{ "claims": [ { "claim": "Under Assumption 1, with heterogeneous effects and $c>0$, any estimator of the population average causal effect that is consistent at rate $n^{-1/2}$ and asymptotically normal is superefficient for the data-adaptive target: $\\sqrt{n}(\\hat\\theta-\\theta_{F_n})=o_p(1)$.", "claim_id": "P017-H1", "related_corrections": [ "P017-E001", "P017-E002", "P017-E003", "P017-N001" ], "tied_to": "Proposition 3 (with Assumption 1 and Definitions 1-2); abstract; introduction; Section 3" }, { "claim": "The data-adaptive target $\\theta_{F_n}$ is claimed to be a local average causal effect in every realization: there is a nonnegative weighting $w_{F_n}$ with $\\theta_{F_n}=\\mathrm{E}_G[w_{F_n}\\tau]/\\mathrm{E}_G[w_{F_n}]$, so the target is the average effect of at least one subpopulation.", "claim_id": "P017-H2", "related_corrections": [ "P017-E001" ], "tied_to": "Proposition 1; Definition 2; abstract and introduction ('a data-adaptive local average causal effect')" }, { "claim": "The data-adaptive target converges to the population average causal effect at rate $n^{-1/2}$: $\\sqrt{n}(\\theta_{F_n}-\\mathrm{E}_G[\\tau])=O_p(1)$.", "claim_id": "P017-H3", "related_corrections": [], "tied_to": "Proposition 2" }, { "claim": "Under Assumption 1, if $\\sqrt{n}(\\hat\\theta-\\mathrm{E}_G[\\tau])=Z_n+\\varepsilon_n$, where $Z_n\\sim N(0,\\sigma^2)$ with $0<\\sigma^2<\\infty$ and $\\mathrm{E}[\\varepsilon_n^2]=o(n^{-1/2})$, the mean square error of $\\hat\\theta$ for the data-adaptive target is $o(n^{-1})$.", "claim_id": "P017-H4", "related_corrections": [ "P017-E001", "P017-E002", "P017-E004" ], "tied_to": "Proposition 4 and its displayed normal-tail calculation; introduction, last sentence" }, { "claim": "Adapting the target on the basis of efficiency can deliver extraordinary precision gains; without further restrictions the gains in statistical certainty 'may be essentially unbounded', and in the support-containment case the estimator has zero error for the adaptive target in every realization.", "claim_id": "P017-H5", "related_corrections": [ "P017-E004", "P017-N001" ], "tied_to": "Proposition 3 and its proof ($o_p(a_n)$ for every positive deterministic sequence $(a_n)$); Corollary 1; abstract; Section 3" }, { "claim": "The efficiency gain can come at the cost of a target of little intrinsic interest and a subpopulation whose composition is unknown to the researcher; restrictions on adaptive targets may therefore be warranted.", "claim_id": "P017-H6", "related_corrections": [], "tied_to": "Section 3, second paragraph; Definition 2; Proposition 1 and the paragraph after its proof; Corollary 1" } ], "date_published": "2026-10-25", "findings": [ { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "historical_lean_attribution", "exhibits": [ { "alt": "Definition 2 with its display of three cases, the two sentences after it, Proposition 1 and the paragraph that follows. Red boxes mark the second and third cases of the display, 'E_G[tau] + c : theta-hat - E_G[tau] > c' and 'E_G[tau] - c : theta-hat - E_G[tau] < -c'; the words 'across all F_n' in Proposition 1; the words 'a weighted mean can obtain any value in the interval defined by the infimum and supremum of its distribution's support'; and the words 'across all realizations'.", "caption": "Definition 2, Proposition 1 and the paragraph that follows it, PDF p. 2. The marks show the two cases of Definition 2 in which the target equals $\\mathrm{E}_G[\\tau]+c$ or $\\mathrm{E}_G[\\tau]-c$, the words “across all $F_n$” in Proposition 1, the fact given as its proof, and the words “across all realizations”.", "url": "https://pmaronow.github.io/llm-audit/works/P017/exhibits/P017-E001.png" }, { "alt": "Chart with the exponent k of the weighting tau to the power k on the horizontal axis, at the five equally spaced labels 0, 1, 2, 10 and 100 (spacing is not to scale), and the value on the scale of the effect on the vertical axis, labelled from 0 to 1. Five black dots show the weighted means 1/2, 2/3, 3/4, 11/12 and 101/102, which rise toward 1. A dashed blue horizontal line at 1 is labelled as the printed target on the event with radius c equal to 1/2; the last dot lies just below it. A solid orange horizontal line at 3/4 is labelled as the target on the event with radius d equal to 1/4; the dot for k equal to 2 lies on it. The title reads 'Weighted means of tau, uniform on (0, 1), approach the printed target 1 and do not reach it'.", "caption": "The counterexample to Proposition 1, with $\\tau$ uniform on $(0,1)$ and $c=1/2$. On the event that every observation is an untreated unit with outcome $-1$, the printed target is $1$ (dashed line). The weightings $w=\\tau^k$ with $k=0,1,2,10,100$ have weighted means $1/2$, $2/3$, $3/4$, $11/12$ and $101/102$; by the derivation in the text, no admissible weighting has weighted mean $1$. With radius $d=1/4$ the target on the same event is $3/4$ (solid line), which the weighting $\\tau^2$ attains. All values are exact.", "url": "https://pmaronow.github.io/llm-audit/works/P017/exhibits/P017-E001-endpoint.png" } ], "formal_release": null, "formal_statements": [ { "id": "P017-T1", "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T1" }, { "id": "P017-T2", "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T2" }, { "id": "P017-T3", "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T3" }, { "id": "P017-T4", "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T4" }, { "id": "P017-T9", "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T9" } ], "headline": "Proposition 1 fails when the target of Definition 2 is an endpoint of the support of $\\tau$ that has probability zero; with a fixed clipping radius $0\\hl{c}\\\\ \\mathrm{E}_G[\\tau]-\\hl{c} & :\\ \\hat\\theta-\\mathrm{E}_G[\\tau]<-\\hl{c}\\end{cases},\n\\]\nwhere, as in Assumption 1, $c=\\min(\\sup(\\operatorname{Supp}_G[\\tau])-\\mathrm{E}_G[\\tau],\\ \\mathrm{E}_G[\\tau]-\\inf(\\operatorname{Supp}_G[\\tau]))$.\n\nThe target parameter adapts naturally to the closest value in an interval surrounding $\\mathrm{E}_G[\\tau]$, where the width of the interval is defined by the support of $\\tau$. We formalize how each $\\theta_{F_n}$ is a local average treatment effect.\n\nProposition 1: There exists a nonnegative weighting associated with each empirical distribution $F_n$, $w_{F_n}$, such that \\hl{across all $F_n$}, $\\theta_{F_n}=\\frac{\\mathrm{E}_G[w_{F_n}\\tau]}{\\mathrm{E}_G[w_{F_n}]}$.\n\nA proof of Proposition 1 follows directly from the fact that \\hl{a weighted mean can obtain any value in the interval defined by the infimum and supremum of its distribution's support}. Proposition 1 asserts that \\hl{across all realizations}, the target parameter $\\theta_{F_n}$ corresponds to an average causal effect for at least one subpopulation.", "corrected": "Definition 2: Let $c$ be as in Assumption 1, and fix a finite radius $d$ such that $0\\hl{d}\\\\ \\mathrm{E}_G[\\tau]-\\hl{d} & :\\ \\hat\\theta-\\mathrm{E}_G[\\tau]<-\\hl{d}\\end{cases}.\n\\]\n\nProposition 1: Suppose that $\\tau$ has finite expectation and that Assumption 1 holds. There exists a nonnegative weighting associated with each empirical distribution $F_n$, $w_{F_n}$, with $0<\\mathrm{E}_G[w_{F_n}]<\\infty$ and $\\mathrm{E}_G[|w_{F_n}\\tau|]<\\infty$, such that across all $F_n$, $\\theta_{F_n}=\\frac{\\mathrm{E}_G[w_{F_n}\\tau]}{\\mathrm{E}_G[w_{F_n}]}$.\n\nA proof of Proposition 1 follows from the fact that a weighted mean with nonnegative weights can obtain \\hl{every value strictly between} the infimum and supremum of its distribution's support, \\hl{and obtains a finite endpoint of the support if and only if that endpoint has positive probability}. Since $d0$, then the weighting $w=\\mathbf{1}\\{\\tau=t\\}/p_0$ satisfies $\\mathrm{E}_G[w]=1$ and $\\mathrm{E}_G[w\\tau]=t$. Suppose next that $p_-=\\Pr_G(\\tau0$ and $p_+=\\Pr_G(\\tau>t)>0$, and set\n\\[\nm_-=\\frac{\\mathrm{E}_G[\\tau\\mathbf{1}\\{\\taut\\}]}{p_+},\\qquad a=\\frac{m_+-t}{m_+-m_-}.\n\\]\nThen $m_-t\\}\n\\]\nis bounded and nonnegative. Computing the two expectations, I find $\\mathrm{E}_G[w]=a+(1-a)=1$ and $\\mathrm{E}_G[w\\tau]=am_-+(1-a)m_+=t$. Conversely, suppose that $\\Pr_G(\\tau=t)=0$ and $\\Pr_G(\\tau>t)=0$, so that $\\tau0$ almost surely, I have $w=0$ almost surely, which contradicts $\\mathrm{E}_G[w]>0$. The case $\\Pr_G(\\tau0,\\quad\\text{or}\\quad \\Pr_G(\\tau0\\ \\text{ and }\\ \\Pr_G(\\tau>t)>0.\n\\]\nEvery $t$ strictly between the infimum and the supremum of the support of $\\tau$ satisfies the second condition. A finite endpoint of the support satisfies neither condition unless it has positive probability.\n\n\\emph{The counterexample.} Let $\\tau$ be uniform on $(0,1)$, let $B$ equal $-1$ or $1$ with probability $1/2$ each, and let $T$ be a fair treatment assignment, with $\\tau$, $B$ and $T$ independent. Let $Y(0)=B$, $Y(1)=B+\\tau$ and $Y=TY(1)+(1-T)Y(0)$. Then $\\mathrm{E}_G[\\tau]=1/2$, the support of $\\tau$ is $[0,1]$ and $c=1/2$, so that Definition 2 clips $\\hat\\theta$ to $[0,1]$. For $n$ independent units, I take the inverse-probability estimator $\\hat\\theta=n^{-1}\\sum_{i=1}^nX_i$ with $X_i=2T_iY_i-2(1-T_i)Y_i$. The summands are bounded, with mean $1/2$ and variance $53/12$. By the central limit theorem, I have $\\sqrt n(\\hat\\theta-1/2)\\Rightarrow N(0,53/12)$. On the event that $(T_i,Y_i)=(0,-1)$ for all $i$, every $X_i$ equals $2$, and therefore $\\hat\\theta=2$ and $\\theta_{F_n}=1$. Since $\\tau<1$ almost surely, no admissible weighting has weighted mean $1$, by the characterization above.\n\\begin{audittable}{lr}{The counterexample to Proposition 1, with $\\tau$ uniform on $(0,1)$. All values are exact.}\nQuantity & Value \\\\\n$\\mathrm{E}_G[\\tau]$ and $c$ & $1/2$ and $1/2$ \\\\\nMean and variance of $X_i$ & $1/2$ and $53/12$ \\\\\nProbability that $(T_i,Y_i)=(0,-1)$ for every $i$ & $4^{-n}$ \\\\\nEstimate $\\hat\\theta$ on this event & $2$ \\\\\nPrinted target $\\theta_{F_n}$ on this event & $1$ \\\\\nTarget on this event with radius $d=1/4$ & $3/4$ \\\\\n\\end{audittable}\nThe Lean proof covers the following statement: for an integrable weighting $w\\ge0$ with positive expectation, on a population in which $\\tau<1$ almost surely and $w\\tau$ is integrable, the weighted mean is less than $1$. It also covers the uniform distribution on $(0,1)$, for which the value obtained by clipping the estimate $2$ to $[0,1]$ has no such representation. The Lean statements also specify the Gaussian limit, the probability $4^{-n}$ of this event at every $n\\ge1$, and the identity expressing the estimator as an integral against the empirical distribution of the observed pairs. A second example has a failure event of probability $16^{-n}$: its effect is discrete, with values that accumulate at $-1$ and $1$ and do not reach them.", "problem": "Proposition 1 asserts that, for every empirical distribution $F_n$, the target $\\theta_{F_n}$ of Definition 2 equals $\\mathrm{E}_G[w\\tau]/\\mathrm{E}_G[w]$ for some nonnegative weighting $w$. Definition 2 clips $\\hat\\theta$ to the closed interval $[\\mathrm{E}_G[\\tau]-c,\\ \\mathrm{E}_G[\\tau]+c]$. When $c$ is finite, at least one end of this interval is the infimum or the supremum of the support of $\\tau$, and the target equals that value whenever $\\hat\\theta$ lies at or beyond it. A weighted mean with nonnegative weights attains every value strictly between the infimum and the supremum of the support, but it attains a finite endpoint only if the endpoint has positive probability. The fact given as the proof of Proposition 1 includes the endpoints without this condition. If, for example, $\\tau$ is bounded and has a continuous distribution, the endpoint has probability zero. On the realizations where $\\hat\\theta$ lies at or beyond it, the target is a value that no weighting attains, and it is not the average causal effect of any subpopulation. These realizations have probability tending to zero as $n\\to\\infty$, but in the example below they have positive probability at every sample size. The abstract, the introduction and the discussion describe the target as a local average causal effect, and that description depends on Proposition 1.", "what_survives": "The conclusions of Propositions 2, 3 and 4 and the equality $\\Pr(\\hat\\theta=\\theta_{F_n})=1$ of Corollary 1 hold for the target as printed, since they can be derived without Proposition 1. Proposition 1 holds as printed when $\\tau$ has finite support or support unbounded in both directions, and for the printed target the weighted representation exists with probability tending to one. The statement that the printed target is a local average causal effect in every realization does not survive; in particular, under the hypothesis of Corollary 1 the common value of $\\hat\\theta$ and $\\theta_{F_n}$ need not be a weighted average of causal effects. With the radius $d$ the target differs from the printed one on the realizations with $|\\hat\\theta-\\mathrm{E}_G[\\tau]|>d$. No new assumption on $G$ is needed, the conclusions of Propositions 2, 3 and 4 hold with $d$ in place of $c$, and the claim that $\\hat\\theta$ is superefficient for a data-adaptive local average causal effect stands." }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-E001", "work": { "arxiv_id": "1601.01413", "doi": "10.1515/jci-2016-0007", "title": "Data-Adaptive Causal Effects and Superefficiency", "url": "https://pmaronow.github.io/llm-audit/works/P017/" }, "work_id": "P017" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "The five lines of the proof of Proposition 3. Red boxes mark the words '(theta-tilde minus theta_Fn) is o_p(a_n) for any positive sequence (a_n)'; the expression 'Pr(theta-tilde minus theta_Fn not equal to 0) = 2 Phi(-c root n / sigma)', which runs over a line break; and the term 'o_p(a_n)' in the last line, 'Thus theta-hat minus theta_Fn = o_p(a_n) + o_p(n to the -1/2) = o_p(n to the -1/2), yielding the result'.", "caption": "Proof of Proposition 3, PDF p. 2. The marks show the claim that $(\\tilde\\theta-\\theta_{F_n})$ is $o_p(a_n)$ for any positive sequence, the equality $\\Pr(\\tilde\\theta-\\theta_{F_n}\\ne0)=2\\Phi(-c\\sqrt n/\\sigma)$, and the term $o_p(a_n)$ in the last line, where the claim is used.", "url": "https://pmaronow.github.io/llm-audit/works/P017/exhibits/P017-E002.png" }, { "alt": "Proposition 4 and the whole of its proof, including a display of three lines. Red boxes mark the words 'Within c of E_G[tau], the squared deviation is zero' and the first line of the display, 'E_G[(theta-tilde minus theta_Fn) squared] is at most 2 times the integral from c to infinity of x squared times the normal density with variance sigma squared over n', which is printed without a differential.", "caption": "Proposition 4 and its proof, PDF p. 3. The marks show the sentence “Within $c$ of $\\mathrm{E}_G[\\tau]$, the squared deviation is zero” and the first line of the display, which bounds $\\mathrm{E}_G[(\\tilde\\theta-\\theta_{F_n})^2]$ by a Gaussian tail integral.", "url": "https://pmaronow.github.io/llm-audit/works/P017/exhibits/P017-E002-b.png" }, { "alt": "Chart with two rows on a logarithmic horizontal axis labelled from 10 to the -7 to 1. In the row labelled 'Probability Pr(theta-tilde minus theta_Fn not equal to 0)', an open blue diamond at 5.7 times 10 to the -7 is joined by a thin line to a filled orange circle at 1. In the row labelled 'Mean square E[(theta-tilde minus theta_Fn) squared]', an open blue diamond at 6.2 times 10 to the -7 is joined to a filled orange circle at 1.6 times 10 to the -3. The legend says that the diamonds are the values as printed, 2 Phi(-c root n / sigma) and the tail integral, and that the circles are the actual values in the example. The title reads 'In the example at n = 25, the printed probability and the printed bound are far below the actual values'.", "caption": "The counterexample at $n=25$, with $\\mathrm{E}_G[\\tau]=0$, $c=\\sigma=1$ and $\\hat\\theta=\\tilde\\theta+1/n$, on a logarithmic scale. The printed probability $2\\Phi(-5)\\approx5.7\\times10^{-7}$ stands against the actual value $1$, and the printed tail integral $\\approx6.2\\times10^{-7}$ against the actual mean square $\\approx1.6\\times10^{-3}$. The mean square values are from numerical integration. They are consistent with the exact bounds: the tail integral is at most $15/15625$, and the mean square is at least $21/15625$.", "url": "https://pmaronow.github.io/llm-audit/works/P017/exhibits/P017-E002-printed-vs-actual.png" } ], "formal_release": null, "formal_statements": [ { "id": "P017-T5", "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T5" }, { "id": "P017-T6", "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T6" }, { "id": "P017-T7", "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T7" }, { "id": "P017-T8", "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T8" }, { "id": "P017-T11", "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T11" }, { "id": "P017-T14", "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T14" }, { "id": "P017-T15", "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T15" } ], "headline": "The proofs of Propositions 3 and 4 clip the normal part $\\tilde\\theta$ where Definition 2 clips $\\hat\\theta$; the probability equality, the rate claimed for $\\tilde\\theta-\\theta_{F_n}$ and the displayed bound on the mean square are false in general, and both propositions hold as stated by a direct clipping argument.", "id": "P017-E002", "location": "Proof of Proposition 3, PDF p. 2, the claim that $(\\tilde\\theta-\\theta_{F_n})$ is $o_p(a_n)$ and the equality that ends in $2\\Phi(-c\\sqrt n/\\sigma)$; proof of Proposition 4, PDF p. 3, the sentence “Within $c$ of $\\mathrm{E}_G[\\tau]$, the squared deviation is zero” and the first line of the display", "pdf_page": 2, "printed_page": null, "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P017-T11", "P017-T14", "P017-T15", "P017-T5", "P017-T6", "P017-T7", "P017-T8" ], "text": { "as_printed": "\\emph{Proof of Proposition 3, PDF p.~2.} Proof: Decompose $\\hat\\theta$ into $\\tilde\\theta=N(\\mathrm{E}_G[\\tau],\\sigma^2/n)$ and $u=o_p(n^{-1/2})$, so that $\\hat\\theta=\\tilde\\theta+u$. Since \\hl{$(\\tilde\\theta-\\theta_{F_n})$ is $o_p(a_n)$ for any positive sequence $(a_n)$}, the rate of convergence of $\\hat\\theta$ is at worst governed by the bound ensured by $u$'s $o_p(n^{-1/2})$ convergence. To prove the claim, note that for any positive $\\varepsilon$, $\\Pr\\left(\\frac{|\\tilde\\theta-\\theta_{F_n}|}{a_n}\\ge\\varepsilon\\right)\\le\\Pr\\left(\\tilde\\theta-\\theta_{F_n}\\ne0\\right)\\hl{=2\\Phi(-c\\sqrt n/\\sigma)}$, where $\\Phi(.)$ denotes the standard Normal CDF. Since $\\lim_{n\\to\\infty}2\\Phi(-c\\sqrt n/\\sigma)=0$, \\hl{$(\\tilde\\theta-\\theta_{F_n})$ is $o_p(a_n)$}. Thus $\\hat\\theta-\\theta_{F_n}=\\hl{o_p(a_n)}+o_p(n^{-1/2})=o_p(n^{-1/2})$, yielding the result.\n\n\\emph{Proof of Proposition 4, PDF p.~3, from its second sentence.} To obtain the rate of convergence of the mean square error of $\\tilde\\theta$, we integrate over its squared deviation from the target parameter. \\hl{Within $c$ of $\\mathrm{E}_G[\\tau]$, the squared deviation is zero}, thus we need only integrate over the squared deviation over the tails of the normal distribution. To ease calculations, we obtain an upper bound by integrating over the squared deviation from $\\mathrm{E}_G[\\tau]$, rather than from $\\theta_{F_n}$:\n\\[\n\\begin{aligned}\n\\mathrm{E}_G[(\\tilde\\theta-\\theta_{F_n})^2]&\\hl{\\le}2\\int_c^\\infty x^2\\frac{\\sqrt n}{\\sigma\\sqrt{2\\pi}}e^{-\\frac{x^2n}{2\\sigma^2}}\\\\\n&=c\\sigma\\sqrt{\\frac{2}{\\pi}}\\frac{e^{-\\frac{c^2n}{2\\sigma^2}}}{\\sqrt n}+2\\sigma^2\\frac{\\Phi\\left(\\frac{-c\\sqrt n}{\\sigma}\\right)}{n}\\\\\n&=o(n^{-1}).\n\\end{aligned}\n\\]", "corrected": "Let $\\mu=\\mathrm{E}_G[\\tau]$. For finite $c$, let $C(x)=\\min(\\mu+c,\\max(\\mu-c,x))$ and $r(x)=x-C(x)$, so that $\\theta_{F_n}=C(\\hat\\theta)$ and $\\hat\\theta-\\theta_{F_n}=r(\\hat\\theta)$. If $c=+\\infty$, then $\\theta_{F_n}=\\hat\\theta$ and both conclusions are immediate.\n\n\\emph{Proof of Proposition 3.} Since $\\hat\\theta$ is root-$n$ consistent, $\\Pr(|\\hat\\theta-\\mu|>c)\\to0$. The residual $r(\\hat\\theta)$ is zero whenever $|\\hat\\theta-\\mu|\\le c$. Hence for any positive sequence $(a_n)$ and any positive $\\varepsilon$,\n\\[\n\\Pr\\left(\\frac{|\\hl{\\hat\\theta}-\\theta_{F_n}|}{a_n}>\\varepsilon\\right)\\le\\hl{\\Pr\\left(|\\hat\\theta-\\mu|>c\\right)}\\longrightarrow0.\n\\]\nThus $\\hat\\theta-\\theta_{F_n}$ is $o_p(a_n)$ for any positive sequence $(a_n)$, and the choice $a_n=n^{-1/2}$ yields the result.\n\n\\emph{Proof of Proposition 4.} Let $\\hat\\theta=\\tilde\\theta+\\varepsilon/\\sqrt n$ with $\\tilde\\theta=N(\\mu,\\sigma^2/n)$. Since $r$ is 1-Lipschitz, $|r(\\hat\\theta)|\\le|r(\\tilde\\theta)|+|\\varepsilon|/\\sqrt n$, and the inequality $(A+B)^2\\le2A^2+2B^2$ gives\n\\[\n\\mathrm{E}[(\\hat\\theta-\\theta_{F_n})^2]\\le2\\,\\mathrm{E}[\\hl{r(\\tilde\\theta)^2}]+\\hl{\\frac{2\\,\\mathrm{E}[\\varepsilon^2]}{n}}.\n\\]\nWithin $c$ of $\\mu$, the residual $r(\\tilde\\theta)$ is zero, and elsewhere $r(\\tilde\\theta)^2\\le(\\tilde\\theta-\\mu)^2$. Hence only the tails of the normal distribution contribute:\n\\[\n\\begin{aligned}\n\\mathrm{E}[\\hl{r(\\tilde\\theta)^2}]&\\le2\\int_c^\\infty x^2\\frac{\\sqrt n}{\\sigma\\sqrt{2\\pi}}e^{-\\frac{x^2n}{2\\sigma^2}}\\,dx\\\\\n&=c\\sigma\\sqrt{\\frac{2}{\\pi}}\\frac{e^{-\\frac{c^2n}{2\\sigma^2}}}{\\sqrt n}+2\\sigma^2\\frac{\\Phi\\left(\\frac{-c\\sqrt n}{\\sigma}\\right)}{n}\\\\\n&=o(n^{-1}).\n\\end{aligned}\n\\]\nThe same expectation is at most $\\mathrm{E}[(\\tilde\\theta-\\mu)^4]/c^2=3\\sigma^4/(c^2n^2)$. Since $\\mathrm{E}[\\varepsilon^2]=o(n^{-1/2})$, it follows that\n\\[\n\\mathrm{E}[(\\hat\\theta-\\theta_{F_n})^2]\\le\\frac{6\\sigma^4}{c^2n^2}+\\frac{2\\,\\mathrm{E}[\\varepsilon^2]}{n}=o(n^{-1}).\n\\]", "evidence": "I give an exact counterexample to the three steps and derive both propositions in the corrected proofs below.\n\nLet the effects equal $-1$ or $1$ with probability $1/2$ each, so that $\\mathrm{E}_G[\\tau]=0$ and $c=1$, and let $X_1,\\ldots,X_n$ be independent $N(0,1)$ observations. Set $\\tilde\\theta=\\bar X_n$ and $\\hat\\theta=\\tilde\\theta+1/n$. Then $\\sigma=1$ and $u=1/n=o_p(n^{-1/2})$. Further, $\\sqrt n\\,\\hat\\theta=N(0,1)+\\varepsilon$ with $\\varepsilon=n^{-1/2}$ and $\\mathrm{E}[\\varepsilon^2]=1/n=o(n^{-1/2})$, and the assumptions of both propositions hold. The difference $\\tilde\\theta-\\theta_{F_n}$ equals $-1/n$ when $|\\hat\\theta|\\le1$, and outside this event it vanishes only if $\\tilde\\theta=1$, which has probability zero. I therefore have $\\Pr(\\tilde\\theta-\\theta_{F_n}\\ne0)=1$ for all $n$, while the printed value is $2\\Phi(-\\sqrt n)=\\Pr(|\\tilde\\theta|>1)\\le1/n$. For $n\\ge2$, the event $|\\tilde\\theta|\\le1/2$ has probability at least $1-4/n$, and on it $\\tilde\\theta-\\theta_{F_n}=-1/n$. It follows that $n(\\tilde\\theta-\\theta_{F_n})\\to-1$ in probability, and $\\tilde\\theta-\\theta_{F_n}$ is not $o_p(a_n)$ for $a_n=1/n$. At $n=25$, using the same event, I obtain\n\\[\n\\mathrm{E}[(\\tilde\\theta-\\theta_{F_n})^2]\\ge\\frac{1}{625}\\cdot\\frac{21}{25}=\\frac{21}{15625},\n\\]\nwhile the printed tail integral equals $\\mathrm{E}[\\tilde\\theta^2\\mathbf{1}\\{|\\tilde\\theta|>1\\}]$, which is at most $\\mathrm{E}[\\tilde\\theta^6]=15/15625$. The printed inequality fails by at least $6/15625$.\n\\begin{audittable}{lll}{The printed calculations in the counterexample, at $n=25$ in the first three rows. The mean square values are from numerical integration; decimal values are rounded.}\nQuantity & As printed & In the example \\\\\n$\\Pr(\\tilde\\theta-\\theta_{F_n}\\ne0)$ & $2\\Phi(-5)\\approx5.7\\times10^{-7}$ & $1$ \\\\\n$\\mathrm{E}[(\\tilde\\theta-\\theta_{F_n})^2]$, decimal value & at most $6.2\\times10^{-7}$ & $1.6\\times10^{-3}$ \\\\\n$\\mathrm{E}[(\\tilde\\theta-\\theta_{F_n})^2]$, exact bound & at most $15/15625$ & at least $21/15625$ \\\\\nLimit in probability of $n(\\tilde\\theta-\\theta_{F_n})$ & $0$ & $-1$ \\\\\n\\end{audittable}\nThe corrected proofs use three facts. First, Definition 1 implies that $\\sqrt n(\\hat\\theta-\\mathrm{E}_G[\\tau])=O_p(1)$, and hence that $\\hat\\theta$ is consistent. Second, the function $r(x)=x-C(x)$, where $C$ is the clip to the interval, is continuous and piecewise linear with slopes $1$, $0$ and $1$, so that $|r(y)-r(x)|\\le|y-x|$ for all $x$ and $y$. Third, a variable $W\\sim N(0,\\sigma^2/n)$ has $\\mathrm{E}[W^4]=3\\sigma^4/n^2$. Note that the argument for Proposition 4 does not use independence of $\\tilde\\theta$ and $\\varepsilon$.", "problem": "Definition 2 defines $\\theta_{F_n}$ as the clip of the estimator $\\hat\\theta=\\tilde\\theta+u$ to the interval $[\\mathrm{E}_G[\\tau]-c,\\ \\mathrm{E}_G[\\tau]+c]$. Both proofs compute with the normal part $\\tilde\\theta$ as though $\\theta_{F_n}$ were the clip of $\\tilde\\theta$. On the event $|\\hat\\theta-\\mathrm{E}_G[\\tau]|\\le c$ the target equals $\\hat\\theta$, which implies that $\\tilde\\theta-\\theta_{F_n}=-u$, and $u$ is not zero in general. In the proof of Proposition 3, the probability $\\Pr(\\tilde\\theta-\\theta_{F_n}\\ne0)$ need not equal $2\\Phi(-c\\sqrt n/\\sigma)$, and $\\tilde\\theta-\\theta_{F_n}$ need not be $o_p(a_n)$ for every positive sequence $(a_n)$. In the proof of Proposition 4, the squared deviation need not vanish when $|\\tilde\\theta-\\mathrm{E}_G[\\tau]|\\le c$; when $|\\hat\\theta-\\mathrm{E}_G[\\tau]|\\le c$, it equals $u^2$. The Gaussian tail integral does not bound $\\mathrm{E}[(\\tilde\\theta-\\theta_{F_n})^2]$ in general. The three steps do hold for some remainders, for instance a remainder that is zero when $\\tilde\\theta$ is in the interval and moves $\\tilde\\theta$ away from the interval otherwise. They do not follow from the assumptions of the two propositions. Both propositions are true as stated, but the printed proofs do not establish them.", "what_survives": "Propositions 3 and 4 hold as stated under their printed assumptions, with no further assumption, and the superefficiency and mean square rates stated in the abstract, the introduction and the discussion hold for the Definition 2 target. Proposition 3 holds for any consistent estimator of $\\mathrm{E}_G[\\tau]$, and then $\\hat\\theta-\\theta_{F_n}=o_p(a_n)$ for any positive sequence $(a_n)$. The closed-form evaluation of the Gaussian tail integral is correct, and the intermediate conclusion $\\mathrm{E}[(\\tilde\\theta-\\theta_{F_n})^2]=o(n^{-1})$ is true, although the displayed bound does not establish it. Definition 2, Proposition 2 and Corollary 1 do not use these steps. The corrected proofs also apply to the target of \\fid{P017-E001}, with a fixed finite radius $00$, the probability of this event tends to one. No result depends on the sentence." }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-E003", "work": { "arxiv_id": "1601.01413", "doi": "10.1515/jci-2016-0007", "title": "Data-Adaptive Causal Effects and Superefficiency", "url": "https://pmaronow.github.io/llm-audit/works/P017/" }, "work_id": "P017" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Corollary 1, which ends 'Then Pr(theta-hat = theta_Fn) = 1', followed by three lines that begin 'A proof of Corollary 1 follows by noting that Pr(|theta-hat minus E_G[tau]| is at most c) = 0, and applying Definition 2'. A red box marks '= 0'.", "caption": "Corollary 1 and the first lines of the paragraph that follows it, PDF p. 2. The mark shows the value $0$ printed for $\\Pr(|\\hat\\theta-\\mathrm{E}_G[\\tau]|\\le c)$.", "url": "https://pmaronow.github.io/llm-audit/works/P017/exhibits/P017-E004.png" }, { "alt": "Four lines of the proof of Proposition 4, followed by a display of three lines. A red box marks the integral on the right of the first line of the display, '2 times the integral from c to infinity of x squared times root n over sigma root 2 pi times e to the power minus x squared n over 2 sigma squared'. No dx follows the integrand.", "caption": "Proof of Proposition 4, PDF p. 3: the four lines before the display, and the display. The marked integral is printed without a differential.", "url": "https://pmaronow.github.io/llm-audit/works/P017/exhibits/P017-E004-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P017-T10", "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T10" }, { "id": "P017-T11", "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T11" } ], "headline": "In the proof of Corollary 1, the probability $\\Pr(|\\hat\\theta-\\mathrm{E}_G[\\tau]|\\le c)$ is $1$ and not $0$; in the proof of Proposition 4, the Gaussian integral lacks its differential $dx$.", "id": "P017-E004", "location": "Sentence after Corollary 1, PDF p. 2 (“A proof of Corollary 1 follows by noting that …”); first line of the display in the proof of Proposition 4, PDF p. 3", "pdf_page": 2, "printed_page": null, "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P017-T10", "P017-T11" ], "text": { "as_printed": "\\emph{After Corollary 1, PDF p.