{ "claims": [ { "claim": "Under Bernoulli assignment with arbitrary, unspecified interference, the average marginalized effect $\\operatorname{AME}(d;\\eta)$ is well defined and identified by randomization, and the Horvitz–Thompson contrast of circle averages is unbiased for it.", "claim_id": "P004-H1", "related_corrections": [], "tied_to": "Proposition 1; eqs. (4)-(14); Supplement A.1.1" }, { "claim": "Under the Bernoulli design, bounded outcomes, local interference (C3) and node spacing (C4), and with scaled variances bounded below, the Horvitz–Thompson and Hajek estimators are $\\sqrt{N}$-consistent and asymptotically normal.", "claim_id": "P004-H2", "related_corrections": [ "P004-E007", "P004-E011", "P004-E012", "P004-N013", "P004-N014", "P004-N015" ], "tied_to": "Propositions 2 and 3; Lemmas A.2-A.5 of Supplement A" }, { "claim": "With the uniform kernel and cutoff $h(d)$, the spatial HAC variance estimator gives asymptotically conservative Wald intervals for the Hajek estimator under a homophily condition (C5); the SAH bound estimator does so without C5; a positive semidefinite version of the HAC estimator is weakly larger.", "claim_id": "P004-H3", "related_corrections": [ "P004-E013", "P004-E016", "P004-E024", "P004-E025", "P004-E026", "P004-N004", "P004-N014" ], "tied_to": "Proposition 4; eqs. (19)-(22); Supplement A.4 and A.7" }, { "claim": "When effects from different intervention nodes are additive, the AME does not depend on the assignment design and equals the average of the node-specific additive distance effects.", "claim_id": "P004-H4", "related_corrections": [], "tied_to": "Section 6.1; eqs. (23)-(25)" }, { "claim": "The paper presents smoothed AMEs in equation (26), including disk and donut averages, and sharp-null randomization tests. In its simulations, pointwise rejection rates are close to $5\\%$ under the null and approach one at distances with nonzero effects; the supremum-statistic test rejects in $5.6\\%$ of null simulations and all additive and interactive simulations at $N=64$.", "claim_id": "P004-H5", "related_corrections": [ "P004-E005", "P004-E006", "P004-E022", "P004-E030" ], "tied_to": "Sections 6.2-6.3; eq. (26); Section 7, Figure 9 and footnotes 22-23; Supplement B.1" }, { "claim": "The paper extends the framework to observational studies: the propensity-weighted Horvitz–Thompson estimator is unbiased under C6, and Section 6.4 claims a complete inferential theory for the IPW estimator with a logistic propensity model in Appendix A.6, including linearization, asymptotic normality, and HAC and SAH variance estimators.", "claim_id": "P004-H6", "related_corrections": [ "P004-E001", "P004-E002", "P004-E003", "P004-E004", "P004-E014", "P004-E015", "P004-N016" ], "tied_to": "Section 6.4; Assumption C6; Supplement A.1.3 and A.6 (Lemma A.6, Proposition 4, eqs. (36)-(43))" }, { "claim": "Under its near-epoch dependence, spacing and variance assumptions, the paper extends $\\sqrt{N}$-consistency, asymptotic normality and consistency or conservativeness of the HAC estimator with a growing bandwidth to the Hajek estimator beyond local interference (C3).", "claim_id": "P004-H7", "related_corrections": [ "P004-E017", "P004-E018", "P004-E019", "P004-E020", "P004-E021", "P004-N016", "P004-N018" ], "tied_to": "Section 6.5; footnote 5; Supplement Appendix C (Lemmas C.1-C.10, Propositions 5 and 6)" }, { "claim": "In simulations the Hajek estimator's MSE falls with the number of nodes; HAC intervals cover well at small distances and need the effective-degrees-of-freedom adjustment at large distances; the SAH interval is extremely conservative; the positive-semidefinite version costs little length; the recommended procedure is the positive-semidefinite HAC estimator with the effective-degrees-of-freedom adjustment.", "claim_id": "P004-H8", "related_corrections": [ "P004-E024", "P004-E025", "P004-E026", "P004-E027", "P004-N003", "P004-N004" ], "tied_to": "Section 7; Figures 5-8; Supplement B.1-B.3 (Figures 1-9); Supplement A.7" }, { "claim": "The Uganda reanalysis reports reduced deforestation inside treated villages, no indication of substantial leakage up to about $15$ km, and a randomization test on the summed $0$–$5$ km estimates, with statistic $-0.034$ and a $p$-value around $0.065$.", "claim_id": "P004-H9", "related_corrections": [ "P004-E007", "P004-E008", "P004-E029", "P004-N005", "P004-N006", "P004-N007", "P004-N009", "P004-N011", "P004-N022", "P004-N023" ], "tied_to": "Section 8.1; Figures 10-11; footnotes 24-25" }, { "claim": "Reanalysis of Costa Rica's protected areas as an observational study (233 treated and 522 untreated tiles, logistic propensity score on three covariates): deforestation diminished within protected areas and nearby, with no indication of substantial leakage.", "claim_id": "P004-H10", "related_corrections": [ "P004-E001", "P004-E009", "P004-E023", "P004-E029", "P004-N011" ], "tied_to": "Section 8.2; Figures 12-13; Section 6.4; Supplement A.6" } ], "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": "Crop of the start of Proposition 4. After the words We have the following linearization results and Proposition 4. Define, a two-line display defines tau-hat-L sub IPW of d as the average of O sub i prime beta sub 1,N minus the average of O sub i prime beta sub 0,N, plus the average of Z sub i times mu sub i of Y and d minus O sub i prime beta sub 1,N, over p of O sub i given theta-zero, minus the average of 1 minus Z sub i times mu sub i of Y and d minus O sub i prime beta sub 0,N, over 1 minus p of O sub i given theta-zero. The two last sums have the upper limit n. Then: Under C2, C3, C4, C6, and C1, equation (36): tau-hat-L sub IPW of d minus tau-hat sub IPW of d equals o-p of N to the minus one half. A red outline surrounds equation (36).", "caption": "Supplement A, Proposition 4 of Appendix A.6, p. 15. The outline marks equation (36), which asserts that the printed linear form and the IPW estimator differ by $o_p(N^{-1/2})$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E001.png" }, { "alt": "Crop of equations (42) and (43) with the text between them. Equation (42) expands the average of Z sub i mu sub i over p of O sub i given theta-hat-MLE as the same average at theta-zero, minus the average of Z sub i mu sub i over p of O sub i given theta-zero, times 1 minus p of O sub i given theta-zero, times O sub i prime, times theta-hat-MLE minus theta-zero, plus a second-order term. Equation (43) states that H inverse times the average of Z sub i mu sub i over p of O sub i given theta-zero, times p of O sub i given theta-zero times 1 minus p of O sub i given theta-zero times O sub i, minus beta sub 1,N, is o-p of 1. One red outline surrounds the factor 1 minus p times O sub i prime in (42). A second surrounds the factor p times 1 minus p times O sub i in (43).", "caption": "Supplement A, proof of Proposition 4, equations (42) and (43), p. 16. The outlines mark the weight $(1-p(O_i\\mid\\theta_0))O_i'$ in the first-order term of (42) and the different weight $p(O_i\\mid\\theta_0)(1-p(O_i\\mid\\theta_0))O_i$ in (43).", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E001-b.png" }, { "alt": "Crop of one line of text and a displayed equation. The text reads: Hence (42) becomes. The equation states that