{
 "claims": [
  {
   "claim": "With the studentized Anderson-Rubin statistic, the randomization-based confidence set (7), and its regression-adjusted version (11), is finite-sample exact under a constant additive treatment effect, and in particular under the sharp null of no effect.",
   "claim_id": "P002-H1",
   "related_corrections": [
    "P002-E001"
   ],
   "tied_to": "No numbered result: the opening of Sections 3.3 and 4.4 (citing Imbens & Rosenbaum 2005), equations (3)-(7) and (11); Supplement Lemma 3"
  },
  {
   "claim": "Without covariates the same confidence set is asymptotically valid for the LATE under heterogeneous effects, uniformly over a parameter space that allows an arbitrarily weak instrument (bounded standardised fourth moments, a complier covariance condition, treatment fractions bounded away from 0 and 1); the limiting infimum of coverage is exactly $1-\\alpha$.",
   "claim_id": "P002-H2",
   "related_corrections": [
    "P002-E001",
    "P002-E002",
    "P002-E004",
    "P002-E005",
    "P002-E006",
    "P002-E014"
   ],
   "tied_to": "Theorem 1; Assumptions 1-4; Remark 3; Supplement Sections B-C (Lemmas 2-13, Theorems 4-6)"
  },
  {
   "claim": "The regression-adjusted confidence set (interacted OLS adjustment) has the same finite-sample and uniform asymptotic guarantees under moment and eigenvalue conditions on the covariates; the adjusted Wald estimator equals the coefficient of the corresponding interacted IV regression.",
   "claim_id": "P002-H3",
   "related_corrections": [
    "P002-E001",
    "P002-E002",
    "P002-E003",
    "P002-E007",
    "P002-E008",
    "P002-E009",
    "P002-E010",
    "P002-E013",
    "P002-E014",
    "P002-N009"
   ],
   "tied_to": "Theorem 2; Assumption 5; Remark 4; Supplement Sections D-E (Lemmas 14-20, Theorem 7) and I"
  },
  {
   "claim": "New algorithms recover the Monte Carlo confidence set exactly over the whole real line, without a grid: the index of the simulated AR function that attains the quantile changes only at intersections of two AR functions, which are roots of quartics, and Algorithm 2 skips intersections that do not involve the current quantile function; the same computation serves the regression-adjusted statistic.",
   "claim_id": "P002-H4",
   "related_corrections": [
    "P002-E011",
    "P002-E012",
    "P002-E014",
    "P002-E015",
    "P002-E016",
    "P002-E020",
    "P002-N002",
    "P002-N015",
    "P002-N016",
    "P002-N019",
    "P002-N022"
   ],
   "tied_to": "Lemma 1; Theorem 3; Algorithms 1-2 (main text) and 3-4 (Supplement G); Figure 1; Supplement Sections F-H"
  },
  {
   "claim": "Algorithm 4, the covariate version of Algorithm 2, is much faster than Algorithm 3: with $10^4$ draws it finishes in under an hour where Algorithm 3 takes hours, and with $10^3$ draws both are quick, which makes calculations with $10^4$ draws feasible.",
   "claim_id": "P002-H5",
   "related_corrections": [
    "P002-N002",
    "P002-N003",
    "P002-N011",
    "P002-N012",
    "P002-N018"
   ],
   "tied_to": "Section 5.3; Section 6 (p. 19); Table 1, rows 9-12; Supplement Tables 4-5, 'Median Time' rows"
  },
  {
   "claim": "In simulations with $n=100$ the randomization sets have coverage close to 95% in all seven designs, 2SLS intervals under-cover badly when compliance is low (0.70 at 5% compliance), and the normal-approximation AR and tF sets under-cover in some designs; results do not change materially between $10^3$, $5\\times10^3$ and $10^4$ draws.",
   "claim_id": "P002-H6",
   "related_corrections": [
    "P002-E018",
    "P002-E019",
    "P002-N012",
    "P002-N013",
    "P002-N018"
   ],
   "tied_to": "Section 6; Table 1; Supplement Tables 3-5"
  },
  {
   "claim": "For the male subgroup of Project STAR the randomization sets broadly agree with the 2SLS intervals, except that for SFSP and second-year GPA the 2SLS interval excludes zero (and the normal-approximation AR test gives $p=0.04$) while the randomization set includes zero, despite a high reported first-stage $F$.",
   "claim_id": "P002-H7",
   "related_corrections": [],
   "tied_to": "Section 7; Table 2"
  },
  {
   "claim": "In the six get-out-the-vote experiments the randomization sets are close to the 2SLS intervals and adjusting for prior turnout shortens them; when the Bridgeport sample is subsampled to smaller size and lower compliance the randomization sets become longer relative to 2SLS, 2SLS coverage departs from nominal, and the randomization sets keep nominal coverage.",
   "claim_id": "P002-H8",
   "related_corrections": [
    "P002-E017",
    "P002-N014",
    "P002-N015"
   ],
   "tied_to": "Section 7 (pointer on p. 20); Supplement Sections K-L; Supplement Tables 6-12"
  }
 ],
 "date_published": "2026-10-25",
 "findings": [
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Item 3 of Lemma 20 with display (E.60), whose square root contains var-star of tau-tilde-star-adj, then item 4 with display (E.61), a limit of P-star of the event that sigma-hat-squared-adj-star over var-star of tau-hat-star-adj minus 1 is at least kappa in absolute value. A red outline marks tau-hat in the denominator of (E.61). The proof paragraph follows and ends: The proof of (E.61) uses Lemma 19 and is similar to the proof of (E.18), so we omit it.",
     "caption": "Supplement, p. 42 (PDF p. 22 of the Supplement), items 3 and 4 of Lemma 20 and its proof. Outlined: $\\widehat\\tau$, printed with a hat in the denominator of (E.61); the line above, (E.60), has $\\tilde\\tau^*_{\\mathrm{adj}}$ under the square root.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E003.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P002-T6",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T6"
    },
    {
     "id": "P002-T7",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T7"
    }
   ],
   "headline": "In (E.61) the denominator should be the reassignment variance of the infeasible statistic $\\tilde\\tau^*_{\\mathrm{adj}}$, as in (E.58)–(E.60), not that of the feasible statistic $\\widehat\\tau^*_{\\mathrm{adj}}$.",
   "id": "P002-E003",
   "location": "Supplement, Lemma 20, item 4, display (E.61), p. 42 (PDF p. 22 of the Supplement)",
   "pdf_page": 22,
   "printed_page": "42",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P002-T6",
    "P002-T7"
   ],
   "text": {
    "as_printed": "4. For every $\\kappa>0$,\n\\[\n\\lim_{k\\to\\infty}\\mathbb P^*\\left(\\left|\\frac{\\widehat\\sigma^2_{\\mathrm{adj},*}(\\beta_{n_k})}{\\operatorname{var}^*\\big(\\hl{\\widehat\\tau}^*_{\\mathrm{adj}}(\\beta_{n_k})\\big)}-1\\right|\\ge\\kappa\\right)=0,\\qquad\\text{(E.61)}\n\\]\nFrom the proof: ``The proof of (E.61) uses Lemma 19 and is similar to the proof of (E.18), so we omit it.''",
    "corrected": "Lemma~20, item~4. Let $(\\theta_n)_{n=1}^\\infty$ be a sequence of models with covariates such that $\\theta_n\\in\\Xi_n$ for all $n\\in\\mathbb N$, and denote $\\beta_n=\\bar\\beta(\\theta_n)$. Let $(\\theta_{n_k})_{k=1}^\\infty$ be any of its subsequences. Then, almost surely with respect to the randomness in $Z$, for every $\\kappa>0$,\n\\[\n\\lim_{k\\to\\infty}\\mathbb P^*\\left(\\left|\\frac{\\widehat\\sigma^2_{\\mathrm{adj},*}(\\beta_{n_k})}{\\operatorname{var}^*\\big(\\hl{\\tilde\\tau}^*_{\\mathrm{adj}}(\\beta_{n_k})\\big)}-1\\right|\\ge\\kappa\\right)=0.\\qquad\\text{(E.61)}\n\\]\nHere $\\Xi_n=\\Xi_n(k,\\delta,\\tilde\\varepsilon,A,B,r)$ is the parameter space of Theorem~2, and $\\tilde\\tau^*_{\\mathrm{adj}}$ is the infeasible statistic of (D.4) under reassignment, whose regression coefficients are those of the regression on the full observed population and are held fixed across reassignments. Its variance $\\operatorname{var}^*(\\tilde\\tau^*_{\\mathrm{adj}})$ is given by (D.6). The limit $k\\to\\infty$ is as printed.",
    "evidence": "The evidence is an exact counterexample to the printed statement and a derivation of the corrected one. Recall that $\\widehat\\sigma^2_{\\mathrm{adj},*}=RSS_1/n_1^2+RSS_0/n_0^2$, where $RSS_a$ is the residual sum of squares of the regression in arm $a$ after reassignment.\n\n\\emph{Counterexample.} Let $k=1$, $\\delta=0$, $\\tilde\\varepsilon=1/4$, $A=16$, $B=5$ and $r=1/3$. For even $n=2m\\ge8$, let $t_j=(2j-(m-1))/(m-1)$ for $j=0,\\ldots,m-1$, $\\eta_n=2^{-n^2}$ and $v_m=m^{-1}\\sum_j t_j^2$. The population has two units for each $j$, indexed by $s\\in\\{1,-1\\}$, with\n\\[\nx_{s,j}=s+\\eta_nt_j,\\qquad y_{s,j}(0)=y_{s,j}(1)=t_j+\\frac{1}{10}\\big(t_j^2-v_m\\big),\\qquad d_{s,j}(0)=0,\\qquad d_{s,j}(1)=1,\n\\]\nand $n_1=n_0=m$ under complete randomization. Every unit is a complier, the local average treatment effect is $\\beta=0$, Assumptions 1, 2, 4 and 5 hold with the constants above, and every arm regression has full rank. On the reassignment that treats the whole block $s=1$, both fitted slopes equal $1/\\eta_n$ and $\\widehat\\tau^*_{\\mathrm{adj}}=-2/\\eta_n$; on its complement, $\\widehat\\tau^*_{\\mathrm{adj}}=2/\\eta_n$. Exchanging the two arms changes the sign of $\\widehat\\tau^*_{\\mathrm{adj}}$, so its mean over reassignments is zero, and\n\\[\n\\operatorname{var}^*\\big(\\widehat\\tau^*_{\\mathrm{adj}}\\big)\\ge\\frac{8}{\\binom{n}{m}\\eta_n^2}\\ge8\\cdot2^{2n^2-n}.\n\\]\nSince the fit with zero intercept and zero slope is available in each arm and $|y_i|\\le11/10$, every reassignment has $\\widehat\\sigma^2_{\\mathrm{adj},*}\\le121/(25n)<5/n$. Combining the two bounds, I find that the ratio in (E.61) is at most $(5/(8n))\\,2^{n-2n^2}<1/2$ on every reassignment. This implies that\n\\[\n\\mathbb P^*\\left(\\left|\\frac{\\widehat\\sigma^2_{\\mathrm{adj},*}}{\\operatorname{var}^*\\big(\\widehat\\tau^*_{\\mathrm{adj}}\\big)}-1\\right|\\ge\\frac12\\right)=1\n\\]\nfor every even $n\\ge8$ and every original assignment, while (E.61) asserts that this probability tends to $0$.\n\n\\begin{audittable}{lrrrr}{Exact enumeration of the 70 reassignments at $n=8$ in the construction above, for three values of $\\eta$. The decimals are rounded displays of exact rational values.}\n$\\eta$ & $\\operatorname{var}^*(\\widehat\\tau^*_{\\mathrm{adj}})$ & Largest $\\widehat\\sigma^2_{\\mathrm{adj},*}$ & Largest ratio in (E.61) & $\\mathbb P^*(|\\text{ratio}-1|\\ge1/2)$ \\\\\n$1/100$ & $1143.27$ & $0.27776$ & $2.43\\times10^{-4}$ & $1$ \\\\\n$1/1000$ & $114286.13$ & $0.27778$ & $2.43\\times10^{-6}$ & $1$ \\\\\n$2^{-64}$ & $3.889\\times10^{37}$ & $0.27778$ & $7.14\\times10^{-39}$ & $1$ \\\\\n\\end{audittable}\n\n\\emph{Derivation of the corrected statement,} taking Lemma~19 as given. I write $e_i$ for the residuals of the regression on the full observed population, and I set $H=n^{-1}\\sum_{i=1}^ne_i^2$ and $V=\\operatorname{var}^*(\\tilde\\tau^*_{\\mathrm{adj}})$. By (D.6), I have $V=(1/n_1+1/n_0)\\,nH/(n-1)$. Let $Q_a$ and $h_a$ be the averages of $\\tilde x_i\\tilde x_i'$ and of $\\tilde x_ie_i$ over the units reassigned to arm $a$. When $Q_a$ is invertible, I obtain from the normal equations of least squares that\n\\[\n\\frac{RSS_a}{n_a}=\\frac{1}{n_a}\\sum_{i\\in\\mathcal T^*(a)}e_i^2-h_a'Q_a^{-1}h_a.\n\\]\nLet $Q=n^{-1}\\sum_{i=1}^n\\tilde x_i\\tilde x_i'$. Its smallest eigenvalue is at least $\\min\\{1,\\tilde\\varepsilon\\}$ by Assumption~5(iii) and~(v). Using Assumption~5(iv), the variance formula for sampling without replacement and the identity $\\sum_{i=1}^n\\tilde x_ie_i=0$, I find that $Q_a-Q\\to0$ and $\\|h_a\\|_2^2/H\\to0$ in $\\mathbb P^*$-probability. By Lemma~19 and Chebyshev's inequality, I have $n_a^{-1}\\sum_{i\\in\\mathcal T^*(a)}e_i^2/H\\to1$ in probability. Then $RSS_a/(n_aH)\\to1$ in probability for each $a\\in\\{0,1\\}$, and I conclude that\n\\[\n\\frac{\\widehat\\sigma^2_{\\mathrm{adj},*}}{V}=\\frac{n-1}{n}\\cdot\\frac{n_1^{-1}\\,RSS_1/(n_1H)+n_0^{-1}\\,RSS_0/(n_0H)}{n_1^{-1}+n_0^{-1}}\\longrightarrow1\n\\]\nin probability, which is the corrected (E.61).",
    "problem": "Lemma~20 concerns a sequence of models with covariates $\\theta_n\\in\\Xi_n$, and its claims hold almost surely with respect to the randomness in $Z$. The symbols $\\mathbb P^*$ and $\\operatorname{var}^*$ refer to reassignment. The feasible statistic $\\widehat\\tau^*_{\\mathrm{adj}}$ estimates the regression coefficients anew in each arm for every reassignment (equation (9) of the article). The infeasible statistic $\\tilde\\tau^*_{\\mathrm{adj}}$ of (D.4) holds them fixed at the coefficients of the regression on the full observed population, and its variance is stated in (D.6). Items 1--3 of the lemma, (E.58)--(E.60), are normalized by $\\sqrt{\\operatorname{var}^*(\\tilde\\tau^*_{\\mathrm{adj}})}$, and the proof of Theorem~2 needs item~4 on the same scale. As printed, item~4 asserts instead that $\\widehat\\sigma^2_{\\mathrm{adj},*}$ is consistent for the variance of the feasible statistic. This is a different claim, and it is false under the stated assumptions. Assumption~5 controls the feasible statistic in probability, but it does not control what rare reassignments with ill-conditioned arm regressions contribute to its second moment.",
    "what_survives": "With the tilde in the denominator, (E.60) and (E.61) are on the same scale, and Slutsky's theorem, applied conditionally, gives the normal critical values that this step of the proof of Theorem~2 uses. The correction adds no assumption and changes neither the test statistic, nor the confidence set, nor the coverage statement. Equations (9)--(11) of the article, (F.1)--(F.4) and Algorithms 3 and 4 use residual sums of squares and need no change. Theorem~1 and the unadjusted procedure do not use Lemma~20. The counterexample does not contradict the coverage conclusion of Theorem~2, since all its treatment effects are zero."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E003",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Crop of two displays and the text between them. In the identity that begins with one over n1 times the sum of Z i times v-hat i squared, a red box marks the last term inside the large bracket, 'minus beta n mu d(1)'. In the numerator of the expectation in (C.17), a red box marks the same term.",
     "caption": "Supplement, proof of Lemma 7, p. 26 (PDF p. 6): the identity before (C.17), and (C.17). The marked term $-\\beta_n\\mu_{d(1)}$ should carry a plus sign in both displays.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E004.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P002-T10",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T10"
    },
    {
     "id": "P002-T11",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T11"
    }
   ],
   "headline": "In the identity before (C.17) and in (C.17), the centering term is $-\\mu_{y(d(1))}+\\beta_n\\mu_{d(1)}$; the sign of $\\beta_n\\mu_{d(1)}$ is printed wrong.",
   "id": "P002-E004",
   "location": "Supplement, proof of Lemma 7, p. 26 (PDF p. 6): the identity before (C.17) and the numerator of (C.17)",
   "pdf_page": 6,
   "printed_page": "26",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P002-T10",
    "P002-T11"
   ],
   "text": {
    "as_printed": "We have the identity\n\\[\n\\frac1{n_1}\\sum_{i=1}^nZ_i\\widehat v_i^2(1)=\\frac1{n_1}\\sum_{i=1}^nZ_iv_i^2(1)-\\left(\\frac1{n_1}\\sum_{i=1}^nZ_i\\bigl(y_i(\\hl{d}(1))-\\beta_nd_i(1)\\bigr)-\\mu_{y(d(1))}\\;\\hl{-}\\;\\beta_n\\mu_{d(1)}\\right)^2,\n\\]\nwhich we use to expand the numerator in (C.16), creating two terms. The second term in this expansion converges to zero in probability, by Markov's inequality and the estimate\n\\[\n\\begin{aligned}\nE\\left[\\frac{\\bigl(n_1^{-1}\\sum_{i=1}^nZ_i\\bigl(y_i(\\hl{d}(1))-\\beta_nd_i(1)\\bigr)-\\mu_{y(d(1))}\\;\\hl{-}\\;\\beta_n\\mu_{d(1)}\\bigr)^2}{S_1^2(\\beta_n)}\\right]&=\\frac{n_0}{n_1n(n-1)}\\frac{\\sum_{i=1}^nv_i^2(1)}{S_1^2(\\beta_n)}\\\\\n&=O\\left(\\frac{n_0}{n_1n}\\right)=o(1).\\qquad\\text{(C.17)}\n\\end{aligned}\n\\]",
    "corrected": "We have the identity\n\\[\n\\frac1{n_1}\\sum_{i=1}^nZ_i\\widehat v_i^2(1)=\\frac1{n_1}\\sum_{i=1}^nZ_iv_i^2(1)-\\left(\\frac1{n_1}\\sum_{i=1}^nZ_i\\bigl(y_i(\\hl{d_i}(1))-\\beta_nd_i(1)\\bigr)-\\mu_{y(d(1))}\\;\\hl{+}\\;\\beta_n\\mu_{d(1)}\\right)^2,\n\\]\nand the estimate\n\\[\n\\begin{aligned}\nE\\left[\\frac{\\bigl(n_1^{-1}\\sum_{i=1}^nZ_i\\bigl(y_i(\\hl{d_i}(1))-\\beta_nd_i(1)\\bigr)-\\mu_{y(d(1))}\\;\\hl{+}\\;\\beta_n\\mu_{d(1)}\\bigr)^2}{S_1^2(\\beta_n)}\\right]&=\\frac{n_0}{n_1n(n-1)}\\frac{\\sum_{i=1}^nv_i^2(1)}{S_1^2(\\beta_n)}\\\\\n&=O\\left(\\frac{n_0}{n_1n}\\right)=o(1).\\qquad\\text{(C.17)}\n\\end{aligned}\n\\]\nHere $\\mu_{y(d(1))}$ and $\\mu_{d(1)}$ are the population means of $y_i(d_i(1))$ and $d_i(1)$.",
    "evidence": "The evidence is a derivation and an exact example. Set $\\mu_w=\\mu_{y(d(1))}-\\beta_n\\mu_{d(1)}$ and $\\bar w_1=n_1^{-1}\\sum_iZ_iw_i$. For a treated unit, $\\widehat v_i(1)=w_i-\\bar w_1$ by equation (3) of the article and $v_i(1)=w_i-\\mu_w$ by (B.3). Expanding the square, I obtain\n\\[\n\\frac1{n_1}\\sum_{i=1}^nZ_i\\widehat v_i^2(1)=\\frac1{n_1}\\sum_{i=1}^nZ_iv_i^2(1)-(\\bar w_1-\\mu_w)^2.\n\\]\nUnder complete randomization, I have $E[\\bar w_1]=\\mu_w$ and\n\\[\n\\operatorname{var}(\\bar w_1)=\\frac{n_0}{n_1n(n-1)}\\sum_{i=1}^nv_i^2(1),\n\\]\nwhich is the right side of (C.17) after division by $S_1^2(\\beta_n)$. For the example, set $n=4$, $n_1=2$ and $\\beta_n=2$, let all units be compliers with $y(0)=(-1,-1,1,1)$ and $y(1)=y(0)+2$, and treat the first two units. The left side of the identity is $0$. The printed right side is $1-(-5)^2=-24$, and the corrected right side is $1-(-1)^2=0$.",
    "problem": "The quantity that is centered is $w_i=y_i(d_i(1))-\\beta_nd_i(1)$, whose population mean is $\\mu_{y(d(1))}-\\beta_n\\mu_{d(1)}$. By (B.3), I have $v_i(1)=w_i-\\mu_{y(d(1))}+\\beta_n\\mu_{d(1)}$. The term subtracted in the bracket is then $\\mu_{y(d(1))}-\\beta_n\\mu_{d(1)}$. With the printed sign the bracket is the error of the treated sample mean minus $2\\beta_n\\mu_{d(1)}$. The identity is then false, and the expectation of the numerator in (C.17) exceeds the printed value by $4\\beta_n^2\\mu_{d(1)}^2$, which need not vanish. Both displays also print $y_i(d(1))$ for $y_i(d_i(1))$.",
    "what_survives": "With the corrected sign the identity holds and (C.17) holds with its printed right side. Lemma 7 and the rest of its proof are unchanged, and no assumption is added."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E004",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Crop of the statement of Lemma 12 and the two lines of its proof. A red box marks the numerator of the ratio, sigma hat star of beta n k, printed without a square, above the denominator var star of tau hat star of beta n k.",
     "caption": "Supplementary Material, p. 30 (PDF p. 10), Lemma 12. The marked numerator carries no square, while the denominator is a variance.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E006.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P002-T14",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T14"
    },
    {
     "id": "P002-T15",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T15"
    }
   ],
   "headline": "The numerator of the ratio in Lemma 12 is the estimated variance $\\widehat\\sigma^{*2}(\\beta_{n_k})$, and not the estimated standard deviation $\\widehat\\sigma^{*}(\\beta_{n_k})$.",
   "id": "P002-E006",
   "location": "Supplementary Material, statement of Lemma 12, p. 30 (PDF p. 10 of the supplement)",
   "pdf_page": 10,
   "printed_page": "30",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P002-T14",
    "P002-T15"
   ],
   "text": {
    "as_printed": "Lemma 12. Consider a sequence $(\\theta_n)_{n=1}^\\infty$ such that $\\theta_n\\in\\Theta_n$. Let $\\beta_n=\\bar\\beta(\\theta_n)$. Let $(\\theta_{n_k})_{k=1}^\\infty$ be any of its subsequences. Then for every $\\kappa>0$, we have\n\\[\n\\mathbb P^*\\left(\\left|\\frac{\\hl{\\widehat\\sigma^*(\\beta_{n_k})}}{\\operatorname{var}^*(\\widehat\\tau^*(\\beta_{n_k}))}-1\\right|\\ge\\kappa\\right)=o_{a.s.}(1).\n\\]",
    "corrected": "Lemma 12. Consider a sequence $(\\theta_n)_{n=1}^\\infty$ such that $\\theta_n\\in\\Theta_n$. Let $\\beta_n=\\bar\\beta(\\theta_n)$. Let $(\\theta_{n_k})_{k=1}^\\infty$ be any of its subsequences. Then for every $\\kappa>0$, we have\n\\[\n\\mathbb P^*\\left(\\left|\\frac{\\hl{\\widehat\\sigma^{*2}(\\beta_{n_k})}}{\\operatorname{var}^*(\\widehat\\tau^*(\\beta_{n_k}))}-1\\right|\\ge\\kappa\\right)=o_{a.s.}(1).\n\\]",
    "evidence": "The evidence is an exact example in which the printed statement fails, and a derivation of the corrected one. Let $n$ be even and $n_1=n_0=n/2$. Suppose that every unit is a complier with $y_i(1)=y_i(0)$, that half of these outcomes equal $-1$ and half equal $1$, and that $\\beta_n=0$. Let $q$ be the mean outcome in a reassigned treatment group, so that the control mean is $-q$. Computing the three quantities directly, I find\n\\[\n\\operatorname{var}^*(\\widehat\\tau^*(0))=\\frac1{4(n-1)},\\qquad\\widehat\\sigma^{*2}(0)=\\frac{1-q^2}{4n},\\qquad E^*q^2=\\frac1{n-1}.\n\\]\nThe printed ratio equals $2(n-1)n^{-1/2}(1-q^2)^{1/2}$, which diverges in $\\mathbb P^*$-probability for every original assignment. The ratio with the squared numerator is $R=(n-1)(1-q^2)/n$. Since $0\\le R\\le1$ and $E^*(1-R)=2/n$, Markov's inequality implies that $\\mathbb P^*(|R-1|\\ge\\kappa)\\le2/(n\\kappa)$. With $\\kappa=1/2$ and $n=8,32,128,512$, the exact probability for the printed ratio equals $1$ up to rounding error in each case, and for the corrected ratio it is $0.029$, $4.9\\times10^{-5}$, $6.5\\times10^{-17}$ and $1.6\\times10^{-63}$.\n\nIn general, let $a_i=Y_i-\\beta_nD_i$ minus its mean, $S^2=(n-1)^{-1}\\sum_ia_i^2$ and $M^2=\\max_ia_i^2$. Lemma 11 implies that $M^2/(nS^2)\\to0$ for almost every sequence of original assignments. In a reassigned arm of size $m$, the mean of the $a_i^2$ has expectation $(n-1)S^2/n$ and variance at most a constant times $M^2S^2/n$, and the squared arm mean has expectation $O(S^2/n)$. By Chebyshev's and Markov's inequalities, the centred second moment in each arm divided by $S^2$ tends to $1$ in $\\mathbb P^*$-probability. Substituting into the definition of $\\widehat\\sigma^{*2}$ on p.~6 of the article and comparing with (C.25), I obtain the corrected statement.",
    "problem": "The ratio divides an estimated standard deviation by a variance. In the balanced, bounded-outcome example below, the denominator is of order $1/n$ and $\\widehat\\sigma^*$ is of order $n^{-1/2}$, so the printed ratio is of order $n^{1/2}$ and does not approach $1$. The sentence that introduces the lemma, the parallel statement in Lemma 7 and the use of the lemma to studentize in Lemma 13 all concern the ratio of the estimated variance to the variance.",
    "what_survives": "Lemma 13 and the studentized conditional central limit theorem use the ratio of variances and are unchanged. The correction needs no further assumption, and the coverage results are unaffected."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E006",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Crop of display (E.14) with the line before it and the two lines after it. A red box marks the equals sign between 'max over i of the squared norm of x tilde i' and '1 plus the sum over s of max over i of x s i squared'.",
     "caption": "Supplementary Material, p. 35 (PDF p. 15), display (E.14). The marked sign equates the maximum of $\\|\\tilde x_i\\|_2^2$ with a sum of coordinatewise maxima.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E007.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P002-T16",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T16"
    },
    {
     "id": "P002-T17",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T17"
    }
   ],
   "headline": "The middle relation in (E.14) is an inequality: the maximum over units of a sum is at most the sum of the maxima, and equality can fail.",
   "id": "P002-E007",
   "location": "Supplementary Material, proof of Lemma 16, display (E.14), p. 35 (PDF p. 15 of the supplement)",
   "pdf_page": 15,
   "printed_page": "35",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P002-T16",
    "P002-T17"
   ],
   "text": {
    "as_printed": "For (E.11), for all $s\\in[k]$, Assumption 5-(iv) implies that\n\\[\n\\max_{i\\in[n]}|x_{si}|\\le(An)^{1/4},\\qquad\\max_{i\\in[n]}\\|\\tilde x_i\\|_2^2\\;\\hl{=}\\;1+\\sum_{s=1}^k\\max_{i\\in[n]}|x_{si}|^2\\lesssim n^{1/2},\\qquad\\text{(E.14)}\n\\]\nsince $k$ does not depend on $n$.",
    "corrected": "For (E.11), for all $s\\in[k]$, Assumption 5-(iv) implies that\n\\[\n\\max_{i\\in[n]}|x_{si}|\\le(An)^{1/4},\\qquad\\max_{i\\in[n]}\\|\\tilde x_i\\|_2^2\\;\\hl{\\le}\\;1+\\sum_{s=1}^k\\max_{i\\in[n]}|x_{si}|^2\\lesssim n^{1/2},\\qquad\\text{(E.14)}\n\\]\nsince $k$ does not depend on $n$.",
    "evidence": "The evidence is a derivation and an exact example. Since $\\tilde x_i=[1,x_i']'$, for every $i$ I have\n\\[\n\\|\\tilde x_i\\|_2^2=1+\\sum_{s=1}^k|x_{si}|^2\\le1+\\sum_{s=1}^k\\max_{j\\in[n]}|x_{sj}|^2,\n\\]\nand I obtain the inequality by taking the maximum over $i$. By Assumption 5-(iv), I have $\\max_i|x_{si}|\\le(An)^{1/4}$. The right-hand side is then at most $1+k(An)^{1/2}$, which is $O(n^{1/2})$ because $k$ is fixed. For the example, let $k=2$ and $n=4$, with the centred rows $(1,0)$, $(-1,0)$, $(0,1)$ and $(0,-1)$. The left-hand side equals $2$ and the right-hand side equals $3$.",
    "problem": "Since different coordinates can attain their maxima at different units, the two sides of the marked relation differ in general. Only the upper bound is used in the proof.",
    "what_survives": "The bound $\\max_i\\|\\tilde x_i\\|_2^2\\lesssim n^{1/2}$, claim (E.11) and Lemma 16 are unchanged."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E007",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Crop of the start of the proof of claim (E.15) in Lemma 17: four lines of text, a display, two lines of text, and displays (E.21) to (E.23). Two red boxes mark the left-hand side 'tau hat adj of beta n minus tau tilde adj of beta n', once in the display before (E.21) and once in (E.21).",
     "caption": "Supplementary Material, p. 36 (PDF p. 16), the display before (E.21) and displays (E.21)–(E.23). The marked left-hand sides are the difference whose sign is opposite to that of the right-hand sides.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E008.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P002-T18",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T18"
    }
   ],
   "headline": "The difference $\\widehat\\tau_{\\mathrm{adj}}(\\beta_n)-\\tilde\\tau_{\\mathrm{adj}}(\\beta_n)$ is the negative of the right-hand side printed before (E.21) and in (E.21)–(E.23); the right-hand sides equal $\\tilde\\tau_{\\mathrm{adj}}(\\beta_n)-\\widehat\\tau_{\\mathrm{adj}}(\\beta_n)$.",
   "id": "P002-E008",
   "location": "Supplementary Material, proof of Lemma 17, the display before (E.21) and displays (E.21)–(E.23), p. 36 (PDF p. 16 of the supplement)",
   "pdf_page": 16,
   "printed_page": "36",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P002-T18"
   ],
   "text": {
    "as_printed": "We begin with (E.15). By the definitions of $\\widehat\\tau_{\\mathrm{adj}}(\\beta)$ and $\\tilde\\tau_{\\mathrm{adj}}(\\beta_n)$,\n\\[\n\\hl{\\widehat\\tau_{\\mathrm{adj}}(\\beta_n)-\\tilde\\tau_{\\mathrm{adj}}(\\beta_n)}=\\frac1{n_1}\\sum_{i=1}^nZ_ix_i'\\big(\\widehat\\gamma(1,\\beta_n)-\\gamma(1,\\beta_n)\\big)-\\frac1{n_0}\\sum_{i=1}^n(1-Z_i)x_i'\\big(\\widehat\\gamma(0,\\beta_n)-\\gamma(0,\\beta_n)\\big).\n\\]\nRecall the notation $x_i^\\circ=[0,x_i']'\\in\\mathbb R^{k+1}$ and $\\tilde x_i=[1,x_i']'\\in\\mathbb R^{k+1}$. Using (E.2), the previous expression becomes\n\\[\n\\begin{aligned}\n&{}\\hl{\\widehat\\tau_{\\mathrm{adj}}(\\beta_n)-\\tilde\\tau_{\\mathrm{adj}}(\\beta_n)}&&\\text{(E.21)}\\\\\n&\\quad=\\frac1{n_1}\\sum_{i=1}^nZ_ix_i^{\\circ\\prime}\\Big(\\frac1{n_1}\\sum_{i=1}^nZ_i\\tilde x_i\\tilde x_i'\\Big)^{-1}\\frac1{n_1}\\sum_{i=1}^nZ_i\\tilde x_i\\varepsilon_i(1,\\beta_n)&&\\text{(E.22)}\\\\\n&\\quad\\quad-\\frac1{n_0}\\sum_{i=1}^n(1-Z_i)x_i^{\\circ\\prime}\\Big(\\frac1{n_0}\\sum_{i=1}^n(1-Z_i)\\tilde x_i\\tilde x_i'\\Big)^{-1}\\frac1{n_0}\\sum_{i=1}^n(1-Z_i)\\tilde x_i\\varepsilon_i(0,\\beta_n).&&\\text{(E.23)}\n\\end{aligned}\n\\]",
    "corrected": "We begin with (E.15). By the definitions of $\\widehat\\tau_{\\mathrm{adj}}(\\beta)$ and $\\tilde\\tau_{\\mathrm{adj}}(\\beta_n)$,\n\\[\n\\hl{\\tilde\\tau_{\\mathrm{adj}}(\\beta_n)-\\widehat\\tau_{\\mathrm{adj}}(\\beta_n)}=\\frac1{n_1}\\sum_{i=1}^nZ_ix_i'\\big(\\widehat\\gamma(1,\\beta_n)-\\gamma(1,\\beta_n)\\big)-\\frac1{n_0}\\sum_{i=1}^n(1-Z_i)x_i'\\big(\\widehat\\gamma(0,\\beta_n)-\\gamma(0,\\beta_n)\\big).\n\\]\nRecall the notation $x_i^\\circ=[0,x_i']'\\in\\mathbb R^{k+1}$ and $\\tilde x_i=[1,x_i']'\\in\\mathbb R^{k+1}$. Using (E.2), the previous expression becomes\n\\[\n\\begin{aligned}\n&{}\\hl{\\tilde\\tau_{\\mathrm{adj}}(\\beta_n)-\\widehat\\tau_{\\mathrm{adj}}(\\beta_n)}&&\\text{(E.21)}\\\\\n&\\quad=\\frac1{n_1}\\sum_{i=1}^nZ_ix_i^{\\circ\\prime}\\Big(\\frac1{n_1}\\sum_{i=1}^nZ_i\\tilde x_i\\tilde x_i'\\Big)^{-1}\\frac1{n_1}\\sum_{i=1}^nZ_i\\tilde x_i\\varepsilon_i(1,\\beta_n)&&\\text{(E.22)}\\\\\n&\\quad\\quad-\\frac1{n_0}\\sum_{i=1}^n(1-Z_i)x_i^{\\circ\\prime}\\Big(\\frac1{n_0}\\sum_{i=1}^n(1-Z_i)\\tilde x_i\\tilde x_i'\\Big)^{-1}\\frac1{n_0}\\sum_{i=1}^n(1-Z_i)\\tilde x_i\\varepsilon_i(0,\\beta_n).&&\\text{(E.23)}\n\\end{aligned}\n\\]\nEquivalently, the left-hand sides can be kept as printed and every right-hand side negated.",
    "evidence": "The evidence is a derivation from (9) of the article and (D.4). Subtracting (D.4) from (9), the terms $Y_i-\\beta_nD_i$ cancel, and I obtain\n\\[\n\\widehat\\tau_{\\mathrm{adj}}(\\beta_n)-\\tilde\\tau_{\\mathrm{adj}}(\\beta_n)=-\\frac1{n_1}\\sum_{i=1}^nZ_ix_i'\\big(\\widehat\\gamma(1,\\beta_n)-\\gamma(1,\\beta_n)\\big)+\\frac1{n_0}\\sum_{i=1}^n(1-Z_i)x_i'\\big(\\widehat\\gamma(0,\\beta_n)-\\gamma(0,\\beta_n)\\big).\n\\]\nEquations (E.2) and (E.3) express each slope error as the inverse of the arm design matrix applied to the sum of $\\tilde x_i\\varepsilon_i$ over the arm, with a positive sign. Substituting these expressions, I find (E.22) with a minus sign and (E.23) with a plus sign when the left-hand side is kept.",
    "problem": "Both statistics subtract the fitted covariate term from the adjusted outcome, $\\widehat\\tau_{\\mathrm{adj}}$ with the estimated slopes $\\widehat\\gamma$ and $\\tilde\\tau_{\\mathrm{adj}}$ with the population slopes $\\gamma$. Their difference carries a minus sign in the treated arm and a plus sign in the control arm, which is the opposite of what is printed. Substituting (E.2) and (E.3) does not change a sign, and (E.21)--(E.23) carry the same reversal.",