~2.} A proof of Corollary 1 follows by noting that $\\Pr(|\\hat\\theta-\\mathrm{E}_G[\\tau]|\\le c)=\\hl{0}$, and applying Definition 2.\n\n\\emph{Proof of Proposition 4, PDF p.~3, first line of the display. No differential is printed.}\n\\[\n\\mathrm{E}_G[(\\tilde\\theta-\\theta_{F_n})^2]\\le2\\int_c^\\infty x^2\\frac{\\sqrt n}{\\sigma\\sqrt{2\\pi}}e^{-\\frac{x^2n}{2\\sigma^2}}\n\\]", "corrected": "A proof of Corollary 1 follows by noting that $\\Pr(|\\hat\\theta-\\mathrm{E}_G[\\tau]|\\le c)=\\hl{1}$, and applying Definition 2.\n\n\\emph{Proof of Proposition 4, the integral in the first line of the display.}\n\\[\n2\\int_c^\\infty x^2\\frac{\\sqrt n}{\\sigma\\sqrt{2\\pi}}e^{-\\frac{x^2n}{2\\sigma^2}}\\hl{\\,dx}\n\\]", "evidence": "The evidence is a check of the page and a derivation. Let $\\mu=\\mathrm{E}_G[\\tau]$. The hypothesis $c\\ge\\max(\\sup(\\operatorname{Supp}[\\hat\\theta])-\\mu,\\ \\mu-\\inf(\\operatorname{Supp}[\\hat\\theta]))$ implies that $\\inf(\\operatorname{Supp}[\\hat\\theta])\\ge\\mu-c$ and $\\sup(\\operatorname{Supp}[\\hat\\theta])\\le\\mu+c$. Then $|\\hat\\theta-\\mu|\\le c$ almost surely, and by Definition 2, I have $\\Pr(\\hat\\theta=\\theta_{F_n})=1$. For the integral, set $s=\\sigma/\\sqrt n$ and $a=c/s$, and let $\\phi$ be the standard normal density. Substituting $x=sz$ and integrating by parts, I obtain\n\\[\n2\\int_c^\\infty x^2\\frac{\\sqrt n}{\\sigma\\sqrt{2\\pi}}e^{-\\frac{x^2n}{2\\sigma^2}}\\,dx=2s^2\\int_a^\\infty z^2\\phi(z)\\,dz=2s^2\\bigl(a\\phi(a)+\\Phi(-a)\\bigr),\n\\]\nwhich is the second line of the printed display. Numerical integration agrees with the printed closed form at three choices of $c$, $\\sigma$ and $n$.", "problem": "The hypothesis of Corollary 1 places the support of $\\hat\\theta$ inside the interval $[\\mathrm{E}_G[\\tau]-c,\\ \\mathrm{E}_G[\\tau]+c]$, so that the event $|\\hat\\theta-\\mathrm{E}_G[\\tau]|\\le c$ has probability one. The proof sentence assigns it probability zero, which contradicts the corollary and the sentence that follows the proof. In the proof of Proposition 4, the integral is printed without $dx$. Both are misprints. In the same display the expectation carries a subscript $G$ that the statement of Proposition 4 and the sentence after the display omit. This is a difference of notation and not an error, since the sampling expectation is the one induced by $G$.", "what_survives": "Corollary 1 and the closed-form evaluation of the integral are correct as printed, and no result changes. The inequality in which the integral appears is a separate matter, treated in \\fid{P017-E002}." }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-E004", "work": { "arxiv_id": "1601.01413", "doi": "10.1515/jci-2016-0007", "title": "Data-Adaptive Causal Effects and Superefficiency", "url": "https://pmaronow.github.io/llm-audit/works/P017/" }, "work_id": "P017" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Three lines of text that begin 'In short, Proposition 3 demonstrates that the probability that theta-hat falls inside the support of the effect distribution converges to one quickly; conditional on this event, then estimation error is zero'. A red box marks the word 'quickly'.", "caption": "Paragraph after the proof of Proposition 3, PDF p. 2, first three lines. The marked word attributes a fast rate to the probability that the estimation error is zero.", "url": "https://pmaronow.github.io/llm-audit/works/P017/exhibits/P017-N001.png" } ], "formal_release": null, "formal_statements": [ { "id": "P017-T12", "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T12" }, { "id": "P017-T13", "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T13" } ], "headline": "Under the assumptions of Proposition 3, the probability that $\\hat\\theta$ equals its target tends to one, but at no particular rate; the word “quickly” should be deleted.", "id": "P017-N001", "location": "Paragraph after the proof of Proposition 3, PDF p. 2, first sentence, the word “quickly”", "pdf_page": 2, "printed_page": null, "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P017-T12", "P017-T13" ], "text": { "as_printed": "In short, Proposition 3 demonstrates that the probability that $\\hat\\theta$ falls inside the support of the effect distribution converges to one \\hl{quickly}; conditional on this event, then estimation error is zero (as the target parameter takes on the value as the estimator with probability one).", "corrected": "In short, Proposition 3 demonstrates that the probability that $\\hat\\theta$ falls \\hl{inside the interval $[\\mathrm{E}_G[\\tau]-c,\\ \\mathrm{E}_G[\\tau]+c]$} converges to one \\del{quickly}; on this event, estimation error is zero (as the target parameter takes on the same value as the estimator with probability one).\n\nIn precise terms: if Assumption 1 holds and $\\hat\\theta$ is a root-$n$ consistent and asymptotically normal estimator of $\\mathrm{E}_G[\\tau]$, then $\\Pr(|\\hat\\theta-\\mathrm{E}_G[\\tau]|\\le c)\\to1$, and on this event $\\hat\\theta=\\theta_{F_n}$. No rate of convergence of this probability follows from these assumptions. The interval wording is explained in \\fid{P017-E003}.", "evidence": "The evidence is an exact counterexample. Let $(p_n)$ be any sequence in $(0,1]$ that tends to zero. Let the effect $\\tau$ equal $-1$ or $1$ with probability $1/4$ each and be uniform on $[-1,1]$ otherwise, so that $\\mathrm{E}_G[\\tau]=0$, the support of $\\tau$ is $[-1,1]$ and $c=1$. Let the data contain independent observations $X_i\\sim N(0,1)$ and $U_i$ uniform on $[0,1)$. Let $B_n$ be the indicator of the event that the fractional part of $\\sum_{i=1}^nU_i$ is less than $p_n$. I have $\\Pr(B_n=1)=p_n$, and $B_n$ is independent of $\\bar X_n$. Let\n\\[\nA_n=\\max\\{-1/2,\\min(\\bar X_n,1/2)\\},\\qquad\\hat\\theta=A_n+10B_n.\n\\]\nThe estimator $\\hat\\theta$ is a symmetric function of the sample. I have $\\sqrt n\\,\\hat\\theta=\\sqrt n\\,\\bar X_n+\\varepsilon_n$, where $\\sqrt n\\,\\bar X_n\\sim N(0,1)$ and $\\Pr(\\varepsilon_n\\ne0)\\le4/n+p_n\\to0$, and Definition 1 holds with $\\sigma^2=1$. If $B_n=0$, then $\\hat\\theta=A_n\\in[-1/2,1/2]$ and $\\theta_{F_n}=\\hat\\theta$. If $B_n=1$, then $\\hat\\theta\\ge19/2$, $\\theta_{F_n}=1$ and the error is at least $17/2$. I conclude that\n\\[\n\\Pr(\\hat\\theta\\ne\\theta_{F_n})=\\Pr(|\\hat\\theta|>1)=p_n.\n\\]\nFor $p_n=1/(1+\\lfloor\\log_2n\\rfloor)$, this probability is $1/10$ at $n=1000$ and $1/40$ at $n=10^{12}$, and it is not $O(n^{-\\alpha})$ for any $\\alpha>0$. The conclusion of Proposition 3 holds in the example, since $\\Pr(\\sqrt n|\\hat\\theta-\\theta_{F_n}|>\\delta)\\le p_n$ for all $\\delta>0$.", "problem": "The sentence follows a proof in which the expression $2\\Phi(-c\\sqrt n/\\sigma)$ appears, and in that context ``quickly'' says that the probability of zero estimation error tends to one at a fast rate. No rate follows from the assumptions of Proposition 3. Definition 1 requires only that $\\sqrt n(\\hat\\theta-\\mathrm{E}_G[\\tau])=N(0,\\sigma^2)+o_p(1)$, and the $o_p(1)$ term may be large on an event whose probability tends to zero arbitrarily slowly. For a decomposition $\\hat\\theta=\\tilde\\theta+u$ with $\\tilde\\theta=N(\\mathrm{E}_G[\\tau],\\sigma^2/n)$, by the union bound, I have\n\\[\n\\Pr\\left(|\\hat\\theta-\\mathrm{E}_G[\\tau]|>c\\right)\\le2\\Phi\\left(-\\frac{c\\sqrt n}{2\\sigma}\\right)+\\Pr\\left(|u|>c/2\\right),\n\\]\nand the condition $u=o_p(n^{-1/2})$ does not control the rate of the second term. The sentence prints no specific rate. If ``quickly'' is read as meaning only that the probability tends to one, then the sentence asserts no rate and needs no correction.", "what_survives": "Proposition 3 is unaffected: its conclusion uses only that the probability of the event $|\\hat\\theta-\\mathrm{E}_G[\\tau]|\\le c$ tends to one. No later result uses a rate for this probability. Proposition 4 makes its own stronger assumption on the remainder, and its conclusion holds under that assumption; the example above does not satisfy it." }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-N001", "work": { "arxiv_id": "1601.01413", "doi": "10.1515/jci-2016-0007", "title": "Data-Adaptive Causal Effects and Superefficiency", "url": "https://pmaronow.github.io/llm-audit/works/P017/" }, "work_id": "P017" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Two entries of the reference list. Entry 9 reads 'Balzer LB, Petersen ML, van der Laan MJ. Targeted estimation and inference for the sample average treatment effect Bepress, 2015 Berkeley, CA.' Entry 10 reads 'van der Laan MJ, Hubbard AE, Pajouh SK. Statistical inference for data adaptive target parameters. Princeton, NJ: Bepress, 2013.' A red box marks 'Princeton, NJ'.", "caption": "References 9 and 10, PDF p. 4. The mark shows the place of publication printed for reference 10. Reference 9, also from Bepress, has Berkeley, CA.", "url": "https://pmaronow.github.io/llm-audit/works/P017/exhibits/P017-N003.png" }, { "alt": "The first entry of the reference list, 'Crump RK, Hotz VJ, Imbens GW, Mitnik OA. Dealing with limited overlap in estimation of average treatment effects. Biometrika 2009.', and the first line of the second entry. A red box marks 'Biometrika 2009.'", "caption": "Reference 1, PDF p. 3. The marked words are all that the entry prints after the title: the journal and the year, with no volume, issue or pages.", "url": "https://pmaronow.github.io/llm-audit/works/P017/exhibits/P017-N003-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "Reference 10 is Working Paper 314 of the U.C. Berkeley Division of Biostatistics Working Paper Series, published in Berkeley, CA, and not in Princeton, NJ; reference 1 lacks its volume, issue and pages, which are 96(1):187–199.", "id": "P017-N003", "location": "References 1 (PDF p. 3) and 10 (PDF p. 4)", "pdf_page": 3, "printed_page": null, "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "1. Crump RK, Hotz VJ, Imbens GW, Mitnik OA. Dealing with limited overlap in estimation of average treatment effects. \\hl{Biometrika 2009.}\n\n10. van der Laan MJ, Hubbard AE, Pajouh SK. Statistical inference for data adaptive target parameters. \\hl{Princeton, NJ}: Bepress, 2013.", "corrected": "1. Crump RK, Hotz VJ, Imbens GW, Mitnik OA. Dealing with limited overlap in estimation of average treatment effects. Biometrika 2009\\hl{;96(1):187--199}.\n\n10. van der Laan MJ, Hubbard AE, Pajouh SK. Statistical inference for data adaptive target parameters. \\hl{U.C. Berkeley Division of Biostatistics Working Paper Series, Working Paper 314. Berkeley, CA}: Bepress, 2013.", "evidence": "The evidence is a check of the page against public bibliographic sources. The page of the series for the working paper (\\url{https://biostats.bepress.com/ucbbiostat/paper314/}) reads ``U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 314'', with the date June 2013. A later article by Balzer, Petersen and van der Laan cites the same report as ``Berkeley: Division of Biostatistics, University of California; 2013''. The publisher's page for the article of Crump, Hotz, Imbens and Mitnik has Biometrika 96(1):187--199, 2009. The words quoted in Section 3 appear on p.~188 in the indexed text of a copy in the journal's format that one of the authors has posted. I did not obtain the working paper itself or the publisher's full text of the article. The sources do not show at what stage the place of publication was misprinted.", "problem": "Reference 10 names Princeton, NJ as the place of publication. The item is Working Paper 314 of the U.C. Berkeley Division of Biostatistics Working Paper Series, dated June 2013, and its place of publication is Berkeley, CA. Reference 1 has the journal and the year only, although Section 3 cites its p.~188. Both entries still identify the works, and no result depends on them.", "what_survives": "The discussion and the results are unaffected. The year 2009, the citation of p.~188 and the quotation in Section 3 are correct as printed." }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-N003", "work": { "arxiv_id": "1601.01413", "doi": "10.1515/jci-2016-0007", "title": "Data-Adaptive Causal Effects and Superefficiency", "url": "https://pmaronow.github.io/llm-audit/works/P017/" }, "work_id": "P017" } ], "lean": { "applicable": true, "guide": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/docs/works/P017.md", "repository": "https://github.com/pmaronow/llm-audit-lean", "targets": [ { "declarations": [ { "claim_revision_id": "catalog:95167029a56fe4108e934b2259ac20441ea41531415a8a692c02357d355fd05b", "conditional_on": [], "credit_reason": "Witness for the printed Proposition 1 and its cited support-interval fact; all T1(a)–(f) components are exported. 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Proposition 1 and its proof fact, PDF p. 2; Assumption 1, PDF p. 1. The uniform-effect IID inverse-probability estimator is asymptotically normal and has positive-probability realizations clipping to either unattainable endpoint.

