884387b88b0fe3fdabd8c06ca6d", "conditional_on": [], "credit_reason": "Permits tied maximizing selections; every true-support sample in its witness has distinct equally maximizing vectors, violating the carried unique-argmax convention.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.printed_subset_full_vector_consistency_refuted", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/WP001bB1.lean#L180", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:d940884ab29209bbc6a55e93b8dec24f663b2884387b88b0fe3fdabd8c06ca6d", "conditional_on": [], "credit_reason": "Its nonidentified inactive-component countermodel also permits tied maximizers and supplies no refutation retaining the estimator definition's uniqueness condition.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.printed_general_subset_mixture_refuted", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/WP001bB1.lean#L100", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:d940884ab29209bbc6a55e93b8dec24f663b2884387b88b0fe3fdabd8c06ca6d", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T121", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T121" }, "html": { "body": "

Location: Section 5.2.5, p. 199 (PDF p. 217), second paragraph; Definitions 5.1.1 and 5.2.2; unique arg max convention, Glossary p. 285. The printed claim is false: in this compact two-component mixture on three outcomes, the sample likelihood has a unique maximizer for every sample of every positive size, so the book's unique arg max estimator is defined everywhere. The first component alone characterizes the true law, but this estimator cannot be consistent for every true full vector (1,b), where the unused component's parameter b ranges over the unit circle. The refuted assertion does not assume identification of the full vector. Failure of consistency refutes the consistency-and-asymptotic-efficiency conjunction; no separate definition or refutation of efficiency is needed. The usual measure-theoretic conditions are assumed; all sample statistics are measurable.

", "headline": "Section 5.2.5, the 'any subset' sentence on consistency and asymptotic efficiency of the mixture ML estimator, as printed", "location": "p. 199 (PDF p. 217), second paragraph" }, "id": "P001-T121", "text": { "body": "Location: Section 5.2.5, p. 199 (PDF p. 217), second paragraph; Definitions 5.1.1 and 5.2.2; unique arg max convention, Glossary p. 285. The printed claim is false: in this compact two-component mixture on three outcomes, the sample likelihood has a unique maximizer for every sample of every positive size, so the book's unique arg max estimator is defined everywhere. The first component alone characterizes the true law, but this estimator cannot be consistent for every true full vector (1,b), where the unused component's parameter b ranges over the unit circle. The refuted assertion does not assume identification of the full vector. Failure of consistency refutes the consistency-and-asymptotic-efficiency conjunction; no separate definition or refutation of efficiency is needed. The usual measure-theoretic conditions are assumed; all sample statistics are measurable.", "headline": "Section 5.2.5, the 'any subset' sentence on consistency and asymptotic efficiency of the mixture ML estimator, as printed", "location": "p. 199 (PDF p. 217), second paragraph" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T121" }, { "declarations": [ { "claim_revision_id": "catalog:ee73e55cbba9d55f00f55da6373742dd3beaa3f79dcda93f8c50c8142765bf5e", "conditional_on": [], "credit_reason": "The twelve-atom law satisfies the actual conditional-independence null and a full-rank population design; the failure is nonlinear conditional expectation, not singular-sample inverse or zero-denominator conventions.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.printed_placebo_f_test_size_refuted", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/PlaceboFTest.lean#L950", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:ee73e55cbba9d55f00f55da6373742dd3beaa3f79dcda93f8c50c8142765bf5e", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T124", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T124" }, "html": { "body": "

The printed claim is false: At nominal level one half, an iid conditional-independence placebo null has limiting rejection probability one, even when rejections require nonsingular regressions, positive residual risk and positive residual degrees of freedom. Location: p. 270 (PDF p. 288).