    "what_survives": "The proof uses only the absolute value of this difference, and claim (E.15) is stated for $|\\tilde\\tau_{\\mathrm{adj}}(\\beta_{n_k})-\\widehat\\tau_{\\mathrm{adj}}(\\beta_{n_k})|$. The bounds that follow, the other claims of Lemma 17 and the results on the confidence sets are unchanged."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E008",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Crop of display (E.41): a column vector with entries phi(a, beta_n) minus phi-hat(a, beta_n) and gamma(a, beta_n) minus gamma-hat(a, beta_n), set equal to the inverse of an arm average of x-tilde_i x-tilde_i-prime times an arm average of x-tilde_i epsilon(a, beta_n). Red boxes mark the equality sign and the term epsilon(a, beta_n).",
     "caption": "Supplementary Material, p. 39 (PDF p. 19 of the supplement): display (E.41). The right side, which begins after the marked equality sign, lacks a minus sign, and the marked residual $\\varepsilon(a,\\beta_n)$ lacks its index $i$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E009.png"
    },
    {
     "alt": "Crop of the sentence 'Next, (E.39) is, after applying (E.41),' followed by a display: the number 2 times a product of three arm averages, the middle one inverted. A red box marks the leading 2.",
     "caption": "Supplementary Material, p. 40 (PDF p. 20 of the supplement): the display after (E.46), which evaluates the cross term (E.39). The marked coefficient $2$ should be $-2$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E009-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P002-T19",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T19"
    }
   ],
   "headline": "Two signs in the proof of Lemma 18 are wrong: the right side of (E.41) lacks a minus sign, and the cross term displayed after (E.46) has coefficient $-2$, not $2$; the residual in (E.41) also lacks its index $i$.",
   "id": "P002-E009",
   "location": "Supplementary Material, proof of Lemma 18: display (E.41), p. 39 (PDF p. 19 of the supplement), and the unnumbered display after (E.46), p. 40 (PDF p. 20 of the supplement)",
   "pdf_page": 19,
   "printed_page": "39",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P002-T19"
   ],
   "text": {
    "as_printed": "Recalling the definition of $\\tilde x_i$, we have by a representation analogous to (E.2) that, asymptotically almost surely,\n\\[\n\\begin{bmatrix}\\varphi(a,\\beta_n)-\\widehat\\varphi(a,\\beta_n)\\\\ \\gamma(a,\\beta_n)-\\widehat\\gamma(a,\\beta_n)\\end{bmatrix}\\hl{=}\\left(\\frac{1}{n_a}\\sum_{i\\in\\mathcal T(a)}\\tilde x_i\\tilde x_i'\\right)^{-1}\\left(\\frac{1}{n_a}\\sum_{i\\in\\mathcal T(a)}\\tilde x_i\\hl{\\varepsilon}(a,\\beta_n)\\right).\\qquad\\text{(E.41)}\n\\]\nNext, (E.39) is, after applying (E.41),\n\\[\n\\hl{2}\\left(\\frac{1}{n_a}\\sum_{i\\in\\mathcal T(a)}\\varepsilon_i(a,\\beta_n)\\tilde x_i'\\right)\\left(\\frac{1}{n_a}\\sum_{i\\in\\mathcal T(a)}\\tilde x_i\\tilde x_i'\\right)^{-1}\\left(\\frac{1}{n_a}\\sum_{i\\in\\mathcal T(a)}\\tilde x_i\\varepsilon_i(a,\\beta_n)\\right).\n\\]",
    "corrected": "Recalling the definition of $\\tilde x_i$, we have by a representation analogous to (E.2) that, asymptotically almost surely,\n\\[\n\\begin{bmatrix}\\varphi(a,\\beta_n)-\\widehat\\varphi(a,\\beta_n)\\\\ \\gamma(a,\\beta_n)-\\widehat\\gamma(a,\\beta_n)\\end{bmatrix}=\\hl{-}\\left(\\frac{1}{n_a}\\sum_{i\\in\\mathcal T(a)}\\tilde x_i\\tilde x_i'\\right)^{-1}\\left(\\frac{1}{n_a}\\sum_{i\\in\\mathcal T(a)}\\tilde x_i\\hl{\\varepsilon_i}(a,\\beta_n)\\right).\\qquad\\text{(E.41)}\n\\]\nThe identity holds whenever the matrix $n_a^{-1}\\sum_{i\\in\\mathcal T(a)}\\tilde x_i\\tilde x_i'$ is invertible, which the proof shows to hold asymptotically almost surely.\n\nNext, (E.39) is, after applying (E.41),\n\\[\n\\hl{-2}\\left(\\frac{1}{n_a}\\sum_{i\\in\\mathcal T(a)}\\varepsilon_i(a,\\beta_n)\\tilde x_i'\\right)\\left(\\frac{1}{n_a}\\sum_{i\\in\\mathcal T(a)}\\tilde x_i\\tilde x_i'\\right)^{-1}\\left(\\frac{1}{n_a}\\sum_{i\\in\\mathcal T(a)}\\tilde x_i\\varepsilon_i(a,\\beta_n)\\right).\n\\]\nThe sign in (E.47) may be changed to match; its statement holds with either sign.",
    "evidence": "The evidence is a derivation from the least-squares normal equations. Fix an arm $a$ and write\n\\[\nG=\\frac{1}{n_a}\\sum_{i\\in\\mathcal T(a)}\\tilde x_i\\tilde x_i',\\qquad h=\\frac{1}{n_a}\\sum_{i\\in\\mathcal T(a)}\\tilde x_i\\varepsilon_i(a,\\beta_n).\n\\]\nFor $i\\in\\mathcal T(a)$ the regressand is $w_i(a,\\beta_n)=\\tilde x_i'\\psi+\\varepsilon_i(a,\\beta_n)$. The normal equations are\n\\[\n0=\\frac{1}{n_a}\\sum_{i\\in\\mathcal T(a)}\\tilde x_i\\bigl(w_i(a,\\beta_n)-\\tilde x_i'\\widehat\\psi\\bigr)=h-G(\\widehat\\psi-\\psi).\n\\]\nWhen $G$ is invertible, the normal equations imply that $\\widehat\\psi-\\psi=G^{-1}h$ and $\\psi-\\widehat\\psi=-G^{-1}h$, in agreement with (E.2)--(E.3). Since $x_i'(\\widehat\\gamma-\\gamma)+\\widehat\\varphi-\\varphi=\\tilde x_i'(\\widehat\\psi-\\psi)$, I find that the term (E.39) equals $-2h'(\\widehat\\psi-\\psi)=-2h'G^{-1}h$.\n\nThe positive sign in (E.47) is not a further error. That display states that a quantity is $o_{a.s.}(1)$, and a quantity is $o_{a.s.}(1)$ exactly when its negative is.",
    "problem": "Let $\\psi=(\\varphi,\\gamma')'$ denote the population coefficients and $\\widehat\\psi$ their least-squares estimates in arm $a$. The least-squares identity (E.2)--(E.3) of Lemma 14 gives the difference $\\widehat\\psi-\\psi$, estimate minus population. Display (E.41) prints the same right side for the difference $\\psi-\\widehat\\psi$, which has the opposite sign. The term (E.39) carries the factor $-2/n_a$. When the correct identity is substituted into it, the resulting product has coefficient $-2$, and the display after (E.46) prints $2$.",
    "what_survives": "The bounds (E.44), (E.46) and (E.47) and the conclusion (E.37) of Lemma 18 are unaffected, because they use only the fact that these quantities tend to zero. Only the two displays change."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E009",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "The display that follows the words Then Assumption 5-(v): a quadratic form in the difference of psi-hat(a, beta-n) and psi-star(beta-n) is bounded below by epsilon-tilde plus an almost surely vanishing term, times a squared norm. Red outlines mark epsilon-tilde and the term psi-hat(a) inside the norm. Below, the text continues: Using (E.11), we find that (E.57) is bounded by, with a further display.",
     "caption": "Supplement, p. 42 (PDF p. 22 of the Supplement), first display, in the proof of Lemma 19. Outlined: the constant $\\tilde\\varepsilon$, and $\\widehat\\psi(a)$ printed without its argument $\\beta_n$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E010.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P002-T8",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T8"
    },
    {
     "id": "P002-T9",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T9"
    }
   ],
   "headline": "The lower bound on the quadratic form holds with the constant $\\min\\{1,\\tilde\\varepsilon\\}$ in place of $\\tilde\\varepsilon$, because the matrix includes the intercept.",
   "id": "P002-E010",
   "location": "Supplement, proof of Lemma 19, p. 42 (PDF p. 22 of the Supplement), first display, after “Then Assumption 5-(v),”",
   "pdf_page": 22,
   "printed_page": "42",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P002-T8",
    "P002-T9"
   ],
   "text": {
    "as_printed": "Then Assumption 5-(v),\n\\[\n\\begin{aligned}\n&\\big[\\widehat\\psi(a,\\beta_n)-\\psi^*(\\beta_n)\\big]'\\left(\\frac{1}{n_a}\\sum_{i\\in\\mathcal T(a)}\\tilde x_i\\tilde x_i'\\right)\\big[\\widehat\\psi(a,\\beta_n)-\\psi^*(\\beta_n)\\big]\\\\\n&\\qquad\\qquad\\ge\\big(\\hl{\\tilde\\varepsilon}+o_{a.s.}(1)\\big)\\,\\|\\hl{\\widehat\\psi(a)}-\\psi^*(\\beta_n)\\|_2^2.\n\\end{aligned}\n\\]",
    "corrected": "Then Assumption 5-(v),\n\\[\n\\begin{aligned}\n&\\big[\\widehat\\psi(a,\\beta_n)-\\psi^*(\\beta_n)\\big]'\\left(\\frac{1}{n_a}\\sum_{i\\in\\mathcal T(a)}\\tilde x_i\\tilde x_i'\\right)\\big[\\widehat\\psi(a,\\beta_n)-\\psi^*(\\beta_n)\\big]\\\\\n&\\qquad\\qquad\\ge\\big(\\hl{\\min\\{1,\\tilde\\varepsilon\\}}+o_{a.s.}(1)\\big)\\,\\|\\widehat\\psi(a\\hl{,\\beta_n})-\\psi^*(\\beta_n)\\|_2^2.\n\\end{aligned}\n\\]\nHere $a\\in\\{0,1\\}$, $\\tilde x_i=[1,x_i']'$, and $\\tilde\\varepsilon>0$ is the constant of Assumption~5(v), on which no upper bound is imposed. The bound uses Assumption~5(iii), $\\sum_{i=1}^nx_i=0_k$, and (E.40) together with Assumption~5(v).",
    "evidence": "The evidence is a derivation, with a counterexample to the printed step. Let $Q_n=n^{-1}\\sum_{i=1}^n x_ix_i'$. By Assumption~5(iii), I have $n^{-1}\\sum_{i=1}^n\\tilde x_i\\tilde x_i'=\\operatorname{diag}(1,Q_n)$, and $\\lambda_{\\min}(Q_n)\\ge\\tilde\\varepsilon$ by Assumption~5(v). The smallest eigenvalue of $\\operatorname{diag}(1,Q_n)$ is $\\min\\{1,\\lambda_{\\min}(Q_n)\\}$, which is at least $\\min\\{1,\\tilde\\varepsilon\\}$. By (E.40), the matrix $n_a^{-1}\\sum_{i\\in\\mathcal T(a)}\\tilde x_i\\tilde x_i'$ equals $\\operatorname{diag}(1,Q_n)+o_{a.s.}(1)$. Weyl's inequality then implies that its smallest eigenvalue is at least $\\min\\{1,\\tilde\\varepsilon\\}+o_{a.s.}(1)$, and the corrected bound follows for every vector in place of $\\widehat\\psi(a,\\beta_n)-\\psi^*(\\beta_n)$.\n\nFor the counterexample, suppose that there is a single covariate, whose values alternate between $2$ and $-2$, so that $Q_n=4$, and let $\\tilde\\varepsilon=2$ and $v=(1,0)'$. Then\n\\[\nv'\\operatorname{diag}(1,4)\\,v=1<2=\\tilde\\varepsilon\\,\\|v\\|_2^2.\n\\]\nThe matrix $\\operatorname{diag}(1,Q_n)$ does not satisfy the bound with the constant $\\tilde\\varepsilon$. This does not contradict Lemma~19.",
    "problem": "Here $\\tilde x_i=[1,x_i']'$ is the covariate vector with an intercept. Assumption~5(v) bounds below by $\\tilde\\varepsilon$ the smallest eigenvalue of $n^{-1}\\sum_{i=1}^n x_ix_i'$, the second-moment matrix of the covariates alone. Since the covariates are centred by Assumption~5(iii), the matrix $n^{-1}\\sum_{i=1}^n\\tilde x_i\\tilde x_i'$ has the additional eigenvalue $1$ from the intercept, and its smallest eigenvalue is bounded below by $\\min\\{1,\\tilde\\varepsilon\\}$. Assumption~5 allows $\\tilde\\varepsilon>1$, so the printed constant is justified only when $\\tilde\\varepsilon\\le1$. The norm on the right-hand side also omits the argument $\\beta_n$ of $\\widehat\\psi$.",
    "what_survives": "The step needs only a lower bound by a fixed positive constant, and $\\min\\{1,\\tilde\\varepsilon\\}$ is such a constant. The bound on (E.57), Lemma~19 and the results that rest on it hold under the assumptions as stated."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E010",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "page_check",
   "exhibits": [
    {
     "alt": "Crop of Theorem 7: 'For every alpha in (0, 1)', followed by display (E.36), a double limit of a supremum over beta and a supremum over theta in Theta_n with beta-bar(theta) equal to beta, of the probability that the absolute value of Delta_adj(beta) exceeds z_(1 - alpha/2) minus kappa, bounded by alpha. A red box marks Theta_n in the subscript of the inner supremum. The proof below reads 'Given Lemma 17, the proof of (E.36) is nearly identical to the proof of Theorem 6, so we omit it.'",
     "caption": "Supplementary Material, p. 39 (PDF p. 19 of the supplement): Theorem 7 and the first lines of its proof. The marked symbol $\\Theta_n$ under the inner supremum of (E.36) should be $\\Xi_n$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E013.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P002-T21",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T21"
    }
   ],
   "headline": "The inner supremum of (E.36) is taken over $\\Theta_n$, the parameter space of the theory without covariates; it should be taken over $\\Xi_n$, the parameter space of the regression-adjusted theory. If the population size ranges over all positive integers, the inner limit should also be a limit superior.",
   "id": "P002-E013",
   "location": "Supplementary Material, Theorem 7, display (E.36), p. 39 (PDF p. 19 of the supplement), inner supremum",
   "pdf_page": 19,
   "printed_page": "39",
   "reaches": [],
   "related_entries": [],
   "revisions": [
    {
     "classification_after": {
      "class": "error",
      "severity": "Minor"
     },
     "classification_before": {
      "class": "error",
      "severity": "Minor"
     },
     "note": "The wording was revised; the classification is unchanged. How this came to light: stating Theorem 7 in Lean showed that the display asserts an ordinary inner limit, and that with $r=1/2$ the model classes are empty at every odd population size. Admissible models exist whose rejection probabilities stay above one half along the sizes divisible by four, for every $\\kappa$ near zero, while the supremum is zero at odd sizes. Why it changed: the correction had said that only the symbol changes and that Theorem 2 is unaffected; neither assurance holds as stated. The corrected display now uses a limit superior, which the argument of the paper supports. P. M. Aronow delegated this decision to the model session that runs the audit, which adopted the amendment recommended by the two checks; this note was written by that session."
    }
   ],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P002-T21"
   ],
   "text": {
    "as_printed": "Theorem 7. For every $\\alpha\\in(0,1)$,\n\\[\n\\lim_{\\kappa\\to0^+}\\lim_{n\\to\\infty}\\sup_{\\beta\\in\\mathbb R}\\ \\sup_{\\{\\theta\\in\\hl{\\scriptstyle\\Theta_n}:\\,\\bar\\beta(\\theta)=\\beta\\}}\\mathbb P_\\theta\\bigl(\\lvert\\Delta_{\\mathrm{adj}}(\\beta)\\rvert>z_{1-\\alpha/2}-\\kappa\\bigr)\\le\\alpha.\\qquad\\text{(E.36)}\n\\]",
    "corrected": "Theorem 7. For every $\\alpha\\in(0,1)$,\n\\[\n\\lim_{\\kappa\\to0^+}\\hl{\\limsup_{n\\to\\infty}}\\sup_{\\beta\\in\\mathbb R}\\ \\sup_{\\{\\theta\\in\\hl{\\scriptstyle\\Xi_n}:\\,\\bar\\beta(\\theta)=\\beta\\}}\\mathbb P_\\theta\\bigl(\\lvert\\Delta_{\\mathrm{adj}}(\\beta)\\rvert>z_{1-\\alpha/2}-\\kappa\\bigr)\\le\\alpha.\\qquad\\text{(E.36)}\n\\]\nHere $\\Xi_n=\\Xi_n(k,\\delta,\\tilde\\varepsilon,A,B,r)$ is the parameter space of models that satisfy Assumptions 1, 2, 4 and 5, with the constants fixed in the statement of Theorem 2, as in Section D.1 of the Supplementary Material. The population size ranges over all positive integers, and an empty supremum of probabilities is zero. The outer limit exists because the inner limit superior is nondecreasing in $\\kappa$ and lies in $[0,1]$.",
    "evidence": "The evidence is a check of the pages: the definitions of $\\Theta_n$ (p.~7) and of $\\Xi_n$ (p.~11) in the article, and Section D.1 (p.~31) and the statement of Lemma 17 (p.~35) in the Supplementary Material. I checked the four-cell example by exact enumeration of the assignment distribution for $n=4$, $8$ and $12$.",
    "problem": "The space $\\Theta_n$ collects the models without covariates that satisfy Assumptions 1--4 (p.~7 of the article). Theorem 7 concerns the regression-adjusted statistic $\\Delta_{\\mathrm{adj}}$. Section D.1 of the Supplementary Material fixes $\\Xi_n=\\Xi_n(k,\\delta,\\tilde\\varepsilon,A,B,r)$, the parameter space of Theorem 2, for Sections D and E, and the proof of Theorem 7 rests on Lemma 17, which is stated for sequences $\\theta_n\\in\\Xi_n$.\n\nThere is also an issue with the inner limit. Assumption 4 (p.~7 of the article) allows $r=1/2$, which forces $n_1=n/2$, so both $\\Theta_n$ and $\\Xi_n$ are empty at every odd population size. The paper neither restricts the size index to feasible sizes nor says what value an empty supremum of probabilities takes. If that value is zero, the inner limit can fail to exist, even for $\\kappa$ arbitrarily close to zero. Take $m$ units in each of the four cells $(x_i,z_i)\\in\\{-1,1\\}^2$, all compliers, with $y_i(0)=y_i(1)=z_i$ and $n_1=n/2$; these models belong to $\\Xi_n$ with $k=1$, $\\delta=0$, $\\tilde\\varepsilon=1/2$ and $A=B=2$. For $\\alpha=1/2$ and every $0<\\kappa<z_{1-\\alpha/2}$, the rejection probability tends to $2\\{1-\\Phi(z_{1-\\alpha/2}-\\kappa)\\}>1/2$ along the sizes divisible by four, where $\\Phi$ is the standard normal distribution function, while the supremum is zero at every odd size. The argument of Lemma 17 and Theorem 6 bounds the limit superior; it does not show that the ordinary inner limit exists. Restricting the index to feasible sizes removes this example but would need its own argument for existence.",
    "what_survives": "The model-space correction and the conservative bound on the adjusted tail are retained in the limit-superior form below. That the ordinary inner limits exist is not established."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E013",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Crop of the statement of Theorem 3. A red box marks the words 'in the Supplementary Material' that follow 'Without covariates and under Assumption 6'.",
     "caption": "Theorem 3, p. 17. The marked phrase places Assumption 6 in the Supplementary Material; it is stated in Section 5.2 of the article, p. 14.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E014.png"
    },
    {
     "alt": "Crop of the last line of a displayed inequality and the sentence after it. A red box marks 'Assumption 5-(ii)' in 'where the inequality follows from Assumption 5-(ii). Therefore, recalling (D.5),'.",
     "caption": "Supplementary Material, p. 35 (PDF p. 15), proof of Lemma 16. The marked pointer should be to Assumption 5-(i); the printed inequality follows for $\\delta\\ge0$, but needs $\\delta_+=\\max\\{\\delta,0\\}$ in place of $\\delta$ over the full stated range.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E014-b.png"
    },
    {
     "alt": "Crop of display (E.14) and the two lines after it. A red box marks '(using Assumption 5-(iv))' in 'Claim (E.12) can be proved as in Lemma 4 (using Assumption 5-(iv)), and claim (E.13) can be proved as in Lemma 6'.",
     "caption": "Supplementary Material, p. 35 (PDF p. 15), end of the proof of Lemma 16. The marked pointer is given for claim (E.12), which concerns residuals and follows from Assumption 5-(ii).",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E014-c.png"
    },
    {
     "alt": "Crop of the first lines of Section I, Algebra for Remark 4, to display (I.1) and the sentence after it. Red boxes mark '(4.12)' in 'We first show that (4.12) satisfies', '(4.8)' in 'introduced below equation (4.8)', the two right-hand sides of display (I.1), 'tau hat y,1 minus beta tau hat d,1' and 'tau hat y,0 minus beta tau hat d,0', and the four symbols 'tau hat y,1, tau hat y,0, tau hat d,1, and tau hat d,0' in the sentence 'were introduced in Remark 4'.",
     "caption": "Supplementary Material, p. 45 (PDF p. 25), start of Section I. The marked pointers (4.12) and (4.8) are to equations (12) and (8) of the article, and the marked symbols $\\widehat\\tau_{y,z}$, $\\widehat\\tau_{d,z}$ are the slopes that Remark 4 calls $\\widehat\\gamma_{y,z}$, $\\widehat\\gamma_{d,z}$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E014-d.png"
    },
    {
     "alt": "Crop of the first line of printed page 46 and the first line of the display after it. A red box marks 'tau hat of beta' in 'The quantity tau hat of beta can be rewritten as'; the display below begins 'tau hat adj of beta equals'.",
     "caption": "Supplementary Material, p. 46 (PDF p. 26). The marked symbol lacks the subscript of the statistic $\\widehat\\tau_{\\mathrm{adj}}(\\beta)$ that the display defines.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E014-e.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P002-T24",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T24"
    },
    {
     "id": "P002-T28",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T28"
    }
   ],
   "headline": "Several cross-references point to the wrong place: the Supplementary Material cites equations of the article by section-prefixed numbers that the article does not use, Theorem 3 places Assumption 6 in the Supplementary Material, two steps in the proof of Lemma 16 cite the wrong parts of Assumption 5, its covariance and variance bounds can fail for allowed negative $\\delta$, and displays (I.1) and (I.2) write the regression slopes of Remark 4 as $\\widehat\\tau$.",
   "id": "P002-E014",
   "location": "Theorem 3, p. 17; Supplementary Material, pp. 23, 25, 32, 34, 35, 42, 43, 45, 46 and 48 (PDF pp. 3, 5, 12, 14, 15, 22, 23, 25, 26 and 28 of the supplement)",
   "pdf_page": 17,
   "printed_page": "17",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P002-T24",
    "P002-T28"
   ],
   "text": {
    "as_printed": "\\begin{itemize}\n\\item Theorem 3, p.~17: ``Without covariates and under Assumption 6 \\hl{in the Supplementary Material}, the confidence set CS returned by Algorithm 1 or 2 is equal to that in (14).''\n\\item Supplementary Material, pointers to equations of the article: ``following \\hl{(2.2)}'' (p.~23); ``recalling the definitions in \\hl{(3.3)}'' (p.~25); ``between expressions \\hl{(4.9)} and (D.4)'' (p.~32); ``similar to \\hl{(5.15)} and \\hl{(5.16)}'' (p.~42); ``the definition of $\\Delta_{\\mathrm{adj}}(\\beta)$ from \\hl{(4.10)}'' (p.~43); ``the definition of $\\mathcal I$ in \\hl{(5.17)}'', ``the confidence set defined in \\hl{(5.14)}'', ``We first show that \\hl{(4.12)} satisfies'' and ``introduced below equation \\hl{(4.8)}'' (p.~45); ``generated according to \\hl{(6.18)} and \\hl{(6.19)}'' and ``generated according to \\hl{(6.20)} and \\hl{(6.21)}'' (p.~48).\n\\item Lemma 16, p.~34:\n\\[\n\\operatorname{var}\\big(\\widetilde\\tau_{\\mathrm{adj}}(\\beta_{n_k})\\big)\\ge(1-\\hl{\\delta})\\frac{n_0}{n_1n}S_1^2(\\beta_{n_k})+(1-\\hl{\\delta})\\frac{n_1}{n_0n}S_0^2(\\beta_{n_k}).\\qquad\\text{(E.10)}\n\\]\nIts proof, p.~35, uses\n\\[\n\\begin{aligned}\n\\sum_{i=1}^n\\big(\\varepsilon_i(1,\\beta_n)-\\varepsilon_i(0,\\beta_n)\\big)^2\n&=\\sum_{i=1}^n\\varepsilon_i^2(1,\\beta_n)+\\sum_{i=1}^n\\varepsilon_i^2(0,\\beta_n)-2\\sum_{i=1}^n\\varepsilon_i(1,\\beta_n)\\varepsilon_i(0,\\beta_n)\\\\\n&\\le\\Big(1+\\frac{\\hl{\\delta}n_0}{n_1}\\Big)\\sum_{i=1}^n\\varepsilon_i^2(1,\\beta_n)+\\Big(1+\\frac{\\hl{\\delta}n_1}{n_0}\\Big)\\sum_{i=1}^n\\varepsilon_i^2(0,\\beta_n),\n\\end{aligned}\n\\]\nfollowed by ``where the inequality follows from \\hl{Assumption 5-(ii)}'', and ends with ``Claim (E.12) can be proved as in Lemma 4 (using \\hl{Assumption 5-(iv)})''.\n\\item Section I, p.~45:\n\\[\n\\widehat\\gamma(1,\\beta)=\\hl{\\widehat\\tau_{y,1}}-\\beta\\,\\hl{\\widehat\\tau_{d,1}},\\qquad\\widehat\\gamma(0,\\beta)=\\hl{\\widehat\\tau_{y,0}}-\\beta\\,\\hl{\\widehat\\tau_{d,0}},\\qquad\\text{(I.1)}\n\\]\n``where $\\widehat\\tau_{y,1}$, $\\widehat\\tau_{y,0}$, $\\widehat\\tau_{d,1}$, and $\\widehat\\tau_{d,0}$ were introduced in Remark 4.'' The same symbols recur in (I.2) and in the text after it.\n\\item Section I, p.~46: ``The quantity $\\hl{\\widehat\\tau(\\beta)}$ can be rewritten as'', followed by a display for $\\widehat\\tau_{\\mathrm{adj}}(\\beta)$.\n\\end{itemize}",
    "corrected": "\\begin{itemize}\n\\item Theorem 3: ``Without covariates and under Assumption 6, the confidence set CS returned by Algorithm 1 or 2 is equal to that in (14).'' The reference to the Supplementary Material after Assumption 7 stays.\n\\item In the Supplementary Material, replace (2.2), (3.3), (4.8), (4.9), (4.10), (4.12), (5.14), (5.15), (5.16), (5.17), (6.18), (6.19), (6.20) and (6.21) by \\hl{(2), (3), (8), (9), (10), (12), (14), (15), (16), (17), (18), (19), (20) and (21)}.\n\\item Lemma 16 and its proof: define $\\delta_+=\\max\\{\\delta,0\\}$ and replace $\\delta$ by $\\hl{\\delta_+}$ in the covariance inequality, the ensuing variance bounds including (E.10), and subsequent uses of this lower bound. The pointer sentences become ``where the inequality follows from \\hl{Assumption 5-(i)}'' and ``Claim (E.12) can be proved as in Lemma 4 (using \\hl{Assumption 5-(ii)})''.\n\\item Section I:\n\\[\n\\widehat\\gamma(1,\\beta)=\\hl{\\widehat\\gamma_{y,1}}-\\beta\\,\\hl{\\widehat\\gamma_{d,1}},\\qquad\\widehat\\gamma(0,\\beta)=\\hl{\\widehat\\gamma_{y,0}}-\\beta\\,\\hl{\\widehat\\gamma_{d,0}},\\qquad\\text{(I.1)}\n\\]\n``where $\\widehat\\gamma_{y,1}$, $\\widehat\\gamma_{y,0}$, $\\widehat\\gamma_{d,1}$, and $\\widehat\\gamma_{d,0}$ were introduced in Remark 4.'' The same replacement applies in (I.2) and in the text after it.\n\\item Section I, p.~46: ``The quantity $\\hl{\\widehat\\tau_{\\mathrm{adj}}(\\beta)}$ can be rewritten as''.\n\\end{itemize}",
    "evidence": "I checked the pages and the bounds, including an exact counterexample. The targets of the equation pointers are:\n\\begin{audittable}{lll}{Pointers in the Supplementary Material to equations of the article.}\nPrinted page of the supplement & Printed pointer & Equation of the article \\\\\n23 & (2.2) & (2) \\\\\n25 & (3.3) & (3) \\\\\n32 & (4.9) & (9) \\\\\n42 & (5.15), (5.16) & (15), (16) \\\\\n43 & (4.10) & (10) \\\\\n45 & (5.17), (5.14), (4.12), (4.8) & (17), (14), (12), (8) \\\\\n48 & (6.18), (6.19), (6.20), (6.21) & (18), (19), (20), (21)\n\\end{audittable}\nFor the first step of Lemma 16, let $U=\\sum_i\\varepsilon_i^2(1,\\beta_n)$, $W=\\sum_i\\varepsilon_i^2(0,\\beta_n)$ and $C=\\sum_i\\varepsilon_i(1,\\beta_n)\\varepsilon_i(0,\\beta_n)$. The residuals have mean zero, and Assumption 5-(i) implies that $C\\ge-\\delta(UW)^{1/2}$. Set $\\delta_+=\\max\\{\\delta,0\\}$. Then $C\\ge-\\delta_+(UW)^{1/2}$, and since $\\delta_+\\ge0$ and $2(UW)^{1/2}\\le(n_0/n_1)U+(n_1/n_0)W$, I have\n\\[\nU+W-2C\\le\\Big(1+\\frac{\\delta_+n_0}{n_1}\\Big)U+\\Big(1+\\frac{\\delta_+n_1}{n_0}\\Big)W.\n\\]\nThis yields (E.10) with $1-\\delta_+$ in place of $1-\\delta$. To see why the replacement is needed, take $n=6$, $n_1=2$, $n_0=4$, $\\delta=-9/10$, all compliers, $x=(1,-1,1,-1,0,0)$ and $y(0)=y(1)=(1,1,-1,-1,0,0)$. These satisfy the stated assumptions with $r=1/3$, $\\widetilde\\varepsilon=1/2$, $A=1$ and $B=2$, and give $\\overline\\beta=0$ and $U=W=C=4$: the printed inequality reads $0\\le-1$, and (E.10) claims the variance $3/5$ is at least $19/30$. For (E.12), Assumption 5-(ii) gives $\\max_i\\varepsilon_i^2(a,\\beta_n)\\le\\{\\sum_i\\varepsilon_i^4(a,\\beta_n)\\}^{1/2}\\le B^2n^{1/2}\\sigma^2_{\\varepsilon(a,\\beta_n)}$, and $n^{-1}\\sum_i\\varepsilon_i^2(a,\\beta_n)=(n-1)\\sigma^2_{\\varepsilon(a,\\beta_n)}/n$. The ratio in (E.12) is then at most $B^2n^{3/2}/(n-1)$, which is $o(n)$. The use of Assumption 5-(iv) for (E.14) is correct. For (I.1), the outcome $Y_i-\\beta D_i$ is linear in $(Y_i,D_i)$ and the regressions share one design. The fitted slope is then $\\widehat\\gamma_{y,a}-\\beta\\widehat\\gamma_{d,a}$.",
    "problem": "The article numbers its equations (1) to (21), and the numbers with a section prefix that the Supplementary Material cites do not exist in it. Assumption 6 is stated in Section 5.2 of the article, p.~14; the Supplementary Material states only Assumption 7. In the proof of Lemma 16, the inequality that leads to (E.10) uses a bound on the sum of products of the residuals $\\varepsilon_i(1,\\beta_n)\\varepsilon_i(0,\\beta_n)$, which is the subject of the correlation bound 5-(i) and not of the fourth-moment bound 5-(ii). Changing this pointer alone is insufficient: the printed inequality and (E.10) can fail when $\\delta<0$, which Assumption 5 permits. Claim (E.12) concerns the largest squared residual, which is the subject of 5-(ii) and not of the covariate bound 5-(iv). Remark 4 introduces the slopes $\\widehat\\gamma_{y,1}$, $\\widehat\\gamma_{y,0}$, $\\widehat\\gamma_{d,1}$ and $\\widehat\\gamma_{d,0}$, and no symbols $\\widehat\\tau_{y,z}$ or $\\widehat\\tau_{d,z}$. Each intended target can be recovered from the content.",
    "what_survives": "The definitions and regression algebra are unaffected. The variance lower bound (E.10) and its uses need $1-\\delta_+$, which remains strictly positive throughout the stated range $\\delta<1$."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E014",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Three lines of text that introduce Algorithm 1, then the four-line display (H.1) for the set CS-i. The first line compares the absolute value of Delta squared with the absolute value of Delta-star squared with index m-i. One red outline encloses eta-hat-star with subscript 1 minus alpha, evaluated at (beta, Z), in the second line and again in the third line. The last line reads CS-star intersected with the interval from beta-i to beta-(i+1). Below: where the second equality is justified by Lemma 1. Hence we have.",
     "caption": "Supplement, p. 45 (PDF p. 25 of the Supplement), display (H.1) in the proof of Theorem 3. Outlined: the critical value $\\widehat\\eta^*_{1-\\alpha}(\\beta,Z)$ in the second and third lines, printed without a square.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E012.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P002-T23",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T23"
    }
   ],
   "headline": "In the second and third lines of (H.1) the critical value $\\widehat\\eta^*_{1-\\alpha}(\\beta,Z)$ should be squared, since it is compared with the squared statistic.",
   "id": "P002-E012",
   "location": "Supplement, Section H.2, display (H.1), second and third lines, p. 45 (PDF p. 25 of the Supplement)",
   "pdf_page": 25,
   "printed_page": "45",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P002-T23"
   ],
   "text": {
    "as_printed": "\\[\n\\begin{aligned}\n\\mathrm{CS}_i&=\\big\\{\\beta:|\\Delta^2(\\beta)|\\le|\\Delta^{*2}_{m_i}(\\beta)|,\\ \\beta\\in[\\beta_i,\\beta_{i+1}]\\big\\}\\\\\n&=\\big\\{\\beta:|\\Delta^2(\\beta)|\\le\\hl{\\widehat\\eta^*_{1-\\alpha}(\\beta,Z)},\\ \\beta\\in[\\beta_i,\\beta_{i+1}]\\big\\}\\\\\n&=\\big\\{\\beta:|\\Delta^2(\\beta)|\\le\\hl{\\widehat\\eta^*_{1-\\alpha}(\\beta,Z)}\\big\\}\\cap[\\beta_i,\\beta_{i+1}]\\\\\n&=\\mathrm{CS}^*\\cap[\\beta_i,\\beta_{i+1}],\\qquad\\qquad\\text{(H.1)}\n\\end{aligned}\n\\]\nwhere the second equality is justified by Lemma 1.",
    "corrected": "\\[\n\\begin{aligned}\n\\mathrm{CS}_i&=\\big\\{\\beta:|\\Delta^2(\\beta)|\\le|\\Delta^{*2}_{m_i}(\\beta)|,\\ \\beta\\in[\\beta_i,\\beta_{i+1}]\\big\\}\\\\\n&=\\big\\{\\beta:|\\Delta^2(\\beta)|\\le\\big[\\widehat\\eta^*_{1-\\alpha}(\\beta,Z)\\big]\\hl{{}^{2}},\\ \\beta\\in[\\beta_i,\\beta_{i+1}]\\big\\}\\\\\n&=\\big\\{\\beta:|\\Delta^2(\\beta)|\\le\\big[\\widehat\\eta^*_{1-\\alpha}(\\beta,Z)\\big]\\hl{{}^{2}}\\big\\}\\cap[\\beta_i,\\beta_{i+1}]\\\\\n&=\\mathrm{CS}^*\\cap[\\beta_i,\\beta_{i+1}],\\qquad\\qquad\\text{(H.1)}\n\\end{aligned}\n\\]\nwhere the second equality is justified by Lemma 1. Here $i\\in\\{0,\\ldots,K\\}$, the intervals $[\\beta_i,\\beta_{i+1}]$ are those of Step~2 of Algorithm~1, $m_i$ is the index selected in Step~3, $\\mathrm{CS}_i$ is the set returned in Step~4, $\\widehat\\eta^*_{1-\\alpha}(\\beta,Z)$ is the critical value defined in (13) of the article, and $\\mathrm{CS}^*$ is the confidence set (14).",