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The uniform-effect IID inverse-probability estimator is asymptotically normal and has positive-probability realizations clipping to either unattainable endpoint.", "headline": "Proposition 1 and the fact cited as its proof, as printed (the witness is the validator’s construction and is not in the paper)", "location": "PDF p. 2 (no page number is printed); Assumption 1 on PDF p. 1" }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T1" }, { "declarations": [ { "claim_revision_id": "catalog:db1ed030273efdbfba35c1c1b4fff37711dca541bb19686630713c87683ab1cd", "conditional_on": [], "credit_reason": "Judged against the characterization component identified by row_says. 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the fact cited after Proposition 1, PDF p. 2, corrected. An admissible weighted mean is attainable exactly at an atom or between positive masses on both sides, with all interior, endpoint and exterior support consequences. The prescribed normalized atom indicator and two-region mixture are bounded, nonnegative, admissible, and have weight mean one and weighted-effect mean equal to the target; the conditional means straddle the target and the specified mixing coefficient is strictly between zero and one.

", "headline": "The fact cited as the proof of Proposition 1, as corrected (which values a nonnegatively weighted mean attains)", "location": "PDF p. 2 (no page number is printed)" }, "id": "P017-T2", "text": { "body": "the fact cited after Proposition 1, PDF p. 2, corrected. An admissible weighted mean is attainable exactly at an atom or between positive masses on both sides, with all interior, endpoint and exterior support consequences. The prescribed normalized atom indicator and two-region mixture are bounded, nonnegative, admissible, and have weight mean one and weighted-effect mean equal to the target; the conditional means straddle the target and the specified mixing coefficient is strictly between zero and one.", "headline": "The fact cited as the proof of Proposition 1, as corrected (which values a nonnegatively weighted mean attains)", "location": "PDF p. 2 (no page number is printed)" }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T2" }, { "declarations": [ { "claim_revision_id": "catalog:186874f6ad96643824a7de2227d9ff3c933fe8830333f134b916b529322aa267", "conditional_on": [], "credit_reason": "Credit is for the corrected T3, not a proof of the printed all-realizations T9. Measurability is the ordinary random-variable requirement, not an additional moment or support restriction.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.Wave1.corrected_proposition1_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/Wave1/Gap/GP017B2.lean#L440", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:186874f6ad96643824a7de2227d9ff3c933fe8830333f134b916b529322aa267", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P017-T3", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P017-T3" }, "html": { "body": "