", "headline": "Section 7.2.8 (Placebo Testing), the FFF-test passage, as printed", "location": "p. 270 (PDF p. 288)" }, "id": "P001-T124", "text": { "body": "The printed claim is false: At nominal level one half, an iid conditional-independence placebo null has limiting rejection probability one, even when rejections require nonsingular regressions, positive residual risk and positive residual degrees of freedom. Location: p. 270 (PDF p. 288).", "headline": "Section 7.2.8 (Placebo Testing), the $F$-test passage, as printed", "location": "p. 270 (PDF p. 288)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T124" }, { "declarations": [ { "claim_revision_id": "catalog:ee8ef16645c9bb26a48cce008f1e0a24f4f764520e916fcd5339ae265a43d4be", "conditional_on": [], "credit_reason": "MemLp at exponent 2 is equivalent to the book’s finite mean/variance convention on a probability space; the zero-variance extension is explicitly in footnote 10.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.weak_law_finite_variance", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4049", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:ee8ef16645c9bb26a48cce008f1e0a24f4f764520e916fcd5339ae265a43d4be", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T125", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T125" }, "html": { "body": "

The iid sample mean converges in probability to its expectation with a finite second moment, including zero variance. Location: p. 100 (PDF p. 118). Definition 3.2.6: p. 99 (PDF p. 117).

", "headline": "Theorem 3.2.8 (Weak Law of Large Numbers)", "location": "p. 100 (PDF p. 118). Definition 3.2.6: p. 99 (PDF p. 117)" }, "id": "P001-T125", "text": { "body": "The iid sample mean converges in probability to its expectation with a finite second moment, including zero variance. Location: p. 100 (PDF p. 118). Definition 3.2.6: p. 99 (PDF p. 117).", "headline": "Theorem 3.2.8 (Weak Law of Large Numbers)", "location": "p. 100 (PDF p. 118). Definition 3.2.6: p. 99 (PDF p. 117)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T125" }, { "declarations": [ { "claim_revision_id": "catalog:30fb4d90d3d538e67003cd7572d8c9d77e25d59204574d30e1d4c656735b2f1b", "conditional_on": [], "credit_reason": "GaussianReal takes variance as its second argument; the positive-variance premise is in the target itself.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.central_limit_finite_variance", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4073", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:30fb4d90d3d538e67003cd7572d8c9d77e25d59204574d30e1d4c656735b2f1b", "conditional_on": [], "credit_reason": "This is the auxiliary finite standardization component, not a separate assertion of distributional convergence.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.standardized_mean_fixed_sample_moments", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6648", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:30fb4d90d3d538e67003cd7572d8c9d77e25d59204574d30e1d4c656735b2f1b", "conditional_on": [], "credit_reason": "The denominator is the sample mean’s true sampling standard deviation, exactly Definition 3.2.23; n=0 is irrelevant to the limit.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.central_limit_standardized_mean", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6692", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:30fb4d90d3d538e67003cd7572d8c9d77e25d59204574d30e1d4c656735b2f1b", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T126", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T126" }, "html": { "body": "

Both printed CLT forms hold for iid data of finite positive variance, with the finite standardization identity and moments. Location: p. 109 (PDF p. 127). Definition 3.2.23: p. 109. Definition 3.2.22: p. 108 (PDF p. 126).

", "headline": "Theorem 3.2.24 (Central Limit Theorem)", "location": "p. 109 (PDF p. 127). Definition 3.2.23: p. 109. Definition 3.2.22: p. 108 (PDF p. 126)" }, "id": "P001-T126", "text": { "body": "Both printed CLT forms hold for iid data of finite positive variance, with the finite standardization identity and moments. Location: p. 109 (PDF p. 127). Definition 3.2.23: p. 109. Definition 3.2.22: p. 108 (PDF p. 126).", "headline": "Theorem 3.2.24 (Central Limit Theorem)", "location": "p. 109 (PDF p. 127). Definition 3.2.23: p. 109. Definition 3.2.22: p. 108 (PDF p. 126)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T126" }, { "declarations": [ { "claim_revision_id": "catalog:ca04a3d7ab67a9a508c199e4288b1e812541bed749322ebc0663a7890e9ecd5b", "conditional_on": [], "credit_reason": "Global continuity is present in the quoted preceding sentence and the target’s all-x reading; I do not use the record’s suggested point-specific variant to narrow that quoted claim.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.rectangular_kernel_consistency", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4312", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:ca04a3d7ab67a9a508c199e4288b1e812541bed749322ebc0663a7890e9ecd5b", "conditional_on": [], "credit_reason": "An auxiliary finite-sample identity; the selected theorem already supplies every asserted target conclusion.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.rectangular_kernel_fixed_sample_moments", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L5429", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:ca04a3d7ab67a9a508c199e4288b1e812541bed749322ebc0663a7890e9ecd5b", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T127", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T127" }, "html": { "body": "

The rectangular-kernel density estimator is pointwise consistent for a continuous density when positive bandwidths tend to zero and n times bandwidth tends to infinity. Location: p. 122 (PDF p. 140); proof sketch on p. 123 (PDF p. 141); the continuity condition is stated on p. 121 (PDF p. 139).