    "evidence": "The evidence is a derivation. By Lemma~1, there exists an index $m_i$ such that, on $[\\beta_i,\\beta_{i+1}]$, I have $\\widehat\\eta^*_{1-\\alpha}(\\beta,Z)=|\\Delta^*_{m_i}(\\beta)|$, so that $|\\Delta^{*2}_{m_i}(\\beta)|=\\big[\\widehat\\eta^*_{1-\\alpha}(\\beta,Z)\\big]^2$. For nonnegative $u$ and $q$, the inequality $u\\le q$ holds if and only if $u^2\\le q^2$. With the squares in place, the four sets in (H.1) are therefore equal. Without them the comparison is a different one: if $|\\Delta(\\beta)|=2$ and $\\widehat\\eta^*_{1-\\alpha}(\\beta,Z)=3$, then (14) accepts $\\beta$ because $2\\le3$, while the printed lines reject it because $4>3$.",
    "problem": "By (13) of the article, $\\widehat\\eta^*_{1-\\alpha}(\\beta,Z)$ is an empirical quantile of the simulated absolute values $|\\Delta^*_m(\\beta)|$, and the confidence set $\\mathrm{CS}^*$ of (14) compares $|\\Delta(\\beta)|$ with it. The first line of (H.1) compares the squared statistic with a squared simulated statistic. The second and third lines keep the square on $\\Delta$ and omit it on the critical value. As printed, the sets in these two lines do not equal $\\mathrm{CS}^*\\cap[\\beta_i,\\beta_{i+1}]$ in general.",
    "what_survives": "Algorithm~1 and the display on p.~15 of the article already use the squared comparison. With the squares restored, (H.1) holds and the union over $i$ equals $\\mathrm{CS}^*$, which is the part of the proof of Theorem~3 that concerns Algorithm~1."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E012",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Crop of Algorithm 2 from 'Set e i equal to min of C i and record the pair of the interval from e i minus 1 to e i and m i' to Step 6. Red boxes mark, in Step 5, the pair 'interval from e k to e k plus 1, m k' and, in the next line, the interval 'e k to e k plus 1' at the end of the definition of CS k.",
     "caption": "Algorithm 2, end of Step 4 and Steps 5–6, p. 16. Step 4 records $\\{[e_{i-1},e_i],m_i\\}$ (first line of the crop); the marked pair and interval in Step 5 are shifted by one position.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E011.png"
    },
    {
     "alt": "Crop of Steps 2 to 4 of Algorithm 4. A red box marks 'm i' in 'Step 3: Set e 0 equal to minus infinity and let m i be any element in I of beta 1 minus 1'.",
     "caption": "Supplementary Material, p. 44 (PDF p. 24), Algorithm 4, Step 3. The marked index is $m_i$, where Algorithm 2 has $m_1$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E011-b.png"
    },
    {
     "alt": "Crop of the last two paragraphs of the proof of Theorem 3. Red boxes mark 'm k in I of beta for all beta in the interval from e k to V k', 'CS i equals CS star intersected with the interval from e i to e i plus 1', and the union sign with limits i equal to 0 and n I.",
     "caption": "Supplementary Material, p. 45 (PDF p. 25), Section H.2, last paragraph. The three marked expressions use the indexing of Step 5.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E011-c.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P002-T22",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T22"
    },
    {
     "id": "P002-T28",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T28"
    }
   ],
   "headline": "Step 5 of Algorithms 2 and 4 and the proof of Theorem 3 in Section H.2 label the recorded pairs one position off: Step 4 records the pairs $([e_{i-1},e_i],m_i)$, and Step 3 of Algorithm 4 should set $m_1$.",
   "id": "P002-E011",
   "location": "Algorithm 2, Steps 3–6, p. 16; Supplementary Material, Algorithm 4, Steps 3 and 5, p. 44 (PDF p. 24 of the supplement), and Section H.2, last paragraph, p. 45 (PDF p. 25)",
   "pdf_page": 16,
   "printed_page": "16",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P002-T22",
    "P002-T28"
   ],
   "text": {
    "as_printed": "Algorithm 2, p.~16: ``Step 5: For each interval-index pair in $\\hl{([e_k,e_{k+1}],m_k)}$ recorded in the previous steps: Find $\\mathrm{CS}_k=\\{\\beta:|\\Delta^2(\\beta)|\\leqslant|\\Delta^{*2}_{m_k}(\\beta)|,\\ \\beta\\in\\hl{[e_k,e_{k+1}]}\\}$.'' Step 5 of Algorithm 4 has the same indexing.\n\nAlgorithm 4, Supplementary Material p.~44: ``Step 3: Set $e_0=-\\infty$ and let $\\hl{m_i}$ be any element in $\\mathcal I(\\beta_1-1)$.''\n\nSection H.2, Supplementary Material p.~45: ``For $k\\notin\\{0,n_I\\}$, we have by construction that $[e_k,e_{k+1}]=[e_k,V_k]\\cup[V_k,e_{k+1}]$. We have $\\hl{m_k\\in\\mathcal I(\\beta)\\text{ for all }\\beta\\in[e_k,V_k]}$ by construction and Lemma 1. If $V_k=e_{k+1}$ then we are done. If $V_k\\ne e_{k+1}$, on $(V_k,e_{k+1})$, there is no distinct AR function that intersects the AR function that realizes the $1-\\alpha$ quantile. Hence $m_k\\in\\mathcal I(\\beta)$ for all $\\beta\\in[V_k,e_{k+1})$ and by continuity $m_k\\in I(e_{k+1})$ if $e_{k+1}<\\infty$. By the same argument as in (H.1), the set $\\mathrm{CS}_i$ returned in Step 5 of Algorithm 2 satisfies $\\hl{\\mathrm{CS}_i=\\mathrm{CS}^*\\cap[e_i,e_{i+1}]}$, and hence $\\hl{\\bigcup_{i=0}^{n_I}}\\mathrm{CS}_i=\\mathrm{CS}^*$.''",
    "corrected": "Algorithms 2 and 4: ``Step 5: For each interval-index pair $\\hl{([e_{k-1},e_k],m_k)}$, $k=1,\\ldots,n_I$, recorded in Step 4, where $e_{n_I}=\\infty$ for the last pair: Find $\\mathrm{CS}_k=\\{\\beta:|\\Delta^2(\\beta)|\\leqslant|\\Delta^{*2}_{m_k}(\\beta)|,\\ \\beta\\in\\hl{[e_{k-1},e_k]}\\}$.'' Step 6 is unchanged.\n\nAlgorithm 4: ``Step 3: Set $e_0=-\\infty$ and let $\\hl{m_1}$ be any element in $\\mathcal I(\\beta_1-1)$.''\n\nSection H.2: ``For $\\hl{1\\le k\\le n_I-2}$, we have by construction that $[e_k,e_{k+1}]=[e_k,V_k]\\cup[V_k,e_{k+1}]$. We have $\\hl{m_{k+1}}\\in\\mathcal I(\\beta)$ for all $\\beta\\in[e_k,V_k]$ by construction and Lemma 1. If $V_k=e_{k+1}$ then we are done. If $V_k\\ne e_{k+1}$, on $(V_k,e_{k+1})$, there is no distinct AR function that intersects the AR function that realizes the $1-\\alpha$ quantile. Hence $\\hl{m_{k+1}}\\in\\mathcal I(\\beta)$ for all $\\beta\\in[V_k,e_{k+1})$ and by continuity $\\hl{m_{k+1}}\\in I(e_{k+1})$ if $e_{k+1}<\\infty$. By the same argument as in (H.1), the set $\\mathrm{CS}_i$ returned in Step 5 of Algorithm 2 satisfies $\\mathrm{CS}_i=\\mathrm{CS}^*\\cap\\hl{[e_{i-1},e_i]}$, and hence $\\hl{\\bigcup_{i=1}^{n_I}}\\mathrm{CS}_i=\\mathrm{CS}^*$.''",
    "evidence": "The evidence is a check of the pages and a trace of the algorithm. The first recorded pair is $([e_0,e_1],m_1)$. After the first finite intersection, Step 4 chooses $m_2$ in $\\mathcal I((e_1+V_1)/2)$ or in $\\mathcal I(e_1+1)$, and the second recorded pair is $([e_1,e_2],m_2)$ or $([e_1,\\infty),m_2)$. Each iteration records one pair, including the last. There are $n_I$ pairs indexed $1,\\ldots,n_I$, and the union in Step 6 is correct. Both branches of Step 4 assign $m_{i+1}$, and no index is left undefined. In the research code, the routine for Algorithm 2 pairs the $i$th recorded index with the interval whose left endpoint is the $i$th entry of $(-\\infty,e_1,e_2,\\ldots)$ and whose right endpoint is the $i$th entry of $(e_1,e_2,\\ldots,\\infty)$, which is the pairing of Step 4. I read this part of the code and did not run it for this point.",
    "problem": "Step 3 chooses $m_1$ for the interval $(-\\infty,e_1]$. In iteration $i$, Step 4 records $([e_{i-1},e_i],m_i)$, or $([e_{i-1},\\infty),m_i)$ when it exits, and chooses $m_{i+1}$ for the next interval. The paragraph below the algorithm says the same: the index $m_{i+1}$ realizes the quantile on $[e_i,e_{i+1}]$. Step 5 instead pairs $m_k$ with $[e_k,e_{k+1}]$, on which the recorded index is $m_{k+1}$, and it leaves $[e_0,e_1]$ without an index. The last paragraph of Section H.2 uses the indexing of Step 5 for the index and for $\\mathrm{CS}_i$, and takes a union from $i=0$, while Step 6 takes it from $i=1$. Step 5 says to process the pairs ``recorded in the previous steps'', and those pairs are correct. An implementation that used the displayed formula of Step 5 in place of the recorded pairs could compare with the wrong simulated AR function.",
    "what_survives": "The algorithm determined by the recorded pairs, the argument of Section H.2 once its indices are aligned, Lemma 1 and Theorem 3 are unchanged. The correction is one of labels."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E011",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of the start of Algorithm 2: the title, the Require line, Step 1 and Step 2. A red box marks the two lines 'If J is empty, choose any beta in R and m in I of beta, return CS equal to the set of beta such that the absolute value of Delta squared of beta is at most the absolute value of Delta star squared m of beta, and break'.",
     "caption": "Algorithm 2, Step 1, p. 16. The marked instruction is the comparison that the code omits.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E015.png"
    },
    {
     "alt": "Listing of lines 188 to 202 of the function AR_permutation_test_poly2_cpp. A red box marks lines 194 to 200: an if statement on intersects_list counter equal to 0, a print statement, a comment saying that the observed AR curve must lie below the alpha-quantile AR curve because the Wald estimate must always be in the interval, the assignment CS_region_output=c(-Inf,Inf), and return(CS_region_output).",
     "caption": "Research code, 2_AR_inversion_algo2.R, lines 188–202 (a listing of the replication code, with line numbers added). The marked branch returns the whole real line when no intersection is found.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E015-b.png"
    },
    {
     "alt": "Listing of lines 104 to 110 of the function AR_algo2. A red box marks lines 107 to 110: an if statement on length of intersections_grid equal to 0, a message 'No intersections found - returning full real line', and return(list(c(-Inf, Inf))).",
     "caption": "The rilate package, R/AR_algo2.R, lines 104–110 (a listing of the replication code, with line numbers added). The marked branch returns the whole real line.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E015-c.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "When no two simulated AR functions intersect, both implementations of Algorithm 2 in the replication code return the whole real line without the comparison of the observed AR function with the quantile function that Step 1 prescribes.",
   "id": "P002-E015",
   "location": "Algorithm 2, Step 1, p. 16, as implemented in the replication code: research code, 2_AR_inversion_algo2.R, lines 194–200; rilate package, R/AR_algo2.R, lines 107–110",
   "pdf_page": null,
   "printed_page": "16",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "Algorithm 2, Step 1, p.~16: ``If $\\mathcal J$ is empty, choose any $\\beta\\in\\mathbb R$ and $m\\in\\mathcal I(\\beta)$, return $\\mathrm{CS}=\\{\\beta:|\\Delta^2(\\beta)|\\leqslant|\\Delta^{*2}_m(\\beta)|\\}$ and break.''\n\nResearch code, \\texttt{2\\_AR\\_inversion\\_algo2.R}, lines 194, 197, 198 and 200:\n\n\\texttt{if (intersects\\_list\\$counter==0)\\{}\n\n\\texttt{\\#if there is no intersections amont the AR curves. The observed AR curve must lies below the alpha-quantile AR curve becasue the Wald-estimate must always be the interval}\n\n\\texttt{\\hl{CS\\_region\\_output=c(-Inf,Inf)}}\n\n\\texttt{return(CS\\_region\\_output)}\n\nThe \\texttt{rilate} package, \\texttt{R/AR\\_algo2.R}, lines 107 and 109:\n\n\\texttt{if (length(intersections\\_grid) == 0) \\{}\n\n\\texttt{\\hl{return(list(c(-Inf, Inf)))}}",
    "corrected": "The printed text needs no change. When $\\mathcal J$ is empty, the code should choose an index $m$ that attains the simulated quantile at any finite $\\beta_0$ and \\hl{return $\\{\\beta\\in\\mathbb R:\\Delta^2(\\beta)\\le\\Delta_m^{*2}(\\beta)\\}$}, with its endpoints, in place of $(-\\infty,\\infty)$. In the \\texttt{rilate} package the result should be returned in the usual list, with \\texttt{confidence\\_set} and the p-value that the function has already computed.",
    "evidence": "The evidence is an exact counterexample run through both unmodified routines. Let $n=6$, $n_1=n_0=3$, $Y=(0,1,2,3,4,5)$, $D=(1,0,1,0,1,0)$ and $Z=(1,1,1,0,0,0)$, and let the $M=1000$ simulated assignments all equal $(1,0,1,0,1,0)$, at confidence level $0.95$. The data are consistent with exclusion, monotonicity, two compliers and a local average treatment effect of zero, and Assumption 6 holds for all $20$ assignments. A direct computation gives\n\\[\n\\Delta^2(\\beta)=\\frac{3(\\beta+9)^2}{4(\\beta^2+3)},\\qquad\\Delta_m^{*2}(\\beta)=\\frac{9(\\beta+1)^2}{16}\\quad\\text{for every }m.\n\\]\nSince all simulated AR functions coincide, $\\mathcal J$ is empty. Clearing the positive denominators, I have\n\\[\n\\Delta^2(\\beta)\\le\\Delta_m^{*2}(\\beta)\\iff(\\beta^2-9)\\big(3(\\beta+1)^2+32\\big)\\ge0\\iff|\\beta|\\ge3,\n\\]\nso (14) equals $(-\\infty,-3]\\cup[3,\\infty)$. At $\\beta=0$ the observed value $81/4$ exceeds the critical value $9/16$. Both routines return the whole real line. The package returns it as a bare list, without the fields \\texttt{confidence\\_set} and \\texttt{p\\_value} of its documented result, while its p-value at $\\beta=0$ is $1/1001$. In a second example, with $Y=(0,1,3,2,4,7)$ and the same $D$, $Z$ and simulated assignments, (14) again equals $(-\\infty,-3]\\cup[3,\\infty)$, while both routines again return the whole real line.",
    "problem": "When the simulated AR functions have no intersection, Lemma 1 implies that one index $m$ realizes the quantile on all of $\\mathbb R$, and (14) is the set where the observed AR function is at most the simulated one with index $m$. This set need not be the whole real line. The comment in the research code argues that the observed AR function must lie below the quantile function because the Wald estimate belongs to the set. The Wald estimate belongs to the set because $\\Delta$ vanishes there, and this says nothing about other values of $\\beta$.",
    "what_survives": "Step 1 of Algorithm 2 as printed, Lemma 1 and Theorem 3 are correct. Since the whole real line contains the intended set, coverage cannot fall on a run that reaches this branch; rejections and precision are lost. The example uses $1000$ identical simulated assignments, an event of probability $20^{-1000}$, and no reported result is shown to have reached the branch. The branch is also reached when the fixed tolerance of \\fid{P002-N015} leaves the grid of intersections empty."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E015",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of the end of Algorithm 1, from the last lines of Step 3 to Step 5. A red box marks the less-than-or-equal sign in Step 4, 'Find CS i equal to the set of beta such that the absolute value of Delta squared of beta is at most the absolute value of Delta star squared m i of beta, beta in the interval from beta i to beta i plus 1'.",
     "caption": "Algorithm 1, Step 4, p. 15. The marked relation is the weak inequality with which the paper accepts a value of $\\beta$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E016.png"
    },
    {
     "alt": "Listing of lines 244 to 254 of the function compare_AR. A red box marks line 252, 'return(AR1 < AR2)', under the comment 'Return TRUE if observed is below permutation'.",
     "caption": "The rilate package, R/find_intervals.R, lines 244–254 (a listing of the replication code, with line numbers added). The marked line is the strict comparison of the observed with the simulated AR statistic.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E016-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "The routine of the replication code that builds an acceptance region from two AR functions tests the intervals between their intersections with a strict inequality; it returns nothing when the two functions coincide and it drops isolated points of equality, although the confidence sets of the paper accept with $\\le$.",
   "id": "P002-E016",
   "location": "Algorithm 1, Step 4, p. 15 (also equation (14), p. 12, and Algorithm 2, Step 5, p. 16), as implemented in the replication code: rilate package, R/find_intervals.R, lines 228 and 252; research code, 2_AR_inversion_algo1.R, lines 413 and 426",
   "pdf_page": 15,
   "printed_page": "15",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "Algorithm 1, Step 4, p.~15: ``For each interval $\\{[\\beta_i,\\beta_{i+1}]\\}_{i=0}^K$: Find $\\mathrm{CS}_i=\\{\\beta:|\\Delta^2(\\beta)|\\;\\leqslant\\;|\\Delta^{*2}_{m_i}(\\beta)|,\\ \\beta\\in[\\beta_i,\\beta_{i+1}]\\}$.''\n\nThe \\texttt{rilate} package, \\texttt{R/find\\_intervals.R}, line 252 (line 228 is the same; the research code has the same comparison):\n\n\\texttt{return(AR1 \\hl{<} AR2)}",
    "corrected": "The printed text needs no change. In the code, the interval routine should return $\\{\\beta:N_1(\\beta)Q_2(\\beta)-N_2(\\beta)Q_1(\\beta)\\;\\hl{\\le}\\;0\\}$: the whole real line when the polynomial $P=N_1Q_2-N_2Q_1$ is identically zero, and otherwise the closures of the intervals on which $P<0$ together with every real zero of $P$, kept as a one-point interval $[r,r]$ where it is not already an endpoint. These pieces must be kept by the later intersection and merging steps. The strict comparison may remain for the interior of the intervals.",
    "evidence": "The evidence is a set of exact counterexamples run through the unmodified code of both sources. For the coefficient vector $c_1=(1,0,0,1,0,1)$, which represents $\\beta^2/(1+\\beta^2)$, both routines return an empty result when $c_1$ is compared with itself, where the acceptance set is $\\mathbb R$. They also return an empty result when $c_1$ is compared with $c_0=(0,0,0,1,0,1)$, where $\\beta^2/(1+\\beta^2)\\le0$ holds exactly at $\\beta=0$. For $c_1$ against $(0,0,1,1,0,1)$ they return $[-1,1]$, which is correct.\n\nBoth cases arise from data. Consider eight units with $Y=(0,1,2,3,0,1,2,3)$ and $D=Z=(1,1,1,1,0,0,0,0)$. Assumption 6 holds for all $70$ assignments of four units. The observed assignment and its complement both have $\\Delta^2(\\beta)=8\\beta^2/5$, and the assignment $W=(1,0,0,1,1,0,0,1)$ and its complement have squared AR function identically $0$. Let the $M=20$ simulated assignments be $Z$, $1-Z$, and nine copies of the pair $W$, $1-W$. At level $0.95$ the quantile function of (13) is the observed AR function, so (14) is $\\mathbb R$. Both \\texttt{AR\\_algo1} and \\texttt{AR\\_algo2} of the package return an empty confidence set. When only the interval routine is replaced by one that keeps the two cases, both return $(-\\infty,\\infty)$.",
    "problem": "The routine computes the real zeros of $P=N_1Q_2-N_2Q_1$ for the observed and the simulated AR function, evaluates the strict comparison at one point of each interval between consecutive zeros, and returns the intervals where the comparison holds, with their endpoints. Ordinary intersections are handled correctly, because the endpoints are included. Two cases are lost. If the two functions are identical, $P$ has no isolated zero, the strict comparison fails, and the routine returns nothing, where the set $\\{\\beta:\\Delta^2(\\beta)\\le\\Delta_m^{*2}(\\beta)\\}$ is the whole real line. If the two functions touch at a point without crossing, the comparison fails on both sides and the point is dropped. Note that Assumption 6 does not exclude identical functions. With arms of equal size, an assignment and its complement have the same squared AR function, and a simulated assignment can equal the observed one.",
    "what_survives": "The definitions (13) and (14), the steps of Algorithms 1 and 2 as printed, and the theory are correct. No reported table entry or application result is shown to change. How often the two cases occur in the designs of the paper is not established, and the application was not recomputed."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E016",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Top of page 20 of the article with the running head. A red outline marks the sentence: In the Supplementary Material we report an additional set of results based on the Get Out the Vote experiment in Green et al. (2003). The paragraph on Project STAR follows.",
     "caption": "Article, p. 20 (PDF p. 20), first two lines. Outlined: the sentence that points to the get-out-the-vote application in the Supplementary Material, which the script 1_data_analysis_GGN2003.R computes.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E017.png"
    },
    {
     "alt": "Monospaced listing of code lines 115 to 132, headed ORIGINAL CODE TEXT (not a paper screenshot). A red outline marks line 122, which stores a first-stage F statistic computed by coeftest from fs1. A second red outline marks the statement on line 127 that assigns a linear model to fs1.",
     "caption": "Lines 115–132 of 1_data_analysis_GGN2003.R in the replication files, rendered as text (not a page of the article). Outlined: line 122, which uses fs1, and the statement on line 127 that first creates it.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E017-code.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "The script for the get-out-the-vote application uses the first-stage regression fs1 on line 122, five lines before line 127 creates it, so the first city stops with an error; line 122 should be deleted.",
   "id": "P002-E017",
   "location": "Replication code, 1_data_analysis_GGN2003.R, lines 122 and 127; the article points to this application on p. 20 (PDF p. 20), and its results are Supplement Tables 6 and 7, pp. 49–50",
   "pdf_page": 20,
   "printed_page": "20",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "Replication code (not a passage of the article):\n\nline 2: \\texttt{rm(list=ls())}\n\nline 122: \\texttt{output\\_list\\_temp[['first\\_stage\\_F']] = coeftest(\\hl{fs1},param=c('assignment'),vcov. = vcovHC(\\hl{fs1},type=\"HC0\"))['assignment','t value']$\\texttt{\\textasciicircum}$2}\n\nline 127: \\texttt{fs1 = lm(D\\_observed $\\texttt{\\textasciitilde}$ assignment + xz1 + xz0, data = data\\_temp2)}\n\nline 128: \\texttt{output\\_list\\_temp[['first\\_stage\\_F\\_cov']] = coeftest(fs1,param=c('assignment'),vcov. = vcovHC(fs1,type=\"HC0\"))['assignment','t value']$\\texttt{\\textasciicircum}$2}",
    "corrected": "Delete line 122 of \\texttt{1\\_data\\_analysis\\_GGN2003.R}. Keep lines 127--128, which fit \\texttt{fs1} and compute \\texttt{first\\_stage\\_F\\_cov}, and lines 151--153, which fit \\texttt{fs2} and compute \\texttt{first\\_stage\\_F\\_no\\_cov} and \\texttt{compliance\\_rate}. The F statistic printed in Tables~6 and~7 is the one without covariates, which lines 151--152 compute from \\texttt{fs2}. The deletion repairs the order of the statements only; a complete run still needs the paths, the packages and the six-city data.",
    "evidence": "The evidence is an exact counterexample obtained by running the code, and a recalculation. With the Bridgeport data supplied in the replication files (1307 voters), running the unchanged lines 109--113 and then the unchanged line 122 produces the error \\texttt{object 'fs1' not found}, and nothing is stored. With line 122 omitted, lines 127--128 and 151--153 run for Bridgeport and for Columbus. The saved output of the application in the replication files is an empty list. Exact rational arithmetic on the two city files reproduces the entries of Tables~6 and~7 that require no simulation.\n\n\\begin{audittable}{lrrrr}{Entries of Supplement Tables~6 and~7 recomputed in exact rational arithmetic from the two city files. The columns headed by a city give the recomputed values, with decimals rounded to six places; the columns headed by a table give the printed values.}\nQuantity & Bridgeport & Table 6 & Columbus & Table 7 \\\\\nSample size & 1307 & 1307 & 2424 & 2424 \\\\\nCompliance rate & $2/7=0.285714$ & 0.286 & $53/377=0.140584$ & 0.141 \\\\\nFirst-stage F, no covariate & $1274/5=254.8$ & 254.8 & $19981/108=185.009259$ & 185.0 \\\\\nPoint estimate, no covariate & 0.163425 & 0.163 & 0.103971 & 0.104 \\\\\nPoint estimate, with covariate & 0.173315 & 0.173 & 0.111099 & 0.111 \\\\\n\\end{audittable}\n\nThe whole script could not be run. It also needs directory paths that exist only on the machine where it was written, the \\texttt{ivreg} package, and the data file for the six cities, which is not among the replication files. Without those, a run stops near the top of the script, at the statement on line 7 that sets the working directory.",
    "problem": "The script clears its workspace on line 2 and then loops over the six cities, beginning with Bridgeport. Inside the loop, line 122 computes a first-stage F statistic from the fitted model \\texttt{fs1}, but no earlier statement creates \\texttt{fs1}; its first assignment is on line 127. When R reaches line 122 for the first city, it stops with the error \\texttt{object 'fs1' not found}. The loop then does not finish Bridgeport, does not reach the other five cities or the randomization-based confidence sets, and does not reach the statement on line 164 that saves the results. As released, the script cannot regenerate Supplement Tables~6 and~7. The entry that line 122 would store, \\texttt{first\\_stage\\_F}, is used nowhere else in the replication files.",
    "what_survives": "No printed number is shown to be wrong, and the defect does not show what state the code was in when the tables were produced. For Bridgeport and Columbus, the sample sizes, compliance rates, first-stage F statistics and both point estimates recompute to the printed values. The randomization-based confidence sets and the other four cities were not recomputed; the two-city 2SLS intervals were recalculated separately (see \\fid{P002-N014}). The theory, the simulation scripts and the STAR application do not use this line."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E017",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of six lines of the caption of Table 1. A red box marks the words '(ii) the nominal 95 per cent 2SLS intervals from the IV regression', which are followed on the next line by the formula Y ~ 1 + D + X + X squared, with instruments 1, Z, X, X squared.",
     "caption": "Caption of Table 1, p. 19 (PDF p. 19). The marked words describe the 2SLS comparison as the interval from an IV regression that includes the covariate $X$ and its square.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E019.png"
    },
    {
     "alt": "Listing of source code with line numbers, under the comment 'tsls wald CI'. A red box marks line 76, which sets denominator to the mean of D_observed among units with assignment 1 minus the mean among units with assignment 0. A red box marks the condition abs(denominator)<1e-8 in line 78. A red box marks tsls_t_CI_lo=-Inf,tsls_t_CI_up=Inf in line 80. In the else branch, line 84 calls ivreg with formula Y_observed ~ D_observed+x1+x2 | assignment+x1+x2 and line 87 computes a 95 per cent confidence interval with HC0 standard errors.",
     "caption": "Source code of the research repository, not a page of the article: 2_main_function_with_covariates.R, lines 69–92. The marked line 76 computes the unadjusted difference in treatment rates, the marked condition on line 78 tests it, and the marked assignment on line 80 stores the whole real line; the regression with covariates on line 84 is fitted only otherwise.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E019-code.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "The simulation code stores the covariate-adjusted 2SLS comparison interval as the whole real line whenever the unadjusted difference in treatment rates between the two arms is zero, although the adjusted regression can then still be fitted; the exception should depend on whether the adjusted regression can be computed.",
   "id": "P002-E019",
   "location": "Research repository RandomizaionIV_ResearchCode (commit 4d919c4): 2_main_function_with_covariates.R, lines 76–80. The code is not printed in the article; the comparison that it computes is described in the caption of Table 1, p. 19 (PDF p. 19).",
   "pdf_page": 19,
   "printed_page": "19",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "\\texttt{2\\_main\\_function\\_with\\_covariates.R}, lines 76, 78, 80, 82 and 84:\n\n\\texttt{denominator = mean(data\\_temp[which(data\\_temp[,'assignment']==1),'D\\_observed'])- mean(data\\_temp[which(data\\_temp[,'assignment']==0),'D\\_observed'])}\n\n\\texttt{if (\\hl{abs(denominator)<1e-8})\\{}\n\n\\texttt{output=c(output, tsls\\_t\\_CI\\_lo=-Inf,tsls\\_t\\_CI\\_up=Inf,tsls=NA,Fstat=NA)}\n\n\\texttt{\\}else\\{}\n\n$\\texttt{iv\\_result=ivreg( formula=Y\\_observed\\textasciitilde{}D\\_observed+x1+x2|assignment+x1+x2, data = data\\_tsls)}$\n\nCaption of Table 1, p.~19: ``Rows 1--4 report the coverage rates of (i) our randomization-based confidence sets, (ii) the nominal 95\\% 2SLS intervals from the IV regression $Y\\sim 1+D+X+X^2\\mid 1,Z,X,X^2$, (iii) confidence sets based on the AR statistic with a normal critical value using the same IV regression, and (iv) the tF confidence intervals from Lee et al. (2022) with the same IV regression.''",
    "corrected": "\\texttt{2\\_main\\_function\\_with\\_covariates.R}, lines 76--80: \\hl{the condition on the unadjusted difference is replaced by a test of whether the adjusted regression of line 84 can be computed}, on the same observations and with the same constant and covariates as the fit. In the notation above, the regression can be computed when $C$ has rank three, $z'M_Cz>0$ and $z'M_Cd\\ne0$. In floating-point arithmetic these conditions are tested with a rank tolerance scaled to the matrices. The Wald interval is computed whenever the conditions hold, and another interval is stored only when they fail, under a stated rule. The 2SLS rows of Table 1 and of Table 3 in the Supplementary Material are computed again before any entry is replaced.",
    "evidence": "The evidence is a pair of exact counterexamples, each run through the unchanged code.\n\nIn the first example, eight units have covariate $X=(-3,-2,-1,0,1,2,3,4)$, assignments $z=(1,1,1,1,0,0,0,0)$, treatments $d=(1,1,0,0,1,1,0,0)$ and outcomes $y=2d+e$, where $e=(14,-22,-3,11,-1,3,2,-4)/6000$. Let $x_1$ and $x_2$ be $X$ and $X^2$ minus their means, as in the simulation script. In each arm two of the four units take the treatment, and the unadjusted difference is zero. The residual of $z$ after projection on $C$ is $M_Cz=(-7,1,9,17,-17,-9,-1,7)/42$, so that $z'M_Cz=10/21$ and $z'M_Cd=-16/21$. The first-stage coefficient of the adjusted regression is $-8/5$. The 2SLS coefficient is exactly $2$, and its HC0 variance is $661/512000000$. The $95\\%$ Wald interval is approximately $[1.998,2.002]$ with the normal critical value and $[1.997,2.003]$ with the Student critical value. The data are consistent with a monotone compliance pattern, in which units $1$, $2$, $7$ and $8$ are the compliers, and with potential outcomes $y_i(0)=y_i(1)=y_i$. The LATE is then zero, and the interval excludes it.\n\nThe second example has $100$ units with the counts of the design L5\\%: $48$ always-takers, $47$ never-takers and $5$ compliers, with $50$ units in each arm. The treated arm holds $22$ always-takers, $24$ never-takers and $4$ compliers, and $26$ units take the treatment in each arm. The first-stage coefficient of the adjusted regression is $-85037548/309802841$, the 2SLS coefficient is $2$, and the $95\\%$ Wald interval is approximately $[1.9996,2.0004]$ while the LATE is zero. On both examples the code returns $(-\\infty,\\infty)$ and does not call the regression.\n\nIn the design L5\\%, with $50$ of $100$ units assigned to treatment, the unadjusted difference is zero with probability\n\\[\n\\frac{971765151348859865}{13607964513309364889}\\approx0.0714.\n\\]\nThis is the probability that the exception applies. It is not the size of any change in a coverage rate.",