the average of Z sub i mu sub i over p of O sub i given theta-hat-MLE equals the average of Z sub i mu sub i over p of O sub i given theta-zero, minus the average of Z sub i minus p of O sub i given theta-zero, over p of O sub i given theta-zero, times O sub i prime beta sub 1,N, plus o-p of N to the minus one half. A red outline surrounds the second term on the right, from its minus sign to beta sub 1,N.", "caption": "Supplement A, proof of Proposition 4, top of p. 17. The outline marks the correction term that is substituted into (42).", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E001-c.png" }, { "alt": "Crop of a line of text, a heading and a displayed inequality. The text reads: A similar homophily condition is needed for the HAC variance estimation. C 2. For all sample size N. The inequality states that one over N times the sum over i of tau sub i minus the difference of O sub i prime beta sub 1,N and O sub i prime beta sub 0,N, times the sum over j in B of i and d of tau sub j minus the difference of O sub i prime beta sub 1,N and O sub i prime beta sub 0,N, is at least 0. Red outlines surround the two symbols O sub i prime in the second factor.", "caption": "Supplement A, condition C 2, p. 14. The outlines mark the two places in the second factor where $O_i'$ is printed and $O_j'$ is needed.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E001-d.png" }, { "alt": "Crop of five lines of text. They read: In practice, p of O sub i is unknown to researchers. In such cases, parametric methods, such as logistic regression, and nonparametric methods, such as the sieve estimator in Hirano, Imbens and Ridder (2003), can be used to estimate the propensity score. In Appendix A.6, we develop a complete inferential theory for the inverse probability weighted (IPW) estimator where propensity scores are modeled using a logistic model. A red outline surrounds the line that begins we develop a complete inferential theory.", "caption": "Main text, Section 6.4, p. 759 (PDF p. 16). The outline marks the line that announces the theory of Appendix A.6.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E001-e.png" }, { "alt": "Two panels under the title: Counterexample to (36): the printed variance is about half the variance of the IPW estimator, and an interval built on it covers less often than its nominal level. Left panel, three bars on an axis labelled N times the variance: a blue bar of height 1753/750000 = 0.00234 for N times the variance of the printed linear form, an orange bar of height 3/625 = 0.00480 for N times the variance of the IPW estimator to first order, and a grey bar of height 10369/750000 = 0.01383 for the variance of their difference scaled by the square root of N. The panel is headed: Equation (36) asserts that the third bar is 0. Right panel, a line chart of limiting coverage in percent against the nominal level of the interval in percent, from 80 to 99: a grey straight line for the nominal level and a blue curve below it for the limiting coverage with the printed variance, with marked points 74.9 percent at nominal 90 and 82.9 percent at nominal 95.", "caption": "The counterexample to (36). Left: $N$ times the variance of the printed linear form ($1753/750000$) and of the first-order form of the IPW estimator ($3/625$), and the variance of $\\sqrt N$ times their difference ($10369/750000$), which (36) asserts to vanish in the limit. Right: the limiting coverage $2\\Phi(z\\sqrt{1753/3600})-1$ of a normal interval for the IPW estimator whose variance estimate converges to the printed variance, against its nominal level.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E001-expansion.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T1", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T1" }, { "id": "P004-T2", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T2" }, { "id": "P004-T4", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T4" }, { "id": "P004-T35", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T35" } ], "headline": "Equation (36) can fail when the assignment probabilities differ across nodes: to first order, estimating the logistic coefficient changes the IPW estimator by $-N^{-1}\\sum_i(Z_i-p_i)O_i'\\gamma_N$ with $\\gamma_N=H_N^{-1}N^{-1}\\sum_i[(1-p_i)m_{i1}+p_im_{i0}]O_i$, and the linear form, the variance, the two variance estimators and the coverage statements of Proposition 4 of A.6 are rebuilt on this term.", "id": "P004-E001", "location": "Supplement A, Appendix A.6: Proposition 4 and equation (36), p. 15; its proof, equations (42) and (43), p. 16, and the display after “Hence (42) becomes”, p. 17; condition C 2, p. 14. Announced in Section 6.4, p. 759 (PDF p. 16)", "pdf_page": 15, "printed_page": "15", "reaches": [ { "effect": "implementation_differs_from_paper", "effect_text": "the code implements something other than the printed method", "explanation": null, "package": "SpatialEffect", "recorded_as": null, "to": "package", "url": null, "where": null, "work_id": "P004" } ], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P004-T1", "P004-T2", "P004-T35", "P004-T4" ], "text": { "as_printed": "\\emph{Main text, Section 6.4, p.~759.} In Appendix A.6, we develop \\hl{a complete inferential theory} for the inverse probability weighted (IPW) estimator where propensity scores are modeled using a logistic model.\n\n\\emph{Supplement A, p.~14.} For brevity, we denote $p(O_i\\mid\\theta_0)=p_0(O_i)$. We define the following coefficients:\n\\[\n\\beta_{1,N}=\\Bigl(\\frac1N\\sum_{i=1}^{N}p_0(O_i)(1-p_0(O_i))O_iO_i'\\Bigr)^{-1}\\Bigl(\\frac1N\\sum_{i=1}^{N}\\mathrm E\\bigl[\\mu_i(1;d)\\,p_0(O_i)(1-p_0(O_i))O_i\\bigr]\\Bigr),\n\\]\nand,\n\\[\n\\beta_{0,N}=\\Bigl(\\frac1N\\sum_{i=1}^{N}p_0(O_i)(1-p_0(O_i))O_iO_i'\\Bigr)^{-1}\\Bigl(\\frac1N\\sum_{i=1}^{N}\\mathrm E\\bigl[\\mu_i(0;d)\\,p_0(O_i)(1-p_0(O_i))O_i\\bigr]\\Bigr).\n\\]\n\n\\emph{Supplement A, p.~15.} Proposition 4. Define:\n\\[\n\\begin{aligned}\n\\widehat\\tau^{\\mathrm{L}}_{\\mathrm{IPW}}(d)={}&\\frac1N\\sum_{i=1}^{N}O_i'\\beta_{1,N}-\\frac1N\\sum_{i=1}^{N}O_i'\\beta_{0,N}\\\\\n&+\\frac1N\\sum_{i=1}^{n}\\frac{Z_i\\bigl(\\mu_i(\\mathbf Y;d)-O_i'\\beta_{1,N}\\bigr)}{p(O_i\\mid\\theta_0)}-\\frac1N\\sum_{i=1}^{n}\\frac{(1-Z_i)\\bigl(\\mu_i(\\mathbf Y;d)-O_i'\\beta_{0,N}\\bigr)}{1-p(O_i\\mid\\theta_0)}.\n\\end{aligned}\n\\]\nUnder C2, C3, C4, C6, and C1,\n\\[\n\\widehat\\tau^{\\mathrm{L}}_{\\mathrm{IPW}}(d)-\\widehat\\tau_{\\mathrm{IPW}}(d)=\\hl{o_p(N^{-\\frac12})}.\n\\]\n\n\\emph{Supplement A, proof, pp.~16--17, equation (43) and the next display.}\n\\[\nH^{-1}\\Bigl(\\frac1N\\sum_{i=1}^{N}\\frac{Z_i\\mu_i(\\mathbf Y;d)}{p(O_i\\mid\\theta_0)}\\,p(O_i\\mid\\theta_0)(1-p(O_i\\mid\\theta_0))O_i\\Bigr)-\\beta_{1,N}=o_p(1).\n\\]\nHence (42) becomes\n\\[\n\\frac1N\\sum_{i=1}^{N}\\frac{Z_i\\mu_i(\\mathbf Y;d)}{p(O_i\\mid\\widehat\\theta_{\\mathrm{MLE}})}=\\frac1N\\sum_{i=1}^{N}\\frac{Z_i\\mu_i(\\mathbf Y;d)}{p(O_i\\mid\\theta_0)}\\hl{-\\frac1N\\sum_{i=1}^{N}\\frac{Z_i-p(O_i\\mid\\theta_0)}{p(O_i\\mid\\theta_0)}O_i'\\beta_{1,N}}+o_p(N^{-\\frac12}).\n\\]\n\n\\emph{Supplement A, p.~14, condition C~2.} For all sample size $N$,\n\\[\n\\frac1N\\sum_{i=1}^{N}\\Bigl(\\tau_i(d;\\eta)-\\bigl(O_i'\\beta_{1,N}-O_i'\\beta_{0,N}\\bigr)\\Bigr)\\sum_{j\\in\\mathcal B(i;d)}\\Bigl(\\tau_j(d;\\eta)-\\bigl(\\hl{O_i'}\\beta_{1,N}-\\hl{O_i'}\\beta_{0,N}\\bigr)\\Bigr)\\ge0.