Proposition 1 and its gloss, PDF p. 2, corrected. The original clipped target is representable exactly when strictly between the effect-support bounds or at a support endpoint with positive mass. A strictly inside estimate equals its target and satisfies the full chain of clipping and support bounds. Infinite radius, atoms at binding clipping endpoints, and almost surely finite-valued effects give representation on every realization. Under root-n asymptotic normality, both the strictly inside and representation probabilities tend to one.

", "headline": "Proposition 1, as corrected with Definition 2 unchanged (the representation holds except at a support endpoint without mass, hence with probability tending to one)", "location": "PDF p. 2 (no page number is printed)" }, "id": "P017-T3", "text": { "body": "Proposition 1 and its gloss, PDF p. 2, corrected. The original clipped target is representable exactly when strictly between the effect-support bounds or at a support endpoint with positive mass. A strictly inside estimate equals its target and satisfies the full chain of clipping and support bounds. Infinite radius, atoms at binding clipping endpoints, and almost surely finite-valued effects give representation on every realization. Under root-n asymptotic normality, both the strictly inside and representation probabilities tend to one.", "headline": "Proposition 1, as corrected with Definition 2 unchanged (the representation holds except at a support endpoint without mass, hence with probability tending to one)", "location": "PDF p. 2 (no page number is printed)" }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T3" }, { "declarations": [ { "claim_revision_id": "catalog:d28464e9960ff489316679e4eaf15326a8aaefb7a075e1d3f789d00fcb76a0fe", "conditional_on": [], "credit_reason": "The fixed smaller radius precedes all real estimate values. Both finite and infinite original radius are allowed, and no estimator consistency is assumed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.interior_radius_representation", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/P017_Audit.lean#L708", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:d28464e9960ff489316679e4eaf15326a8aaefb7a075e1d3f789d00fcb76a0fe", "conditional_on": [], "credit_reason": "Its component is supplied by the selected composite declaration; no separate application is needed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.interior_radius_support", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/P017_Audit.lean#L672", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:d28464e9960ff489316679e4eaf15326a8aaefb7a075e1d3f789d00fcb76a0fe", "conditional_on": [], "credit_reason": "General finite companions, without any asymptotic claim. The outer-probability extension agrees with ordinary sampling probabilities for all intended measurable estimators.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.interior_radius_finite_companions", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/P017_Audit.lean#L727", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:d28464e9960ff489316679e4eaf15326a8aaefb7a075e1d3f789d00fcb76a0fe", "conditional_on": [], "credit_reason": "Only a pointwise witness is credited here; a constant estimator at the exported x gives failure of radius-d identity under the old radius-c support bound. No root-n-normal estimator counterexample is claimed by this row.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.smaller_radius_changes_target", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/P017_Audit.lean#L741", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:d28464e9960ff489316679e4eaf15326a8aaefb7a075e1d3f789d00fcb76a0fe", "conditional_on": [], "credit_reason": "General finite support corollary; specializing μ to the effect mean and 0Proposition 1 and Definition 2, PDF p. 2, corrected. A positive finite radius smaller than the population radius always exists and makes every clipped estimate representable, with the repaired finite companions and support corollary and the obstruction to substituting a larger radius.