    "problem": "The code computes \\texttt{denominator}, the difference between the two arms in the mean of the observed treatment. No covariate enters it, and I call it the unadjusted difference. When its absolute value is below $10^{-8}$, the code stores the interval $(-\\infty,\\infty)$, stores \\texttt{NA} for the estimate and for the first-stage $F$ statistic, and does not call the IV regression. The regression that the caption describes, and that the code fits otherwise, includes a constant, the covariate and its square. A zero unadjusted difference does not make this regression singular. Let $C=[\\mathbf 1,x_1,x_2]$ have rank three, let $M_C=I-C(C'C)^{-1}C'$, and let $z$, $d$ and $y$ be the vectors of assignments, treatments and outcomes. The 2SLS coefficient of the treatment is $z'M_Cy/z'M_Cd$, and it is defined whenever $z'M_Cd\\ne0$. The code therefore replaces some finite Wald intervals by the whole real line. Since the whole line always covers the true value and has infinite length, the exception can raise the coverage rate and the median length reported for 2SLS and cannot lower them.",
    "what_survives": "The randomization-based confidence sets, their coverage rates, lengths and computation times, and the AR and tF comparisons do not pass through this exception. Tables 4 and 5 of the Supplementary Material have no 2SLS rows, and the 2SLS intervals of the applications are computed by separate calls. No entry of Table 1 or of Table 3 in the Supplementary Material is shown to change, and the size of any change is not established. Since the exception cannot lower the coverage rate reported for 2SLS, the coverage below the nominal level that Table 1 reports for 2SLS at low compliance rates does not come from it."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E019",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Figure 1 of the article: a plot of log AR squared against a transformed beta axis, with four dashed curves labelled AR_sim1 to AR_sim4, one dot-dash curve labelled AR_obs, a solid quantile curve, and four vertical lines labelled e_i, V_i, S_i and e_i+1. A red box marks the label AR_obs. Below the plot is the caption, in which a red box marks the words 'Illustration of Algorithm 2'.",
     "caption": "Figure 1 and its caption, p. 17 (PDF p. 17). The marked label AR_obs names the dot-dash curve, which the script computes from the unadjusted statistic; the dashed curves and the solid quantile curve come from the regression-adjusted statistic. The marked words of the caption present the figure as an illustration of Algorithm 2.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E020.png"
    },
    {
     "alt": "Listing of source code with line numbers. In line 103 of 4_plot_AR.R a red box marks AR_coef in the assignment AR_obs_coef = AR_coef(data_temp,N1,N0). In line 104 a red box marks AR_permutation_test_calculate_coef_adj in the assignment to AR_sim_coef. Below, lines of 2_AR_inversion3.R show that AR_coef constructs the coefficients without covariates, that AR_coef_adj constructs them with covariates, and that AR_permutation_test_calculate_coef_adj returns AR_coef_adj(data_table, N1, N0), which is marked by a red box in line 126.",
     "caption": "Source code of the research repository, not a page of the article: 4_plot_AR.R, lines 100–104 and 108–109, above the definitions that these lines call in 2_AR_inversion3.R. The marked call on line 103 uses the function without covariates, and the marked call on line 104 uses a function that returns AR_coef_adj, the function with covariates.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E020-code.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "The script that draws Figure 1 computes the observed curve from the unadjusted AR statistic and the four simulated curves from the regression-adjusted statistic, so the region below the quantile curve in the figure is not the confidence set (14); line 104 should call the unadjusted function, as Algorithm 2 requires.",
   "id": "P002-E020",
   "location": "Figure 1 and its caption, p. 17 (PDF p. 17), with the description of the figure on p. 16 (PDF p. 16). Research repository RandomizaionIV_ResearchCode (commit 4d919c4): 4_plot_AR.R, block “Plot for Paper”, lines 103–104. The code is not printed in the article.",
   "pdf_page": 17,
   "printed_page": "17",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "Caption of Figure 1, p.~17: ``Fig. 1. Illustration of Algorithm 2: the dashed lines represent the AR functions from simulated assignments, while the dot-dash line is the AR function from the observed data; the solid curve is the quantile function. We use four simulated assignments and the 75th percentile; the vertical lines denote intersections of the simulated AR functions.''\n\nPage 16: ``The confidence set is the region where the dot-dash curve lies below the solid curve.''\n\n\\texttt{4\\_plot\\_AR.R}, lines 103--104:\n\n\\texttt{AR\\_obs\\_coef = AR\\_coef(data\\_temp,N1,N0)}\n\n\\texttt{AR\\_sim\\_coef = pbsapply(1:nsim\\_permu,AR\\_permutation\\_test\\_calculate\\_coef\\hl{\\_adj},data\\_table=data\\_temp,N1=N1,N0=N0,assignment\\_random=zsim)}",
    "corrected": "\\texttt{4\\_plot\\_AR.R}, line 104 (line 103 is kept):\n\n\\texttt{AR\\_sim\\_coef = pbsapply(1:nsim\\_permu,\\hl{AR\\_permutation\\_test\\_calculate\\_coef},data\\_table=data\\_temp,N1=N1,N0=N0,assignment\\_random=zsim)}\n\nThe figure is then drawn again, and the labels $e_i$, $V_i$, $S_i$ and $e_{i+1}$ are checked against the new crossings. Alternatively, line 103 may call \\texttt{AR\\_coef\\_adj}; the caption and the description on p.~16 should then refer to Algorithm 4 in the Supplementary Material and to its regression-adjusted confidence set.",
    "evidence": "The evidence is an exact counterexample and an exact calculation on the inputs of the figure.\n\nLet eight units have outcomes $Y=(0,1,2,4,0,2,1,3)$, treatments $D=(0,1,1,1,0,0,1,0)$, assignments $Z=(1,1,1,1,0,0,0,0)$ and one covariate $x=(-3,-1,1,3,-3,-1,1,3)$. For this assignment the squared unadjusted statistic and the squared adjusted statistic are\n\\[\n\\frac{4(2\\beta-1)^2}{6\\beta^2-10\\beta+55}\n\\qquad\\text{and}\\qquad\n\\frac{10(2\\beta-1)^2}{10\\beta^2+22\\beta+21}.\n\\]\nBoth denominators are positive for every real $\\beta$. At $\\beta=0$ the two values are $4/55$ and $10/21$. The two functions of the script therefore give different curves for the same data and the same assignment.\n\nWhen I run the block ``Plot for Paper'' with its own seed, the curves and their crossings agree by eye with those of the published figure. The block generates $30$ units, of which $15$ are assigned to treatment, and four simulated assignments. At $\\beta=-89/100$, exact rational arithmetic on these inputs gives the following values of the squared statistics.\n\n\\begin{audittable}{lr}{Inputs of Figure 1 at $\\beta=-0.89$: squared AR statistics, rounded to four decimals. The figure draws the second and third rows.}\nQuantity & Value \\\\\n75th percentile of the four simulated curves, unadjusted & $0.8140$ \\\\\nObserved curve, unadjusted & $1.5902$ \\\\\n75th percentile of the four simulated curves, adjusted & $1.6715$ \\\\\nObserved curve, adjusted & $3.2551$ \\\\\n\\end{audittable}\n\nThe rule of the figure compares the second row with the third and includes $\\beta=-0.89$. The rule (14) compares the second row with the first, the regression-adjusted rule compares the fourth with the third, and both exclude this value.",
    "problem": "Algorithm 2 and the set (14) compare the observed AR statistic with the same statistic computed under each simulated assignment. The data that the script generates include three covariates. Line 103 computes the observed curve with \\texttt{AR\\_coef}, which ignores the covariates and returns the unadjusted statistic of equation (4). Line 104 computes the four simulated curves with \\texttt{AR\\_permutation\\_test\\_calculate\\_coef\\_adj}, which calls \\texttt{AR\\_coef\\_adj} and returns the regression-adjusted statistic of equation (10). The two statistics are different functions of $\\beta$, and no change of normalization turns one into the other. The dot-dash curve of Figure 1 is therefore not the observed counterpart of the dashed curves. The region where it lies below the solid curve is neither the set (14) of Algorithm 2 nor the regression-adjusted set of Algorithm 4 in the Supplementary Material.",
    "what_survives": "The four simulated curves all use the adjusted statistic. Their intersections, their order, the 75th-percentile curve and the illustration of the skipped intersection $S_i$ do not involve the observed curve and are unaffected. Lemma 1, Algorithms 1--4 and Theorem 3 are unaffected, and no table or application result is shown to depend on this block of the script. In the simulation and application scripts that I read, the routines in use apply one statistic to the observed and the simulated assignments. What changes is the dot-dash curve of Figure 1 and, with it, the region that p.~16 describes as the confidence set."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E020",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of Theorem 1 and Remark 3. Red boxes mark '= 1 minus alpha.' at the end of the displayed limit in Theorem 1, the words 'is equal to 1 minus alpha' in the first sentence of Remark 3, and the words 'is exactly 1 minus alpha' in its last sentence.",
     "caption": "Theorem 1 and Remark 3, p. 8 (PDF p. 8). The marked terms are the equality in the theorem and the two phrases of the remark that assert it.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E001.png"
    },
    {
     "alt": "Crop of Theorem 2. A red box marks '= 1 minus alpha.' at the end of the displayed limit of the infimum of coverage over the parameter space Xi n.",
     "caption": "Theorem 2, p. 11 (PDF p. 11). The marked term is the equality with $1-\\alpha$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E001-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P002-T4",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T4"
    },
    {
     "id": "P002-T5",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T5"
    },
    {
     "id": "P002-T29",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T29"
    },
    {
     "id": "P002-T30",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T30"
    }
   ],
   "headline": "The limiting infimum of coverage in Theorems 1 and 2 is at least $1-\\alpha$; it does not equal $1-\\alpha$ for every admitted choice of the constants.",
   "id": "P002-E001",
   "location": "Theorem 1 and Remark 3, p. 8 (PDF p. 8); Theorem 2, p. 11 (PDF p. 11)",
   "pdf_page": 8,
   "printed_page": "8",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P002-T29",
    "P002-T30",
    "P002-T4",
    "P002-T5"
   ],
   "text": {
    "as_printed": "Theorem 1. Fix $\\alpha\\in(0,1)$, $\\delta<1$, $r\\in(0,1/2]$ and $A>0$. Let $\\Theta_n(\\delta,r,A)$ be the parameter space of models that satisfy Assumptions 1--4 with these constants. Then\n\\[\n\\lim_{n\\to\\infty}\\,\\inf_{\\beta\\in\\mathbb R}\\,\\inf_{\\{\\theta\\in\\Theta_n(\\delta,r,A):\\,\\bar\\beta(\\theta)=\\beta\\}}\\mathbb P_\\theta\\{\\beta\\in I^*_{1-\\alpha}(Z)\\}\\;\\hl{=}\\;1-\\alpha.\n\\]\n\nRemark 3. The infimum of the coverage rates of the confidence sets \\hl{is equal to} $1-\\alpha$ because some constant-additive-effect models belong to the set $\\Theta_n(\\delta,r,A)$. In the design-based setting, the variance is not identified, and a conservative variance bound must instead be estimated (Aronow et al., 2014; Imbens \\& Rubin, 2015). The asymptotic coverage rate is bounded below by $1-\\alpha$ for many data-generating processes, and \\hl{is exactly} $1-\\alpha$ when the data are generated under a constant-additive-effect model.\n\nTheorem 2. Fix $k\\in\\mathbb N$, $\\alpha\\in(0,1)$, $r\\in(0,1/2]$, $\\delta<1$ and $\\tilde\\varepsilon,A,B>0$. Let $\\Xi_n(k,\\delta,\\tilde\\varepsilon,A,B,r)$ be the parameter space of models that satisfy Assumptions 1, 2, 4 and 5 with these constants. Then\n\\[\n\\lim_{n\\to\\infty}\\,\\inf_{\\beta\\in\\mathbb R}\\,\\inf_{\\{\\theta\\in\\Xi_n(k,\\delta,\\tilde\\varepsilon,A,B,r):\\,\\bar\\beta(\\theta)=\\beta\\}}\\mathbb P_\\theta\\{\\beta\\in I^{\\mathrm{adj},*}_{1-\\alpha}(Z)\\}\\;\\hl{=}\\;1-\\alpha.\n\\]",
    "corrected": "Theorem 1. Fix $\\alpha\\in(0,1)$, $\\delta<1$, $r\\in(0,1/2]$ and $A>0$. Let $\\Theta_n(\\delta,r,A)$ be the parameter space of models that satisfy Assumptions 1--4 with these constants. Then, \\hl{along the sample sizes $n$ for which $\\Theta_n(\\delta,r,A)$ is nonempty, provided these sample sizes are unbounded,}\n\\[\n\\hl{\\liminf_{n\\to\\infty}}\\,\\inf_{\\beta\\in\\mathbb R}\\,\\inf_{\\{\\theta\\in\\Theta_n(\\delta,r,A):\\,\\bar\\beta(\\theta)=\\beta\\}}\\mathbb P_\\theta\\{\\beta\\in I^*_{1-\\alpha}(Z)\\}\\;\\hl{\\ge}\\;1-\\alpha.\n\\]\n\nRemark 3. The \\hl{limiting} infimum of the coverage rates of the confidence sets \\hl{is at least} $1-\\alpha$. \\hl{It is equal to $1-\\alpha$ when the set $\\Theta_n(\\delta,r,A)$ contains a sequence of models whose coverage tends to $1-\\alpha$, for instance constant-additive-effect models with positive variances $\\sigma^2_{w(a)}$ that satisfy every assumption with the fixed constants. When the constants force $\\sigma^2_{w(a)}=0$, the coverage is one.} In the design-based setting, the variance is not identified, and a conservative variance bound must instead be estimated (Aronow et al., 2014; Imbens \\& Rubin, 2015).\n\nTheorem 2. Fix $k\\in\\mathbb N$, $\\alpha\\in(0,1)$, $r\\in(0,1/2]$, $\\delta<1$ and $\\tilde\\varepsilon,A,B>0$. Let $\\Xi_n(k,\\delta,\\tilde\\varepsilon,A,B,r)$ be the parameter space of models that satisfy Assumptions 1, 2, 4 and 5 with these constants. Then, \\hl{along the sample sizes $n$ for which this parameter space is nonempty, provided these sample sizes are unbounded,}\n\\[\n\\hl{\\liminf_{n\\to\\infty}}\\,\\inf_{\\beta\\in\\mathbb R}\\,\\inf_{\\{\\theta\\in\\Xi_n(k,\\delta,\\tilde\\varepsilon,A,B,r):\\,\\bar\\beta(\\theta)=\\beta\\}}\\mathbb P_\\theta\\{\\beta\\in I^{\\mathrm{adj},*}_{1-\\alpha}(Z)\\}\\;\\hl{\\ge}\\;1-\\alpha.\n\\]\n\nThe last two sentences of the proof of Theorem 1 (Supplement, p.~31), which pass from the inequality to the equality, are deleted. I do not give a condition on the constants of Theorem 2 under which equality holds.",
    "evidence": "The evidence is an exact counterexample to each equality, with a derivation that covers every model in the parameter space. Let $n\\ge2$ and $v\\in\\mathbb R^n$, and set $S_2=\\sum_i v_i^2$ and $S_4=\\sum_i v_i^4$. Then\n\\[\nnS_4-S_2^2=\\sum_{i<j}(v_i^2-v_j^2)^2\\ge0 .\n\\]\nIf $(S_4/n)^{1/4}\\le(1/2)\\{S_2/(n-1)\\}^{1/2}$, then $16(n-1)^2S_4\\le nS_2^2$. Combining the two inequalities, I obtain\n\\[\n\\{16(n-1)^2-n^2\\}S_2^2\\le0,\\qquad 16(n-1)^2-n^2=15(n-2)^2+28(n-2)+12>0,\n\\]\nwhich implies that $S_2=0$.\n\nFor Theorem 1, set $\\delta=0$, $r=1/4$ and $A=1/2$. By Assumption 3(ii) and the previous display with $v_i=w_i(a)-\\mu_{w(a)}$, I have $w_i(a)=c_a$ for all $i$ and each $a\\in\\{0,1\\}$. Under exclusion and monotonicity, $w_i(1)-w_i(0)$ equals $y_i(1)-y_i(0)-\\bar\\beta$ for a complier and $0$ for every other unit, so that $n(c_1-c_0)=\\sum_{i\\in\\mathcal C}\\{y_i(1)-y_i(0)\\}-|\\mathcal C|\\bar\\beta=0$. Then $Y_i-\\bar\\beta D_i$ is the same constant for every unit and every assignment. The observed statistic and every randomization statistic are $0/0=0$, the critical value is $0$, and $\\bar\\beta\\in I^*_{1-\\alpha}(Z)$. For every $\\beta\\in\\mathbb R$ and every $n\\ge2$, there exists a model in the parameter space such that $\\bar\\beta=\\beta$: all units are compliers, $y_i(0)=0$, $y_i(1)=\\beta$ and $n_1=\\lfloor n/2\\rfloor$. I conclude that the infimum in Theorem 1 is $1$ for every $n\\ge2$. Enumerating all assignments in rational arithmetic for $n=4,6,8$ (6, 20 and 70 assignments), I find coverage $1$ in each case.\n\nFor Theorem 2, set $k=1$, $\\delta=0$, $\\tilde\\varepsilon=1/4$, $A=4$, $B=1/2$ and $r=1/4$. The population regression residuals have mean zero. By Assumption 5(ii) and the same argument, they vanish, so that $w_i(a)=\\varphi_a+\\gamma_ax_i$. Since the covariates are centered, $\\varphi_a$ is the mean of $w(a)$, and the identity used for Theorem 1 implies that $\\varphi_1=\\varphi_0$. When the regression in each arm has full rank, the fitted residuals are zero and the adjusted numerator is the difference of the two fitted intercepts, which is $0$. Then $\\bar\\beta\\in I^{\\mathrm{adj},*}_{1-\\alpha}(Z)$, since the critical value is nonnegative. By Chebyshev's inequality for the sample mean and the sample second moment of the covariate, the probability that the regression in either arm is rank deficient is at most $1548/(n-1)$ for every model in the parameter space. For every $\\beta$ and every $n\\ge4$, there exists a model in the parameter space such that $\\bar\\beta=\\beta$: $x_i=(2i-n-1)/n$, all units are compliers, $y_i(0)=0$, $y_i(1)=\\beta$ and $n_1=\\lfloor n/2\\rfloor$. The infimum in Theorem 2 lies between $1-1548/(n-1)$ and $1$, and it tends to $1$.\n\nSeparately, Assumption 5(iv)--(v) imply that $\\tilde\\varepsilon\\le A^{1/2}$, so that $\\Xi_n$ is empty for some admitted constants (for example $A=4$ and $\\tilde\\varepsilon=3$).",
    "problem": "Both theorems assert that the smallest coverage over the parameter space converges to $1-\\alpha$ for every admitted choice of the constants. For some admitted choices the limit is $1$. Assumption 3(ii) bounds the standardized fourth moment of each vector $w(a)=(w_i(a))_{i=1}^n$, where $w_i(a)=y_i\\{d_i(a)\\}-\\bar\\beta d_i(a)$, by the constant $A$. Assumption 5(ii) bounds the standardized fourth moment of the population regression residuals by $B$. For a vector that is not constant, the standardized fourth moment is at least $\\{(n-1)/n\\}^{1/2}$. A constant below $1$ then admits only vectors with zero variance when $n$ is large. For $A=1/2$ this holds for every $n\\ge2$, and every model in $\\Theta_n(\\delta,r,1/2)$ has constant $w(1)$ and $w(0)$. The statistic at the true effect is $0/0$, which the Supplement (p.~23) sets to $0$, and the confidence set contains the true effect under every assignment. Coverage is $1$ for every model, and the infimum is $1$.\n\nThe proof of Theorem 1 ends (Supplement, p.~31) by passing from the lower bound to the equality, since ``a sequence of constant-additive-effect models belongs to our parameter set''. This step needs constant-additive-effect models with positive variances whose coverage tends to $1-\\alpha$, and for these constants the parameter space contains none. The last sentence of Remark 3 fails in the same way, since the models with $y_i(0)=0$ and $y_i(1)=\\beta$ have a constant additive effect and coverage $1$. Note that coverage is above $1-\\alpha$ in both counterexamples. They do not show that coverage can fall below $1-\\alpha$.",
    "what_survives": "This error does not affect the lower bound on coverage. The estimand, the confidence sets (7) and (11), their finite-sample validity under a constant additive effect, the algorithms and Theorem 3 do not use the equality. The simulations and the applications do not use it either, and none was recomputed for this point. Theorems 1 and 2 now state a lower bound, Remark 3 asserts equality only when a suitable sequence of models is exhibited, and Remark 5 extends the lower bound."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E001",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of the statement of Lemma 5: the variance of tau-hat at beta n is at least (1 minus delta) times the sum of n0 over n n1 times S1 squared and n1 over n n0 times S0 squared. A red box marks the factor (1 minus delta).",
     "caption": "Supplement, Lemma 5, p. 23 (PDF p. 3). The marked factor is $1-\\delta$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E002.png"
    },
    {
     "alt": "Crop of the chain of inequalities (C.3) to (C.6) and the two lines of text below it. Red boxes mark the coefficients delta n0 over n1 and delta n1 over n0 in line (C.4), and the phrase 'the elementary inequality 2ab less than or equal to a squared plus b squared' in the sentence that explains the steps.",
     "caption": "Supplement, proof of Lemma 5, p. 24 (PDF p. 4), steps (C.3)–(C.6). The marked coefficients of (C.4) come from multiplying $2ab\\le a^2+b^2$ by $\\delta$, and the marked phrase is the justification given for the step.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E002-b.png"
    },
    {
     "alt": "Crop of the opening of Lemma 16 and its first claim: the variance of tau-tilde-adj is at least (1 minus delta) n0 over n1 n times script S1 squared plus (1 minus delta) n1 over n0 n times script S0 squared, numbered (E.10). Red boxes mark the two factors (1 minus delta).",
     "caption": "Supplement, Lemma 16, claim 1, bound (E.10), p. 34 (PDF p. 14). The marked factors are $1-\\delta$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E002-c.png"
    },
    {
     "alt": "Crop of Theorem 1. A red box marks 'delta less than 1' in the list of constants 'Fix alpha in (0,1), delta less than 1, r in (0, 1/2] and A greater than 0'.",
     "caption": "Theorem 1, p. 8 (PDF p. 8). The marked condition is the only restriction on $\\delta$. Theorem 2 (p. 11) and Assumptions 3 and 5 state the same condition.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E002-d.png"
    },
    {
     "alt": "Two panels, each with a horizontal axis of variance and three rows. Left, Lemma 5: the printed lower bound 65/4752 (about 0.0137; orange square) lies to the right of the exact variance 5/396 (about 0.0126; black diamond on a vertical line), and the corrected lower bound 65/7128 (about 0.0091; blue circle) lies to its left. Right, bound (E.10): printed lower bound 39/55 (about 0.7091), exact variance 36/55 (about 0.6545), corrected lower bound 26/55 (about 0.4727), in the same arrangement.",
     "caption": "Exact variance, printed lower bound and corrected lower bound for the population with $n=12$ and $\\delta=-1/2$ described in the Evidence section above, for Lemma 5 (left) and for (E.10) (right). In both panels the printed bound is above the exact variance and the corrected bound is below it. The variances are computed by enumerating all 66 assignments, and the lower bounds from the displayed formulas.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E002-bounds.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P002-T1",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T1"
    },
    {
     "id": "P002-T2",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T2"
    },
    {
     "id": "P002-T3",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T3"
    }
   ],
   "headline": "The variance lower bounds of Lemma 5 and (E.10) hold with the factor $1-\\max\\{\\delta,0\\}$; with the printed factor $1-\\delta$ they fail for negative $\\delta$, which Theorems 1 and 2 admit.",
   "id": "P002-E002",
   "location": "Supplement, Lemma 5, p. 23 (PDF p. 3), and its proof, steps (C.3)–(C.8), p. 24 (PDF p. 4); Lemma 16, claim 1, bound (E.10), p. 34 (PDF p. 14), and its proof, p. 35 (PDF p. 15)",
   "pdf_page": 3,
   "printed_page": "23",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P002-T1",
    "P002-T2",
    "P002-T3"
   ],
   "text": {
    "as_printed": "Lemma 5. For all $\\theta_n\\in\\Theta_n$, let $\\beta_n=\\bar\\beta(\\theta_n)$. Then\n\\[\n\\operatorname{var}\\bigl(\\widehat\\tau(\\beta_n)\\bigr)\\ge(1-\\hl{\\delta})\\left(\\frac{n_0}{nn_1}S_1^2(\\beta_n)+\\frac{n_1}{nn_0}S_0^2(\\beta_n)\\right).\n\\]\n\nProof of Lemma 5, p.~24:\n\\[\n\\begin{aligned}\n\\cdots&\\le\\sigma^2_{w(1),\\mathcal C}+\\sigma^2_{w(0),\\mathcal C}+2\\delta\\sigma_{w(1),\\mathcal C}\\cdot\\sigma_{w(0),\\mathcal C}&&\\text{(C.3)}\\\\\n&{}\\hl{\\le}\\;\\sigma^2_{w(1),\\mathcal C}+\\sigma^2_{w(0),\\mathcal C}+\\frac{\\delta n_0}{n_1}\\sigma^2_{w(1),\\mathcal C}+\\frac{\\delta n_1}{n_0}\\sigma^2_{w(0),\\mathcal C}&&\\text{(C.4)}\n\\end{aligned}\n\\]\nIn (C.3) we used Assumption 3-(i), in (C.4) we used the elementary inequality $2ab\\le a^2+b^2$, \\ldots\n\nLemma 16, claim 1. We have\n\\[\n\\operatorname{var}\\bigl(\\tilde\\tau_{\\mathrm{adj}}(\\beta_{n_k})\\bigr)\\ge(1-\\hl{\\delta})\\frac{n_0}{n_1n}\\mathcal S_1^2(\\beta_{n_k})+(1-\\hl{\\delta})\\frac{n_1}{n_0n}\\mathcal S_0^2(\\beta_{n_k}).\\qquad\\text{(E.10)}\n\\]",
    "corrected": "Let $\\delta_+=\\max\\{\\delta,0\\}$.\n\nLemma 5. For all $\\theta_n\\in\\Theta_n$, let $\\beta_n=\\bar\\beta(\\theta_n)$. Then\n\\[\n\\operatorname{var}\\bigl(\\widehat\\tau(\\beta_n)\\bigr)\\ge(1-\\hl{\\delta_+})\\left(\\frac{n_0}{nn_1}S_1^2(\\beta_n)+\\frac{n_1}{nn_0}S_0^2(\\beta_n)\\right).\n\\]\n\nLemma 16, claim 1. Consider a sequence $(\\theta_n)_{n=1}^\\infty$ such that $\\theta_n\\in\\Xi_n$ and let $\\beta_n=\\bar\\beta(\\theta_n)$. Let $(\\theta_{n_k})_{k=1}^\\infty$ be any of its subsequences. We have\n\\[\n\\operatorname{var}\\bigl(\\tilde\\tau_{\\mathrm{adj}}(\\beta_{n_k})\\bigr)\\ge(1-\\hl{\\delta_+})\\frac{n_0}{n_1n}\\mathcal S_1^2(\\beta_{n_k})+(1-\\hl{\\delta_+})\\frac{n_1}{n_0n}\\mathcal S_0^2(\\beta_{n_k}).\\qquad\\text{(E.10)}\n\\]\n\nIn the two proofs, $\\delta$ is replaced by $\\delta_+$ from (C.3) to (C.8) and throughout the proof of (E.10). The same replacement is made in the constants of the steps that use these bounds: Lemma 6 for Lemma 5, and (E.13) and (E.26)--(E.27) for (E.10). Alternatively, the two lemmas can be stated for $0\\le\\delta<1$ and applied with $\\delta=0$ to the models with $\\delta<0$.",
    "evidence": "The evidence is an exact counterexample and a derivation of the corrected bounds. Set $n=12$, $n_1=2$, $n_0=10$, $\\delta=-1/2$, $r=1/6$, $A=B=2$, $k=1$ and $\\tilde\\varepsilon=1/2$. Every unit is a complier, with $d_i(0)=0$, $d_i(1)=1$ and $y_i(1)=y_i(0)=y_i$, and the four pairs $(x_i,y_i)=(-1,-1),(-1,1),(1,-1),(1,1)$ each occur three times. Then $\\bar\\beta=0$ and $w_i(1)=w_i(0)=y_i$. The complier covariance and both complier variances are $12/11$, so that Assumption 3(i) reads $12/11\\ge6/11$. The remaining conditions of Assumptions 1--5 hold, the population regression coefficients are zero, and the residuals are $y_i$. Enumerating the 66 assignments in rational arithmetic, I obtain the following values.\n\\begin{audittable}{lrrr}{Population with $n=12$, $n_1=2$, $n_0=10$ and $\\delta=-1/2$: exact variance and the two lower bounds.}\nBound & Exact variance & Printed lower bound & Corrected lower bound \\\\\nLemma 5 & $5/396\\approx0.01263$ & $65/4752\\approx0.01368$ & $65/7128\\approx0.00912$ \\\\\n(E.10) & $36/55\\approx0.6545$ & $39/55\\approx0.7091$ & $26/55\\approx0.4727$ \\\\\n\\end{audittable}\nFor this population the right side of (C.3) is $12/11$ and the right side of (C.4) is $-36/55$. In the proof of (E.10), the left side of the first inequality is $0$ and its right side is $-36/5$. Repeating the four pairs $3m$ times, I obtain a population with $n=12m$ that satisfies the same conditions, and for it the right side of (E.10) is $13/12$ times the variance for every $m\\ge1$. The failure persists along a sequence with $n\\to\\infty$.\n\nFor the corrected bounds, set $\\delta_+=\\max\\{\\delta,0\\}$. Since standard deviations are nonnegative, Assumption 3(i) implies that $\\sigma_{w(1),w(0),\\mathcal C}\\ge-\\delta_+\\sigma_{w(1),\\mathcal C}\\sigma_{w(0),\\mathcal C}$. Since $\\delta_+\\ge0$ and both coefficients are at least $1$, steps (C.3)--(C.8) hold with $\\delta_+$ in place of $\\delta$, and I have\n\\[\nS_{10}^2(\\beta_n)\\le\\Bigl(1+\\frac{\\delta_+n_0}{n_1}\\Bigr)S_1^2(\\beta_n)+\\Bigl(1+\\frac{\\delta_+n_1}{n_0}\\Bigr)S_0^2(\\beta_n).\n\\]\nInserting this bound into (B.1), which reads $\\operatorname{var}(\\widehat\\tau(\\beta))=S_1^2(\\beta)/n_1+S_0^2(\\beta)/n_0-S_{10}^2(\\beta)/n$, I obtain\n\\[\n\\begin{aligned}\n\\operatorname{var}\\bigl(\\widehat\\tau(\\beta_n)\\bigr)&\\ge\\Bigl(\\frac1{n_1}-\\frac1n-\\frac{\\delta_+n_0}{nn_1}\\Bigr)S_1^2(\\beta_n)+\\Bigl(\\frac1{n_0}-\\frac1n-\\frac{\\delta_+n_1}{nn_0}\\Bigr)S_0^2(\\beta_n)\\\\\n&=(1-\\delta_+)\\Bigl(\\frac{n_0}{nn_1}S_1^2(\\beta_n)+\\frac{n_1}{nn_0}S_0^2(\\beta_n)\\Bigr).\n\\end{aligned}\n\\]\nIn the last equality, I used that $n=n_0+n_1$. The same computation with the regression residuals, Assumption 5(i) and (D.5) gives (E.10) with $\\delta_+$.",
    "problem": "Theorems 1 and 2 and Assumptions 3 and 5 require only that $\\delta<1$, and negative values are admitted. For $\\delta<0$, the covariance condition of Assumption 3(i), $\\sigma_{w(1),w(0),\\mathcal C}\\ge-\\delta\\sigma_{w(1),\\mathcal C}\\sigma_{w(0),\\mathcal C}$, says that the correlation of $w(1)$ and $w(0)$ among compliers is at least $-\\delta$ when both complier variances are positive. For $-1\\le\\delta<0$ every population with a constant additive effect satisfies it, since then $w_i(1)=w_i(0)$. For negative $\\delta$ both lower bounds can fail. The proof of Lemma 5 passes from (C.3) to (C.4) by multiplying the inequality $2ab\\le a^2+b^2$, with $a=(n_0/n_1)^{1/2}\\sigma_{w(1),\\mathcal C}$ and $b=(n_1/n_0)^{1/2}\\sigma_{w(0),\\mathcal C}$, by $\\delta$. The direction of the inequality is preserved only if $\\delta\\ge0$. Steps (C.6) and (C.8) also use that the coefficients $1+\\delta n_0/n_1$ and $1+\\delta n_1/n_0$ are nonnegative, which can fail for negative $\\delta$. The first inequality in the proof of (E.10), on p.~35, makes the same step for the regression residuals. When $\\delta<0$, the printed proofs of Theorems 1 and 2 rely on bounds that need not hold, and the printed right sides are $1-\\delta$ times the corrected right sides.",
    "what_survives": "Every model admitted by the original assumptions is still admitted, since a model that satisfies the covariance condition with a negative $\\delta$ also satisfies it with $\\delta=0$. The corrected factor $1-\\delta_+$ is positive, which is what the later steps of the two proofs use. This error does not by itself weaken the coverage conclusions of Theorems 1 and 2 (for the equality asserted there, see \\fid{P002-E001}). For $0\\le\\delta<1$ nothing changes. The repair leaves the confidence-set definitions, algorithms, simulations and applications unchanged."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E002",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Crop of the end of the proof of Lemma 11 around display (C.31). One red box marks the fraction n 0 n 1 over n in the first line of the display. A second red box marks the whole second line, which reads: greater than or equal up to a constant to the squared difference of the arm means.",