\n\\]", "corrected": "\\emph{Condition C~1(iv).} $\\Theta_0\\subset\\mathbb R^k$ is a fixed compact set, and $\\theta_0$ is an \\hl{interior point} of $\\Theta_0$ (\\fid{P004-E003}).\n\n\\emph{Condition C~2.} For all sample sizes $N$, with $\\delta_i=O_i'(\\beta_{1,N}-\\beta_{0,N})$,\n\\[\n\\frac1N\\sum_{i=1}^{N}\\sum_{j\\in\\mathcal B(i;d)}\\bigl(\\tau_i(d;\\eta)-\\delta_i\\bigr)\\bigl(\\tau_j(d;\\eta)-\\hl{\\delta_j}\\bigr)\\ge0 .\n\\]\nHere $\\mathcal B(i;d)$ is the neighbourhood (12), which contains $i$ (see also \\fid{P004-E015}).\n\n\\emph{Proposition 4 of A.6.} Write $p_i=p(O_i\\mid\\theta_0)$, $m_{ia}=E[\\mu_i(a;d)]$ and $H_N=N^{-1}\\sum_ip_i(1-p_i)O_iO_i'$. Define\n\\[\n\\hl{\\gamma_N=H_N^{-1}\\,\\frac1N\\sum_{i=1}^{N}\\bigl[(1-p_i)m_{i1}+p_im_{i0}\\bigr]O_i},\\qquad L_i=O_i'\\gamma_N,\n\\]\n\\[\n\\hl{f_{i1}=p_i\\delta_i+p_i(1-p_i)L_i},\\qquad\\hl{f_{i0}=-(1-p_i)\\delta_i+p_i(1-p_i)L_i},\n\\]\n\\[\n\\widehat\\tau^{L}_{IPW}(d)=\\frac1N\\sum_{i=1}^{N}\\delta_i+\\frac1N\\sum_{i=1}^{N}T_i,\\qquad T_i=\\frac{Z_i\\bigl(\\mu_i(\\mathbf Y;d)-\\hl{f_{i1}}\\bigr)}{p_i}-\\frac{(1-Z_i)\\bigl(\\mu_i(\\mathbf Y;d)-\\hl{f_{i0}}\\bigr)}{1-p_i}.\n\\]\nUnder C2, C3, C4, C6 and C~1 with item (iv) as above,\n\\[\n\\widehat\\tau^{L}_{IPW}(d)-\\widehat\\tau_{IPW}(d)=o_p(N^{-\\frac12}).\n\\]\nThe variance $V_N=V(\\widehat\\tau^{L}_{IPW}(d))$ is\n\\[\n\\begin{aligned}\nV_N={}&\\frac{1}{N^2}\\sum_{i=1}^{N}\\Bigl\\{\\frac{E\\bigl[(\\mu_i(\\mathbf 1;d)-f_{i1})^2\\bigr]}{p_i}+\\frac{E\\bigl[(\\mu_i(\\mathbf 0;d)-f_{i0})^2\\bigr]}{1-p_i}\\Bigr\\}\\\\\n&+\\frac{1}{N^2}\\sum_{i;\\,j\\in\\mathcal B(i;d),\\,j\\ne i}\\ \\sum_{a=0}^{1}\\sum_{b=0}^{1}(-1)^{a+b}E\\bigl[\\bigl(\\mu_i(\\mathbf a_i,\\mathbf b_j;d)-f_{ia}\\bigr)\\bigl(\\mu_j(\\mathbf a_i,\\mathbf b_j;d)-f_{jb}\\bigr)\\bigr]\\\\\n&-\\frac{1}{N^2}\\sum_{i=1}^{N}\\sum_{j\\in\\mathcal B(i;d)}(\\tau_i-\\delta_i)(\\tau_j-\\delta_j).\n\\end{aligned}\n\\]\nIf \\hl{$N\\times V_N$} is uniformly bounded below by a positive constant for all large $N$, then\n\\[\n\\frac{\\widehat\\tau_{IPW}(d)-\\mathrm{AME}(d;\\eta)}{\\sqrt{V_N}}\\xrightarrow{d}N(0,1).\n\\]\nLet $\\widehat p_i=p(O_i\\mid\\widehat\\theta_{MLE})$, $\\widehat H=N^{-1}\\sum_i\\widehat p_i(1-\\widehat p_i)O_iO_i'$, and\n\\[\n\\widehat\\gamma=\\widehat H^{-1}\\,\\frac1N\\sum_{i=1}^{N}\\Bigl[\\frac{Z_i\\mu_i(\\mathbf Y;d)(1-\\widehat p_i)}{\\widehat p_i}+\\frac{(1-Z_i)\\mu_i(\\mathbf Y;d)\\,\\widehat p_i}{1-\\widehat p_i}\\Bigr]O_i .\n\\]\nLet $\\widehat\\beta_{1,N}$ and $\\widehat\\beta_{0,N}$ be as in footnote 9 of the supplement, $\\widehat\\delta_i=O_i'(\\widehat\\beta_{1,N}-\\widehat\\beta_{0,N})$ and $\\widehat L_i=O_i'\\widehat\\gamma$, and let $\\widehat f_{i1}$, $\\widehat f_{i0}$ and $\\widehat T_i$ be defined as $f_{i1}$, $f_{i0}$ and $T_i$ with $\\widehat p_i$, $\\widehat\\delta_i$ and $\\widehat L_i$ in place of $p_i$, $\\delta_i$ and $L_i$. Define\n\\[\n\\begin{aligned}\n\\widehat V^{obs}_{HAC}(d)&=\\frac{1}{N^2}\\sum_{i=1}^{N}\\sum_{j\\in\\mathcal B(i;d)}\\widehat T_i\\widehat T_j,\\\\\n\\widehat V^{obs}_{SAH}(d)&=\\frac{1}{N^2}\\sum_{i=1}^{N}c_i(d)\\Bigl[\\frac{Z_i\\bigl(\\mu_i(\\mathbf Y;d)-\\widehat f_{i1}\\bigr)^2}{\\widehat p_i^{\\,2}}+\\frac{(1-Z_i)\\bigl(\\mu_i(\\mathbf Y;d)-\\widehat f_{i0}\\bigr)^2}{(1-\\widehat p_i)^2}\\Bigr].\n\\end{aligned}\n\\]\nThen, for each $\\hl{0<}\\alpha<1/2$,\n\\[\n\\liminf_{N\\to\\infty}\\Pr\\Bigl(\\bigl|\\widehat\\tau_{IPW}(d)-\\mathrm{AME}(d;\\eta)\\bigr|\\le z_{1-\\alpha/2}\\sqrt{\\widehat V^{obs}_{SAH}(d)}\\Bigr)\\ge1-\\alpha,\n\\]\nand, additionally under C~2 as corrected above, the same inequality holds with $\\hl{\\max\\{\\widehat V^{obs}_{HAC}(d),0\\}}$ in place of $\\widehat V^{obs}_{SAH}(d)$.\nWhen \\hl{both assignment arms are nonempty}, the estimator $\\widehat V^{obs}_{HAC}(d)$ is the (2,2) entry of the sandwich printed on p.~15 when the residual there is taken to be\n\\[\n\\widehat\\epsilon_i=\\frac{N_1}{N}\\,\\frac{Z_i\\bigl(\\mu_i(\\mathbf Y;d)-\\widehat f_{i1}\\bigr)}{\\widehat p_i}+\\frac{N_0}{N}\\,\\frac{(1-Z_i)\\bigl(\\mu_i(\\mathbf Y;d)-\\widehat f_{i0}\\bigr)}{1-\\widehat p_i}.\n\\]\nIn $\\widehat V^{obs}_{SAH}(d)$ the residuals $\\mu_i(\\mathbf Y;d)-\\widehat f_{ia}$ are not weighted before they are squared, so that the inverse probability enters once, squared (see \\fid{P004-E002}).\n\n\\emph{Main text, Section 6.4, p.~759.} The sentence stands with this proposition in place of the printed one.", "evidence": "The evidence is an exact counterexample to (36), computed in rational arithmetic, and a derivation of the corrected statement.\n\n\\emph{The expansion of the IPW estimator.} For fixed outcomes and assignments,\n\\[\n\\nabla_\\theta\\,\\frac{Z_iR_i}{p(O_i\\mid\\theta)}=-\\frac{Z_iR_i\\,(1-p(O_i\\mid\\theta))}{p(O_i\\mid\\theta)}\\,O_i,\\qquad\\nabla_\\theta\\Bigl[-\\frac{(1-Z_i)R_i}{1-p(O_i\\mid\\theta)}\\Bigr]=-\\frac{(1-Z_i)R_i\\,p(O_i\\mid\\theta)}{1-p(O_i\\mid\\theta)}\\,O_i.\n\\]\nSince $Z_i$ is independent of the other assignments, I have $E[Z_iR_i]=p_im_{i1}$ and $E[(1-Z_i)R_i]=(1-p_i)m_{i0}$. The expected gradient of $\\widehat\\tau_{IPW}$ at $\\theta_0$ is therefore $-D_N$, where $D_N=N^{-1}\\sum_i[(1-p_i)m_{i1}+p_im_{i0}]O_i$. Under C2, C3, C4, C6 and C~1, the summands are bounded and each shares assignments with a bounded number of others, which implies that the gradient differs from $-D_N$ by $O_p(N^{-1/2})$. The second derivatives are bounded near $\\theta_0$. Inserting the expansion $\\widehat\\theta_{MLE}-\\theta_0=H_N^{-1}S_N+O_p(N^{-1})$ from the logistic score equation, with $\\theta_0$ an interior point (\\fid{P004-E003}), I obtain $\\widehat\\tau_{IPW}=\\widehat\\tau_{HT}-S_N'\\gamma_N+O_p(N^{-1})$ with $\\gamma_N=H_N^{-1}D_N$.\n\n\\emph{The counterexample.} Let $N=3K$. The nodes are the points $(10i,0)$, $i=1,\\dots,N$, and the outcome points are the same points, so that at $d=0$ the circle average of a node is the outcome at its own point. The nodes have the types $t=-1,0,1$, repeated in this order. A node of type $t$ has $O_t=(1,t)'$ and the outcome $q_t$ at its own point under every assignment, with $q_{-1}=1$, $q_0=21/20$ and $q_1=1$. The assignments are independent with $\\theta_0=(0,\\log2)'$ and $\\Theta_0=[-2,2]^2$. Then $p_t=2^t/(1+2^t)$, that is, $1/3$, $1/2$ and $2/3$. Every hypothesis of the proposition holds. The outcomes are bounded. There is no interference, and $\\mathcal B(i;0)=\\{i\\}$. The nodes are 10 apart, the probabilities lie in $(1/10,9/10)$, the Gram matrix of the confounders is $\\operatorname{diag}(1,2/3)$, and $\\theta_0$ is an interior point of $\\Theta_0$. Further, $\\tau_i=0$ and $\\beta_{1,N}=\\beta_{0,N}$, which implies that the sum in C~2 is 0, in the printed form and in the corrected one. The variance condition of the proposition also holds, since $V_P>0$ below. A direct computation gives\n\\[\nH_N=\\operatorname{diag}\\Bigl(\\frac{25}{108},\\frac{4}{27}\\Bigr),\\qquad\\beta_{1,N}=\\beta_{0,N}=\\Bigl(\\frac{509}{500},0\\Bigr)',\\qquad\\gamma_N=\\Bigl(\\frac{549}{125},0\\Bigr)'.