", "headline": "Definition 2 and Proposition 1, as corrected to keep the representation in every realization (a fixed clipping radius 0<d<c0 < d < c0<d<c)", "location": "PDF p. 2 (no page number is printed)" }, "id": "P017-T4", "text": { "body": "Proposition 1 and Definition 2, PDF p. 2, corrected. A positive finite radius smaller than the population radius always exists and makes every clipped estimate representable, with the repaired finite companions and support corollary and the obstruction to substituting a larger radius.", "headline": "Definition 2 and Proposition 1, as corrected to keep the representation in every realization (a fixed clipping radius $0 < d < c$)", "location": "PDF p. 2 (no page number is printed)" }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T4" }, { "declarations": [ { "claim_revision_id": "catalog:6face510ceb5aa1c090b1c51e63147d7d5db750caa1e8a6761cbae12b87c35f7", "conditional_on": [], "credit_reason": "The witness refutes the two identified steps concerning tilde minus the actual target, rather than Proposition 3 or the all-rate behavior of the actual clipping error; its sample mean is a statistic of the observed empirical X distribution.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.Wave1.normal_shift_probability_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/Wave1/Gap/GLP017B4.lean#L351", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:6face510ceb5aa1c090b1c51e63147d7d5db750caa1e8a6761cbae12b87c35f7", "conditional_on": [], "credit_reason": "Its component is supplied by the selected composite declaration; no separate application is needed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.shifted_clip_zero_iff", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/P017_Audit.lean#L1300", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:6face510ceb5aa1c090b1c51e63147d7d5db750caa1e8a6761cbae12b87c35f7", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P017-T5", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P017-T5" }, "html": { "body": "

proof of Proposition 3, PDF p. 2. The shifted normal sample mean has nonzero intermediate error with probability one and violates the printed every-positive-sequence assertion, while its actual clipping error vanishes on the stated central event.

", "headline": "Proof of Proposition 3, as printed: the equality Pr⁡(θ~−θFn≠0)=2Φ(−cn/σ)\\Pr(\\tilde\\theta - \\theta_{F_n} \\ne 0) = 2\\Phi(-c\\sqrt{n}/\\sigma)Pr(θ~−θFn​​=0)=2Φ(−cn​/σ) and the claim that (θ~−θFn)(\\tilde\\theta - \\theta_{F_n})(θ~−θFn​​) is op(an)o_p(a_n)op​(an​) for any positive sequence (the witness is the validator’s construction and is not in the paper)", "location": "PDF p. 2 (no page number is printed)" }, "id": "P017-T5", "text": { "body": "proof of Proposition 3, PDF p. 2. The shifted normal sample mean has nonzero intermediate error with probability one and violates the printed every-positive-sequence assertion, while its actual clipping error vanishes on the stated central event.", "headline": "Proof of Proposition 3, as printed: the equality $\\Pr(\\tilde\\theta - \\theta_{F_n} \\ne 0) = 2\\Phi(-c\\sqrt{n}/\\sigma)$ and the claim that $(\\tilde\\theta - \\theta_{F_n})$ is $o_p(a_n)$ for any positive sequence (the witness is the validator’s construction and is not in the paper)", "location": "PDF p. 2 (no page number is printed)" }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T5" }, { "declarations": [ { "claim_revision_id": "catalog:17881f415804f11f8d47cb8671fa54bcd0642250bc3aa1ed87b3092b6a8a9c37", "conditional_on": [], "credit_reason": "The model satisfies the printed Proposition 4 decomposition and moment condition. The 6/15625 positive gap concerns the intermediate difference, and does not contradict the proposition's MSE limit.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.Wave1.normal_shift_mse_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/Wave1/Gap/GP017B2.lean#L887", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:17881f415804f11f8d47cb8671fa54bcd0642250bc3aa1ed87b3092b6a8a9c37", "conditional_on": [], "credit_reason": "This row exports the scalar ranges and central-event algebra only; R119-03 supplies the probability and expectation certificates.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.shifted_25_central", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/P017_Audit.lean#L1307", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:17881f415804f11f8d47cb8671fa54bcd0642250bc3aa1ed87b3092b6a8a9c37", "conditional_on": [], "credit_reason": "Its component is supplied by the selected composite declaration; no separate application is needed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.normal_shift_remainder_ms", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/P017_Audit.lean#L1279", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:17881f415804f11f8d47cb8671fa54bcd0642250bc3aa1ed87b3092b6a8a9c37", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P017-T6", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P017-T6" }, "html": { "body": "

proof of Proposition 4, PDF p. 3. The shifted normal estimator satisfies the remainder condition but at size 25 has positive central intermediate error and a mean-square gap above the printed tail bound, while both true mean-square limits still hold.

", "headline": "Proof of Proposition 4, as printed: “Within ccc of EG[τ]\\mathrm{E}_G[\\tau]EG​[τ], the squared deviation is zero” and the first inequality of the display (the witness is the validator’s construction and is not in the paper)", "location": "PDF p. 3 (no page number is printed)" }, "id": "P017-T6", "text": { "body": "proof of Proposition 4, PDF p. 3. The shifted normal estimator satisfies the remainder condition but at size 25 has positive central intermediate error and a mean-square gap above the printed tail bound, while both true mean-square limits still hold.", "headline": "Proof of Proposition 4, as printed: “Within $c$ of $\\mathrm{E}_G[\\tau]$, the squared deviation is zero” and the first inequality of the display (the witness is the validator’s construction and is not in the paper)", "location": "PDF p. 3 (no page number is printed)" }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T6" }, { "declarations": [ { "claim_revision_id": "source-claim-import-2026-11-08:P017-T7:1", "conditional_on": [], "credit_reason": "The population integrability and measurability are the finite-mean random-effect assumptions, and RootNNormal is the packet's accepted distributional reading. 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proof of Proposition 3, PDF p. 2, corrected. With the finite population mean and positive, possibly infinite effect radius of Assumption 1, the actual clipping error has its exact zero region and exceedance-event inclusion and probability bound at every positive sample size. Consistency alone gives every fixed-radius tail limit, the clipping-failure probability limit, o_p(a_n) actual error for every deterministic sequence positive at n ≥ 1, and root-n actual error o_p(1). Definition 1 additionally gives root-n boundedness in probability and consistency, hence all these limits. At infinite radius the target is the estimate and both the actual error and its root-n scaling are identically zero. The clipping-event deduction requires no normal decomposition or rate for the clipping-failure probability. The usual measure-theoretic conditions are assumed.

", "headline": "Proof of Proposition 3, as repaired (the clipping-event bound for the actual estimator; the every-positive-sequence rate for θ^−θFn\\hat\\theta - \\theta_{F_n}θ^−θFn​​)", "location": "PDF p. 2 (no page number is printed)" }, "id": "P017-T7", "text": { "body": "proof of Proposition 3, PDF p. 2, corrected. With the finite population mean and positive, possibly infinite effect radius of Assumption 1, the actual clipping error has its exact zero region and exceedance-event inclusion and probability bound at every positive sample size. Consistency alone gives every fixed-radius tail limit, the clipping-failure probability limit, o_p(a_n) actual error for every deterministic sequence positive at n ≥ 1, and root-n actual error o_p(1). Definition 1 additionally gives root-n boundedness in probability and consistency, hence all these limits. At infinite radius the target is the estimate and both the actual error and its root-n scaling are identically zero. The clipping-event deduction requires no normal decomposition or rate for the clipping-failure probability. The usual measure-theoretic conditions are assumed.", "headline": "Proof of Proposition 3, as repaired (the clipping-event bound for the actual estimator; the every-positive-sequence rate for $\\hat\\theta - \\theta_{F_n}$)", "location": "PDF p. 2 (no page number is printed)" }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T7" }, { "declarations": [ { "claim_revision_id": "catalog:458a0253bb61505280dcd900f25970abf44c31e7eb56b5ca083a9c979088f206", "conditional_on": [], "credit_reason": "Every finite certificate in the assigned component is exported, including both squared-error bounds, the fourth moment, the residual positive-part identity and the correctly attributed auxiliary tail identity. 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proof of Proposition 4, PDF p. 3, corrected. The Lipschitz clipping and residual inequalities give the finite Gaussian fourth-moment bounds, both mean-square limits, and zero actual error at infinite radius. The usual measure-theoretic conditions are assumed.