     "caption": "Supplementary Material, p. 29 (PDF p. 9), display (C.31). The marked factor $n_0n_1/n$ belongs to the bound for $S_1^{*2}(\\beta_n)$ and cancels in the bound for the variance; the marked second line then lacks a factor $1/n$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E005.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P002-T12",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T12"
    },
    {
     "id": "P002-T13",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T13"
    }
   ],
   "headline": "Display (C.31) omits the factor $1/n_1+1/n_0$ of (C.25): the lower bound for the randomization variance is $\\pi^2(1-\\pi)^2\\Delta_n^2/(n-1)$, which is of order $\\Delta_n^2/n$ and not of order $\\Delta_n^2$.",
   "id": "P002-E005",
   "location": "Supplementary Material, proof of Lemma 11, display (C.31), p. 29 (PDF p. 9 of the supplement)",
   "pdf_page": 9,
   "printed_page": "29",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P002-T12",
    "P002-T13"
   ],
   "text": {
    "as_printed": "Recalling (C.25), this shows that\n\\[\n\\begin{aligned}\n\\operatorname{var}^*(\\widehat\\tau^*(\\beta_n))&\\ge\\frac{\\pi^2(1-\\pi)^2}{n-1}\\,\\hl{\\frac{n_0n_1}{n}}\\,\\Big(\\widehat\\mu_{y(d(1))}-\\widehat\\mu_{y(d(0))}-\\beta_n\\big(\\widehat\\mu_{d(1)}-\\widehat\\mu_{d(0)}\\big)\\Big)^2\\\\\n&{}\\hl{\\gtrsim\\Big(\\widehat\\mu_{y(d(1))}-\\widehat\\mu_{y(d(0))}-\\beta_n\\big(\\widehat\\mu_{d(1)}-\\widehat\\mu_{d(0)}\\big)\\Big)^2}.\\qquad\\text{(C.31)}\n\\end{aligned}\n\\]",
    "corrected": "Recalling (C.25), this shows that\n\\[\n\\operatorname{var}^*(\\widehat\\tau^*(\\beta_n))\\ge\\frac{\\pi^2(1-\\pi)^2}{n-1}\\,\\Delta_n^2\\;\\hl{\\gtrsim\\frac1n}\\,\\Delta_n^2,\\qquad\\Delta_n=\\widehat\\mu_{y(d(1))}-\\widehat\\mu_{y(d(0))}-\\beta_n\\big(\\widehat\\mu_{d(1)}-\\widehat\\mu_{d(0)}\\big).\\qquad\\text{(C.31)}\n\\]\nThe first inequality holds for every $n\\ge2$ and every data set, with $\\pi=n_1/n$. The second uses Assumption 4.",
    "evidence": "The evidence is a derivation and an exact example. Let $w_i=Y_i-\\beta_nD_i$ and $\\bar w=n^{-1}\\sum_{i=1}^nw_i$. By (C.25), (C.26) and the decomposition on p.~29, I have\n\\[\n\\operatorname{var}^*(\\widehat\\tau^*(\\beta_n))=\\frac{\\pi^2(1-\\pi)^2}{n-1}\\Big(\\frac1{n_1}+\\frac1{n_0}\\Big)\\sum_{i=1}^n(w_i-\\bar w)^2\\ge\\frac{\\pi^2(1-\\pi)^2}{n-1}\\cdot\\frac{n}{n_0n_1}\\cdot\\frac{n_0n_1}{n}\\,\\Delta_n^2=\\frac{\\pi^2(1-\\pi)^2}{n-1}\\,\\Delta_n^2 .\n\\]\nSince Assumption 4 keeps $\\pi$ away from $0$ and $1$, the right-hand side is at least a positive constant times $\\Delta_n^2/n$. For the example, let $n=8$, $n_1=n_0=4$ and $\\beta_n=0$, with observed outcomes $(-1,-1,-1,-1,1,1,1,1)$, where the first four units are treated. Then $\\Delta_n=-2$ and $\\pi^2(1-\\pi)^2=1/16$. The variance of $\\widehat\\tau^*(0)$ over the $70$ assignments of four units to treatment is $1/28$. The right-hand side of the first inequality in (C.31) equals $1/14$, which exceeds the variance, and the corrected bound equals $1/28$.",
    "problem": "Let $\\Delta_n=\\widehat\\mu_{y(d(1))}-\\widehat\\mu_{y(d(0))}-\\beta_n(\\widehat\\mu_{d(1)}-\\widehat\\mu_{d(0)})$. The display before (C.31) decomposes the total sum of squares of $Y_i-\\beta_nD_i$ into the sums of squares within the two arms and the term $(n_0n_1/n)\\Delta_n^2$ between the arms. Dropping the sums within the arms bounds the quantity $S_1^{*2}(\\beta_n)$ of (C.26) from below by $\\pi^2(1-\\pi)^2(n_0n_1/n)\\Delta_n^2/(n-1)$, which is the right-hand side of the first inequality in (C.31). That side is therefore a lower bound for $S_1^{*2}(\\beta_n)$ and not for the variance. By (C.25) the variance is $S_1^{*2}(\\beta_n)(1/n_1+1/n_0)$, and $1/n_1+1/n_0=n/(n_0n_1)$ cancels the factor $n_0n_1/n$. The second bound in (C.31) is then too large by a factor of order $n$.",
    "what_survives": "Lemma 11 holds as stated. The estimate after (C.32) needs the between-arm contribution $\\Delta_n^2/\\{n^2\\operatorname{var}^*(\\widehat\\tau^*(\\beta_n))\\}$ to be $O(1/n)$. This holds by the corrected bound, and the last term displayed on p.~29 already has that order. Nothing else in the paper changes."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E005",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of five lines of text beginning 'where W_2i is a standard normal variable. We further centre the average treatment effect'. A red box marks the words 'for the complier group' at the start of the second line, which continues 'so that the LATE is exactly zero.'",
     "caption": "Section 6, p. 18 (PDF p. 18), first lines: the sentence that follows equation (19). The marked words name the complier group as the group in which the treatment effects are centred.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E018.png"
    },
    {
     "alt": "Listing of source code with line numbers. Line 73 is the comment 'calibrate the effects to match exactly'. Red boxes mark lines 75 and 76 under the comment 'For always-takers', which compute factor_at and subtract it from the second column of outcome_raw, and lines 79 and 80 under the comment 'For never-takers', which do the same with factor_nt. Lines 83 and 84, under the comment 'For compliers', compute factor_c and subtract it, and are not marked.",
     "caption": "Source code of the research repository, not a page of the article: 2_gen_data.R, lines 71–84. The marked lines subtract the mean drawn effect from the treated potential outcomes of the always-takers (lines 75–76) and of the never-takers (lines 79–80); lines 83–84 do the same for the compliers.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-E018-code.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "The description of the first simulation design states that the treatment effects are centred in the complier group and omits that the generator also centres them within the always-takers and within the never-takers; the sentence should describe all three centrings.",
   "id": "P002-E018",
   "location": "Section 6, p. 18 (PDF p. 18), first lines: the sentence after equation (19); equations (18) and (19) are on p. 17. Research repository RandomizaionIV_ResearchCode (commit 4d919c4): 2_gen_data.R, lines 73–84. The code is not printed in the article.",
   "pdf_page": 18,
   "printed_page": "18",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "``where $W_{2i}$ is a standard normal variable. We further centre the average treatment effect \\hl{for the complier group} so that the LATE is exactly zero.''\n\n\\texttt{2\\_gen\\_data.R}, lines 75--76 and 79--80:\n\n\\texttt{factor\\_at = mean(outcome\\_raw[1:N\\_at,2]-outcome\\_raw[1:N\\_at,3]) -tau\\_at}\n\n\\texttt{outcome\\_raw[1:N\\_at,2] = outcome\\_raw[1:N\\_at,2] - factor\\_at}\n\n\\texttt{factor\\_nt = mean(outcome\\_raw[(N\\_at+1):(N\\_at+N\\_nt),2]-outcome\\_raw[(N\\_at+1):(N\\_at+N\\_nt),3]) -tau\\_nt}\n\n\\texttt{outcome\\_raw[(N\\_at+1):(N\\_at+N\\_nt),2] = outcome\\_raw[(N\\_at+1):(N\\_at+N\\_nt),2] -factor\\_nt}",
    "corrected": "``where $W_{2i}$ is a standard normal variable. We further centre the treatment effects \\hl{separately within the always-taker, never-taker and complier groups, so that the average treatment effect of each group is exactly zero; in particular,} the LATE is exactly zero.''\n\nEquations (18) and (19) then describe the draws before this centring.",
    "evidence": "The evidence is an exact example, run through the unchanged generator. I write $e_i=0.1W_{2i}$ for the drawn effect of unit $i$ and $\\bar e_g$ for the mean of the drawn effects in group $g$. The printed steps give $y_i(1)=y_i(0)+e_i-\\mathbf{1}\\{i\\in\\mathcal C\\}\\bar e_{\\mathcal C}$, and the generator gives $y_i(1)=y_i(0)+e_i-\\bar e_g$ for $i$ in group $g$. The two populations have the same $y_i(0)$. Their observed outcomes differ by $-\\bar e_{\\mathcal A}$ for each always-taker and agree for every other unit. Suppose that $n_1$ units are assigned to treatment and $n_0$ to control, and that $a_1$ of the $m_{\\mathcal A}$ always-takers are assigned to treatment. Then the difference in mean outcomes between the two arms changes by\n\\[\n-\\bar e_{\\mathcal A}\\left(\\frac{a_1}{n_1}-\\frac{m_{\\mathcal A}-a_1}{n_0}\\right).\n\\]\nFor the example, suppose that there are $100$ units, of which $25$ are always-takers, $25$ never-takers and $50$ compliers, and that every $W_{1i}$ equals $0$. Let the drawn effects have mean $1/4$ among the always-takers and $-1/8$ among the never-takers, and let them equal $-1/8$ for $25$ compliers and $3/8$ for the other $25$. Suppose further that the $25$ always-takers and the $25$ compliers whose drawn effect is $3/8$ are assigned to treatment. The difference in mean outcomes is $-3/4$ under the printed steps and $-7/8$ under the generator. Every unit assigned to treatment takes it and no other unit does. The Wald estimate then changes by the same amount, $-1/8$. The LATE is exactly zero in both populations. The generator, run on these draws, returns the second population.\n\nThe example does not show that any published coverage rate, length or computation time is wrong, and it does not measure how far the simulation results would move under the printed steps.",
    "problem": "Someone who builds the first design from equations (18) and (19) and from this sentence centres the drawn treatment effects of the compliers and leaves those of the always-takers and never-takers as drawn. The generator does more. In each of the three groups it subtracts the mean drawn effect of the group from the treated potential outcomes, so that the average effect is exactly zero in every group. The two procedures give different populations. Since an always-taker takes the treatment under either assignment, the additional centring changes the observed outcomes of the always-takers. The additional centring of the never-takers changes only a potential outcome that is never observed. The printed sentence is true as far as it goes: the complier effects are centred, and the LATE is exactly zero.",
    "what_survives": "Under both procedures the LATE is exactly zero, the average effect of assignment is zero and the treatment effects are heterogeneous. The theorems and the algorithms are unaffected. The simulation scripts call the generator as it stands, so correcting the sentence requires no new simulation. I could not determine which version of the code produced each published entry. If the centring of the always-takers were removed from the generator instead, the summaries for that design would have to be computed again."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-E018",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of the second half of Step 4 of Algorithm 2, from 'Else: Set e i equal to min of C i and record' to 'Set m i plus 1 to be any element in I of e i plus V i over 2'. A red box marks the two lines 'If E i is empty: Set m i plus 1 to be any element of I of e i plus 1'.",
     "caption": "Algorithm 2, Step 4, p. 16. The marked branch chooses a new index after the last intersection; the code omits this choice.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N002.png"
    },
    {
     "alt": "Listing of lines 287 to 297 of the function AR_permutation_test_poly2_cpp. An if statement tests whether ind_int equals the length of intersections_grid, with the comment 'if the intersection is the last point, break'. A red box marks line 290, 'indices = c(indices,s)', followed by break. In the else branch, line 297 sets s to find_quantile_R of ind_int+1.",
     "caption": "Research code, 2_AR_inversion_algo2.R, lines 287–297 (a listing of the replication code, with line numbers added). At the last intersection the marked line records the current index s for the final interval; the call that selects a new index is made only in the other branch, on line 297.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N002-b.png"
    },
    {
     "alt": "Listing of lines 177 to 187 of the function AR_algo2. Under the comment 'Check if we reached the end', an if statement tests whether ind_int equals the length of intersections_grid. A red box marks line 179, 'indices <- c(indices, s)', followed by break. After the if block, lines 185 to 187 set s to find_quantile_index of ind_int + 1.",
     "caption": "The rilate package, R/AR_algo2.R, lines 177–187 (a listing of the replication code, with line numbers added). The marked line records the current index for the final interval.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N002-c.png"
    },
    {
     "alt": "Line chart with a strip below it. The horizontal axis is beta from minus 10 to 24; the vertical axis is the squared AR statistic from 0 to 5.2. A blue curve, the squared critical value used in (14), has two tall peaks, falls to about 2.5 at beta equal to 8, and then stays near 2.8. A black curve, the observed AR function, falls from about 4.6 near beta equal to minus 5 to zero at beta equal to 4 and rises slowly to about 1.8 at beta equal to 24. A dashed orange curve, the AR function kept by the code, starts at beta equal to 8, where a dotted vertical line marks the last intersection, and falls towards zero; an open circle marks where the black and orange curves meet at beta equal to 12. At beta equal to 16 three dots are labelled 128/45 on the blue curve, 18/13 on the black curve and 32/57 on the orange curve. In the strip, a blue bar shows the set (14), from minus 4 to minus 2 and from 2 to the right edge, and an orange bar shows the returned set, from minus 4 to minus 2 and from 2 to 12.",
     "caption": "The eight-unit example at $M=1000$. Upper panel: the observed squared AR function, the squared critical value used in (14), and the simulated AR function that the code keeps beyond the last intersection of two simulated AR functions, at $\\beta=8$. Lower panel: the set (14) and the set returned by the code. To the right of $\\beta=12$ the observed function lies above the kept function and below the squared critical value, so the code rejects values that (14) accepts.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N002-last-intersection.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "When the intersection reached by the loop of Algorithm 2 is the largest of all intersections, both implementations in the replication code keep the previous quantile index for the final interval instead of choosing $m_{i+1}\\in\\mathcal I(e_i+1)$; the returned set can then differ from (14) to the right of the last intersection and can be bounded where (14) is unbounded.",
   "id": "P002-N002",
   "location": "Algorithm 2, Step 4, p. 16 (the same step in Algorithm 4, Supplementary Material p. 44), as implemented in the replication code: research code, 2_AR_inversion_algo2.R, lines 287–292 (and 103–108); rilate package, R/AR_algo2.R, lines 178–180",
   "pdf_page": null,
   "printed_page": "16",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Moderate",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "Algorithm 2, Step 4, p.~16: ``Set $e_i=\\min(\\mathcal C_i)$ and record $\\{[e_{i-1},e_i],m_i\\}$. Set $\\mathcal E_i=\\{j\\in\\mathcal J,j>e_i\\}$. If $\\mathcal E_i$ is empty: Set $m_{i+1}$ to be any element of $\\mathcal I(e_i+1)$.''\n\nResearch code, \\texttt{2\\_AR\\_inversion\\_algo2.R}, lines 287, 290 and 292, where \\texttt{s} is the current quantile index:\n\n\\texttt{if (ind\\_int == length(intersections\\_grid))\\{}\n\n\\texttt{\\hl{indices = c(indices,s)}}\n\n\\texttt{break}\n\nThe \\texttt{rilate} package, \\texttt{R/AR\\_algo2.R}, lines 178--180:\n\n\\texttt{if (ind\\_int == length(intersections\\_grid)) \\{}\n\n\\texttt{\\hl{indices <- c(indices, s)}}\n\n\\texttt{break}",
    "corrected": "The printed text needs no change. In the code, the branch for the last intersection should select the index for the final interval before recording it. In the research code:\n\n\\texttt{if (ind\\_int == length(intersections\\_grid))\\{}\n\n\\texttt{\\hl{s = as.integer(find\\_quantile\\_R(ind\\_int+1,AR\\_sim\\_coef,intersections\\_grid,alpha=1-alpha,tol=tol))}}\n\n\\texttt{indices = c(indices,s)}\n\n\\texttt{break}\n\nIn the \\texttt{rilate} package:\n\n\\texttt{if (ind\\_int == length(intersections\\_grid)) \\{}\n\n\\texttt{\\hl{s <- as.integer(find\\_quantile\\_index(ind\\_int + 1L, AR\\_sim\\_coef, intersections\\_grid, alpha = 1 - alpha, tol = tol))}}\n\n\\texttt{indices <- c(indices, s)}\n\n\\texttt{break}\n\nThe second wrapper of the research code, written in R only (lines 103--108), needs the same line with its fixed significance level $0.05$. The index already recorded for the preceding interval is kept.",
    "evidence": "The evidence is an exact counterexample, run through the unmodified code of both sources, and a derivation. I take eight units, four of them treated in every assignment, with the observed assignment $Z$ and four other assignments $A$, $B$, $H$ and $L$:\n\\begin{audittable}{rrrrrrr}{Data of the example: outcome, treatment received, the observed assignment and four assignments used for the simulation.}\n$Y$ & $D$ & $Z$ & $A$ & $B$ & $H$ & $L$ \\\\\n$3$ & $1$ & $1$ & $1$ & $1$ & $1$ & $1$ \\\\\n$3$ & $0$ & $1$ & $1$ & $0$ & $0$ & $0$ \\\\\n$-1$ & $1$ & $1$ & $0$ & $1$ & $1$ & $1$ \\\\\n$-1$ & $1$ & $1$ & $0$ & $0$ & $1$ & $0$ \\\\\n$1$ & $1$ & $0$ & $1$ & $1$ & $1$ & $0$ \\\\\n$1$ & $0$ & $0$ & $1$ & $0$ & $0$ & $1$ \\\\\n$-3$ & $0$ & $0$ & $0$ & $1$ & $0$ & $1$ \\\\\n$-3$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$\n\\end{audittable}\nFor an assignment $j$ let $t_j(\\beta)$ be the difference of the arm means of $Y-\\beta D$, and let $T(\\beta)=2(\\beta-1)^2+38$. The squared AR function of $j$ is $F_j=16t_j^2/(T-2t_j^2)$, with positive denominator, and\n\\[\nt_Z=2-\\tfrac\\beta2,\\qquad t_A=4,\\qquad t_B=-\\tfrac\\beta2,\\qquad t_H=1-\\beta,\\qquad t_L=0 .\n\\]\nSince $F_j$ is increasing in $t_j^2$, every comparison reduces to one between squares of these linear functions. The simulated AR functions intersect at $-8,-3,0,2/3,1,2,5$ and $8$, and the largest intersection, at $\\beta=8$, is a crossing of $F_A$ and $F_B$. Further,\n\\[\nF_Z\\le F_A\\iff-4\\le\\beta\\le12,\\qquad F_Z\\le F_B\\iff\\beta\\ge2,\\qquad F_Z\\le F_H\\iff|\\beta|\\ge2 .\n\\]\nLet $r$ be the rank of the quantile, and let the $M$ simulated assignments be $r-2$ copies of $L$, one copy each of $A$ and $B$, and $M-r$ copies of $H$. The squared critical value at rank $r$ is then $\\min\\{F_H,\\max(F_A,F_B)\\}$, and $\\max(F_A,F_B)$ equals $F_A$ for $|\\beta|\\le8$ and $F_B$ otherwise. For (13), $r=\\lceil0.95M\\rceil$, and I obtain\n\\[\n\\widehat I^*_{0.95}(Z)=[-4,-2]\\cup[2,\\infty).\n\\]\nJust left of $\\beta=8$ the quantile is realized by $A$, and just right of it by $B$. The code keeps $A$ on $(8,\\infty)$ and returns $[-4,-2]\\cup[2,12]$. At $\\beta=16$ the kept function equals $32/57$, the observed function $18/13$ and the squared critical value $128/45$.\n\\begin{audittable}{llll}{Sets returned at $M=1000$ and confidence level $0.95$. The last column is the intended inversion at each routine's own rank; (13) uses rank $950$, while the package uses $951$.}\nRoutine & Rank $r$ & Returned & After the repair below \\\\\nResearch code (C++ path), no covariate & $950$ & $[-4,-2]\\cup[2,12]$ & $[-4,-2]\\cup[2,\\infty)$ \\\\\n\\texttt{rilate}, no covariate & $951$ & $[-4,-2]\\cup[2,12]$ & $[-4,-2]\\cup[2,\\infty)$ \\\\\nResearch code (C++ path), one covariate & $950$ & $[-20,-18]\\cup[-14,-4]$ & $[-20,-18]\\cup[-14,\\infty)$ \\\\\n\\texttt{rilate}, one covariate & $951$ & $[-20,-18]\\cup[-14,-4]$ & $[-20,-18]\\cup[-14,\\infty)$\n\\end{audittable}\nThe rows with a covariate use $32$ units, obtained by repeating each unit four times, with a covariate taking the values $-1,-1,1,1$ within each group of four and with $Y$ replaced by $Y-16D$. The endpoints in these rows agree with the stated values to $10^{-12}$. In these rows the returned set excludes the value $0$, which the intended set contains. With the same construction, the stated sets also hold at $M=5000$ and $M=10^4$, with ranks $4750$ and $9500$ for the research code and $4751$ and $9501$ for the package; the code was run at $M=1000$ only. A second, independent example with eight simulated assignments has set $(-\\infty,-3]\\cup[-1,\\infty)$, and both routines return only $(-\\infty,-3]$.\n\nWith correctly computed intersections and quantile indices, the repair is justified as follows. Under Assumption 6, or Assumption 7 with covariates, the squared AR functions are continuous, and two distinct ones differ by a rational function with at most four real zeros. To the right of the largest intersection no difference changes sign, so the order of the simulated values is constant there. Any index in $\\mathcal I(e_i+1)$ realizes the quantile on the whole final interval, including its left endpoint by continuity.\n\nThe effect of the defect on the reported numbers is not established. The simulated assignments of the published runs are not in the replication files, and neither are the data of the application to Angrist et al.\\ or of four of the six cities. An argument that the branch is reached with probability of order $2/M$ does not hold in general: for four units with $Y=(0,1,2,4)$, $D=(1,1,0,0)$ and two treated, under complete randomization, the code keeps a wrong index with probability at least $1-2\\times10^{-106}$ at $M=1000$. Coverage over $5000$ data sets moves in steps of $0.0002$, and one changed decision can change a printed third decimal.",
    "problem": "Step 4 follows the simulated AR function that currently realizes the quantile, with index $m_i$, to its next intersection $e_i$, and then chooses a new index for the interval to the right of $e_i$. The printed step does this in both cases: when later intersections exist it evaluates $\\mathcal I$ between $e_i$ and the next one, and when $e_i$ is the largest of all intersections it evaluates $\\mathcal I$ at $e_i+1$. The code carries out the first case. In the second case it appends the current index \\texttt{s}, which is $m_i$, for the final interval and leaves the loop. If the crossing changes the function that realizes the quantile, the code compares the observed AR function with the wrong simulated one on $(e_i,\\infty)$. The returned set then differs from the intended inversion where the observed function is strictly above the smaller of the two critical functions and at or below the larger. The difference between the two sets consists either only of added points or only of deleted points, and it can be unbounded.\n\nThe defect is confined to the interval to the right of the largest intersection of simulated AR functions. Its effect on the reported numbers is not known. When the last intersection is a unique simple crossing of two singly occurring functions, the kept function is one rank away from the correct one, but this bounds neither how often the branch is reached nor how far an endpoint moves: in the example below the kept function is exactly one rank away at $M=1000$, $5000$ and $10^4$, and the right endpoint moves from $\\infty$ to $12$.",
    "what_survives": "Step 4 as printed, Lemma 1, the definitions (13)--(17), Theorem 3 and the theory for the confidence sets are unaffected by this code defect. This defect does not change membership at or to the left of the largest intersection of simulated AR functions. A value $\\beta$ to the right of it changes status only if the observed AR function is strictly above the smaller of the kept function and the correct one and at or below the larger; in particular a changed coverage decision at the true value $0$ in Section 6 requires both a negative largest intersection and this bracket at $0$. The p-value of the \\texttt{rilate} package at $\\beta=0$ is computed separately and is unaffected. The procedures that do not use randomization critical values are unaffected. The simulation and application scripts of the research code call the affected routine. I have not established whether any reported entry changes: this applies to the coverage rates and lengths of the randomization sets in Table 1 and Supplementary Tables 3--5 and 12, to their endpoints in Table 2 and Supplementary Tables 6 and 7, and to the relative lengths in Supplementary Table 10."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-N002",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of Table 1 with seven columns, headed L5, L10, L25, L50 and L90 per cent, IR1 and IR2, and twelve rows of coverage rates, median lengths and times. A red box marks the column heading L25 per cent. A second red box marks the last four entries of that column: 50, 7, 28 571 and 2919, in the rows 'Time (s, Algorithm 3, M = 10^3)', 'Time (s, Algorithm 4, M = 10^3)', 'Time (s, Algorithm 3, M = 10^4)' and 'Time (s, Algorithm 4, M = 10^4)'.",
     "caption": "Table 1, p. 19 (PDF p. 19). The marked entries are the four computation times of column L25%. The script selects runs with $20$ compliers for this column, but whether it produced these published values is not established.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N003.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "The script that assembles the computation times of Table 1 fills the column L25% from the output directory of the runs with $20$ compliers; it should read the directory of the runs with $25$ compliers. Whether the published column was assembled in this way is not established.",
   "id": "P002-N003",
   "location": "Table 1, p. 19 (PDF p. 19), rows 9–12, column L25%. Research repository RandomizaionIV_ResearchCode (commit 4d919c4): 3_merge_results_grace2.R, lines 110 and 129. The code is not printed in the article.",
   "pdf_page": 19,
   "printed_page": "19",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "\\texttt{3\\_merge\\_results\\_grace2.R}, line 110, final directory name in the argument \\texttt{path} for \\texttt{output\\_25\\_time}:\n\n\\texttt{sim\\_12172024\\_randomization\\_cov\\_\\hl{20}}\n\nLine 129:\n\n\\texttt{output\\_table\\_temp = cbind(output\\_5\\_time,output\\_10\\_time,output\\_25\\_time,output\\_50\\_time,output\\_90\\_time,output\\_IR2dx\\_time,output\\_IR2dx2\\_time)}\n\n\\begin{audittable}{lr}{Table 1, p.~19: rows 9--12 of column L25\\%.}\nRow & L25\\% \\\\\nTime (s, Algorithm 3, $M=10^3$) & $50$ \\\\\nTime (s, Algorithm 4, $M=10^3$) & $7$ \\\\\nTime (s, Algorithm 3, $M=10^4$) & $28\\,571$ \\\\\nTime (s, Algorithm 4, $M=10^4$) & $2919$ \\\\\n\\end{audittable}\n\nCaption of Table 1: ``Rows 9--12 report the median computation times (in seconds) for constructing randomization-based confidence sets using Algorithms 3 and 4 in the Supplementary Material, with $10^3$ and $10^4$ simulated assignments. The times are obtained separately with 50 additional simulations; all times are rounded to the nearest integer''",
    "corrected": "\\texttt{3\\_merge\\_results\\_grace2.R}, line 110: replace the final directory name in the argument \\texttt{path} for \\texttt{output\\_25\\_time} by\n\n\\texttt{sim\\_12172024\\_randomization\\_cov\\_\\hl{25}}\n\nThe comment on line 108 should also read 25. The four medians are then computed from the timing files of the design with $25$ compliers, and an entry of Table 1 is replaced if it changes. The replication files do not contain the times from which replacement values could be computed.",
    "evidence": "The evidence is an exact check of the code on constructed inputs. I ran the unchanged function and the unchanged lines on $50$ constructed timing files in each directory. The medians of the four times are $(200,20,20000,2000)$ in the directory \\texttt{cov\\_20} and $(250,25,25000,2500)$ in the directory \\texttt{cov\\_25}. The third column of the assembled table is $(200,20,20000,2000)$. When the suffix on line 110 is changed to \\texttt{25}, the third column is $(250,25,25000,2500)$ and the other six columns are unchanged.\n\nThe inputs are constructed, and they are not the times of the paper. The original timing files are not in the replication files, and nothing there identifies the run that produced the published column. I cannot say whether any of the entries $50$, $7$, $28\\,571$ and $2919$ differs from the median for the design with $25$ compliers.",
    "problem": "\\begin{audittable}{rl}{Selected lines of \\texttt{3\\_merge\\_results\\_grace2.R}. An ellipsis replaces the directory prefix in each \\texttt{path}; line numbers are from the source.}\nLine & Source code \\\\\n106 & \\texttt{output\\_10\\_time=extract\\_results3(path='\\ldots/sim\\_12172024\\_randomization\\_cov\\_10',tau=0)} \\\\\n110 & \\texttt{output\\_25\\_time=extract\\_results3(path='\\ldots/sim\\_12172024\\_randomization\\_cov\\_20',tau=0)} \\\\\n114 & \\texttt{output\\_50\\_time=extract\\_results3(path='\\ldots/sim\\_12172024\\_randomization\\_cov\\_50',tau=0)} \\\\\n129 & \\texttt{output\\_table\\_temp = cbind(output\\_5\\_time,output\\_10\\_time,output\\_25\\_time,output\\_50\\_time,output\\_90\\_time,output\\_IR2dx\\_time,output\\_IR2dx2\\_time)} \\\\\n\\end{audittable}\nLine 110 assigns to \\texttt{output\\_25\\_time} the medians that \\texttt{extract\\_results3} computes from the directory whose name ends in \\texttt{cov\\_20}. The timing script with $40$ always-takers, $40$ never-takers and $20$ compliers (\\texttt{1\\_simulation\\_grace\\_cov\\_20\\_nmc1000.R}) writes to that directory. The timing script with $38$ always-takers, $37$ never-takers and $25$ compliers (\\texttt{1\\_simulation\\_grace\\_cov\\_25\\_nmc1000.R}) writes to the directory whose name ends in \\texttt{cov\\_25}. Line 129 places \\texttt{output\\_25\\_time} third among the seven timing columns, which is the position of L25\\% in Table 1. If this block assembled the published column from the outputs of these scripts, its four entries describe the design with $20$ compliers and not the design with $25$.",
    "what_survives": "Line 110 affects neither the coverage rates and lengths of column L25\\%, which the script reads from a separate directory of runs with $25$ compliers, nor the other six timing columns. The other six designs support the comparison of the speeds of Algorithms 3 and 4. No theorem, algorithm or application result uses these times."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-N003",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "clarity_note",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "page_check",
   "exhibits": [
    {
     "alt": "Crop of five lines of text beginning 'quickly. Tables 4 and 5 in the Supplementary Material report coverage rates'. Red boxes mark the letter K in 'randomization draws with K in {1000, 5000, 10 000}' and in 'this range of K'.",
     "caption": "Section 6, p. 19 (PDF p. 19), paragraph after Table 1. The two marked symbols $K$ denote the number of simulated assignments, which the paper elsewhere writes $M$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N004.png"
    },
    {
     "alt": "Crop of the four-line caption of Table 4. Red boxes mark N_1 in 'N_1 = 50', N_0 in 'N_0 = 50', and K in '(K = 1000, 5000, 10000)'.",