\n\\]\nSince the two arms have the same outcome, both linear forms are averages over the nodes of multiples of $Z_i-p_i$. At a node of type $t$, the first-order form of the IPW estimator has the coefficient $a_t$ and the printed linear form has the coefficient $b_t$, where\n\\[\na_t=\\frac{q_t}{p_t(1-p_t)}-O_t'\\gamma_N,\\qquad b_t=\\frac{q_t-O_t'\\beta_{1,N}}{p_t(1-p_t)}.\n\\]\n\\begin{audittable}{crrrr}{The three types of node and the coefficient of $Z_i-p_i$ in the two linear forms.}\nType $t$ & $p_t$ & $q_t$ & IPW estimator, $a_t$ & Printed linear form, $b_t$ \\\\\n$-1$ & $1/3$ & $1$ & $27/250$ & $-81/1000$ \\\\\n$0$ & $1/2$ & $21/20$ & $-24/125$ & $16/125$ \\\\\n$1$ & $2/3$ & $1$ & $27/250$ & $-81/1000$ \\\\\n\\end{audittable}\nNote that the two sets of coefficients have opposite signs. Writing $w_t=p_t(1-p_t)$, I find that $N$ times the variances of the two linear forms and their difference are\n\\[\nV_A=\\frac13\\sum_tw_ta_t^2=\\frac{3}{625},\\qquad V_P=\\frac13\\sum_tw_tb_t^2=\\frac{1753}{750000},\\qquad V_D=\\frac13\\sum_tw_t(a_t-b_t)^2=\\frac{10369}{750000}\n\\]\nfor the first-order form of the IPW estimator, for the printed linear form, and for their difference, respectively. By the expansion above, $\\sqrt N(\\widehat\\tau_{IPW}-\\widehat\\tau^{L}_{IPW})=U_K+o_p(1)$, where $U_K$ is a normalized sum of $K$ independent and identically distributed terms, one for each block of three nodes. Its second moment is $V_D$ and its fourth moment is at most $2205448981/2250000000000$ for every $K$. By the Cauchy--Schwarz inequality, I have\n\\[\n\\Pr\\bigl(U_K^2>V_D/2\\bigr)\\ge\\frac{107516161}{2205448981}\\quad\\text{for every }K\\ge1.\n\\]\nHence the difference in (36) is not $o_p(N^{-1/2})$. Further, $\\sqrt N\\,\\widehat\\tau_{IPW}$ converges in distribution to a normal variable with variance $3/625$, and the AME is 0. Then the ratio that Proposition 4 standardizes by the printed variance has limiting variance $V_A/V_P=3600/1753$, not 1. In this example the printed HAC estimator, multiplied by $N$, converges to $V_P$. A normal interval with critical value $z$ then has limiting coverage $2\\Phi(z\\sqrt{1753/3600})-1$, which is about $0.83$ at the nominal level $0.95$ and about $0.75$ at the nominal level $0.90$.\n\n\\emph{The corrected linear form.} With $L_i=O_i'\\gamma_N$, the fits $f_{i1}$ and $f_{i0}$ of the corrected statement satisfy $f_{i1}-f_{i0}=\\delta_i$ and $f_{i1}/p_i+f_{i0}/(1-p_i)=L_i$. For every assignment,\n\\[\nf_{i1}-f_{i0}-\\frac{Z_if_{i1}}{p_i}+\\frac{(1-Z_i)f_{i0}}{1-p_i}=-(Z_i-p_i)\\Bigl[\\frac{f_{i1}}{p_i}+\\frac{f_{i0}}{1-p_i}\\Bigr]=-(Z_i-p_i)L_i.\n\\]\nSumming over $i$, I obtain $N^{-1}\\sum_i\\delta_i+N^{-1}\\sum_iT_i=\\widehat\\tau_{HT}-S_N'\\gamma_N$ as an identity, with $T_i$ as in the corrected statement, and $E[T_i]=\\tau_i-\\delta_i$. The corrected form of (36) then follows from the expansion of the IPW estimator.\n\n\\emph{The variance estimators.} Let $V_N$ be the variance of $N^{-1}\\sum_iT_i$, and let $W$ be the sum of the products $T_iT_j$ over $i$ and $j\\in\\mathcal B(i;d)$, divided by $N^2$. By C3, the covariance of $T_i$ and $T_j$ vanishes when $j\\notin\\mathcal B(i;d)$, which implies that\n\\[\nE[W]=V_N+\\frac{1}{N^2}\\sum_{i=1}^{N}\\sum_{j\\in\\mathcal B(i;d)}(\\tau_i-\\delta_i)(\\tau_j-\\delta_j).\n\\]\nThe last sum is nonnegative under the corrected C~2, which is the homophily condition of the paper with $O_j$ in its second factor. For the SAH estimator, the inequality $\\operatorname{Cov}(T_i,T_j)\\le(\\operatorname{Var}T_i+\\operatorname{Var}T_j)/2$ and the symmetry of the neighbourhoods give $\\operatorname{Var}(\\sum_iT_i)\\le\\sum_ic_i(d)E[T_i^2]$, with no homophily condition, and $T_i^2$ is the bracket in the corrected SAH estimator. The $T_i$ are bounded, and each depends on the assignments of a bounded number of nodes. By Chebyshev's inequality, $N$ times each of the two sums concentrates around its expectation. The estimated fits $\\widehat f_{ia}$ are consistent uniformly in $i$, since $\\widehat\\theta_{MLE}$, $\\widehat\\gamma$ and $\\widehat\\beta_{a,N}$ have errors of order $N^{-1/2}$. Then replacing $T_i$ by $\\widehat T_i$ changes $N$ times each sum by $o_p(1)$. Finally, a sum of bounded variables whose dependency graph has bounded degree is asymptotically normal when its variance is of order $N$, which is the condition that $NV_N$ is bounded below.\n\n\\emph{Three remarks on the correction.} First, the index in C~2 matters. In a second exact example, with three types on a path, the printed sum of C~2 equals $7/339>0$ and the corrected sum equals $-775/38307<0$. The corrected HAC estimator, multiplied by $N$, then converges to $6053/153228$, below the value $27/452$ of $NV_N$. Second, a positive printed variance does not imply that $V_N$ is positive. With $q=(1,9/8,1)$ in the counterexample, the first-order form of the IPW estimator vanishes identically, while $N$ times the printed variance is $1753/120000$. Third, the fits are not unique. The fits $g_{i1}=p_iO_i'\\gamma_1$ and $g_{i0}=(1-p_i)O_i'\\gamma_0$, with $\\gamma_1=H_N^{-1}N^{-1}\\sum_i(1-p_i)m_{i1}O_i$ and $\\gamma_0=H_N^{-1}N^{-1}\\sum_ip_im_{i0}O_i$, yield the same linear form. With them, however, the bound for the HAC estimator needs homophily of $\\tau_i-g_{i1}+g_{i0}$, which C~2 does not imply: in a third example the sum for $g$ equals $-9/2000$ while the sum of C~2 is 0. The fits $f$ avoid this additional condition.\n\n\\emph{What is not established.} Equation (36) does not fail in every design with unequal probabilities. With two types of node, a saturated logistic model and outcomes that depend only on the type and the arm, (36) holds although the probabilities differ. No published figure, estimate or interval of the observational application has been shown to change, and the corrected conditions have not been checked for that application.", "problem": "Throughout, $d$ is fixed, and I write $R_i=\\mu_i(\\mathbf Y;d)$, $p_i=p(O_i\\mid\\theta_0)$, $m_{ia}=E[\\mu_i(a;d)]$ for $a\\in\\{0,1\\}$, $\\tau_i=m_{i1}-m_{i0}$ and $\\delta_i=O_i'(\\beta_{1,N}-\\beta_{0,N})$. Let\n\\[\nH_N=\\frac1N\\sum_{i=1}^{N}p_i(1-p_i)O_iO_i',\\qquad S_N=\\frac1N\\sum_{i=1}^{N}(Z_i-p_i)O_i,\\qquad\\widehat\\tau_{HT}=\\frac1N\\sum_{i=1}^{N}\\Bigl[\\frac{Z_iR_i}{p_i}-\\frac{(1-Z_i)R_i}{1-p_i}\\Bigr],\n\\]\nso that $\\beta_{a,N}=H_N^{-1}N^{-1}\\sum_ip_i(1-p_i)m_{ia}O_i$. The upper limits $n$ in the two residual sums of the proposition are read as $N$, and ``C1'' in its hypotheses is the condition C~1 of A.6 (see \\fid{P004-N016}).