", "headline": "Proof of Proposition 4, as repaired (the clipping residual is 1-Lipschitz; a mean-square bound for the actual estimator)", "location": "PDF p. 3 (no page number is printed)" }, "id": "P017-T8", "text": { "body": "proof of Proposition 4, PDF p. 3, corrected. The Lipschitz clipping and residual inequalities give the finite Gaussian fourth-moment bounds, both mean-square limits, and zero actual error at infinite radius. The usual measure-theoretic conditions are assumed.", "headline": "Proof of Proposition 4, as repaired (the clipping residual is 1-Lipschitz; a mean-square bound for the actual estimator)", "location": "PDF p. 3 (no page number is printed)" }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T8" }, { "declarations": [ { "claim_revision_id": "catalog:06485e7e355bb9e1ac5a300455f4eec0749ede41e2f7987c6fe2efc422d469b2", "conditional_on": [], "credit_reason": "Its negated universal lacks both the population-sampling law and empirical-functional measurability premises; the existing IID certificate and Mathlib kernel-integral theorem prove the strengthened refutation.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.Wave1.proposition1_printed_empirical_counterexample_printed_negation", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/Wave1/Checks/FidP017Contracts.lean#L58", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:06485e7e355bb9e1ac5a300455f4eec0749ede41e2f7987c6fe2efc422d469b2", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P017-T9", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P017-T9" }, "html": { "body": "

The printed claim is false: the universal empirical-statistic representation assertion fails even with IID full-data draws from G, measurable observed records and an estimator given by a measurable functional of the observed empirical law, integrable effects, positive radius, and asymptotic normality. The counterexample samples independently from the specified population G; IID sampling supplies a valid counterexample to the printed all-realizations assertion. Proposition 1 and its all-realizations gloss, PDF p. 2. The usual measure-theoretic conditions are assumed.

", "headline": "Proposition 1", "location": "PDF p. 2 (no page number is printed)" }, "id": "P017-T9", "text": { "body": "The printed claim is false: the universal empirical-statistic representation assertion fails even with IID full-data draws from G, measurable observed records and an estimator given by a measurable functional of the observed empirical law, integrable effects, positive radius, and asymptotic normality. The counterexample samples independently from the specified population G; IID sampling supplies a valid counterexample to the printed all-realizations assertion. Proposition 1 and its all-realizations gloss, PDF p. 2. The usual measure-theoretic conditions are assumed.", "headline": "Proposition 1", "location": "PDF p. 2 (no page number is printed)" }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T9" }, { "declarations": [ { "claim_revision_id": "catalog:d790eb6751a09a0396c4164839112d0a9656179764f1e815799e96888e44d10b", "conditional_on": [], "credit_reason": "The final zero-probability implication has an impossible antecedent under hcond, already recognized in the blind backtranslation; the independently exported exact event equality supplies the meaningful implication in T10(c). 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The conclusions give c=mu, failure of the corollary support condition, and Pr(hat=target) <= 1-Pr(hat<0) < 1, exactly matching T10(e). The closed half-line agrees with the standing closed-support reading of the printed R+ notation, and the measurable random-variable hypotheses introduce no material restriction. 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Corollary 1, its proof, and its two support remarks, PDF p. 2, corrected. Estimator support inside the clipping interval gives probability one of zero error; full real effect support gives the identity target, while nonnegative effect support and negative estimates give the stated strict loss of equality probability. The usual measure-theoretic conditions are assumed.

", "headline": "Corollary 1, with the probability in its proof sentence corrected from 0 to 1", "location": "PDF p. 2 (no page number is printed)" }, "id": "P017-T10", "text": { "body": "Corollary 1, its proof, and its two support remarks, PDF p. 2, corrected. Estimator support inside the clipping interval gives probability one of zero error; full real effect support gives the identity target, while nonnegative effect support and negative estimates give the stated strict loss of equality probability. The usual measure-theoretic conditions are assumed.", "headline": "Corollary 1, with the probability in its proof sentence corrected from 0 to 1", "location": "PDF p. 2 (no page number is printed)" }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T10" }, { "declarations": [ { "claim_revision_id": "catalog:483c934a80d08ff613a6fd2c638a42af005dea72cf57fd2633b60aa7aa3d9d19", "conditional_on": [], "credit_reason": "The corrected target itself restores dx. Using an open Lebesgue half-line rather than a closed endpoint changes nothing because a singleton is null. The false leading MSE inequality in the printed display is outside this row and this corrected identity target.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.Wave1.normal_tail_identity_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/Wave1/Gap/GP017B1.lean#L151", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:483c934a80d08ff613a6fd2c638a42af005dea72cf57fd2633b60aa7aa3d9d19", "conditional_on": [], "credit_reason": "All positive real powers are explicitly in the target, so the second conjunct is not an uncredited change of claim. Metadata inconsistency: T11.asymptotic is false although the target and this row contain asymptotic limits.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.Wave1.normal_tail_limits_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/Wave1/Gap/GP017B1.lean#L289", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:483c934a80d08ff613a6fd2c638a42af005dea72cf57fd2633b60aa7aa3d9d19", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P017-T11", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P017-T11" }, "html": { "body": "

the normal-tail display in the proof of Proposition 4, PDF p. 3, corrected. The displayed Lebesgue tail integral equals its closed form, the truncated second moment of any random variable with the specified Gaussian law, and the scale-s density/CDF expression. The standardized density integral has the stated integration-by-parts value. The relevant integrands are integrable, and the common value decays faster than every positive power of sample size. At infinite radius the tail set is empty, its integral and truncated moment are zero, and the same power limits hold. The usual measure-theoretic conditions are assumed.

", "headline": "Display in the proof of Proposition 4: the normal tail integral, with its differential restored, and its closed form", "location": "PDF p. 3 (no page number is printed)" }, "id": "P017-T11", "text": { "body": "the normal-tail display in the proof of Proposition 4, PDF p. 3, corrected. The displayed Lebesgue tail integral equals its closed form, the truncated second moment of any random variable with the specified Gaussian law, and the scale-s density/CDF expression. The standardized density integral has the stated integration-by-parts value. The relevant integrands are integrable, and the common value decays faster than every positive power of sample size. At infinite radius the tail set is empty, its integral and truncated moment are zero, and the same power limits hold. The usual measure-theoretic conditions are assumed.", "headline": "Display in the proof of Proposition 4: the normal tail integral, with its differential restored, and its closed form", "location": "PDF p. 3 (no page number is printed)" }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T11" }, { "declarations": [ { "claim_revision_id": "catalog:8c55201c207ce5266d6960bc72ef28edabf7f6abe0a187ab5371b0730f1148db", "conditional_on": [], "credit_reason": "The witness contradicts any uniform polynomial or Gaussian-speed reading, while the paper never states a numerical bound with quickly; the row accurately disclaims a full formal negation of a printed quantitative theorem. For the concrete logarithmic p, failure of SmallRemainderMS distinguishes it from Proposition 4. Ancillary finite identities are supplied by R120-08, not inferred from a hidden proof.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.Wave1.slow_outlier_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/Wave1/Gap/GP017B1b.lean#L578", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8c55201c207ce5266d6960bc72ef28edabf7f6abe0a187ab5371b0730f1148db", "conditional_on": [], "credit_reason": "The row establishes the assigned parametrized witness components, not an assertion that every vanishing p is slow and not a formal negation of the unquantified word quickly. The exact standard-normal law, decomposition and o_P(1) remainder export variance-one asymptotic normality despite RootNNormal itself existentially quantifying its variance. Metadata: T12.asymptotic=false conflicts with its explicit limit assertions.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.Wave1.arbitrary_outlier_sequence_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/Wave1/Gap/GLP017B2.lean#L769", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8c55201c207ce5266d6960bc72ef28edabf7f6abe0a187ab5371b0730f1148db", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'At AronowAudit/P017/P017_Audit.lean:1642, A is the clip of x to [-1/2,1/2], equivalent to the max/min expression in the counterexample. The identities clip(A)=A and clip(A+10)=1 give the zero-error and outlier branches of T12(d), with outlier error (A+10)-1=A+9 and exactly the claimed bounds. 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the sentence after Proposition 3, PDF p. 2. The IID rare-outlier construction realizes every positive vanishing clipping- failure sequence under the stated hypotheses. It is a statistic of the observed empirical distribution and has the stated exact events, normal decomposition, finite moment bounds, central-event probability bounds and probability limits. The logarithmic example has the stated numerical values and slow rates, and fails the mean-square remainder condition. For each prescribed positive vanishing rate r, the explicit sequence min(1, sqrt(r)) equals sqrt(r) eventually, tends to zero, and its ratio to r tends to infinity; the same construction realizes it. These facts refute a universal-rate interpretation of "quickly"; the printed sentence specifies no numerical rate.

", "headline": "Sentence after the proof of Proposition 3, “converges to one quickly” (the witness is the validator’s construction and is not in the paper)", "location": "PDF p. 2 (no page number is printed)" }, "id": "P017-T12", "text": { "body": "the sentence after Proposition 3, PDF p. 2. The IID rare-outlier construction realizes every positive vanishing clipping- failure sequence under the stated hypotheses. It is a statistic of the observed empirical distribution and has the stated exact events, normal decomposition, finite moment bounds, central-event probability bounds and probability limits. The logarithmic example has the stated numerical values and slow rates, and fails the mean-square remainder condition. For each prescribed positive vanishing rate r, the explicit sequence min(1, sqrt(r)) equals sqrt(r) eventually, tends to zero, and its ratio to r tends to infinity; the same construction realizes it. These facts refute a universal-rate interpretation of \"quickly\"; the printed sentence specifies no numerical rate.", "headline": "Sentence after the proof of Proposition 3, “converges to one quickly” (the witness is the validator’s construction and is not in the paper)", "location": "PDF p. 2 (no page number is printed)" }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T12" }, { "declarations": [ { "claim_revision_id": "catalog:68df00a1e7ee30ed9c6b281216e7e86dac9d2acdc81a9e721165d51ac558fa1e", "conditional_on": [], "credit_reason": "The interval is the zero-error region; the theorem does not identify it with possibly disconnected effect support. The separate population and sampling measures and weak convergence match the target standing reading.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.corrected_zero_error_sentence", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/P017_Audit.lean#L1033", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:68df00a1e7ee30ed9c6b281216e7e86dac9d2acdc81a9e721165d51ac558fa1e", "conditional_on": [], "credit_reason": "This is the corrected finite bound T13(d), not the false Gaussian-only formula for an intermediate event in the printed proof. Dependence between the normal part and u is allowed; no convergence rate for the retained remainder term is claimed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.Wave1.normal_plus_remainder_failure_bound_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/Wave1/Gap/GP017B2.lean#L84", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:68df00a1e7ee30ed9c6b281216e7e86dac9d2acdc81a9e721165d51ac558fa1e", "conditional_on": [], "credit_reason": "Its component is supplied by the selected composite declaration; no separate application is needed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.sum_failure_union_bound", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/P017_Audit.lean#L284", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:68df00a1e7ee30ed9c6b281216e7e86dac9d2acdc81a9e721165d51ac558fa1e", "conditional_on": [], "credit_reason": "A fixed population with μ=0 and c=1 is sufficient because T13(c) existentially quantifies the law and estimator after each prescribed p. All required failure-event equalities and asymptotic normality are exported by the declaration.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.Wave1.arbitrary_outlier_sequence_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/Wave1/Gap/GLP017B2.lean#L769", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:68df00a1e7ee30ed9c6b281216e7e86dac9d2acdc81a9e721165d51ac558fa1e", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P017-T13", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P017-T13" }, "html": { "body": "

the sentence after Proposition 3, PDF p. 2, corrected. The clipping interval is exactly the zero-error event with probability tending to one, every prescribed positive vanishing failure rate is realizable, and a finite normal-plus- remainder probability bound retains the uncontrolled remainder probability. The usual measure-theoretic conditions are assumed.