     "caption": "Supplementary Material, p. 48 (PDF p. 28 of the supplement): caption of Table 4. The marked symbols $N_1$ and $N_0$ are the arm sizes $n_1$ and $n_0$, and the marked $K$ is the number of simulated assignments.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N004-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "The number of simulated assignments is written $K$ in the paragraph after Table 1 and in Section J of the Supplementary Material, where the rest of the paper writes $M$; the sizes of the two arms are written $N_1$ and $N_0$ in the tables of the Supplementary Material and in Table 2, where the paper defines $n_1$ and $n_0$.",
   "id": "P002-N004",
   "location": "Section 6, p. 19 (PDF p. 19), paragraph after Table 1; also Supplementary Material, Section J and the captions of Tables 4 and 5, pp. 47–49 (PDF pp. 27–29 of the supplement), and Table 2, p. 21 (PDF p. 21)",
   "pdf_page": 19,
   "printed_page": "19",
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   "related_entries": [],
   "revisions": [],
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   "severity": null,
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   "text": {
    "as_printed": "Page 19: ``Tables 4 and 5 in the Supplementary Material report coverage rates, median lengths and median computation times for different numbers of randomization draws with $\\hl{K}\\in\\{1000,5000,10\\,000\\}$ using Algorithm 4. We do not observe substantial differences across this range of $\\hl{K}$.''\n\nSupplementary Material, caption of Table 4, p.~48: ``Coverage rates, median lengths, and average computation times for randomization-based confidence sets across simulation designs with $\\hl{N_1}=50$ and $\\hl{N_0}=50$ and with varying numbers of randomization draws ($\\hl{K}=1000,5000,10000$).''",
    "corrected": "Page 19: ``\\ldots for different numbers of randomization draws with $\\hl{M}\\in\\{1000,5000,10\\,000\\}$ using Algorithm 4. We do not observe substantial differences across this range of $\\hl{M}$.''\n\nSupplementary Material, Section J and the captions of Tables 4 and 5: $\\hl{n_1}$ and $\\hl{n_0}$ for the arm sizes, and $\\hl{M}=1000,5000,10000$ for the numbers of randomization draws. Table 2: row labels $\\hl{n_1}$ and $\\hl{n_0}$.",
    "intended_meaning": "In these passages $K$ counts the simulated assignments, which Section 5, Algorithms 1--4 and Table 1 denote by $M$. In Section 5.3 and in the algorithms $K$ is the number of intersection points. The symbols $N_1$ and $N_0$ are the numbers of units assigned to treatment and to control, which the paper defines as $n_1$ and $n_0$. Each meaning is stated where the symbol is used, and no reported number depends on the choice of letter."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-N004",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "page_check",
   "exhibits": [
    {
     "alt": "Three entries of the reference list: Kang, Peck and Keele (2018); Keane and Neal; Keele, Small and Grieve (2017). A red outline marks the year (2021) in the entry: Keane, M. P. and Neal, T. (2021). A practical guide to weak instruments. Annu. Rev. Econ. 16, 185 to 212.",
     "caption": "Reference list, p. 22 (PDF p. 22), entry for Keane & Neal with its neighbours. Outlined: the year 2021, printed with volume 16 of Annu. Rev. Econ.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N007.png"
    },
    {
     "alt": "Three entries of the reference list: Moreira (2009); Nelson and Startz, J. Bus. 63; Nelson and Startz (1990), Econometrica 58. A red outline marks the year (1988) in the entry: Nelson, C. and Startz, R. (1988). The distribution of the instrumental variables estimator and its t-ratio when the instrument is a poor one. J. Bus. 63, S125 to 40.",
     "caption": "Reference list, p. 22 (PDF p. 22), the two entries for Nelson & Startz with the entry before them. Outlined: the year 1988, printed with volume 63 of J. Bus.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N007-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "Two references pair the year of a working paper with the volume and pages of the journal article: the Keane & Neal article appeared in 2024, and the Journal of Business article of Nelson & Startz in 1990.",
   "id": "P002-N007",
   "location": "Reference list, p. 22 (PDF p. 22), entries “Keane, M. P. & Neal, T. (2021)” and “Nelson, C. & Startz, R. (1988)”; citations on pp. 3 and 18",
   "pdf_page": 22,
   "printed_page": "22",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "Keane, M. P. \\& Neal, T. (\\hl{2021}). A practical guide to weak instruments. \\emph{Annu. Rev. Econ.} \\textbf{16}, 185--212.\n\nNelson, C. \\& Startz, R. (\\hl{1988}). The distribution of the instrumental variables estimator and its \\emph{t}-ratio when the instrument is a poor one. \\emph{J. Bus.} \\textbf{63}, S125--40.\n\nIn the text: ``Keane \\& Neal (\\hl{2021})'' (p.~3) and ``(Nelson \\& Startz, \\hl{1988}, 1990; Maddala \\& Jeong, 1992; Angrist \\& Kolesár, 2024)'' (p.~18).",
    "corrected": "Keane, M. P. \\& Neal, T. (\\hl{2024}). A practical guide to weak instruments. \\emph{Annu. Rev. Econ.} \\textbf{16}, 185--212.\n\nNelson, C. \\& Startz, R. (\\hl{1990a}). The distribution of the instrumental variables estimator and its \\emph{t}-ratio when the instrument is a poor one. \\emph{J. Bus.} \\textbf{63}, S125--40.\n\nNelson, C. R. \\& Startz, R. (\\hl{1990b}). Some further results on the exact small sample properties of the instrumental variable estimator. \\emph{Econometrica} \\textbf{58}, 967--76.\n\nIn the text: ``Keane \\& Neal (\\hl{2024})'' on p.~3 and ``(Nelson \\& Startz, \\hl{1990a,b}; Maddala \\& Jeong, 1992; Angrist \\& Kolesár, 2024)'' on p.~18. Alternatively, the years 2021 and 1988 may stand if the entries cite the working papers themselves, without the journal volumes and pages.",
    "evidence": "The evidence is a check against the publishers' records. The Annual Reviews page for the article reads ``Vol. 16:185--212 (Volume publication date August 2024)''. The JSTOR contents of the \\emph{Journal of Business} issue are headed ``Vol. 63, No. 1, Jan., 1990, Part 2'' and list the article on pp.~S125--S140. The working papers are CEPAR Working Paper 2021/24, dated 27 September 2021, and NBER Technical Working Paper 69, issued in September 1988.",
    "problem": "Each of the two entries prints the volume and pages of the journal article with the year of an earlier working paper. Volume 16 of \\emph{Annual Review of Economics}, pp.~185--212, was published in 2024. Volume 63 of \\emph{Journal of Business}, no.~1, part~2, pp.~S125--S140, is dated January 1990. The printed years, 2021 and 1988, are those of the working-paper versions, and the citations on pp.~3 and 18 carry them into the text.",
    "what_survives": "Both works remain identifiable from their authors, titles, volumes and pages, and nothing in the paper depends on the years."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-N007",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "page_check",
   "exhibits": [
    {
     "alt": "Four lines of text from the notation section of the Supplement. A red box marks the citation 'van der Vaart (2000)' in the sentence which says that the conventions for stochastic o and O symbols follow that work.",
     "caption": "Supplement, Section A, p. 21 (PDF p. 1). The marked citation has no entry in a reference list.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N008.png"
    },
    {
     "alt": "Five lines of text from the proof of Lemma 13. A red box marks the words 'Lemma 21.2 of van der Vaart (2000)', which the sentence says states that convergence in distribution implies weak convergence of quantile functions.",
     "caption": "Supplement, proof of Lemma 13, p. 30 (PDF p. 10). The same work is cited for a numbered lemma.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N008-c.png"
    },
    {
     "alt": "A displayed identity for alpha-hat, the sentence that begins 'By the OLS representation in Lin (2013)' and the display that follows it. A red box marks the citation 'Lin (2013)'.",
     "caption": "Supplement, Section I, p. 47 (PDF p. 27). The marked citation has no entry in a reference list.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N008-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "The Supplement cites van der Vaart (2000) and Lin (2013), and neither work has an entry in a reference list.",
   "id": "P002-N008",
   "location": "Supplement, Section A, p. 21 (PDF p. 1); proof of Lemma 13, p. 30 (PDF p. 10); Section I, p. 47 (PDF p. 27)",
   "pdf_page": 1,
   "printed_page": "21",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "For stochastic $o$ and $O$ symbols, we follow the conventions in \\hl{van der Vaart (2000)}. (p.~21)\n\nAs a consequence of this convergence in distribution and \\hl{Lemma 21.2 of van der Vaart (2000)}, which states that convergence in distribution implies weak convergence of quantile functions, \\ldots\\ (p.~30)\n\nBy the OLS representation in \\hl{Lin (2013)}, we have: (p.~47)",
    "corrected": "Add a reference list to the Supplement, or two entries to the reference list of the article, with the full references for van der Vaart (2000) and Lin (2013). I do not supply the entries here, since the two works were not consulted.",
    "evidence": "The evidence is a check of the pages and a search of the text of the article and of the Supplement. The names of the two authors occur only in the three citations above. The two works were not consulted, and no statement taken from them was checked.",
    "problem": "The Supplement has no reference list: it ends with Table 12 on p.~52. The reference list of the article (pp.~21--23) has no entry for either work. The first is cited for a numbered lemma that the proof of Lemma 13 uses, and the second for a representation used in Section I.",
    "what_survives": "Every formula, proof and result is unchanged."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-N008",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Crop of a page of mathematics. In display (E.29) a red box marks Z_i x_i-circle-prime at the start of the first sum. In the three-line expansion of tau-tilde-adj(beta_n) below it, four red boxes mark the letter a in the four occurrences of w_i(a, beta_n), and in the last line a red box marks the factor (1 - Z_i), which stands before the summation sign.",
     "caption": "Supplementary Material, p. 37 (PDF p. 17 of the supplement): display (E.29) and the expansion of $\\tilde\\tau_{\\mathrm{adj}}(\\beta_n)$ that follows it. The marked terms are the factor $Z_i$ in the first sum of (E.29), the four indices $a$, and the factor $(1-Z_i)$ outside the last sum.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N009.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P002-T20",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-T20"
    }
   ],
   "headline": "Three slips in the proof of Lemma 17: the first sum of (E.29) should carry $(1-Z_i)$ in place of $Z_i$, the outcomes $w_i(a,\\beta_n)$ of the next display should be $w_i(1,\\beta_n)$ in the treated sums and $w_i(0,\\beta_n)$ in the control sums, and the factor $(1-Z_i)$ of its last line belongs inside the sum.",
   "id": "P002-N009",
   "location": "Supplementary Material, proof of Lemma 17, p. 37 (PDF p. 17 of the supplement): display (E.29) and the display that follows “For the second claim, in (E.16), first notice that”",
   "pdf_page": 17,
   "printed_page": "37",
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   "severity": "Minor",
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   "target_ids": [
    "P002-T20"
   ],
   "text": {
    "as_printed": "A similar argument shows that\n\\[\n\\frac{1}{n_0\\sqrt{\\operatorname{var}(\\tilde\\tau_{\\mathrm{adj}}(\\beta_n))}}\\sum_{i=1}^n\\hl{Z_i}x_i^{\\circ\\prime}\\left(\\frac{1}{n_0}\\sum_{i=1}^n(1-Z_i)\\tilde x_i\\tilde x_i'\\right)^{-1}\\frac{1}{n_0}\\sum_{i=1}^n(1-Z_i)\\tilde x_i\\varepsilon_i(0,\\beta_n)=o_p(1).\\qquad\\text{(E.29)}\n\\]\nThis proves the first claim. For the second claim, in (E.16), first notice that\n\\[\n\\begin{aligned}\n\\tilde\\tau_{\\mathrm{adj}}(\\beta_n)&=\\frac{1}{n_1}\\sum_{i=1}^nZ_i\\bigl(w_i(\\hl{a},\\beta_n)-x_i'\\gamma(1,\\beta_n)\\bigr)-\\frac{1}{n_0}\\sum_{i=1}^n(1-Z_i)\\bigl(w_i(\\hl{a},\\beta_n)-x_i'\\gamma(0,\\beta_n)\\bigr)\\\\\n&=\\frac{1}{n_1}\\sum_{i=1}^nZ_i\\bigl(w_i(\\hl{a},\\beta_n)-\\varphi(1,\\beta_n)-x_i'\\gamma(1,\\beta_n)\\bigr)\\\\\n&\\qquad-\\frac{1}{n_0}\\sum_{i=1}^n(1-Z_i)\\bigl(w_i(\\hl{a},\\beta_n)-\\varphi(0,\\beta_n)-x_i'\\gamma(0,\\beta_n)\\bigr)\\\\\n&=\\frac{1}{n_1}\\sum_{i=1}^nZ_i\\varepsilon_i(1,\\beta_n)-\\frac{1}{n_0}\\hl{(1-Z_i)}\\sum_{i=1}^n\\varepsilon_i(0,\\beta_n),\n\\end{aligned}\n\\]\nwhere we use the fact that \\ldots",
    "corrected": "A similar argument shows that\n\\[\n\\frac{1}{n_0\\sqrt{\\operatorname{var}(\\tilde\\tau_{\\mathrm{adj}}(\\beta_n))}}\\sum_{i=1}^n\\hl{(1-Z_i)}x_i^{\\circ\\prime}\\left(\\frac{1}{n_0}\\sum_{i=1}^n(1-Z_i)\\tilde x_i\\tilde x_i'\\right)^{-1}\\frac{1}{n_0}\\sum_{i=1}^n(1-Z_i)\\tilde x_i\\varepsilon_i(0,\\beta_n)=o_p(1).\\qquad\\text{(E.29)}\n\\]\nThis proves the first claim. For the second claim, in (E.16), first notice that\n\\[\n\\begin{aligned}\n\\tilde\\tau_{\\mathrm{adj}}(\\beta_n)&=\\frac{1}{n_1}\\sum_{i=1}^nZ_i\\bigl(w_i(\\hl{1},\\beta_n)-x_i'\\gamma(1,\\beta_n)\\bigr)-\\frac{1}{n_0}\\sum_{i=1}^n(1-Z_i)\\bigl(w_i(\\hl{0},\\beta_n)-x_i'\\gamma(0,\\beta_n)\\bigr)\\\\\n&=\\frac{1}{n_1}\\sum_{i=1}^nZ_i\\bigl(w_i(\\hl{1},\\beta_n)-\\varphi(1,\\beta_n)-x_i'\\gamma(1,\\beta_n)\\bigr)\\\\\n&\\qquad-\\frac{1}{n_0}\\sum_{i=1}^n(1-Z_i)\\bigl(w_i(\\hl{0},\\beta_n)-\\varphi(0,\\beta_n)-x_i'\\gamma(0,\\beta_n)\\bigr)\\\\\n&=\\frac{1}{n_1}\\sum_{i=1}^nZ_i\\varepsilon_i(1,\\beta_n)-\\frac{1}{n_0}\\sum_{i=1}^n\\hl{(1-Z_i)}\\varepsilon_i(0,\\beta_n).\n\\end{aligned}\n\\]\nHere $\\beta_n=\\bar\\beta(\\theta_n)$ for a sequence $\\theta_n\\in\\Xi_n$, as in the statement of Lemma 17, and the second equality uses $\\varphi(1,\\beta_n)=\\varphi(0,\\beta_n)$.",
    "evidence": "The evidence is a derivation from the definitions of the Supplementary Material. By the definition (D.1), I have $w_i(a,\\beta)=y_i(d_i(a))-\\beta d_i(a)$, so that $Z_i(Y_i-\\beta D_i)=Z_iw_i(1,\\beta)$ and $(1-Z_i)(Y_i-\\beta D_i)=(1-Z_i)w_i(0,\\beta)$. Substituting these identities into the definition (D.4) of $\\tilde\\tau_{\\mathrm{adj}}(\\beta)$, I obtain the first line of the corrected expansion below. The proof shows that $\\varphi(1,\\beta_n)=\\varphi(0,\\beta_n)$. Since $n_1^{-1}\\sum_{i=1}^nZ_i=n_0^{-1}\\sum_{i=1}^n(1-Z_i)=1$, subtracting $\\varphi(1,\\beta_n)$ inside the first average and $\\varphi(0,\\beta_n)$ inside the second leaves the difference unchanged, which gives the second line. The third line follows from the definition $\\varepsilon_i(a,\\beta)=w_i(a,\\beta)-\\varphi(a,\\beta)-x_i'\\gamma(a,\\beta)$.\n\nFor (E.29), Assumption 5(iii) states that $\\sum_{i=1}^nx_i=0$, which implies that $\\sum_{i=1}^nZ_ix_i^{\\circ}=-\\sum_{i=1}^n(1-Z_i)x_i^{\\circ}$. The printed left side of (E.29) is then the negative of the intended one, and the printed statement is true as it stands. The slip is in the expression and not in the limit.",
    "problem": "Display (E.29) is introduced as the control-arm analogue of the display before it, which concerns the treated-arm remainder (E.22). The control-arm remainder (E.23) begins with $n_0^{-1}\\sum_{i=1}^n(1-Z_i)x_i^{\\circ\\prime}$, and (E.29) prints $Z_i$ in that sum. In the next display the index $a$ is not bound: a treated unit reveals $w_i(1,\\beta_n)$ and a control unit reveals $w_i(0,\\beta_n)$. In the last line of that display the factor $(1-Z_i)$ stands outside the sum over $i$.",
    "what_survives": "Lemma 17 and every result that uses it are unaffected. Only the three expressions change."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-N009",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "clarity_note",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "page_check",
   "exhibits": [
    {
     "alt": "The three timing rows of Table 4, labelled Median Time (1000), Median Time (5000) and Median Time (10000), followed by the caption. A red outline marks the word average in the phrase: Coverage rates, median lengths, and average computation times. The caption ends: Median time is reported in seconds, rounded to the nearest integer.",
     "caption": "Supplement, p. 48 (PDF p. 28 of the Supplement), last rows and caption of Table 4. Outlined: the word “average” in the first sentence of the caption; the rows above and the last sentence say “Median”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N011.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "The captions of Supplement Tables 4 and 5 call the computation times “average” in their first sentence, while the row labels and the last sentence call them “median”; the median is meant.",
   "id": "P002-N011",
   "location": "Supplement, captions of Tables 4 and 5, pp. 48–49 (PDF pp. 28–29 of the Supplement)",
   "pdf_page": 28,
   "printed_page": "48–49",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": null,
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "Table 4: Coverage rates, median lengths, and \\hl{average} computation times for randomization-based confidence sets across simulation designs with $N_1=50$ and $N_0=50$ and with varying numbers of randomization draws ($K=1000,5000,10000$). Median time is reported in seconds, rounded to the nearest integer.\n\nRow labels of the same table: ``Median Time (1000)'', ``Median Time (5000)'', ``Median Time (10000)''. The caption of Table~5 has the same wording with $N_1=30$ and $N_0=70$.",
    "corrected": "Table 4: Coverage rates, median lengths, and \\hl{median} computation times for randomization-based confidence sets across simulation designs with $N_1=50$ and $N_0=50$ and with varying numbers of randomization draws ($K=1000,5000,10000$). Median time is reported in seconds, rounded to the nearest integer.\n\nThe caption of Table~5 changes in the same way.",
    "intended_meaning": "The row labels and the last sentence of each caption name the median, and p.~19 of the article describes the same rows as median computation times. The word ``average'' in the first sentence is loose wording for that summary. This note concerns the wording only. Whether the printed times are medians over the simulations is a separate question, treated in \\fid{P002-N012}."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-N011",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of Table 4 with seven columns, headed L5, L10, L25, L50 and L90 per cent, IR1 and IR2, and nine rows: Coverage, Median Length and Median Time, each for 1000, 5000 and 10000 draws. A red box marks the three rows Median Time (1000), Median Time (5000) and Median Time (10000), with entries 9, 8, 8, 11, 8, 9, 8; 511, 511, 472, 640, 563, 552, 529; and 3640, 3651, 3561, 4863, 3904, 3227, 3650. The caption below ends 'Median time is reported in seconds, rounded to the nearest integer.'",
     "caption": "Supplementary Material, p. 48 (PDF p. 28 of the supplement): Table 4 and its caption. The marked rows are the three rows “Median Time”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N012.png"
    },
    {
     "alt": "Crop of Table 5, laid out as Table 4. A red box marks the three rows Median Time (1000), Median Time (5000) and Median Time (10000), with entries 9, 10, 9, 6, 10, 8, 7; 520, 614, 507, 350, 529, 546, 493; and 3297, 3835, 3639, 2670, 3872, 3560, 3303. The caption below ends 'Median time is reported in seconds, rounded to the nearest integer.'",
     "caption": "Supplementary Material, p. 49 (PDF p. 29 of the supplement): Table 5 and its caption. The marked rows are the three rows “Median Time”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N012-b.png"
    },
    {
     "alt": "Listing of source code with line numbers. Line 111 stores the elapsed time of each file in the first column of output3. Red boxes mark median(output1[1]) in line 130, median(output2[1]) in line 132 and median(output3[1]) in line 134. Below, line 171 of extract_results3 reads return(apply(output,2,median)).",
     "caption": "Source code of the research repository, not a page of the article: 3_merge_results_functions.R, lines 110–111 and 130–135 of extract_results2, and lines 170–171 of extract_results3. Each marked expression takes the median of a single element of the matrix of results; the function that assembles the times of Table 1 takes the median of every column.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N012-code.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "The function that assembles the rows “Median Time” of Tables 4 and 5 in the Supplementary Material returns the time of the first replication and not the median over replications; the three lines should take the median of the first column. Whether the published rows were produced by this function is not established.",
   "id": "P002-N012",
   "location": "Supplementary Material, Tables 4 and 5, rows “Median Time”, pp. 48–49 (PDF pp. 28–29 of the supplement). Research repository RandomizaionIV_ResearchCode (commit 4d919c4): 3_merge_results_functions.R, function extract_results2, lines 130, 132 and 134. The code is not printed in the article.",
   "pdf_page": 28,
   "printed_page": "48–49",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "\\texttt{3\\_merge\\_results\\_functions.R}, lines 130, 132 and 134:\n\n\\texttt{K1000=c(median(output1\\hl{[1]}), apply(output1[,2:3],2,mean))}\n\n\\texttt{K5000=c(median(output2\\hl{[1]}), apply(output2[,2:3],2,mean))}\n\n\\texttt{K10000=c(median(output3\\hl{[1]}), apply(output3[,2:3],2,mean))}\n\n\\begin{audittable}{lrrrrrrr}{Supplementary Material, Tables 4 and 5, pp.~48--49: the rows ``Median Time''. Both captions end ``Median time is reported in seconds, rounded to the nearest integer.''}\nRow & L5\\% & L10\\% & L25\\% & L50\\% & L90\\% & IR1 & IR2 \\\\\nTable 4, Median Time (1000) & 9 & 8 & 8 & 11 & 8 & 9 & 8 \\\\\nTable 4, Median Time (5000) & 511 & 511 & 472 & 640 & 563 & 552 & 529 \\\\\nTable 4, Median Time (10000) & 3640 & 3651 & 3561 & 4863 & 3904 & 3227 & 3650 \\\\\nTable 5, Median Time (1000) & 9 & 10 & 9 & 6 & 10 & 8 & 7 \\\\\nTable 5, Median Time (5000) & 520 & 614 & 507 & 350 & 529 & 546 & 493 \\\\\nTable 5, Median Time (10000) & 3297 & 3835 & 3639 & 2670 & 3872 & 3560 & 3303 \\\\\n\\end{audittable}",
    "corrected": "\\texttt{3\\_merge\\_results\\_functions.R}, lines 130, 132 and 134:\n\n\\texttt{K1000=c(median(output1\\hl{[,1]}), apply(output1[,2:3],2,mean))}\n\n\\texttt{K5000=c(median(output2\\hl{[,1]}), apply(output2[,2:3],2,mean))}\n\n\\texttt{K10000=c(median(output3\\hl{[,1]}), apply(output3[,2:3],2,mean))}\n\nIf this function produced the published rows, the six rows are computed again from the elapsed times of all replications, and an entry is replaced if it changes. The replication files do not contain the times from which replacement values could be computed.",
    "evidence": "The evidence is an exact check of the code on constructed inputs. I ran the unchanged function on three constructed replications. In the order in which the files are read, their times are $(9,1,2)$ for $1000$ draws, $(90,10,20)$ for $5000$ draws and $(900,100,200)$ for $10000$ draws. The function returns $9$, $90$ and $900$. The medians are $2$, $20$ and $200$. When the second file is read first, the function returns $1$, $10$ and $100$, and the medians are the same. With the three corrected lines the function returns $2$, $20$ and $200$, and its coverage and length summaries are unchanged.\n\nThe original times of the individual replications are not in the replication files, and nothing there ties this version of the function to the published tables. I cannot say which of the $42$ published entries, if any, differs from its median.",
    "problem": "In R, a single index applied to a matrix selects one element. The expression \\texttt{output1[1]} is the element in the first row and first column of the matrix \\texttt{output1}, which holds the elapsed time of the first file read, and the median of one number is that number. The three timing entries that \\texttt{extract\\_results2} returns are therefore the times of the first replication, whatever the times of the other replications. The script \\texttt{3\\_merge\\_results\\_grace2.R} (lines 73--75 and 87--89) takes these entries for seven designs and two splits of the arms, which makes $42$ entries laid out as the rows ``Median Time'' of Tables 4 and 5.",
    "what_survives": "The error concerns the timing entries alone; the coverage and length summaries of the same function do not use the expressions in question. The times of Table 1 come from a different function, \\texttt{extract\\_results3}, which returns the median of every column. The statement on p.~19 that Algorithm 4 finishes in under an hour with $10^4$ random assignments describes Table 1, in which all seven entries for Algorithm 4 with $M=10^4$ are below $3600$ seconds (the largest is $3163$). Eight of the $14$ entries for $10000$ draws in Tables 4 and 5 exceed $3600$ seconds. This neither shows that they are times of single replications nor contradicts the statement about Table 1."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-N012",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Header and six rows of Table 5 for the seven simulation designs L5, L10, L25, L50, L90, IR1 and IR2. In the row Coverage (10000), a red outline marks 0.949 and 0.950 in columns IR1 and IR2; the row Coverage (5000) above has the same two values. In the row Median Length (10000), a red outline marks 0.425 in column IR2; the row Median Length (5000) above also has 0.425.",
     "caption": "Supplement Table 5, p. 49 (PDF p. 29 of the Supplement), coverage and median-length rows. Outlined: the IR1 and IR2 coverage rates and the IR2 median length in the 10000-draw rows, which equal the entries of the 5000-draw rows above them.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N013.png"
    },
    {
     "alt": "The eight rows of Table 3. In the row Coverage Rate (Rand), a red outline marks 0.952 and 0.951 in columns IR1 and IR2. In the row Median Length (Rand), a red outline marks 0.422 in column IR2. The L columns of these two rows read 0.949, 0.949, 0.954, 0.948, 0.952 and Inf, Inf, 13.667, 1.495, 0.339.",
     "caption": "Supplement Table 3, p. 48 (PDF p. 28 of the Supplement), for the same design with $10^4$ randomization draws. Outlined: the IR1 and IR2 coverage rates and the IR2 median length of the randomization-based sets, which differ from the outlined entries of Table 5.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N013-b.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "Three entries in the 10000-draw rows of Supplement Table 5 repeat the 5000-draw values; with the values of Table 3, the IR1 and IR2 coverage rates are 0.952 and 0.951 and the IR2 median length is 0.422.",
   "id": "P002-N013",
   "location": "Supplement Table 5, p. 49 (PDF p. 29 of the Supplement), rows “Coverage (10000)” and “Median Length (10000)”, columns IR1 and IR2; compare Supplement Table 3, p. 48 (PDF p. 28 of the Supplement)",
   "pdf_page": 29,
   "printed_page": "49",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "\\begin{audittable}{lrrrrrrr}{Supplement Table~5 ($N_1=30$, $N_0=70$), four of its rows.}\nSimulation Design & L5\\% & L10\\% & L25\\% & L50\\% & L90\\% & IR1 & IR2 \\\\\nCoverage (5000) & 0.953 & 0.956 & 0.953 & 0.955 & 0.955 & 0.949 & 0.950 \\\\\nCoverage (10000) & 0.949 & 0.949 & 0.954 & 0.948 & 0.952 & \\hl{0.949} & \\hl{0.950} \\\\\nMedian Length (5000) & Inf & Inf & 14.899 & 1.526 & 0.341 & Inf & 0.425 \\\\\nMedian Length (10000) & Inf & Inf & 13.667 & 1.495 & 0.339 & Inf & \\hl{0.425} \\\\\n\\end{audittable}\n\n\\begin{audittable}{lrrrrrrr}{Supplement Table~3 (the same design, $10^4$ randomization draws), two of its rows.}\nSimulation Design & L5\\% & L10\\% & L25\\% & L50\\% & L90\\% & IR1 & IR2 \\\\\nCoverage Rate (Rand) & 0.949 & 0.949 & 0.954 & 0.948 & 0.952 & 0.952 & 0.951 \\\\\nMedian Length (Rand) & Inf & Inf & 13.667 & 1.495 & 0.339 & Inf & 0.422 \\\\\n\\end{audittable}",
    "corrected": "\\begin{audittable}{lrrrrrrr}{Supplement Table~5, the two 10000-draw rows, with the IR entries of Table~3.}\nSimulation Design & L5\\% & L10\\% & L25\\% & L50\\% & L90\\% & IR1 & IR2 \\\\\nCoverage (10000) & 0.949 & 0.949 & 0.954 & 0.948 & 0.952 & \\hl{0.952} & \\hl{0.951} \\\\\nMedian Length (10000) & Inf & Inf & 13.667 & 1.495 & 0.339 & Inf & \\hl{0.422} \\\\\n\\end{audittable}\n\nIn \\texttt{3\\_merge\\_results\\_grace2.R}, replace \\texttt{[[2]]} by \\texttt{[[3]]} for the two IR designs on lines 83 and 86, and likewise on line 72. If Table~5 was meant to report a separate simulation run, its IR entries should instead be regenerated from the summaries of that run for $10^4$ draws.",
    "evidence": "The evidence is an exact counterexample for the code and a recalculation for the tables. Four synthetic files of results were built in which the 5000-draw sets have coverage $1/4$ and median length $4$, and the sets for $10^4$ draws have coverage $3/4$ and median length $8$. On these files, the unmodified aggregation function \\texttt{extract\\_results2} and the row assignments of the script return $(3/4,3/4,3/4,3/4,3/4,1/4,1/4)$ for the 10000-draw coverage row of Table~5 and $(8,8,8,8,8,4,4)$ for its median-length row. With \\texttt{[[3]]} in the IR positions they return $3/4$ and $8$ in every column. Applying the selectors of the script to the printed summaries, with the 5000-draw rows of Table~5 as element 2 and the randomization rows of Table~3 as element 3, reproduces all 14 printed cells of the two 10000-draw rows. The simulation outputs behind the tables are not among the replication files. The replacement values are taken from Table~3 and are not recomputed from the simulations.",
    "problem": "Table~3 and the 10000-draw rows of Table~5 report the same quantities for the same design, and they agree in the five L columns. In columns IR1 and IR2 they differ, and Table~5 prints the values of its own 5000-draw rows. The script that assembles both tables, \\texttt{3\\_merge\\_results\\_grace2.R}, accounts for this pattern. For the 10000-draw coverage and median-length rows of Table~5, it selects the list element \\texttt{[[3]]}, the summary for $10^4$ draws, for the five L designs and the element \\texttt{[[2]]}, the summary for 5000 draws, for the two IR designs (lines 83 and 86). The rows of Table~3 use \\texttt{[[3]]} throughout (lines 55 and 57). Four cells of Table~5 are selected from the wrong summary. Three of them differ from Table~3 in print; the fourth, the IR1 median length, is Inf in both tables. The median-length row for 10000 draws in Table~4 has the same slip on line 72, with no difference visible at the printed precision.",