\n\nThe first-order term of the expansion (42) has the coefficient $N^{-1}\\sum_iZ_iR_i(1-p_i)O_i'/p_i$, whose expectation is $N^{-1}\\sum_i(1-p_i)m_{i1}O_i'$. Equation (43) concerns a different sum, in which $Z_iR_i/p_i$ is multiplied by $p_i(1-p_i)O_i$ and not by $(1-p_i)O_i$, and it is this sum that converges to $H_N\\beta_{1,N}$. The display after ``Hence (42) becomes'' puts $\\beta_{1,N}$ in place of the coefficient of (42) and moves a factor $1/p_i$ into the score sum. Lemma A.6 expresses $\\widehat\\theta_{MLE}-\\theta_0$ through $H_N^{-1}S_N$, in which every node has the same weight, so no factor $1/p_i$ can arise there. The two expressions agree when all $p_i$ are equal and differ in general. Expanding the definition, I find that the printed linear form is\n\\[\n\\widehat\\tau^{L}_{IPW}=\\widehat\\tau_{HT}-\\frac1N\\sum_{i=1}^{N}(Z_i-p_i)\\Bigl[\\frac{O_i'\\beta_{1,N}}{p_i}+\\frac{O_i'\\beta_{0,N}}{1-p_i}\\Bigr],\n\\]\nwhereas the IPW estimator satisfies $\\widehat\\tau_{IPW}=\\widehat\\tau_{HT}-S_N'\\gamma_N+O_p(N^{-1})$ with $\\gamma_N$ as in the headline. In general the two differ at order $N^{-1/2}$, and then (36) fails. The variance calculation (37)--(40) is based on the printed linear form rather than the correct first-order form of the IPW estimator; the HAC and SAH estimators built on the residuals $R_i-O_i'\\widehat\\beta_{a,N}$ can estimate the wrong quantity, and the normal limit and the two coverage statements for the IPW estimator do not follow. As printed, Appendix A.6 does not supply the inferential theory that Section 6.4 announces.", "what_survives": "The definition of the AME, its identification under independent assignment, and the Horvitz--Thompson estimator with known probabilities (with the weight corrected in \\fid{P004-E004}) are not affected. The theory of Sections 4 and 5 for experiments with a constant probability does not use Proposition 4 of A.6. The IPW estimator remains consistent for the AME. The defect concerns its first-order approximation and the variance derived from it. The printed expansion is correct when the logistic model has an intercept only, since all probabilities are then equal. What changes is the inference for the IPW estimator with a fitted logistic model: the linear form, its variance, the two variance estimators and the conditions are replaced as below. The correction keeps the estimand, the estimator and the homophily condition in its intended form. It adds two regularity conditions, an interior $\\theta_0$ and a positive lower bound on $N$ times the corrected variance." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E001", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the line that defines epsilon-hat_i as N_1 over N times Z_i over p(O_j | theta-hat_MLE) times a treated residual, plus N_0 over N times (1 minus Z_i) over (1 minus p(O_j | theta-hat_MLE)) times a control residual; the sentence 'The formula is similar to (6), except now that the residuals are weighted with the inverse propensity scores'; and the display of V-hat-obs-SAH(d). Red boxes mark the denominators p squared (O_i | theta-hat_MLE) and (1 minus p(O_i | theta-hat_MLE)) squared of the two sums.", "caption": "Supplement A, p. 15, the residual $\\widehat\\epsilon_i$ of A.6 and the observational SAH estimator. The residual in the first line already carries the inverse propensity weights; the two marked denominators divide its square by the squared propensities again.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E002.png" }, { "alt": "Bar chart with three groups, N = 32, N = 64 and the limit as N tends to infinity. The vertical axis is the expected variance estimate divided by the true variance, from 0 to 2.3, with a horizontal line at 1 labelled 'true variance'. Orange bars, as printed: 0.21, 0.30 and 0.32, all far below the line. Blue bars, corrected: 1.07, 1.58 and 2.00, all above the line.", "caption": "The observational SAH estimator, as printed and as corrected, in the example described under Evidence. Each bar is the expected value of the estimator divided by the variance of $\\widehat\\tau_{\\mathrm{IPW}}$, computed exactly at $N=32$ and $N=64$; the third group is the ratio of the limits of $N$ times each quantity. As printed: $0.21$, $0.30$, $0.32$. Corrected: $1.07$, $1.58$, $2.00$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E002-variance.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T3", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T3" }, { "id": "P004-T4", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T4" }, { "id": "P004-T36", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T36" } ], "headline": "The observational SAH estimator divides the square of an inverse-propensity-weighted residual by the squared propensity a second time; the corrected estimator is $N_1^{-2}\\sum_iZ_ic_i(d)\\widehat\\epsilon_i^2+N_0^{-2}\\sum_i(1-Z_i)c_i(d)\\widehat\\epsilon_i^2$.", "id": "P004-E002", "location": "Supplement A, A.6, p. 15 (supplement PDF p. 15): the residual $\\widehat\\epsilon_i$ and the display of $\\widehat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)$", "pdf_page": 15, "printed_page": "15", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P004-T3", "P004-T36", "P004-T4" ], "text": { "as_printed": "``where $\\widehat\\epsilon_i=\\frac{N_1}{N}\\frac{Z_i}{p(\\hl{O_j}|\\widehat\\theta_{\\mathrm{MLE}})}\\big(\\mu_i(\\mathbf Y;d)-O_i'\\widehat\\beta_{1,N}\\big)+\\frac{N_0}{N}\\frac{1-Z_i}{1-p(\\hl{O_j}|\\widehat\\theta_{\\mathrm{MLE}})}\\big(\\mu_i(\\mathbf Y;d)-O_i'\\widehat\\beta_{0,N}\\big)$.$^{9}$ The formula is similar to (6), except now that the residuals are weighted with the inverse propensity scores. Note that the covariates $O_i$ do not appear in the matrix $\\mathbf X$. Similarly, we can similarly define a Sävje, Aronow and Hudgens (2018) type variance estimator as in (20):\n\\[\n\\widehat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)=\\frac1{N^2}\\sum_{i=1}^N\\frac{Z_ic_i(d)\\widehat\\epsilon_i^2}{\\hl{p^2(O_i|\\widehat\\theta_{\\mathrm{MLE}})}}+\\frac1{N^2}\\sum_{i=1}^N\\frac{(1-Z_i)\\,c_i(d)\\widehat\\epsilon_i^2}{\\hl{\\big(1-p(O_i|\\widehat\\theta_{\\mathrm{MLE}})\\big)^2}},\\text{''}\n\\]", "corrected": "``where $\\widehat\\epsilon_i=\\frac{N_1}{N}\\frac{Z_i}{p(\\hl{O_i}|\\widehat\\theta_{\\mathrm{MLE}})}\\big(\\mu_i(\\mathbf Y;d)-O_i'\\widehat\\beta_{1,N}\\big)+\\frac{N_0}{N}\\frac{1-Z_i}{1-p(\\hl{O_i}|\\widehat\\theta_{\\mathrm{MLE}})}\\big(\\mu_i(\\mathbf Y;d)-O_i'\\widehat\\beta_{0,N}\\big)$. \\ldots\\ Similarly, we can similarly define a Sävje, Aronow and Hudgens (2018) type variance estimator as in (20):\n\\[\n\\widehat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)=\\hl{\\frac1{N_1^2}}\\sum_{i=1}^NZ_ic_i(d)\\widehat\\epsilon_i^2+\\hl{\\frac1{N_0^2}}\\sum_{i=1}^N(1-Z_i)\\,c_i(d)\\widehat\\epsilon_i^2,\\text{''}\n\\]\nwhere $N_1=\\sum_{i=1}^NZ_i$ and $N_0=N-N_1$, and the entire contribution from an empty arm is set to $0$. Equivalently, with the unweighted residuals $\\widehat r_{ia}=\\mu_i(\\mathbf Y;d)-O_i'\\widehat\\beta_{a,N}$ and $\\widehat p_i=p(O_i|\\widehat\\theta_{\\mathrm{MLE}})$,\n\\[\n\\widehat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)=\\frac1{N^2}\\sum_{i=1}^Nc_i(d)\\left\\{\\frac{Z_i\\widehat r_{i1}^2}{\\widehat p_i^{\\,2}}+\\frac{(1-Z_i)\\widehat r_{i0}^2}{(1-\\widehat p_i)^2}\\right\\}.