", "headline": "Theorem 3.3.8 (Consistency of the Kernel Density Estimator, Rectangular Kernel)", "location": "p. 122 (PDF p. 140); proof sketch on p. 123 (PDF p. 141); the continuity condition is stated on p. 121 (PDF p. 139)" }, "id": "P001-T127", "text": { "body": "The rectangular-kernel density estimator is pointwise consistent for a continuous density when positive bandwidths tend to zero and n times bandwidth tends to infinity. Location: p. 122 (PDF p. 140); proof sketch on p. 123 (PDF p. 141); the continuity condition is stated on p. 121 (PDF p. 139).", "headline": "Theorem 3.3.8 (Consistency of the Kernel Density Estimator, Rectangular Kernel)", "location": "p. 122 (PDF p. 140); proof sketch on p. 123 (PDF p. 141); the continuity condition is stated on p. 121 (PDF p. 139)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T127" }, { "declarations": [ { "claim_revision_id": "catalog:e93051705a55fc601df0c3c53d8a5050247e6b2ddc136e00dc0134c1a22c7104", "conditional_on": [], "credit_reason": "The record’s note about equality of limiting coverage is stronger than the target’s stated lower bound and does not make the formal conclusion deficient. A full mathematical nonvacuity instance is supplied here, independently of the record’s partial instance certificate.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.normal_approximation_confidence_interval", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L4478", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e93051705a55fc601df0c3c53d8a5050247e6b2ddc136e00dc0134c1a22c7104", "conditional_on": [], "credit_reason": "Supplies deterministic identities when the variance estimate is positive, not an almost-sure endpoint-order guarantee under the retained nonnegative-variance premises.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.studentization_finite_identity_and_coverage", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6027", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:e93051705a55fc601df0c3c53d8a5050247e6b2ddc136e00dc0134c1a22c7104", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T128", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T128" }, "html": { "body": "

Location: Theorem 3.4.2, pp. 125–126 (PDF pp. 143–144); Definition 3.4.1 and footnote 34, p. 124 (PDF p. 142). The usual measure-theoretic conditions are assumed. For an asymptotically normal estimator with finite positive asymptotic variance and a nonnegative sampling-variance estimator consistent at the n scale, the displayed open sets have limiting coverage at least 1-alpha for every alpha in (0,1).

", "headline": "Theorem 3.4.2 (Normal Approximation–Based Confidence Interval)", "location": "p. 125 (PDF p. 143); proof on pp. 125–126 (PDF pp. 143–144). Definition 3.4.1: p. 124 (PDF p. 142)" }, "id": "P001-T128", "text": { "body": "Location: Theorem 3.4.2, pp. 125–126 (PDF pp. 143–144); Definition 3.4.1 and footnote 34, p. 124 (PDF p. 142). The usual measure-theoretic conditions are assumed. For an asymptotically normal estimator with finite positive asymptotic variance and a nonnegative sampling-variance estimator consistent at the n scale, the displayed open sets have limiting coverage at least 1-alpha for every alpha in (0,1).", "headline": "Theorem 3.4.2 (Normal Approximation–Based Confidence Interval)", "location": "p. 125 (PDF p. 143); proof on pp. 125–126 (PDF pp. 143–144). Definition 3.4.1: p. 124 (PDF p. 142)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T128" }, { "declarations": [ { "claim_revision_id": "catalog:0f12d6c48c1cf3064c854921155998f52e7c971675959723fdee64d465e725ee", "conditional_on": [], "credit_reason": "Duplicate or auxiliary consistency route; the selected product-consistency and expected-log-maximum rows supply the retained conclusions.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.ml_consistency_explicit_uniform_regularities", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L7967", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:0f12d6c48c1cf3064c854921155998f52e7c971675959723fdee64d465e725ee", "conditional_on": [], "credit_reason": "The printed lemma uses finite real expected logs under standing assumption S3, so no extended negative-infinite cases are credited. 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T129 reserves 'some requisite regularity conditions' and 'suitable regularity conditions', so its pointwise-WLLN sketch is not an assertion that the named examples alone suffice. Uniform convergence is a concrete sufficient interpretation of this express qualification, and the theorem preserves the claimed consistency conclusion for the actual likelihood estimator. Parameterwise almost-sure positivity at line 8052 also respects the finite-log expectation convention rather than exploiting Lean's totalized logarithm.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.product_maximum_likelihood_consistency", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/P001_Foundations.lean#L6920", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:0f12d6c48c1cf3064c854921155998f52e7c971675959723fdee64d465e725ee", "conditional_on": [], "credit_reason": "Duplicate or auxiliary consistency route; the selected product-consistency and expected-log-maximum rows supply the retained conclusions.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.sample_product_mle_consistent", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/WP001bB2.lean#L11", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:0f12d6c48c1cf3064c854921155998f52e7c971675959723fdee64d465e725ee", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T129", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T129" }, "html": { "body": "