    "what_survives": "The three entries change by $0.003$ and $0.001$ in coverage and by $0.003$ in median length. The remark on p.~19 of the article, that the results do not differ substantially across the numbers of draws, is unchanged. Table~3, the coverage and length rows of Table~1, the theorems, the algorithms and the data applications do not use these selectors."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-N013",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "clarity_note",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "recalculation_from_source",
   "exhibits": [
    {
     "alt": "The last three rows of Table 6 (Wald Est., 2SLS CI, Randomization CI) for Bridgeport, Raleigh and Minneapolis, followed by the first three lines of the caption. A red outline marks the opening words: Table 6: Applications to the Green et al. (2003) dataset.",
     "caption": "Supplement, p. 49 (PDF p. 29 of the Supplement), last rows and caption of Table 6, the first of the five tables concerned. Outlined: the opening words of the caption.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N014.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "Supplement Tables 6–10 cannot be regenerated in full from the replication files, which lack the saved output of the application, the data for four of the six cities and the outputs of the subsampling exercise; no printed entry is shown to be wrong.",
   "id": "P002-N014",
   "location": "Supplement Tables 6–10 and Sections K–L, pp. 49–52 (PDF pp. 29–32 of the Supplement)",
   "pdf_page": 29,
   "printed_page": "49–52",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": null,
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "Table 6: Applications to the Green et al. (2003) dataset for cities Bridgeport, Raleigh, and Minneapolis.\n\nTable 7: Applications to the Green et al. (2003) dataset for cities Detroit, Columbus, and St. Paul.\n\nTable 8: Average sample sizes as a function of the treated-group subsampling rate.\n\nTable 9: Average compliance rates as a function of the treated complier subsampling rate.\n\nTable 10: The median of the difference in length between the randomization-based confidence sets and the 2SLS confidence intervals, relative to the length of the 2SLS confidence intervals.",
    "corrected": "A note with the replication files could read: ``The files include the derived data for Bridgeport and Columbus only. Regenerating Supplement Tables~6--10 also requires the six-city data of Green et al. (2003), the saved results of the six-city analysis with their assignment draws or seeds, and the outputs of the Bridgeport subsampling for each design, with the code and seeds that generated them.'' No change to an entry of the tables is called for.",
    "intended_meaning": "These tables report the application to six cities and Monte Carlo summaries of a subsampling exercise, and nothing in them is shown to be false. The note concerns what the replication files allow. The only saved output of the application is an empty list. The files hold the derived data for Bridgeport and Columbus, but not the data file for the six cities or the data for Raleigh, Minneapolis, Detroit and St.~Paul, and they hold none of the subsampling outputs that the summarizing script reads. The entries that can be recomputed agree with the tables. For Bridgeport and Columbus, the point estimates, the 2SLS intervals, the compliance rates and the first-stage F statistics recompute to the printed rounding. The sample sizes of Table~8 (509.7, 408.0, 306.0) recompute at the printed precision, and the compliance rates of Table~9 (0.167, 0.108, 0.039) are compatible with the expectations under the design. The randomization-based sets, the results for the other four cities and the medians of Table~10 were not recomputed. A separate defect that stops the application script is recorded as \\fid{P002-E017}."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-N014",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of the second half of Remark 7, from 'which, by Assumption 6, is equivalent to' and the displayed product equation to the end of the remark. Red boxes mark the sentence 'Hence, the solution set is the zero set of a polynomial function in beta with degree at most 4.'",
     "caption": "Remark 7, p. 14. The marked sentence is the statement that the code is intended to implement: the intersections are the real zeros of a polynomial of degree at most 4.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N015.png"
    },
    {
     "alt": "Listing of lines 137 to 165 of the function AR_intersection. A red box marks line 140, 'tol <- 1e-8'. A second red box marks lines 146 and 147, an if statement that tests whether the absolute values of v0, v1, v2, v3 and v4 are all below tol and, if so, sets real_roots to an empty vector. The following else-if branches test fewer coefficients against tol and call polyroot on a polynomial of lower degree.",
     "caption": "The rilate package, R/calculate_intersections.R, lines 137–165 (a listing of the replication code, with line numbers added). The marked lines fix the threshold and return no intersection when all five coefficients are below it.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N015-b.png"
    },
    {
     "alt": "Two line charts with logarithmic axes. The horizontal axis is the sample size n and the vertical axis is the largest absolute coefficient over all pairs, labelled from 10 to the minus 10 to 10 to the minus 3. A horizontal black line marks the tolerance 10 to the minus 8, and the region below it is shaded. In each panel a blue line with circles is M equal to 1000 and an orange line with squares is M equal to 10000; both fall steadily as n grows. In the left panel, Columbus composition, the points at n equal to 2424, 4848 and 9696 are filled and above the tolerance, and the points from n equal to 19392 to 155136 are open and below it. In the right panel, equal numbers of the four types, the points are filled up to n equal to 25600; at 51200 the M equal to 1000 point is open and the M equal to 10000 point is filled, and at 102400 both are open.",
     "caption": "The largest absolute coefficient of the intersection polynomials, over all pairs of simulated assignments, against the sample size, for two populations with binary outcome and treatment and balanced assignment. Left: the composition of the Columbus data, replicated $1$ to $64$ times. Right: equal numbers of the four types $(Y,D)$. Below the horizontal line every pair is classified as identical, the code finds no intersection, and its empty-grid branch returns the whole real line. The exact set (14) is a bounded interval at every point shown.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N015-tolerance.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "The replication code uses a fixed absolute threshold of $10^{-8}$ to decide the degree and identity of intersection polynomials, and it can then miss intersections of AR functions and treat distinct AR functions as identical; with binary outcome and treatment and a large sample it returns the whole real line where the confidence set (14) is a bounded interval.",
   "id": "P002-N015",
   "location": "Remark 7, p. 14, and Step 1 of Algorithms 1 and 2, pp. 15–16, as implemented in the replication code: rilate package, R/calculate_intersections.R, lines 140–165; research code, AR_intersection in 2_AR_inversion_algo1.R and find_root in cpp_implementations/auxiliary_functions.cpp",
   "pdf_page": null,
   "printed_page": "14",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Moderate",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "Remark 7, p.~14: ``Hence, the solution set is the zero set of a polynomial function in $\\beta$ with degree at most 4. The solution set may be empty, a discrete set of size at most 4, or the real line. When the solution set is the real line, the two functions agree over the real line and are the same function.''\n\nThe \\texttt{rilate} package, \\texttt{R/calculate\\_intersections.R}, lines 140 and 146--147, where \\texttt{v0}, \\ldots, \\texttt{v4} are the coefficients of the polynomial:\n\n\\texttt{tol <- \\hl{1e-8}}\n\n\\texttt{if (abs(v0)<tol \\& abs(v1)<tol \\& abs(v2)<tol \\& abs(v3)<tol \\& abs(v4)<tol)\\{}\n\n\\texttt{real\\_roots <- c()}\n\nThe branches that follow use the same threshold to choose the degree of the polynomial that is solved. The research code has the same tests with the same threshold.",
    "corrected": "The printed text needs no change. In the code, replace the fixed threshold by \\hl{exact or certified tests of $v_j=0$}, both for the identity of two AR functions and for the degree of $P_{ms}$; keep every nonzero coefficient; and compute all real zeros of $N_mQ_s-N_sQ_m$. The coefficients may be rescaled for the numerical solution, but no term may be dropped. When no pair of simulated AR functions intersects, the code must carry out the comparison of Step 1 of Algorithm 2 (\\fid{P002-E015}).\n\nThe set (14) can also be computed without the pairwise intersections. Under Assumption 6, with $P_{om}=N_oQ_m-N_mQ_o$ and $k=\\lceil(1-\\alpha)M\\rceil$,\n\\[\n\\beta\\in\\widehat I^*_{1-\\alpha}(Z)\\iff\\#\\{m:\\Delta_m^{*2}(\\beta)<\\Delta^2(\\beta)\\}\\le k-1\\iff\\#\\{m:P_{om}(\\beta)\\le0\\}\\ge M-k+1,\n\\]\nso the set is determined by the signs of the \\hl{nonzero} polynomials $P_{om}$ between their real zeros \\hl{and at those zeros}; \\hl{identically zero polynomials count toward the inequality for every $\\beta$}.",
    "evidence": "I use exact counterexamples, run through the unmodified R and C++ code, and recalculations using exact zeros and roots of the calculated AR functions.\n\nFirst, consider the coefficient vectors $(1,0,0,1,0,1)$ and $(1,-2,1,1,0,1)$, which represent $\\beta^2/(\\beta^2+1)$ and $(\\beta-1)^2/(\\beta^2+1)$. Their only intersection is $\\beta=1/2$, and $P=(2\\beta-1)(\\beta^2+1)$. After both vectors are multiplied by $10^{-5}$ each function is unchanged, the coefficients of $P$ are $(-10^{-10},2\\times10^{-10},-10^{-10},2\\times10^{-10},0)$, and the root finders of the \\texttt{rilate} package and of the research code (R and C++) return no intersection. The interval routine of the package then returns the whole real line where the first function is at most the second exactly on $(-\\infty,1/2]$.\n\nSecond, I take eight units with $Y=(0,1,2,3,4,5,6,8)$ and $D=Z=(0,0,0,0,1,1,1,1)$, and $M=40$ simulated assignments consisting of $20$ copies each of $(1,0,0,0,0,1,1,1)$ and $(0,1,0,0,0,1,1,1)$. All denominators are positive, the two simulated AR functions intersect only at $\\beta=5$, and the set (14) at level $0.95$ is $[3,14/3]$. The package returns this interval to within $10^{-12}$. When the outcomes are multiplied by $10^{-7}$, the set (14) is $[3\\times10^{-7},\\,(14/3)\\times10^{-7}]$, the code loses the intersection at $5\\times10^{-7}$, and \\texttt{AR\\_algo2} returns the whole real line.\n\nThird, no rescaling is needed. Let $n$ be a multiple of $8$ such that $n>8$, and consider $n/4$ units of each type $(Y,D)\\in\\{(0,0),(1,0),(0,1),(1,1)\\}$ with $n/2$ treated. Let the observed treatment group contain $0$, $n/8$, $n/8$ and $n/4$ units of these types, the first simulated one $n/8-1$, $n/8+1$, $n/8$ and $n/8$, and the second $n/8-1$, $n/8$, $n/8+1$ and $n/8$, with $20$ copies of each simulated assignment. In the representation used by the code, the numerators are $N_o=(\\beta-1)^2/4$, $N_1=16/n^2$ and $N_2=16\\beta^2/n^2$, the denominators are $Q_j=(\\beta^2+1-N_j)/n>0$, and\n\\[\nP_{12}(\\beta)=\\frac{16}{n^3}\\,(1-\\beta^4),\\qquad\\widehat I^*_{0.95}(Z)=\\Big[1-\\frac8n,\\ \\frac{n}{n-8}\\Big].\n\\]\nThe two simulated AR functions intersect at $\\pm1$. The code discards $P_{12}$ when $16/n^3<10^{-8}$, that is, for $n\\ge1170$. At $n=1168$ it finds both intersections and returns a bounded set. At $n=1176$ the set (14) is $[146/147,\\,147/146]$, and the package returns $(-\\infty,\\infty)$.\n\nFourth, independent simulated assignments lead to the same state. For the pooled composition of the Columbus data ($2075$, $190$, $130$ and $29$ units of the four types) replicated $8$ times, $n=19\\,392$, with balanced assignment and $M=1000$ or $M=10^4$ independent simulated assignments, the largest $|v_j|$ over all pairs is below $10^{-8}$, while (14) equals $[-14/323,\\,50/629]$ and $[-14/323,\\,50/621]$ respectively. At $n=9696$ some pair is still above the threshold. There is no threshold in $n$ alone: for equal numbers of the four types the first tested size is $51\\,200$ at $M=1000$ and $102\\,400$ at $M=10^4$.\n\\begin{audittable}{lll}{The confidence set (14), or (F.4) with the covariate, and the set returned by the code, for the same data and simulated assignments.}\nExample & Set (14) or (F.4) & Returned by the code \\\\\nEight units, outcomes multiplied by $10^{-7}$ & $[3\\times10^{-7},\\,(14/3)\\times10^{-7}]$ & $(-\\infty,\\infty)$ \\\\\nBinary family, $n=1176$, $M=40$ & $[146/147,\\,147/146]$ & $(-\\infty,\\infty)$ \\\\\nColumbus composition, $n=19\\,392$, $M=1000$ & $[-14/323,\\,50/629]$ & $(-\\infty,\\infty)$ \\\\\nColumbus data with covariate, $M=10^4$ (rounded) & $[-0.041574,\\,0.266049]$ & $[-0.041607,\\,0.266049]$\n\\end{audittable}\nThe first and second rows were run through the unmodified code. The fourth used the same root and inversion decisions with optimized storage and grid searches. In the third row the returned set follows from the classification of all pairs, which was computed exactly. In the fourth row the set is the regression-adjusted analogue (F.4) of (14).",
    "problem": "Let $\\Delta_m^{*2}=N_m/Q_m$ be the squared AR function of a simulated assignment $m$, where $N_m$ and $Q_m$ are the quadratics of (16). Under Assumption 6, two such functions meet exactly at the real zeros of $P_{ms}=N_mQ_s-N_sQ_m=\\sum_{j=0}^4v_j\\beta^j$, and they are the same function exactly when all five coefficients $v_j$ are zero. This is Remark 7. The code replaces ``$v_j=0$'' by ``$|v_j|<10^{-8}$'', both to decide whether two functions are the same and to choose the degree of the polynomial that it solves. The coefficients $v_j$ have no fixed scale. Multiplying the coefficient vectors of both functions by $t>0$ multiplies every $v_j$ by $t^2$ and leaves the functions unchanged, and in the binary examples below the $v_j$ decrease as $n$ grows. The code discards nonzero terms that are below the threshold, which can change the set of real zeros. When all five coefficients of a pair are below the threshold, it reports no intersection for that pair. When this happens for every pair, the grid of intersections is empty and the code for Algorithm 2 returns the whole real line (\\fid{P002-E015}).\n\nThe threshold is reached on unrescaled data in the settings of the paper. In recalculations with the same simulated assignments, nonzero terms were discarded in one saved data set for each of the fourteen combinations of design and allocation of Section 6 ($M=1000$) and in the Bridgeport and Columbus data, with and without the covariate. In one recalculation for Columbus without the covariate at $M=1000$, $51\\,705$ pairs of distinct functions are classified as identical and the code reports $621\\,548$ real roots, compared with $977\\,137$ for the full polynomials, counted separately for each pair. Every endpoint returned in these runs still agrees with the exact set (14), or (F.4) with the covariate, to three decimals. The largest error is approximately $0.0000336$, at the lower endpoint for Columbus with the covariate at $M=10^4$, where the code returns approximately $-0.041607$ and the exact value is approximately $-0.041574$. The failure is of a different kind when outcome and treatment are binary and the sample is large: the code returns the whole real line where (14) is a short interval. This occurs from $n=1176$ in a constructed family and at $n=19\\,392$ for the composition of the Columbus data with independent simulated assignments.",
    "what_survives": "Remark 7, Lemma 1, Algorithms 1 to 4 as printed and Theorem 3 use the actual intersections and are not contradicted by this code defect. The definitions of the confidence sets and the finite-sample and asymptotic theory are unaffected. No reported value is shown to change: every endpoint checked in the Section 6 and city recalculations agrees with exact inversion at the three decimals printed in Supplementary Tables 6 and 7. The checks do not cover the application to the data of Angrist et al., the cities Raleigh, Minneapolis, Detroit and St.~Paul, or the Bridgeport subsamples, whose inputs are not in the replication files, and they do not use the simulated assignments of the published runs. The research code also contains a C++ routine in exact integer arithmetic. It tests for exact zero and is not affected. A user who applies the replication code to a large experiment with binary outcome and treatment can obtain the whole real line where (14) is bounded."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-N015",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Listing of the first seventeen lines of the function SolveP3, which solves the cubic x cubed plus a x squared plus b x plus c = 0. A red box marks line 83, 'x[0] = root3(-2 * r);', inside the branch 'if (fabs(q) less than eps)'. Line 78 in the same branch reads 'x[0] = x[1] = x[2] = -a/3;'.",
     "caption": "Research code, cpp_implementations/poly34.cpp, lines 71–87, typeset from the source file with line numbers added; the e-mail addresses in two comments are replaced by <...>. The marked line assigns the real root without subtracting $a/3$. Line 78, for a triple root, includes the shift.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N016.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "In the research code, the branch of the cubic solver for $q=0$ and $r\\ne0$ must subtract $a/3$ from the real root it returns; without this shift an intersection of two AR functions is misplaced.",
   "id": "P002-N016",
   "location": "Research code in the replication files (commit 4d919c4), folder cpp_implementations, file poly34.cpp, function SolveP3, lines 76–87. No passage of the article; the step implemented is the root finding of Remark 7, p. 14",
   "pdf_page": null,
   "printed_page": null,
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "Lines 83--86 of \\texttt{poly34.cpp}, comments omitted:\n\n\\texttt{x[0] = \\hl{root3(-2 * r)};}\n\n\\texttt{x[1] = x[0] * 0.5;}\n\n\\texttt{x[2] = x[0] * sqrt(3.) / 2;}\n\n\\texttt{return(1);}",
    "corrected": "\\texttt{x[0] = root3(-2 * r);}\n\n\\texttt{x[1] = x[0] * 0.5;}\n\n\\texttt{x[2] = x[0] * sqrt(3.) / 2;}\n\n\\texttt{\\hl{x[0] -= a / 3;}}\n\n\\texttt{return(1);}\n\nThe added line corrects the real root, which is the value the intersection routines use. I did not examine the complex pair returned in \\texttt{x[1]} and \\texttt{x[2]}.",
    "evidence": "The evidence is an exact counterexample. The function, compiled as distributed and called with $(a,b,c)=(3,3,2)$, for which $q=0$ and $r=1/2$, returns one real root, exactly $-1$. Since\n\\[\nx^3+3x^2+3x+2=(x+2)(x^2+x+1),\n\\]\nthe only real root is $-2$, and the cubic equals $1$ at $-1$. This cubic arises from data. For a data set of 16 units with binary $D$ and two assignments with 8 treated units each, the squared AR functions are $16(\\beta+3)^2/(15\\beta^2+10\\beta+7)$ and $16(\\beta+1)^2/(15\\beta^2+14\\beta+15)$. Both denominators are positive on the real line. Cross-multiplying, I find the crossing polynomial $1024(\\beta^3+3\\beta^2+3\\beta+2)$. The two functions meet only at $\\beta=-2$, where both equal $16/47$. For this pair the intersection routine passes $(a,b,c)=(3,3,2)$ to the solver and obtains the point $-1$, where the two functions equal $16/3$ and $0$. With the line of the correction inserted, the compiled function returns $-2$.",
    "problem": "The function solves $x^3+ax^2+bx+c=0$ by the substitution $x=y-a/3$, which gives $y^3-3qy+2r=0$ with $q=(a^2-3b)/9$ and $r=(2a^3-9ab+27c)/54$. When $q=0$ and $r\\ne0$, the real root is $y=(-2r)^{1/3}$, and the real root of the original cubic is $x=(-2r)^{1/3}-a/3$. The branch for $|q|<10^{-14}$ returns $(-2r)^{1/3}$, and the other branches of the function subtract $a/3$. The routines that compute the intersections of two AR functions take the returned value as an intersection. When the crossing polynomial of two AR functions is a cubic of this kind, they record a point that is not an intersection and omit the true one. The solver is third-party code included in the research code; its header attributes it to S.~I.~Khashin.",
    "what_survives": "The cross-multiplication argument of the article, Lemma 1, the algorithms as printed and the theorems on coverage are not affected. The R package \\texttt{rilate} finds the roots with \\texttt{polyroot} and returns $-2.0000000000000124$ for this example. The branch is reached only when $q$ vanishes to within $10^{-14}$. No reported number is shown to change, and the simulations and applications were not recomputed with a corrected solver."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P002/#P002-N016",
   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
   },
   "work_id": "P002"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Header and eight rows of Table 1 for the seven simulation designs L5, L10, L25, L50, L90, IR1 and IR2. Red outlines mark the row Coverage rate (2SLS), with entries 0.697, 0.836, 0.922, 0.956, 0.952, 0.840, 0.942, and the row Median length (2SLS), with entries 6.618, 6.943, 2.750, 1.139, 0.298, 3.034, 0.322.",
     "caption": "Table 1, p. 19 (PDF p. 19), coverage and median-length rows. Outlined: the two 2SLS rows, which have the layout of the rows that the aggregation function assembles.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N018.png"
    },
    {
     "alt": "Monospaced listing of code lines 88 to 113, headed ORIGINAL CODE TEXT (not a paper screenshot). The block is labelled K10000. Red outlines mark the word result2 in the two assignments of the 2SLS interval endpoints on lines 100 and 101. Lines 111 and 113 read the time and the randomization interval from result3.",
     "caption": "Lines 88–113 of 3_merge_results_functions.R in the replication files, rendered as text (not a page of the article). Outlined: result2 on lines 100 and 101, in the block for $10^4$ draws; lines 111 and 113 of the same block read result3.",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N018-code.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "In the replication code, the summary built for the runs with $10^4$ randomization draws takes its 2SLS interval from the 5000-draw run; lines 100 and 101 should read result3.",
   "id": "P002-N018",
   "location": "Replication code, 3_merge_results_functions.R, function extract_results2, lines 100–101; the related published rows are “Coverage rate (2SLS)” and “Median length (2SLS)” of Table 1, p. 19 (PDF p. 19)",
   "pdf_page": 19,
   "printed_page": "19",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "Replication code (not a passage of the article), in the block that builds \\texttt{output3}, the summary for $10^4$ draws:\n\nline 100: \\texttt{tsls\\_lo = \\hl{result2}[[1]]['tsls\\_t\\_CI\\_lo']\\$tsls\\_t\\_CI\\_lo}\n\nline 101: \\texttt{tsls\\_up = \\hl{result2}[[1]]['tsls\\_t\\_CI\\_up']\\$tsls\\_t\\_CI\\_up}\n\nline 111: \\texttt{output3[i,1] = result3[[1]]\\$time4['elapsed']}\n\nline 113: \\texttt{permu\\_CI = result3[[1]]\\$permu\\_CI4}\n\nCaption of Table~1: ``Rows 5--8 report the median sizes of these confidence sets. For the randomization-based confidence sets, we use $10^4$ randomization draws.''",
    "corrected": "line 100: \\texttt{tsls\\_lo = \\hl{result3}[[1]]['tsls\\_t\\_CI\\_lo']\\$tsls\\_t\\_CI\\_lo}\n\nline 101: \\texttt{tsls\\_up = \\hl{result3}[[1]]['tsls\\_t\\_CI\\_up']\\$tsls\\_t\\_CI\\_up}\n\nIf this function supplied the 2SLS rows of Table~1 or of Supplement Table~3, those rows should be recomputed from the saved \\texttt{result3} outputs; the size and direction of any change are not known. Alternatively, the code may keep \\texttt{result2}, provided its documentation says that the 2SLS summaries come from the 5000-draw run.",
    "evidence": "The evidence is a check of the code and an exact counterexample. The unmodified function was run on two identical test files in which the 2SLS interval is $[-4,4]$ in \\texttt{result1}, $[1,3]$ in \\texttt{result2} and $[-2,2]$ in \\texttt{result3}, with true value $\\tau=0$. In the third summary it returns 2SLS coverage $0$ and median 2SLS length $2$, which are the values of \\texttt{result2}. The values of \\texttt{result3} are $1$ and $4$. With \\texttt{result3} on lines 100 and 101, the function returns $1$ and $4$, and no other entry of the three summaries changes.\n\n\\begin{audittable}{lrrrr}{Summaries returned for the test files (true value $\\tau=0$). ``As released'' is the unmodified function; ``corrected'' has \\texttt{result3} on lines 100 and 101. Lengths are medians; ``Rand.'' is the randomization-based set.}\nVersion, draws & 2SLS coverage & 2SLS length & Rand. coverage & Rand. length \\\\\nAs released, 5000 & 0 & 2 & 1 & 12 \\\\\nAs released, 10000 & \\hl{0} & \\hl{2} & 1 & 14 \\\\\nCorrected, 5000 & 0 & 2 & 1 & 12 \\\\\nCorrected, 10000 & \\hl{1} & \\hl{4} & 1 & 14 \\\\\n\\end{audittable}\n\nThe scripts' own code for generating and saving the observed data was also run on a population of six units. The three calls produced three different observed assignments, and the data file written for the Anderson--Rubin and tF computations held the third. No entry of Table~1 or of Supplement Table~3 is shown to be wrong, and the replication files do not establish which outputs produced the published rows.",
    "problem": "For each simulated population, the simulation scripts make three calls, with 1000, 5000 and $10^4$ randomization draws, and save the results as \\texttt{result1}, \\texttt{result2} and \\texttt{result3}. Each call draws a new observed assignment. The function \\texttt{extract\\_results2} builds one summary for each number of draws. In the third summary, the 2SLS coverage indicator and the 2SLS interval length are read from \\texttt{result2}, while the computation time and the randomization-based confidence set are read from \\texttt{result3}. The 2SLS coverage and the quantiles of the 2SLS length in the third summary therefore repeat those of the second summary, and use the second run's assignment while the randomization entries beside them use the third run's. The script \\texttt{3\\_merge\\_results\\_grace2.R} reads these entries of the third summary into its 2SLS coverage and median-length rows (lines 48, 50, 56 and 58), which have the layout of the 2SLS rows of Table~1 and of Supplement Table~3. Other indexing slips in the same scripts are recorded as \\fid{P002-N003}, \\fid{P002-N012} and \\fid{P002-N013}.",
    "what_survives": "Under complete randomization, each observed assignment is uniformly distributed over the assignments with the given number of treated units. Then the expected 2SLS coverage and the distribution of the 2SLS interval length are the same whichever of the two runs supplies the interval, and the 2SLS rows estimate the same marginal quantities in either case. This defect does not invalidate the comparison of coverage and length across methods in Table~1. The correction changes only the 2SLS entries of the third summary. Supplement Tables~4 and~5 have no 2SLS rows, and the theoretical results, the algorithms and the data applications do not use this code."
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   "work": {
    "arxiv_id": "2404.18786",
    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
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   "date_published": "2026-10-25",
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     "alt": "Listing of eleven lines of C++ code. Red boxes mark 'round(nsim*(nsim-1)/2/20)' at line 35, the value assigned to the integer variable named progress denominator, and the remainder of count by that variable, written with the percent sign, in the condition of the if statement at line 42, inside two nested for loops over i and j.",
     "caption": "Research code, cpp_implementations/main2.cpp, lines 35–45, typeset from the source file with line numbers added. The marked expression at line 35 is evaluated in integer arithmetic and is zero for fewer than seven simulated assignments. The marked remainder at line 42 then has divisor zero.",
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   "headline": "The progress counter of the research C++ routines takes a remainder by zero when there are 2 to 6 simulated assignments; the condition needs a guard.",
   "id": "P002-N019",
   "location": "Research code in the replication files (commit 4d919c4), folder cpp_implementations: file main2.cpp, function calculate_intersections_adj, lines 35 and 42; file main.cpp, function calculate_intersections, lines 420, 427 and 500. No passage of the article",
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    "as_printed": "Lines 35 and 42 of \\texttt{main2.cpp} (lines 420 and 427 of \\texttt{main.cpp} are the same):\n\n\\texttt{int progress\\_denominator = \\hl{round(nsim*(nsim-1)/2/20)};}\n\n\\texttt{if (\\hl{count \\% progress\\_denominator} == 0)\\{}",
    "corrected": "\\texttt{if (\\hl{progress\\_denominator > 0 \\&\\&} count \\% progress\\_denominator == 0)\\{}\n\nIn \\texttt{main.cpp}, the routine should also return an empty intersection result of the expected shape when there is no intersection, before the vectors are read at line 500. The caller should then use the no-intersection branch of the printed algorithm.",
    "evidence": "The evidence is exact arithmetic on the expression and a run of the code. The routine of \\texttt{main2.cpp}, compiled unchanged with a sanitizer for undefined behavior and called with two simulated assignments, stops at line 42 with a division by zero. Its input was the $6\\times2$ matrix of coefficients computed from six observations, and both denominators are positive on the real line. The progress loops of both routines, compiled separately, stop in the same way for $M=2,\\ldots,6$ and complete for $M=7$ after 21 calls of the root finder. The full routine of \\texttt{main.cpp} was not compiled.",
    "problem": "With $M$ simulated assignments, the expression \\texttt{nsim*(nsim-1)/2/20} is evaluated in integer arithmetic and equals $\\lfloor M(M-1)/40\\rfloor$, which is $0$ for $M\\le6$. For $2\\le M\\le6$ the first pair of assignments evaluates \\texttt{0 \\% 0}, whose behavior is undefined in C++, before any intersection is computed. For $M=1$ the loop is skipped, and the older routine in \\texttt{main.cpp} then reads element 0 of two empty vectors at line 500.",
    "what_survives": "The definitions, theorems and algorithms of the article contain no such step. The article uses $10^3$, $5\\times10^3$ and $10^4$ simulated assignments, for which the divisor is $24975$, $624875$ and $2499750$. No reported result is affected."