\n\\]\nThe two forms agree whenever both arms are nonempty. These displays isolate the double weighting. For the corrected Proposition 4 of A.6, use the fitted values $\\widehat f_{ia}$ of \\fid{P004-E001} in place of $O_i'\\widehat\\beta_{a,N}$ in the residuals.", "evidence": "The evidence is an exact counterexample. Let $N$ be a multiple of $8$. Under independent assignment, a quarter of the nodes have propensity $1/2$ and treated circle averages $a_i=+1$ or $-1$, in equal numbers; the others have propensity $1/10$ and treated circle average $0$. All control circle averages are $0$, there is no interference at $d=0$, and $c_i(0)=1$. The covariates are the two group indicators, for which the logistic model holds, and the parameter space is the compact set on which the fitted propensities lie in $[1/20,19/20]$. The AME equals $0$ and the coefficients $\\beta_{1,N}$ and $\\beta_{0,N}$ equal $0$. Let $S$ be the sum of $a_iZ_i$ over the first group and $\\widehat p_H$ the fitted propensity of that group. Then $\\widehat\\tau_{\\mathrm{IPW}}=S/(N\\widehat p_H)$. A direct computation gives $\\widehat\\tau_{\\mathrm{IPW}}-2S/N=O_p(N^{-1})$ and $N\\operatorname{Var}(\\widehat\\tau_{\\mathrm{IPW}})\\to1/4$. The printed estimator satisfies $N\\widehat V^{\\mathrm{obs}}_{\\mathrm{SAH}}\\to2/25$ in probability, a ratio of $8/25$ to the variance. For the corrected estimator the limit is $1/2$. By an exact calculation over the assignment distribution at $N=32$, the variance of $\\widehat\\tau_{\\mathrm{IPW}}$ is $0.01212$, the expectation of the printed estimator is $0.00256$, and that of the corrected estimator is $0.01298$. At $N=64$ the three values are $0.00483$, $0.00145$ and $0.00764$. The nominal $95\\%$ interval built on the printed estimator has limiting coverage $2\\Phi(1.96\\sqrt{8/25})-1\\approx0.73$.", "problem": "The residual $\\widehat\\epsilon_i$ defined above the display already contains the inverse propensity weight. I write $\\widehat p_i=p(O_i|\\widehat\\theta_{\\mathrm{MLE}})$ and $\\widehat r_{ia}=\\mu_i(\\mathbf Y;d)-O_i'\\widehat\\beta_{a,N}$, and I read the unbound $O_j$ of the residual as $O_i$ (see \\fid{P004-E015}). For a treated node, $\\widehat\\epsilon_i=(N_1/N)\\,\\widehat r_{i1}/\\widehat p_i$. The SAH display divides $\\widehat\\epsilon_i^2$ by $\\widehat p_i^2$ again. Substituting the residual into the display, I obtain\n\\[\n\\widehat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)=\\frac1{N^2}\\sum_{i=1}^Nc_i(d)\\left[\\Big(\\frac{N_1}{N}\\Big)^2\\frac{Z_i\\widehat r_{i1}^2}{\\widehat p_i^{\\,4}}+\\Big(\\frac{N_0}{N}\\Big)^2\\frac{(1-Z_i)\\widehat r_{i0}^2}{(1-\\widehat p_i)^4}\\right],\n\\]\nwhereas the analogue of (20) has $Z_i\\widehat r_{i1}^2/\\widehat p_i^{\\,2}$ and $(1-Z_i)\\widehat r_{i0}^2/(1-\\widehat p_i)^2$ in the brackets. The two differ by the factors $(N_1/(N\\widehat p_i))^2$ and $(N_0/(N(1-\\widehat p_i)))^2$. These factors equal $1$ for an intercept-only logistic model with an interior maximum-likelihood estimate, where $\\widehat p_i=N_1/N$, and they can be far below $1$ when the propensities vary across nodes. The example below shows that the printed estimator need not be a conservative bound for the variance, and the intended IPW interval of Proposition 4(i) of A.6 can cover less often than its nominal level. A.6 defines no unweighted residual for the SAH display, and the page offers no other reading of $\\widehat\\epsilon_i$ there.", "what_survives": "The experimental SAH estimator (20) of the article, which is (24) of Supplement A, uses unweighted residuals with a common $p$; it is a different formula and is unaffected. The experimental simulations, the AME, the point estimators and the randomization tests are also unaffected by this error, and the observational HAC display is not touched by this point. The forest application of Section 8.2 reports HAC intervals, not observational SAH intervals; this error does not establish that any reported interval changes. What changes is the SAH display of A.6 and, with it, the intended IPW interval of Proposition 4(i) there. The corrected display removes the double weighting and adds no assumption. The general interval also requires the corrected linearization and fitted values in \\fid{P004-E001}." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E002", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of items (iv), (v) and (vi) of a condition and of Lemma A.6. Item (iv) reads: Compact Parameter Space: theta-zero in Theta-zero where Theta-zero, a subset of R to the k, is a compact set. Item (vi) defines theta-hat-MLE as the maximizer over theta in Theta-zero of the logistic log-likelihood. Lemma A.6 reads: Under C6 and C1, theta-hat-MLE minus theta-zero is o-p of 1, and theta-hat-MLE minus theta-zero equals the inverse of the average of p times 1 minus p times O sub i O sub i prime, times the average of Z sub i minus p times O sub i, plus o-p of N to the minus one half. In particular, theta-hat-MLE minus theta-zero is O-p of N to the minus one half. One red outline surrounds theta-zero in Theta-zero in item (iv). A second surrounds the term plus o-p of N to the minus one half.", "caption": "Supplement A, condition C 1, items (iv) to (vi), and Lemma A.6, p. 14. The outlines mark the condition on $\\theta_0$ and the remainder of the expansion.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E003.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T9", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T9" }, { "id": "P004-T10", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T10" } ], "headline": "Lemma A.6 needs the true coefficient $\\theta_0$ to be an interior point of the compact set $\\Theta_0$ over which the likelihood is maximized; when $\\theta_0$ lies on the boundary of $\\Theta_0$, the expansion of the estimator fails.", "id": "P004-E003", "location": "Supplement A, Appendix A.6, condition C 1, items (iv) and (vi), and Lemma A.6, p. 14 (C 1 begins on p. 13)", "pdf_page": 14, "printed_page": "14", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P004-T10", "P004-T9" ], "text": { "as_printed": "(iv) Compact Parameter Space: $\\hl{\\theta_0\\in\\Theta_0}$ where $\\Theta_0\\subset\\mathbb R^k$ is a compact set.\n\n(vi) The MLE estimator is used to estimate the coefficient vector $\\theta$:\n\\[\n\\widehat\\theta_{MLE}=\\arg\\max_{\\theta\\in\\Theta_0}\\sum_{i=1}^{N}\\bigl(Z_iO_i'\\theta-\\log(1+\\exp(O_i'\\theta))\\bigr).\n\\]\nLemma A.6. Under C6 and C1, $\\widehat\\theta_{MLE}-\\theta_0=o_p(1)$ and\n\\[\n\\widehat\\theta_{MLE}-\\theta_0=\\Bigl(\\frac1N\\sum_{i=1}^{N}p(O_i\\mid\\theta_0)(1-p(O_i\\mid\\theta_0))O_iO_i'\\Bigr)^{-1}\\frac1N\\sum_{i=1}^{N}\\bigl((Z_i-p(O_i\\mid\\theta_0))O_i\\bigr)\\hl{+o_p(N^{-\\frac12})}.\n\\]\nIn particular, $\\widehat\\theta_{MLE}-\\theta_0=O_p(N^{-\\frac12})$.", "corrected": "(iv) Compact Parameter Space: \\hl{$\\theta_0$ is an interior point of $\\Theta_0$}, where $\\Theta_0\\subset\\mathbb R^k$ is a fixed compact set. That is, there exists a fixed $r>0$ such that the ball $B(\\theta_0,r)$ is contained in $\\Theta_0$.\n\nLemma A.6. Under C6 and C~1 with item (iv) as above, $\\widehat\\theta_{MLE}-\\theta_0=o_p(1)$ and\n\\[\n\\widehat\\theta_{MLE}-\\theta_0=\\Bigl(\\frac1N\\sum_{i=1}^{N}p(O_i\\mid\\theta_0)(1-p(O_i\\mid\\theta_0))O_iO_i'\\Bigr)^{-1}\\frac1N\\sum_{i=1}^{N}(Z_i-p(O_i\\mid\\theta_0))O_i+o_p(N^{-\\frac12}).\n\\]\nIn particular, $\\widehat\\theta_{MLE}-\\theta_0=O_p(N^{-\\frac12})$. The remainder in the display is $O_p(N^{-1})$.", "evidence": "The evidence is an exact counterexample and a derivation of the lemma under the corrected condition.\n\n\\emph{Counterexample.