Actual-product maximum-likelihood estimators are consistent under explicit uniform regularities, and a separate true-model lemma characterizes the maximum expected log likelihood and equality. Location: Theorem 5.2.6: p. 193 (PDF p. 211). Proof sketch: pp. 193–194 (PDF pp. 211–212).

", "headline": "Theorem 5.2.6 (Consistency of the Maximum Likelihood Estimator) and the lemma in its proof sketch", "location": "Theorem 5.2.6: p. 193 (PDF p. 211). Proof sketch: pp. 193–194 (PDF pp. 211–212)" }, "id": "P001-T129", "text": { "body": "Actual-product maximum-likelihood estimators are consistent under explicit uniform regularities, and a separate true-model lemma characterizes the maximum expected log likelihood and equality. Location: Theorem 5.2.6: p. 193 (PDF p. 211). Proof sketch: pp. 193–194 (PDF pp. 211–212).", "headline": "Theorem 5.2.6 (Consistency of the Maximum Likelihood Estimator) and the lemma in its proof sketch", "location": "Theorem 5.2.6: p. 193 (PDF p. 211). Proof sketch: pp. 193–194 (PDF pp. 211–212)" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T129" }, { "declarations": [ { "claim_revision_id": "catalog:a912be936d30b2d194b80d4951d5513578bdce6b4f7e51990c07da067f014fdb", "conditional_on": [], "credit_reason": "The row proves both displayed endpoint population formulas, exact unbiasedness at every positive sample size, and full-sequence consistency in probability. T130 is the estimation theorem; sharpness and the identification inequalities belong to its cited predecessor, Theorem 6.1.3, and are not additional conclusions claimed by this row. Only observable pairs must be IID, as printed; no MCAR or MAR assumption is imposed. Boundedness ensures integrability, probability measures make toReal harmless, and positive respondent mass prevents conditional-mean division by zero. The claim-stage verdict is unchanged after reading the record; no conflicting target kind or status was found.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P001.Wave1.expected_value_bounds_paper_contract", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P001/Wave1/Gap/WP001bB2.lean#L258", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:a912be936d30b2d194b80d4951d5513578bdce6b4f7e51990c07da067f014fdb", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P001-T130", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P001-T130" }, "html": { "body": "

Both printed missing-outcome bound estimators are unbiased at every positive sample size and consistent for their displayed endpoints under iid observable pairs. Location: pp. 211–212 (PDF pp. 229–230); proof on p. 212.