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   "work": {
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    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
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     "caption": "R package rilate, R/AR_algo2.R, lines 26–33, typeset from the source file with line numbers added. The marked words state that the fast algorithm computes the same confidence set and $p$-value as AR_algo1().",
     "url": "https://pmaronow.github.io/llm-audit/works/P002/exhibits/P002-N022.png"
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   "headline": "The fast algorithm of the rilate package does not always return the confidence set of the slow algorithm: in the one case examined it excludes a point that the package’s own test accepts.",
   "id": "P002-N022",
   "location": "R package rilate in the replication files (commit e5085c8), R/AR_algo2.R, lines 28–32, and the saved simulation results in sim_slurm/sim_coverage_power/aggregated/. No passage of the article; the corresponding statement is Theorem 3, p. 17",
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    "as_printed": "Documentation of \\texttt{AR\\_algo2}, lines 28--32:\n\nComputes \\hl{the same Anderson-Rubin confidence set} and randomization p-value as \\texttt{AR\\_algo1()}, but instead of checking every interval it jumps between crossing points (via \\texttt{find\\_crossings\\_with\\_all()}), which is substantially faster. This is the default algorithm. Internal -- dispatched to by \\texttt{run\\_rilate()}, not called directly.",
    "corrected": "The correction is to the code of \\texttt{AR\\_algo2}, and I have not identified the line to change. For the strictly positive denominators in this example, a construction that returns the set of the package's test is the following. Partition the real line at every isolated intersection of the observed AR function with a simulated one; identical functions count as ties throughout. On each open interval and at each point of the partition, count the simulated statistics that are at least as large as the observed one. Keep the intervals and points at which the package's $p$-value exceeds the level of the test, which is $1/20$ in these files. The $p$-value is one plus this count, divided by one plus the number of simulated assignments.\n\nUntil the code is corrected, the documentation should read: Computes the Anderson-Rubin confidence set and randomization p-value by jumping between crossing points, which is substantially faster than checking every interval. \\hl{The confidence set can differ from that of} \\texttt{AR\\_algo1()}.",
    "evidence": "The evidence is an exact counterexample in one case. In the file \\texttt{combo06\\_gamma0.5\\_tau1.0.csv}, case number 5688 has no covariates, 100 units, 50 of them treated, and 1000 simulated assignments. I regenerated it from the seeds in the package's scripts, and both algorithms reproduce the saved widths and $p$-values. The slow algorithm returns $[0.15575,\\,2.09403]$ and the fast algorithm returns $[0.15575,\\,2.07933]$. At $\\beta=417/200$, I find by integer arithmetic that 950 of the 1000 simulated squared statistics are below the observed one, none is equal to it, and 50 are above it. The package takes the 951st order statistic as its critical value, so its test accepts this $\\beta$, with $p$-value $51/1001>1/20$, while the set of the fast algorithm excludes it. All 1001 denominators are positive on the real line. The other 93 flagged cases were not examined, and the line of code responsible was not identified.",
    "problem": "The package implements two algorithms that invert the same randomization test. Its saved simulation results, nine files of 10,000 cases each, flag a difference between the two returned sets in 48 cases without covariates and in 46 cases with covariates, which are 94 distinct cases out of 90,000. In all 90,000 cases the two algorithms agree on whether the true effect is covered, on whether zero is rejected, and on the $p$-value, so the summaries of coverage and power do not show the difference. In 87 of the 94 flagged comparisons the saved distance between the two sets is finite. In the one flagged case that I reconstructed, the two sets differ in their right endpoints by about $0.0147$, and the fast algorithm excludes a point that the package's test accepts.",
    "what_survives": "Theorem 3 and the algorithms as printed are not contradicted. The critical value of (13) is the 950th order statistic, under which $\\beta=417/200$ is rejected. The example concerns the package's rule and not the set (14). The saved summaries of coverage, rejection and $p$-values are unaffected. The designs in these files are not those of Section 6, and no table, simulation or application of the article is shown to change."
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    "doi": "10.1093/biomet/asag010",
    "title": "Randomization-based confidence sets for the local average treatment effect",
    "url": "https://pmaronow.github.io/llm-audit/works/P002/"
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     "body": "<p>Supplement pp. 23-24 (PDF pp. 3-4) and pp. 34-35 (PDF pp. 14-15). The printed claim is false: the twelve-unit population satisfies both parameter spaces, with all the stated moments, OLS coefficients, contrast variances, assignment probabilities and oracle formula. Its variance bounds have right sides 65/4752 and 39/55, exceeding the actual variances 5/396 and 36/55. The repeated-block family preserves both failures with factor 13/12; the displayed negative-delta arithmetic step fails.</p>",
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     "body": "<p>The printed claim is false: Both universally quantified printed variance lower bounds are false. Supplement pp. 23-24 (Supplement PDF pp. 3-4) and pp. 34-35 (Supplement PDF pp. 14-15).</p>",
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     "body": "<p>main p. 8 (Theorem 1 and Remark 3), main p. 11 (Theorem 2), Supplement p. 31 (PDF p. 11), and Assumptions 3(ii), 5(ii). The printed claim is false: the half-moment bound forces collapse. The stated nonempty raw class has all observed and reassigned statistics and quantiles zero, and coverage one. The adjusted class has zero population residuals, exact full-rank fits, the individual 258 and 774 rank-loss bounds and the 1548 coverage bound for every singular-fit completion. The grid moments give every beta fiber; both worst-case coverages tend to one. The six-unit heterogeneous example has 20 accepting assignments and 312 nonzero reassigned squared statistics among its 400 assignment pairs.</p>",
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     "location": "Main p. 8 (PDF p. 8): Theorem 1 and Remark 3; main p. 11 (PDF p. 11): Theorem 2. Moment conditions: Assumption 3(ii), main p. 7, and Assumption 5(ii), main p. 11. Last step of the proof of Theorem 1: Supplement p. 31 (Supplement PDF p. 11)"
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     "body": "main p. 8 (Theorem 1 and Remark 3), main p. 11 (Theorem 2), Supplement p. 31 (PDF p. 11), and Assumptions 3(ii), 5(ii). The printed claim is false: the half-moment bound forces collapse. The stated nonempty raw class has all observed and reassigned statistics and quantiles zero, and coverage one. The adjusted class has zero population residuals, exact full-rank fits, the individual 258 and 774 rank-loss bounds and the 1548 coverage bound for every singular-fit completion. The grid moments give every beta fiber; both worst-case coverages tend to one. The six-unit heterogeneous example has 20 accepting assignments and 312 nonzero reassigned squared statistics among its 400 assignment pairs.",
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     "body": "main pp. 8, 11; Supplement Sections C and E, pp. 23-42 (PDF pp. 3-22). Both corrected uniform liminf lower bounds hold along nonempty sizes, including degenerate models, without a cofinality hypothesis. Nominal worst-case liminf in a fixed class requires a nondegenerate constant-additive-effect sequence in that very class whose coverage tends to nominal. Raw nondegeneracy is baseline variance; adjusted nondegeneracy is population OLS residual variance. Admissible nondegenerate raw constant-effect sequences give nominal coverage and worst-case liminf. The specified degenerate raw and adjusted classes instead have worst-case coverage tending to one.",
     "headline": "Theorem 1, Theorem 2 and Remark 3, corrected",
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     "body": "<p>The printed claim is false: <code class=\"fid\">P002-T6</code>; Supplement p. 42 (Supplement PDF p. 22), Lemma 20, claim 4, display (E.61), with the feasible statistic in the denominator. With k = 1, delta = 0, epsilon = 1/4, A = 16, B = 5 and r = 1/3, the explicit even near-singular family and odd equally spaced extension satisfy the standing assumptions at every size n &gt;= 8. All units comply and the LATE is zero. The even family has the stated centered moments, covariate and residual bounds, common population OLS fit, and distinct covariates, so every arm fit has full rank. The two pure-block reassignments have slopes 1/eta and statistics -2/eta and 2/eta. Arm complementation gives zero expectation. The feasible reassignment variance is at least 8/(choose(n,n/2) eta^2), at least 8*2^(2n^2)/2^n, whereas every variance estimate is between zero and sum(y_i^2)/(n/2)^2 &lt;= 121/(25n) &lt; 5/n. Their ratio is at most 5*2^n/(8n*2^(2n^2)) &lt; 1/2 for every original and reassigned assignment. Consequently the conditional error probability at kappa = 1/2 is identically one along the even subsequence and tends to one, contradicting the printed zero limit. The odd extension has distinct equally spaced covariates in [-1,1] and the same constants. Statements (E.58)-(E.60) are context, not assertions of this counterexample.</p>",
     "headline": "Supplement Lemma 20, claim 4, display (E.61), as printed",
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     "body": "The printed claim is false: P002-T6; Supplement p. 42 (Supplement PDF p. 22), Lemma 20, claim 4, display (E.61), with the feasible statistic in the denominator. With k = 1, delta = 0, epsilon = 1/4, A = 16, B = 5 and r = 1/3, the explicit even near-singular family and odd equally spaced extension satisfy the standing assumptions at every size n >= 8. All units comply and the LATE is zero. The even family has the stated centered moments, covariate and residual bounds, common population OLS fit, and distinct covariates, so every arm fit has full rank. The two pure-block reassignments have slopes 1/eta and statistics -2/eta and 2/eta. Arm complementation gives zero expectation. The feasible reassignment variance is at least 8/(choose(n,n/2) eta^2), at least 8*2^(2n^2)/2^n, whereas every variance estimate is between zero and sum(y_i^2)/(n/2)^2 <= 121/(25n) < 5/n. Their ratio is at most 5*2^n/(8n*2^(2n^2)) < 1/2 for every original and reassigned assignment. Consequently the conditional error probability at kappa = 1/2 is identically one along the even subsequence and tends to one, contradicting the printed zero limit. The odd extension has distinct equally spaced covariates in [-1,1] and the same constants. Statements (E.58)-(E.60) are context, not assertions of this counterexample.",
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     "body": "<p>Supplement p. 42 (Supplement PDF p. 22), Lemma 20, (E.58)-(E.61), with the corrected claim. Fix a positive covariate dimension and admissible constants. For any deterministic strictly increasing sequence of models in Xi whose population OLS residual variance is strictly positive in each arm at every size, and any fixed deterministic strict subsequence, the following hold on one event of probability one, simultaneously for all positive thresholds: the feasible and pooled-coefficient oracle reassignment statistics merge relative to the oracle standard deviation (E.58); the oracle and feasible statistics have standard normal absolute tails on that scale (E.59)-(E.60); and the feasible variance estimator is ratio-consistent for the oracle reassignment variance (corrected E.61). The usual measure-theoretic conditions are assumed. Observed assignments have exact measurable complete-randomization marginals with any coupling across sizes; the conditional reassignment law holds the observed response fixed. Xi includes valid designs with at least two units, monotone receipt, a nonempty complier set, centered covariates and Assumption 5. OLS uses a fixed least-squares minimizer on singular arms. Residual positivity is pointwise in size, with no uniform lower bound. No finite-size positivity of the conditional oracle variance is assumed; the variance quotient is zero at denominator zero, as with real division in Lean.</p>",
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     "body": "Supplement p. 42 (Supplement PDF p. 22), Lemma 20, (E.58)-(E.61), with the corrected claim. Fix a positive covariate dimension and admissible constants. For any deterministic strictly increasing sequence of models in Xi whose population OLS residual variance is strictly positive in each arm at every size, and any fixed deterministic strict subsequence, the following hold on one event of probability one, simultaneously for all positive thresholds: the feasible and pooled-coefficient oracle reassignment statistics merge relative to the oracle standard deviation (E.58); the oracle and feasible statistics have standard normal absolute tails on that scale (E.59)-(E.60); and the feasible variance estimator is ratio-consistent for the oracle reassignment variance (corrected E.61). The usual measure-theoretic conditions are assumed. Observed assignments have exact measurable complete-randomization marginals with any coupling across sizes; the conditional reassignment law holds the observed response fixed. Xi includes valid designs with at least two units, monotone receipt, a nonempty complier set, centered covariates and Assumption 5. OLS uses a fixed least-squares minimizer on singular arms. Residual positivity is pointwise in size, with no uniform lower bound. No finite-size positivity of the conditional oracle variance is assumed; the variance quotient is zero at denominator zero, as with real division in Lean.",
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     "body": "<p>Supplement p. 42 (Supplement PDF p. 22), first display; Assumption 5(v), main p. 11. For every positive even size 2m, the covariate x_i=2(-1)^i is centered, has fourth moment 16, and has covariate Gram [4] with smallest eigenvalue 4, at least 2. Its augmented Gram is diag(1,4) with smallest eigenvalue 1. The unit intercept vector has quadratic form 1&lt;2, refuting the covariate-to-augmented-Gram implication. This is a counterexample to that implication, as asserted by the canonical target.</p>",
     "headline": "Proof of Supplement Lemma 19, first display on Supplement p. 42 (eigenvalue lower bound), as printed",
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     "body": "Supplement p. 42 (Supplement PDF p. 22), first display; Assumption 5(v), main p. 11. For every positive even size 2m, the covariate x_i=2(-1)^i is centered, has fourth moment 16, and has covariate Gram [4] with smallest eigenvalue 4, at least 2. Its augmented Gram is diag(1,4) with smallest eigenvalue 1. The unit intercept vector has quadratic form 1<2, refuting the covariate-to-augmented-Gram implication. This is a counterexample to that implication, as asserted by the canonical target.",
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     "body": "<p>Supplement p. 42 (Supplement PDF p. 22), first display, corrected. Centered covariates give population augmented Gram diag(1,C), smallest eigenvalue min(1,lambda_min(C)) at least min(1,e), and the corresponding quadratic bound for every vector. For Xi sequences with fixed positive dimension and constants, every measurable coupling of the exact complete-randomization marginals satisfies, almost surely along each fixed strictly increasing subsequence, the corrected coefficient-error bound with one vanishing error for both arms. In particular this covers the paper&#x27;s independent coupling.</p>",
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     "body": "Supplement p. 42 (Supplement PDF p. 22), first display, corrected. Centered covariates give population augmented Gram diag(1,C), smallest eigenvalue min(1,lambda_min(C)) at least min(1,e), and the corresponding quadratic bound for every vector. For Xi sequences with fixed positive dimension and constants, every measurable coupling of the exact complete-randomization marginals satisfies, almost surely along each fixed strictly increasing subsequence, the corrected coefficient-error bound with one vanishing error for both arms. In particular this covers the paper's independent coupling.",
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     "body": "<p>Supplement p. 26 (Supplement PDF p. 6), identity before (C.17) and corrected (C.17). For every valid design and every beta, the treated sample-centering identity has +beta times the population receipt mean, and its squared bracket has exact expectation n0*sum(v_i^2)/(n1*n*(n-1)). When S_1 squared is positive, division gives the printed normalized equality, sum(v_i^2)/S_1 squared=(n-1)/(pi^2*(1-pi)^2), and the simplified expectation n0/(n1*n*pi^2*(1-pi)^2). Along increasing Theta sequences under fixed balance and the stated denominator guard this expectation is O(n0/(n1*n)) and tends to zero. The unnormalized finite identities need no variance positivity.</p>",
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     "body": "Supplement p. 26 (Supplement PDF p. 6), identity before (C.17) and corrected (C.17). For every valid design and every beta, the treated sample-centering identity has +beta times the population receipt mean, and its squared bracket has exact expectation n0*sum(v_i^2)/(n1*n*(n-1)). When S_1 squared is positive, division gives the printed normalized equality, sum(v_i^2)/S_1 squared=(n-1)/(pi^2*(1-pi)^2), and the simplified expectation n0/(n1*n*pi^2*(1-pi)^2). Along increasing Theta sequences under fixed balance and the stated denominator guard this expectation is O(n0/(n1*n)) and tends to zero. The unnormalized finite identities need no variance positivity.",
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     "body": "<p>Supplement p. 29 (Supplement PDF p. 9). The eight-unit sign population has variance 1/28 while printed (C.31) demands 1/14, and the corrected bound is attained.</p>",
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     "body": "<p>Supplement p. 29 (Supplement PDF p. 9). The corrected finite reassignment-variance bound holds with the explicit balance constant.</p>",
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     "body": "<p>Supplement p. 30 (Supplement PDF p. 10). The printed claim is false: the admissible balanced-sign family has the stated hypergeometric law, moments, exact variance and ratio formulas, and zero-or-at-least-three support. The printed error probability is one for every original assignment; its ratio divided by the square root of sample size tends to two in probability and diverges. The variance ratio lies in [0,1], has mean deficit 2/n and has the stated vanishing tails.</p>",
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     "body": "Supplement p. 30 (Supplement PDF p. 10). The printed claim is false: the admissible balanced-sign family has the stated hypergeometric law, moments, exact variance and ratio formulas, and zero-or-at-least-three support. The printed error probability is one for every original assignment; its ratio divided by the square root of sample size tends to two in probability and diverges. The variance ratio lies in [0,1], has mean deficit 2/n and has the stated vanishing tails.",
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     "body": "<p>Main p. 10 (PDF p. 10), Remark 4 and (12); Supplement Section I, pp. 45-47 (Supplement PDF pp. 25-27). The adjusted full-randomization confidence set is a finite union of intervals where its ordinary regression inverses exist; it can be nonempty without being a single interval. The adjusted contrast is affine, and its outcome and receipt contrasts are the treatment coefficients of the corresponding interacted fits. The well-defined adjusted Wald ratio has zero contrast and statistic and belongs to the confidence set at every interior level. Under the stated IV invertibility conditions, the IV product identity holds, and a nonzero receipt contrast gives equality of the IV coefficient and the Wald ratio. No positive-variance condition is imposed: zero-over-zero is zero and absolute nonzero-over-zero is infinity.</p>",
     "headline": "Remark 4: the regression-adjusted Wald estimator (12) zeroes the adjusted statistic and equals the coefficient on <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>D</mi><mi>i</mi></msub></mrow><annotation encoding=\"application/x-tex\">D_i</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"katex-base\"><span class=\"katex-strut\" style=\"height:0.8333em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">D</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3117em;\"><span style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"katex-sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">i</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span> in the interacted instrumental variable regression",
     "location": "Main p. 10 (PDF p. 10), Remark 4 and equation (12); algebra in Supplement Section I, pp. 45-47 (Supplement PDF pp. 25-27)"
    },
    "id": "P002-T26",
    "text": {
     "body": "Main p. 10 (PDF p. 10), Remark 4 and (12); Supplement Section I, pp. 45-47 (Supplement PDF pp. 25-27). The adjusted full-randomization confidence set is a finite union of intervals where its ordinary regression inverses exist; it can be nonempty without being a single interval. The adjusted contrast is affine, and its outcome and receipt contrasts are the treatment coefficients of the corresponding interacted fits. The well-defined adjusted Wald ratio has zero contrast and statistic and belongs to the confidence set at every interior level. Under the stated IV invertibility conditions, the IV product identity holds, and a nonzero receipt contrast gives equality of the IV coefficient and the Wald ratio. No positive-variance condition is imposed: zero-over-zero is zero and absolute nonzero-over-zero is infinity.",
     "headline": "Remark 4: the regression-adjusted Wald estimator (12) zeroes the adjusted statistic and equals the coefficient on $D_i$ in the interacted instrumental variable regression",
     "location": "Main p. 10 (PDF p. 10), Remark 4 and equation (12); algebra in Supplement Section I, pp. 45-47 (Supplement PDF pp. 25-27)"
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    "html": {
     "body": "<p>Main p. 14 (PDF p. 14); proof in Supplement Section H.1, p. 45 (Supplement PDF p. 25). Between crossings the empirical quantile index set is constant, and every interior quantile index persists at each finite endpoint.</p>",
     "headline": "Lemma 1",
     "location": "Main p. 14 (PDF p. 14); proof in Supplement Section H.1, p. 45 (Supplement PDF p. 25)"
    },
    "id": "P002-T27",
    "text": {
     "body": "Main p. 14 (PDF p. 14); proof in Supplement Section H.1, p. 45 (Supplement PDF p. 25). Between crossings the empirical quantile index set is constant, and every interior quantile index persists at each finite endpoint.",
     "headline": "Lemma 1",
     "location": "Main p. 14 (PDF p. 14); proof in Supplement Section H.1, p. 45 (Supplement PDF p. 25)"
    },
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      "credit_status": "wrong_credit",
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      "lean_name": "AronowAudit.P002.Wave1.algorithm_correctness_paper",
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    "html": {
     "body": "<p>Main p. 17 (PDF p. 17), Theorem 3; Algorithms 1-2, main pp. 15-16; Algorithms 3-4, Supplement p. 44; proof in Supplement Section H.2, p. 45. For both covariate settings, finite crossing lists and completed runs exist, every completed run using the corrected interval-index pairs returns the empirical confidence set, and with no intersections any selected quantile curve does so.</p>",
     "headline": "Theorem 3",
     "location": "Main p. 17 (PDF p. 17); Algorithm 1, main p. 15 (PDF p. 15); Algorithm 2, main p. 16 (PDF p. 16); Algorithms 3-4, Supplement p. 44 (Supplement PDF p. 24); proof in Supplement Section H.2, p. 45 (Supplement PDF p. 25)"
    },
    "id": "P002-T28",
    "text": {
     "body": "Main p. 17 (PDF p. 17), Theorem 3; Algorithms 1-2, main pp. 15-16; Algorithms 3-4, Supplement p. 44; proof in Supplement Section H.2, p. 45. For both covariate settings, finite crossing lists and completed runs exist, every completed run using the corrected interval-index pairs returns the empirical confidence set, and with no intersections any selected quantile curve does so.",
     "headline": "Theorem 3",
     "location": "Main p. 17 (PDF p. 17); Algorithm 1, main p. 15 (PDF p. 15); Algorithm 2, main p. 16 (PDF p. 16); Algorithms 3-4, Supplement p. 44 (Supplement PDF p. 24); proof in Supplement Section H.2, p. 45 (Supplement PDF p. 25)"
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      "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P002/P002_ConfidenceCompletion.lean:55 negates PrintedTheorem1, defined at :50 as the universal closure of the printed equality limit with the original parameter guards. With delta=0, r=1/4 and A=1/2, the fourth-moment condition forces both LATE-adjusted potential vectors to be the same constant throughout the admissible class, so coverage is one. The all-complier constant-effect population inhabits every beta fiber for n>=2, making the nested infimum at P002/P002_Audit.lean:1608 equal to one without using its empty-class convention. Zero variances are admitted by the printed parameter space, so this is a refutation of that equality, with no material change of claim or parameter domain.'}",
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      "credit_reason": "A variance identity is not needed to transport the existing refutation of the printed coverage limit.",
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    "html": {
     "body": "<p>The printed claim is false: the universal exact nominal-coverage limit of Theorem 1 fails. Main p. 8 (PDF p. 8); proof in Supplement Section C, pp. 23-31 (Supplement PDF pp. 3-11).</p>",
     "headline": "Theorem 1",
     "location": "Main p. 8 (PDF p. 8); proof in Supplement Section C, pp. 23-31 (Supplement PDF pp. 3-11)"
    },
    "id": "P002-T29",
    "text": {
     "body": "The printed claim is false: the universal exact nominal-coverage limit of Theorem 1 fails. Main p. 8 (PDF p. 8); proof in Supplement Section C, pp. 23-31 (Supplement PDF pp. 3-11).",
     "headline": "Theorem 1",
     "location": "Main p. 8 (PDF p. 8); proof in Supplement Section C, pp. 23-31 (Supplement PDF pp. 3-11)"
    },
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     "target_id": "P002-T30"
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    "html": {
     "body": "<p>The printed claim is false: the universal exact nominal-coverage limit of Theorem 2 fails. Main p. 11 (PDF p. 11); proof in Supplement Sections D-E, pp. 31-42 (Supplement PDF pp. 11-22).</p>",
     "headline": "Theorem 2",
     "location": "Main p. 11 (PDF p. 11); proof in Supplement Sections D-E, pp. 31-42 (Supplement PDF pp. 11-22)"
    },
    "id": "P002-T30",
    "text": {
     "body": "The printed claim is false: the universal exact nominal-coverage limit of Theorem 2 fails. Main p. 11 (PDF p. 11); proof in Supplement Sections D-E, pp. 31-42 (Supplement PDF pp. 11-22).",
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     "location": "Main p. 11 (PDF p. 11); proof in Supplement Sections D-E, pp. 31-42 (Supplement PDF pp. 11-22)"
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 "missing_sources": [],
 "reproduction": {
  "agreement": "agrees_in_part",
  "status": "reduced_simulation",
  "targets": [
   {
    "agreement": "within_mc_error",
    "location": "p. 19 (PDF p. 19), rows 1-4; discussion p. 18",
    "method": "I recalculated 2SLS and translated AR coverage over 5,000 replications per design, and randomization coverage over 2,000 replications with 10,000 simulated assignments. The 21 available cells agree within Monte Carlo error. I could not compare the tF row because its implementation was unavailable.",
    "status": "reproduced_reduced",
    "target": "Table 1 coverage"
   },
   {
    "agreement": "within_mc_error",
    "location": "p. 19 (PDF p. 19), rows 5-8",
    "method": "I recalculated 2SLS and translated AR median lengths over 5,000 replications per design. For randomization lengths I used the authors' code with 250 replications and 1,000 simulated assignments. The 21 available cells agree within the stated Monte Carlo comparison intervals, with infinite medians retained; this does not establish agreement at the printed digits, and the intervals omit effects of reducing the assignment count. I could not compare tF.",
    "status": "reproduced_reduced",
    "target": "Table 1 median lengths"
   },
   {
    "agreement": "not_compared",
    "location": "p. 19 (PDF p. 19), rows 9-12 and following discussion",
    "method": "I ran the authors' algorithms three times per design with 1,000 simulated assignments. Algorithm 4 was faster than Algorithm 3 in all 21 runs, and their endpoints agreed. I did not compare the numerical wall-clock times with the table or test the 10,000-draw, under-one-hour claim.",
    "status": "reproduced_reduced",
    "target": "Table 1 computation times"
   },
   {
    "agreement": "agrees_in_part",
    "location": "p. 21 (PDF p. 21), Table 2; p. 20 (PDF p. 20), final paragraph",
    "method": "I compared the cached output with Table 2. All six cached 2SLS intervals and the AR p-value agree to rounding; all six randomization sets differ, by at most about 0.114 at an endpoint. Both versions of the SFSP second-year GPA set contain zero. I could not rerun the application without the STAR data; the cached output does not permit verification of the sample sizes, compliance rates or first-stage F statistics.",
    "status": "inspected_only",
    "target": "Table 2 STAR application and AR p-value"
   }
  ]
 },
 "summary": "The paper constructs confidence sets for the local average treatment effect by inverting a studentized Anderson--Rubin randomization test. Its results are finite-sample validity under a constant additive effect, asymptotic validity under heterogeneous effects with an arbitrarily weak instrument, and algorithms that compute the sets without a grid. 32 entries are classified as errors (2 Moderate, 30 Minor), and 3 are clarity notes. The most consequential is in the code of the replication files, where the fast algorithm can return a bounded set when the set (14) is unbounded (\\fid{P002-N002}). None shows coverage below $1-\\alpha$ for the sets (7) and (11).",
 "what_was_checked": "I checked the following points, which held up. This list is not a statement that the paper is correct, and the absence of a correction elsewhere is not a finding of correctness.\n\\begin{itemize}\n\\item The finite-sample argument. Under a constant additive effect, $Y_i-\\beta D_i$ does not depend on the assignment at the true $\\beta$, and the non-strict inversion gives coverage at least $1-\\alpha$ for the sets (7) and (11). I derived this argument and computed coverage by enumeration for $n=8,9,10$.\n\\item The identity for $\\hat\\tau(\\beta)$ on p.~5, the expectation in Supplement Lemma 3, Remarks 1, 2 and 4 (with Supplement Section I), the coefficients (15)--(16) of the AR function, the variance representation (B.1)--(B.2), and the adjusted expansions (F.1)--(F.4), by algebra and against direct numerical evaluation.\n\\item The steps written out in Supplement Sections C--E, other than those corrected in this audit. I derived each step again. With the corrections, among them \\fid{P002-E002}, the argument written out supports coverage of at least $1-\\alpha$; the steps that the Supplement omits are listed below.\n\\item Lemma 1 and Theorem 3 for the algorithms as Step 4 defines them. An independent implementation of Algorithm 2 returns the same sets as direct inversion on a grid for the Bridgeport and Columbus data.\n\\item Table 1. The coverage of the randomization sets and of 2SLS was simulated again for all seven designs (2000 replications, 1000 simulated assignments) and agrees within Monte Carlo error. Reduced runs based on the research code, for the coverage and median length of the randomization sets, of 2SLS and of the AR sets, agree within Monte Carlo error. The reduced timing benchmark establishes only that Algorithm 4 is faster than Algorithm 3; I did not reproduce the numerical times or the under-one-hour claim at $10^4$ draws.\n\\item Table 2. The printed compliance rates equal $46/108$, $77/100$ and $43/68$. I compared the six 2SLS intervals and the AR $p$-value with the cached output in the replication files; they agree to rounding. Without the STAR data I could not independently verify the compliance rates or rerun the application.\n\\item Supplement Tables 6--7 for Bridgeport and Columbus. From the provided data I recalculated sample sizes, compliance rates, first-stage $F$ statistics, point estimates and 2SLS intervals to the displayed rounding. I did not recompute the randomization sets. In Supplement Tables 8--9, the mean sample sizes recompute to the printed precision and the compliance rates are compatible with the subsampling design (\\fid{P002-N014}).\n\\item The corrected Theorem 2, the corrected Theorem 7 and all four parts of the corrected Lemma 20 are proved in Lean. The precise scope and the additional positive-residual-variance condition for Lemma 20 are described in \\href{#P002-T5}{P002-T5}, \\href{#P002-T21}{P002-T21} and \\href{#P002-T7}{P002-T7}.\n\\end{itemize}\n\nNot checked: Lemma 8, which rests on a cited result; the statements taken from cited works; the tF rows of Table 1, which could not be run; the computation times of Table 1, for which a reduced benchmark confirmed only that Algorithm 4 is faster than Algorithm 3; and Supplement Tables 10--12.\n\nMissing from the replication files: the Project STAR data and the get-out-the-vote data for the four cities other than Bridgeport and Columbus. Without them, the randomization sets of Table 2 and the entries for those four cities in Supplement Tables 6--7 could not be recomputed. The six randomization sets in the cached output for Table 2 differ from the printed ones, by at most about $0.114$ at an endpoint, and both versions of the set for SFSP and second-year GPA contain zero. Also missing are the seeds and the saved cases behind Tables 1, 3--5 and 8--10, and the cached output for the get-out-the-vote application is empty. The two binary files of critical values required by the code were unavailable.",
 "work": {
  "arxiv_id": "2404.18786",
  "audit_complete": true,
  "authors": [
   "P. M. Aronow",
   "Haoge Chang",
   "Patrick Lopatto"
  ],
  "category": "Articles - Theory and Methodology",
  "citation": "P. M. Aronow, Haoge Chang, and Patrick Lopatto (2026). Randomization-based confidence sets for the local average treatment effect. Biometrika 113(2):asag010.",
  "doi": "10.1093/biomet/asag010",
  "edition": {
   "pages_in_audited_file": 23,
   "sha256": "825bec9f474745f834453008c778323437d6865e4a449145d0669e5ade435c3a",
   "source_url": "https://doi.org/10.1093/biomet/asag010",
   "venue": "Biometrika",
   "year": "2026"
  },
  "id": "P002",
  "later_materials": null,
  "title": "Randomization-based confidence sets for the local average treatment effect",
  "type": "Journal article",
  "url": "https://pmaronow.github.io/llm-audit/works/P002/",
  "venue": "Biometrika",
  "year": "2026"
 }
}