} Let $k=1$, $O_i=1$, $\\Theta_0=[0,1]$ and $\\theta_0=0$, and let the $Z_i$ be independent with probability $1/2$. Every item of C~1 holds (with bound 1 on the confounder and Gram matrix 1), and C6 holds with $\\epsilon=1/4$. Let $T=\\sum_i(2Z_i-1)$. The matrix of the lemma equals $1/4$, and the leading term is $2T/N$. The derivative of the log-likelihood at 0 is $T/2$, and the log-likelihood is strictly concave. If $T<0$, it is decreasing on $[0,1]$ and $\\widehat\\theta_{MLE}=0$ exactly. On this event the remainder of the lemma is $-2T/N$, and $\\sqrt N$ times its absolute value exceeds 1 when in addition $T^2>N/4$. Since $E[T^2]=N$ and $E[T^4]=3N^2-2N$, the Cauchy--Schwarz inequality implies that\n\\[\n\\Pr\\bigl(T^2>N/4\\bigr)\\ge\\frac{(3N/4)^2}{3N^2-2N}\\ge\\frac{3}{16},\n\\]\nand the distribution of $T$ is symmetric. I conclude that\n\\[\n\\Pr\\Bigl(\\sqrt N\\,\\bigl|\\widehat\\theta_{MLE}-\\theta_0-2T/N\\bigr|>1\\Bigr)\\ge\\frac{3}{32}\\quad\\text{for every }N\\ge1,\n\\]\nso that the remainder is not $o_p(N^{-1/2})$. The limit of $\\sqrt N\\,\\widehat\\theta_{MLE}$ is $2\\max\\{U,0\\}$ with $U$ standard normal, and the leading term has the limit $2U$.\n\n\\emph{The lemma for an interior $\\theta_0$.} Let $g_N=N^{-1}\\sum_i(Z_i-p_i)O_i$ and $\\delta=\\widehat\\theta_{MLE}-\\theta_0$, and let $\\ell_N$ be the log-likelihood divided by $N$. Under C~1 and C6, I have $g_N=O_p(N^{-1/2})$, and there exists a constant $c>0$ such that the negative Hessian of $\\ell_N$ is at least $c$ times the identity on every segment between two points of $\\Theta_0$. Since $\\theta_0\\in\\Theta_0$, I obtain\n\\[\n0\\le\\ell_N(\\widehat\\theta_{MLE})-\\ell_N(\\theta_0)\\le g_N'\\delta-\\frac c2\\|\\delta\\|^2,\n\\]\nwhich implies that $\\|\\delta\\|\\le2\\|g_N\\|/c=O_p(N^{-1/2})$. This part uses the printed conditions only. If the ball $B(\\theta_0,r)$ lies in $\\Theta_0$, then $\\|\\delta\\|\\frac1{20}.\n\\]\nThe rule with $(1+B)/(P+1)$ rejects when $B\\le4$ and has size $0.0456\\ldots$ in the same design. For the signed estimator with the comparison as printed, the 128 assignments with a negative estimate never reject, and the size is $0.0277\\ldots$, half of the first value.\n\n\\emph{A large number of draws.} With 17 nodes and $y_i=2^{i-17}$ there are $M=65536$ equally likely values of the statistic. For $P=10000$ and $\\alpha=1/20$ the rule rejects when $B\\le500$. If the observed statistic is the $m$th largest value, the rule rejects with probability $f(m/M)$, where $f(q)=\\Pr(\\operatorname{Bin}(10000,q)\\le500)$ is decreasing in $q$ and has integral $501/10001$ over $[0,1]$. Comparing the sum over $m$ with the integral, I find that the size is at least\n\\[\n\\frac{501}{10001}-\\frac{1}{65536}=\\frac{32823535}{655425536}=0.05008\\ldots>\\frac1{20}.\n\\]\n\n\\emph{The corrected rule.} Let $S_0=|T(\\mathbf Y,\\mathbf Z)|$ and $S_p=|T(\\mathbf Y,\\mathbf Z_p)|$ for $p=1,\\dots,P$. Under the sharp null hypothesis the outcomes are fixed, and $S_0,S_1,\\dots,S_P$ are independent and identically distributed. Let $R_i$ be the number of indices $j$ with $S_j\\ge S_i$. Then the corrected $p$-value is $R_0/(P+1)$. For any values and any integer $k$, at most $k$ indices have $R_i\\le k$, since such an index is among the $k$ largest values, ties included. Taking $k=\\lfloor\\alpha(P+1)\\rfloor$ and using that the $P+1$ indices are exchangeable, I have\n\\[\n(P+1)\\Pr\\Bigl(\\frac{R_0}{P+1}\\le\\alpha\\Bigr)=E\\Bigl[\\sum_{i=0}^{P}\\mathbf 1\\{R_i\\le k\\}\\Bigr]\\le k\\le\\alpha(P+1).\n\\]\nApplying the same argument to the event $B\\le\\lfloor\\alpha P\\rfloor$, I obtain $\\Pr(B/P\\le\\alpha)\\le(\\lfloor\\alpha P\\rfloor+1)/(P+1)\\le\\alpha+(1-\\alpha)/(P+1)$.\n\n\\emph{What the examples do not show.} The rule $B/P\\le\\alpha$ is not invalid for every $P$. For $\\alpha=1/20$ and $P=9999$ both rules reject exactly when $B\\le499$. For a fixed finite design the rule becomes conservative when $P$ is large enough: in the eight-node example its size is at most $1997/40000$ for all $P\\ge8{,}000{,}000$. If the printed sentence only asserts that a suitable $P$ exists, these examples do not contradict it. What fails is the guarantee for an arbitrary large $P$.", "problem": "There are two points. First, the indicator compares the absolute value of each simulated statistic with the signed observed statistic. If $T(\\mathbf Y,\\mathbf Z)<0$, then every indicator equals 1, the proportion is 1, and the rule cannot reject. The statistics of the paper are signed. In Section 8.1 the observed statistic is $-0.034$ and the reported $p$-value is about 0.065, which the printed rule cannot produce. The comparison that is meant has $|T(\\mathbf Y,\\mathbf Z)|$ on the right.\n\nSecond, let $B$ be the number of draws with $|T(\\mathbf Y,\\mathbf Z_p)|\\ge|T(\\mathbf Y,\\mathbf Z)|$. The rule that rejects when $B/P\\le\\alpha$ does not have level $\\alpha$ merely because $P$ is large. Its size can exceed $\\alpha$ at a given $P$, by at most $(1-\\alpha)/(P+1)$.", "what_survives": "The definitions of the AME, the point estimators, their limit theory and the confidence intervals do not use this rule. The simulation code in the replication files takes absolute values on both sides and compares the observed statistic with a quantile of the simulated ones, so the printed comparison is not what was computed, and no curve of Figure 9 is shown to be wrong. The separate recalculations in \\fid{P004-N006} do not identify the computation behind the reported $p$-value of Section 8.1. If it is $B/P$ with $P=10{,}000$, then the corrected value exceeds it by at most $1/10001$, and the rejection at the 10\\% level and the non-rejection at the 5\\% level are unchanged. The corrected rule needs no new assumption. The $p$-value increases by $(1-B/P)/(P+1)$." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E005", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of equation (26), sAME(d; eta) equal to the integral over t of AME(t; eta) times K_h((d minus t) over h), and of the six lines of text after it. A red box marks the sentence 'The donut and disk AMEs, defined in Section 3, are special cases with (properly-normalized) uniform kernels.'", "caption": "Section 6.2, p. 757 (PDF p. 14), equation (26) and the paragraph after it. The marked sentence calls the donut and disk AMEs special cases of the smoothed AME (26) with uniform kernels.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E006.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T14", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T14" }, { "id": "P004-T15", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T15" } ], "headline": "The disk and donut AMEs are not special cases of the smoothed AME (26) with uniform kernels: they weight each distance by the mass of outcome points at that distance from each node.", "id": "P004-E006", "location": "Section 6.2, p. 757 (PDF p. 14), the sentence after equation (26)", "pdf_page": 14, "printed_page": "757", "reaches": [ { "effect": "uses_only_what_survives", "effect_text": "relies only on the part that survives", "explanation": null, "package": null, "recorded_as": null, "to": "work", "url": "https://pmaronow.github.io/llm-audit/works/P008/", "where": null, "work_id": "P008" }, { "effect": "implementation_wrong", "effect_text": "the package or replication code has the reported defect", "explanation": null, "package": "SpatialEffect", "recorded_as": null, "to": "package", "url": null, "where": null, "work_id": "P004" } ], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P004-T14", "P004-T15" ], "text": { "as_printed": "``The donut and disk AMEs, defined in Section 3, \\hl{are special cases with (properly-normalized) uniform kernels}.''", "corrected": "``\\hl{The donut and disk AMEs average the node-specific circle effects over their distance bands using the normalized shell-mass measures induced by the spatial measure. For complete Euclidean disks about point interventions under area weighting, these radial weights are proportional to distance; a common uniform radial kernel applies as an identity only when the normalized shell-mass measures coincide with it.