", "headline": "Theorem 6.1.4 (Estimating Sharp Bounds for the Expected Value)", "location": "pp. 211–212 (PDF pp. 229–230); proof on p. 212" }, "id": "P001-T130", "text": { "body": "Both printed missing-outcome bound estimators are unbiased at every positive sample size and consistent for their displayed endpoints under iid observable pairs. Location: pp. 211–212 (PDF pp. 229–230); proof on p. 212.", "headline": "Theorem 6.1.4 (Estimating Sharp Bounds for the Expected Value)", "location": "pp. 211–212 (PDF pp. 229–230); proof on p. 212" }, "url": "https://pmaronow.github.io/llm-audit/works/P001/#P001-T130" } ] }, "missing_sources": [], "reproduction": { "agreement": "agrees_in_part", "status": "selected_recalculation", "targets": [ { "agreement": "within_mc_error", "location": "p. 134 (PDF p. 152), design p. 133 (PDF p. 151)", "method": "I independently simulated the stated uniform design with 1,000,000 replications at each sample size and compared all 18 cells, allowing for rounding and Monte Carlo error.", "status": "recalculated", "target": "Table 3.4.1" }, { "agreement": "within_mc_error", "location": "p. 134 (PDF p. 152)", "method": "I computed the stated Bernoulli design by exact binomial enumeration and compared all 18 cells, allowing for rounding and Monte Carlo error in the published simulation.", "status": "recalculated", "target": "Table 3.4.2" }, { "agreement": "not_compared", "location": "p. 173 (PDF p. 191)", "method": "I inspected the printed table for internal consistency. Recomputing the selected regression estimates and bootstrap standard errors was blocked by missing Quality of Government data and fitting code.", "status": "blocked", "target": "Table 4.4.5" }, { "agreement": "not_compared", "location": "p. 172 (PDF p. 190)", "method": "I inspected the printed figure. I could not compare its country points and fitted curves with a reconstruction because the Quality of Government data and fitting code were unavailable.", "status": "blocked", "target": "Figure 4.4.2" }, { "agreement": "not_compared", "location": "p. 176 (PDF p. 194)", "method": "I inspected the printed figure. I could not compare its fitted surfaces with a reconstruction because the Quality of Government data and fitting code were unavailable.", "status": "blocked", "target": "Figure 4.4.4" }, { "agreement": "not_compared", "location": "p. 233 (PDF p. 251)", "method": "I inspected the printed table for internal consistency. Recomputing the estimates and bootstrap standard errors was blocked by missing Quality of Government data and fitting code; printed-summary arithmetic is covered by the in-text row.", "status": "blocked", "target": "Table 6.3.1" }, { "agreement": "not_compared", "location": "p. 280 (PDF p. 298)", "method": "I inspected the printed table for internal consistency. Recomputing the estimates and bootstrap standard errors was blocked by missing naturalization data and fitting code; printed-summary arithmetic is covered by the in-text row.", "status": "blocked", "target": "Table 7.5.1" }, { "agreement": "agrees_in_part", "location": "pp. 134, 231-232, 277-279 (PDF pp. 152, 249-250, 295-297)", "method": "I compared p. 134 with its uniform simulation table and an independent simulation: the prose disagrees. The bounds, confidence intervals and reported p-values on pp. 231-232 and 277-279 agree with arithmetic from the printed summaries within their precision. No separate unpooled OLS p-value is printed, so that value was not compared.", "status": "recalculated", "target": "In-text numerical checks" } ] }, "summary": "\\emph{Foundations of Agnostic Statistics} (Cambridge University Press, 2019) is a textbook on what can be learned from data without assuming that a parametric model is true. It treats regression and maximum likelihood as plug-in estimators of best approximations, and it separates identification from estimation. 126 entries are classified as errors (8 Moderate, 118 Minor, none Major or Critical), and 24 are clarity notes. The most consequential is \\fid{P001-C07-E011}: the book says that the bootstrap standard errors are likely to be adequate for nearest-neighbor matching, but the bootstrap variance of a matching estimator can be inconsistent even when the estimator has no bias.", "what_was_checked": "I examined the whole book: the front matter, the seven chapters, the two glossaries, the references and the index. The list below states what I examined and what held up. It is not a finding that the rest of the book is correct.