", "headline": "Sentence after the proof of Proposition 3, as repaired (“quickly” deleted; the event is the interval of Definition 2)", "location": "PDF p. 2 (no page number is printed)" }, "id": "P017-T13", "text": { "body": "the sentence after Proposition 3, PDF p. 2, corrected. The clipping interval is exactly the zero-error event with probability tending to one, every prescribed positive vanishing failure rate is realizable, and a finite normal-plus- remainder probability bound retains the uncontrolled remainder probability. The usual measure-theoretic conditions are assumed.", "headline": "Sentence after the proof of Proposition 3, as repaired (“quickly” deleted; the event is the interval of Definition 2)", "location": "PDF p. 2 (no page number is printed)" }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T13" }, { "declarations": [ { "claim_revision_id": "catalog:32e2a4a64b6338cbe888272966749d5be10a3aff40a4e6909d073e947d335388", "conditional_on": [], "credit_reason": "RootNNormal is the explicitly authorized weak-convergence reading of Definition 1, not mere consistency. Positive radius is retained and the conclusion uses the actual estimator-dependent target; no false intermediate proof identity is assumed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.proposition3", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/P017_Audit.lean#L1072", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:32e2a4a64b6338cbe888272966749d5be10a3aff40a4e6909d073e947d335388", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P017-T14", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P017-T14" }, "html": { "body": "

The printed claim is false: P017-T14; Proposition 3 and its printed proof, PDF p. 2. In the admissible shifted-normal example the intermediate error from the actual target fails a positive-sequence little-o assertion, and its nonzero probability is one rather than the asserted Gaussian tail. The proposition's actual root-n error conclusion remains valid under all its original assumptions. The usual measure-theoretic conditions are assumed.

", "headline": "Proposition 3, with its proof as printed", "location": "PDF p. 2 (no page number is printed)" }, "id": "P017-T14", "text": { "body": "The printed claim is false: P017-T14; Proposition 3 and its printed proof, PDF p. 2. In the admissible shifted-normal example the intermediate error from the actual target fails a positive-sequence little-o assertion, and its nonzero probability is one rather than the asserted Gaussian tail. The proposition's actual root-n error conclusion remains valid under all its original assumptions. The usual measure-theoretic conditions are assumed.", "headline": "Proposition 3, with its proof as printed", "location": "PDF p. 2 (no page number is printed)" }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T14" }, { "declarations": [ { "claim_revision_id": "catalog:afaeb032ed10f5c18b6004ebfce1a13eb0e0f93add0d33a42c745855d89381c3", "conditional_on": [], "credit_reason": "NormalRemainder expands to exact Gaussian marginal laws, square-integrable remainders and almost-sure decomposition for every n≥1, without independence. These hypotheses give genuine finite actual-error second moments, avoiding the integral-zero convention; n=0 does not affect the limit.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.Wave1.proposition4_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/Wave1/Gap/GP017B2.lean#L274", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:afaeb032ed10f5c18b6004ebfce1a13eb0e0f93add0d33a42c745855d89381c3", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P017-T15", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P017-T15" }, "html": { "body": "

The printed claim is false: P017-T15; Proposition 4 and its printed proof, PDF p. 3. The shifted-normal example satisfies the exact Gaussian-plus-remainder condition and its moment rate, yet has positive intermediate squared error in the central region at size 25 and an intermediate mean-square error strictly above the printed Gaussian-tail bound. The proposition's actual mean-square conclusion remains valid without independence. The usual measure-theoretic conditions are assumed.

", "headline": "Proposition 4, with its proof as printed", "location": "PDF p. 3 (no page number is printed); introduced on PDF p. 2, last paragraph" }, "id": "P017-T15", "text": { "body": "The printed claim is false: P017-T15; Proposition 4 and its printed proof, PDF p. 3. The shifted-normal example satisfies the exact Gaussian-plus-remainder condition and its moment rate, yet has positive intermediate squared error in the central region at size 25 and an intermediate mean-square error strictly above the printed Gaussian-tail bound. The proposition's actual mean-square conclusion remains valid without independence. The usual measure-theoretic conditions are assumed.", "headline": "Proposition 4, with its proof as printed", "location": "PDF p. 3 (no page number is printed); introduced on PDF p. 2, last paragraph" }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T15" }, { "declarations": [ { "claim_revision_id": "catalog:79565262956f75054c696a88a3522bb0fce63748618c5f68ce5133e7cdadb118", "conditional_on": [], "credit_reason": "RootNNormal is genuine weak convergence with finite positive Gaussian variance; BoundedInProbability has exactly the required ε,M,eventual-n order. Population integrability prevents a zero-by-default integral, measurable pushforwards avoid fallback laws, and the c=0 and c=+∞ cases are included. No claim-stage verdict change.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.proposition2", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/P017_Audit.lean#L1104", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:79565262956f75054c696a88a3522bb0fce63748618c5f68ce5133e7cdadb118", "conditional_on": [], "credit_reason": "Its component is supplied by the selected composite declaration; no separate application is needed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.clipReal_distance", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/P017_Audit.lean#L166", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:79565262956f75054c696a88a3522bb0fce63748618c5f68ce5133e7cdadb118", "conditional_on": [], "credit_reason": "The +∞ branch returns x before any conversion to a real radius, so ENNReal.toReal(+∞)=0 causes no lost case. This supplies the full pointwise auxiliary inequality, not the asymptotic conclusion by itself. No claim-stage verdict change.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P017.clipTarget_distance", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P017/P017_Audit.lean#L182", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:79565262956f75054c696a88a3522bb0fce63748618c5f68ce5133e7cdadb118", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P017-T16", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P017-T16" }, "html": { "body": "

Proposition 2 and its pointwise proof, PDF p. 2. The clipped target minus the population mean, multiplied by sqrt(n), is bounded in probability under root-n asymptotic normality, with zero and infinite radius allowed.

", "headline": "Proposition 2", "location": "PDF p. 2 (no page number is printed)" }, "id": "P017-T16", "text": { "body": "Proposition 2 and its pointwise proof, PDF p. 2. The clipped target minus the population mean, multiplied by sqrt(n), is bounded in probability under root-n asymptotic normality, with zero and infinite radius allowed.", "headline": "Proposition 2", "location": "PDF p. 2 (no page number is printed)" }, "url": "https://pmaronow.github.io/llm-audit/works/P017/#P017-T16" } ] }, "missing_sources": [], "reproduction": { "agreement": "not_applicable", "status": "not_applicable", "targets": [] }, "summary": "This four-page note asks how much precision is gained when the target of inference may depend on the data. It clips a root-$n$ consistent and asymptotically normal estimator of the average causal effect to an interval determined by the support of the effects, takes the clipped value as the target, and shows that the estimator is superefficient for this target. 5 entries are classified as errors (1 Moderate, 4 Minor), and 1 is a clarity note. The most important is \\fid{P017-E001}: this target is not a weighted average of causal effects in every realization.", "what_was_checked": "I read the whole text of the note and every proof. The following were checked and held up.\n\\begin{itemize}\n\\item Assumption 1 holds if and only if $\\tau$ is not constant, where $c=+\\infty$ is allowed.\n\\item Definition 2: the three cases are exhaustive and mutually exclusive, and the target is $\\hat\\theta$ clipped to $[\\mathrm{E}_G[\\tau]-c,\\ \\mathrm{E}_G[\\tau]+c]$.\n\\item Proposition 2 and its proof: $|\\theta_{F_n}-\\mathrm{E}_G[\\tau]|\\le|\\hat\\theta-\\mathrm{E}_G[\\tau]|$ in every realization.\n\\item The statements of Propositions 3 and 4, by the arguments given in \\fid{P017-E002}. Assumption 1 is needed in Proposition 3.\n\\item The statement of Corollary 1, and the remarks that its condition holds when $\\operatorname{Supp}_G[\\tau]=\\mathbb{R}$ and that its conclusion fails when $\\operatorname{Supp}_G[\\tau]=\\mathbb{R}^+$ and $\\Pr(\\hat\\theta<0)>0$.\n\\item The closed-form evaluation of the Gaussian tail integral in the proof of Proposition 4, by integration by parts and by numerical integration.\n\\item Proposition 1 for every target that lies strictly between the infimum and the supremum of the support of $\\tau$ or equals an endpoint of positive probability, and hence in every realization when $\\tau$ has finite support or support unbounded in both directions.\n\\item The statement in the discussion that the gains ``may be essentially unbounded'', which agrees with $\\hat\\theta-\\theta_{F_n}=o_p(a_n)$ for every positive sequence $(a_n)$.\n\\item The cross-references to the numbered assumption, definitions, propositions and corollary, and the citation numbers 1 to 10, which all resolve.\n\\item The page cited for the quotation from Crump et al.\\ in Section 3 and the details of references 1 and 10, against public sources (\\fid{P017-N003}).\n\\end{itemize}\nI did not compare the descriptions of the cited works in the introduction with those works, or references 2 to 9 with outside sources. The note has no tables, figures or appendix; I had no supplement, replication files or correction notice to examine. That no correction is listed here for a passage does not show that the passage is correct.", "work": { "arxiv_id": "1601.01413", "audit_complete": true, "authors": [ "P. M. Aronow" ], "category": "Articles - Theory and Methodology", "citation": "P. M. Aronow (2016). Data-Adaptive Causal Effects and Superefficiency. Journal of Causal Inference 4(2).", "doi": "10.1515/jci-2016-0007", "edition": { "pages_in_audited_file": 4, "sha256": "90a89bc1f401886c4fef400a270ce6200153837d322217746bfe4b68bb47a88e", "source_url": "https://doi.org/10.1515/jci-2016-0007", "venue": "Journal of Causal Inference", "year": "2016" }, "id": "P017", "later_materials": null, "title": "Data-Adaptive Causal Effects and Superefficiency", "type": "Journal article", "url": "https://pmaronow.github.io/llm-audit/works/P017/", "venue": "Journal of Causal Inference", "year": "2016" } }