\n\n\\begin{itemize}\n\\item \\emph{Statements and proofs.} I followed the proofs of the following results step by step. They hold as printed, apart from misprints reported as separate findings: the basic properties of probability and the addition rule (Theorems 1.1.4 and 1.1.7); the Law of Iterated Expectations, the Law of Total Variance and the properties of deviations from the CEF (Theorems 2.2.17 to 2.2.19); the unbiasedness, sampling variance and Chebyshev bound for the sample mean, and the Weak Law of Large Numbers (Theorems 3.2.3 to 3.2.5 and 3.2.8); the OLS solution and the properties of its residuals (Theorems 4.1.4 and 4.1.5); polynomial approximation of the CEF (Theorem 4.3.3); sharp bounds, identification under MCAR and MAR, the propensity score and double robustness for missing data (Theorems 6.1.3, 6.1.6, 6.1.9, 6.1.10, 6.1.12 and 6.2.9); and the corresponding causal results (Theorems 7.1.7, 7.1.13, 7.1.14, 7.1.16, 7.2.5 and 7.2.8).\n\\item \\emph{Worked examples.} I recomputed the following numerical examples exactly; their results hold apart from corrections reported separately: the example of the coin flip and die roll in Chapter 1 ($P(B)=5/24$ and $P(A\\mid B)=3/5$), Example 2.2.23 ($\\beta=10$ and $\\alpha=-5/3$), the regression fitted to the 15 rows of Table 4.1.1 ($3.52$ and $1.04$), and the imputation and matching examples of Chapters 6 and 7 (Examples 6.2.3, 6.2.5 and 7.2.4).\n\\item \\emph{Simulation tables.} I independently reproduced all 18 cells of Table 3.4.1 by simulation with $10^6$ replications for each $n$, with agreement within rounding and Monte Carlo error. I compared all 18 cells of Table 3.4.2 with an exact binomial computation; they agree within rounding and the Monte Carlo error of the published $10^6$ replications.\n\\item \\emph{Empirical tables.} Tables 4.4.5, 6.3.1 and 7.5.1 are internally consistent, and the numbers quoted from them in the text match the tables, apart from findings reported separately.\n\\item \\emph{Front and back matter.} The entries of the Contents and the 36 entries of the list of tables and figures agree with the body in wording and page number. The 18 entries of the Glossary of Common Abbreviations expand correctly and point to the stated items. Every identifiable in-text citation has an entry in the References; the citation on p.~177 lacks its author and cannot be matched to a work (\\fid{P001-C04-N016}). Every entry is cited at least once.\n\\end{itemize}\n\nI did not complete the following checks.\n\n\\begin{itemize}\n\\item I did not have replication code for the book, the Quality of Government data used in Chapters 4 and 6, or the naturalization data of Hainmueller and Hangartner used in Chapter 7. No estimate, standard error or figure drawn from data in Sections 4.4, 6.3 or 7.5 was reproduced.\n\\item I did not have the code that produced the simulated figures. I compared figures with the stated formulas where that was possible.\n\\item I read only a few of the works cited by the book to settle specific points. I did not systematically check bibliographic details against the publications themselves; the individual entries document the specific checks I made.\n\\item I did not check the historical and biographical statements in the front matter against outside sources, except for one spelling in the acknowledgments.\n\\item Six possible errors could not be settled with the sources available. Two concern citations (a year, and several chapter and page references). Two concern whether figures were drawn from the designs stated in the text (Figures 4.3.4, 4.3.5 and 7.3.2). One concerns the definition of the water-access variable in Section 4.4, and one concerns the first year of the sample in Section 7.5.\n\\end{itemize}", "work": { "arxiv_id": null, "audit_complete": true, "authors": [ "P. M. Aronow", "Benjamin T. Miller" ], "category": "Books", "citation": "P. M. Aronow and Benjamin T. Miller (2019). Foundations of Agnostic Statistics. Cambridge University Press.", "doi": "10.1017/9781316831762", "edition": { "pages_in_audited_file": 316, "sha256": "d2d3548a604e7d684b41c7a6bf764c16b883b16491ae5115f433ecbe1ea8f9af", "source_url": "https://doi.org/10.1017/9781316831762", "venue": "Cambridge University Press", "year": "2019" }, "id": "P001", "later_materials": null, "title": "Foundations of Agnostic Statistics", "type": "Book", "url": "https://pmaronow.github.io/llm-audit/works/P001/", "venue": "Cambridge University Press", "year": "2019" } }