}''\n\nPrecisely, for a node $i$ and a set $A$ of distances let $\\nu_i(A)=\\zeta\\{x:d_i(x)\\in A\\}$, and let $\\tau_i(r;\\eta)$ be the effect of node $i$ on the outcome points at distance $r$ (for a measure without atoms, the conditional effect given the distance). For a band $D$ such that $0<\\nu_i(D)<\\infty$ for all $i$,\n\\[\n\\mathrm{AME}_D(\\eta)=\\frac1N\\sum_{i=1}^N\\frac{\\int_D\\tau_i(r;\\eta)\\,\\nu_i(dr)}{\\nu_i(D)}.\n\\]\nFor a complete Euclidean disk about a point node under area measure, the normalized measure $\\nu_i(dr)/\\nu_i(D)$ has density $2r/d^2$ on $D=[0,d]$ and density $2r/(b^2-a^2)$ on an annulus $D=[a,b]$.", "evidence": "The evidence is three exact calculations and a derivation. First, I consider one node at the origin, independent Bernoulli$(1/2)$ assignment, and unit mass at each of the outcome points $(1,0)$, $(0,2)$ and $(-2,0)$, with effects $0$, $1$ and $1$. The circle AMEs at radii $1$ and $2$ are $0$ and $1$, and their uniform average is $1/2$. The disk $[0,2]$ and the donut $[1,2]$ each have AME $2/3$. Second, I consider nodes at $(0,0)$ and $(3,0)$ and unit mass at $(0,4)$, $(3,4)$ and $(3,-4)$, with $Y_x(\\mathbf z)=1+z_1g_1(x)+z_2g_2(x)$, $g_1=(0,1,1)$ and $g_2=(-1,0,0)$. Then $\\mathrm{AME}(4)=\\mathrm{AME}(5)=0$, and every choice of common weights on the two radii gives $0$. The AME of the disk $[0,5]$ and of the donut $[4,5]$ equals $1/6$. The normalized masses at radii $4$ and $5$ are $(1/3,2/3)$ for the first node and $(2/3,1/3)$ for the second. Third, for one point node, area measure on the unit disk and $Y_x(z)=z\\|x\\|$, the circle AME at radius $r$ is $r$. Its uniform average over $[0,1]$ is $1/2$, and the disk AME is $2/3$.\n\nFor the derivation, let $\\delta_i(x)$ be the marginalized effect of node $i$ on the point $x$. By (6), (8) and (9), the effect of node $i$ on a band $D$ is $\\int_{d_i(x)\\in D}\\delta_i(x)\\,\\zeta(dx)$ divided by $\\zeta\\{x:d_i(x)\\in D\\}$. Rewriting the integral over $x$ as an integral over the distance $r=d_i(x)$, I obtain the formula of the corrected passage.", "problem": "By (2), a disk or donut average is the integral of the outcome over the points whose distance to the node lies in a band, divided by the mass of those points under the measure $\\zeta$. Each distance in the band then enters with a weight equal to the mass of outcome points at that distance from the node. Equation (26) averages the circle AMEs over distance with one kernel, common to all nodes. A kernel that is constant on the band gives every distance the same weight, and no normalization of it supplies weights that vary with the distance or with the node. For a point node under area measure, the weight of radius $r$ in the disk $[0,d]$ is $2r/d^2$, not $1/d$.", "what_survives": "Definitions (2), (6), (8) and (9), the inference results of Section 5 and the smoothed quantities (26) and (27) are unaffected. Equations (26) and (27) still define smoothed quantities, and the applications report smooths with a triangular kernel. No reported number depends on the sentence. What is withdrawn is the identity between a disk or donut AME and the smoothed AME (26) with a uniform kernel. When the masses at each distance differ across nodes, the circle AMEs alone need not determine the AME of a disk or donut. The reading of a sum of ring estimates as an area effect in Section 8.1 is the subject of \\fid{P004-E008}." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E006", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of seven lines of the paragraph that begins 'For an interpretable sufficient condition'. A red box marks the sentence 'In this case, h(d) in C3 can be set to 2 d-bar + 2d.'", "caption": "Section 4, p. 752 (PDF p. 9), the paragraph after C3. The marked sentence sets the cutoff of C3 to $2\\bar d+2d$ with no restriction to point nodes.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E007.png" }, { "alt": "Crop of five lines of text. A red box marks the parenthesis '(thus ensuring adequate spacing between nonadjacent polygons)' in the sentence 'For polygon-intervention experiments, one may require that the size of each polygon is larger than a threshold value'.", "caption": "Section 4, p. 753 (PDF p. 10), the end of the paragraph after C4. The marked parenthesis says that a lower bound on polygon size ensures adequate spacing.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E007-b.png" }, { "alt": "Crop of footnotes 24 and 25. A red box marks '2 d-tilde + 2d' in footnote 25, 'To be precise, for AME at distance value d, we set the bandwidth at 2 d-tilde + 2d, as discussed in Section 4.'", "caption": "Footnote 25, p. 763 (PDF p. 20). The marked bandwidth applies the cutoff rule of Section 4 to the village polygons of the first application.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E007-c.png" }, { "alt": "Diagram in the plane, drawn to scale, with horizontal coordinates from 0 to 16 and vertical coordinates from 0 to 3. Two unit squares labelled A and B sit on the horizontal axis, A from 0 to 1 and B from 10 to 11. A double arrow between them is labelled 'd AB = 9'. A long grey rectangle labelled 'C, diameter D = square root of 122, about 11.05' lies above them, from 0 to 11 across and from 2 to 3 in height. A vertical double arrow from the lower edge of C to the lower edge of B is labelled 'effect radius d-bar = 2'. Below the squares, two horizontal bars start at the right edge of A. A short orange bar of length 4 is labelled 'printed cutoff 2 d-bar + 2d = 4: ends before B'. A long blue bar of length about 15.05 is labelled 'corrected cutoff 2 d-bar + 2d + D = 4 + square root of 122, about 15.05: reaches B'. Dashed vertical lines mark the right edge of A and the left edge of B. The title reads: 'Polygon nodes A and B, at distance 9, both depend on the treatment at C: the printed cutoff is 4, and the corrected cutoff is 15.05'.", "caption": "The first counterexample, drawn to scale. The squares $A=[0,1]\\times[0,1]$ and $B=[10,11]\\times[0,1]$ are $9$ apart, and the treatment at the polygon $C=[0,11]\\times[2,3]$ sets the outcome on both of them, within the effect radius $\\bar d=2$. Each bar is a cutoff, drawn as a length from $A$ toward $B$. The printed cutoff $2\\bar d+2d=4$ ends before $B$. The corrected cutoff $2\\bar d+2d+D=4+\\sqrt{122}\\approx15.05$, where $D=\\sqrt{122}$ is the diameter of $C$, reaches $B$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E007-counterexample.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T16", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T16" }, { "id": "P004-T17", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T17" } ], "headline": "The cutoff $h(d)=2\\bar d+2d$ is sufficient for C3 with point nodes; with polygon nodes adding a uniform bound on the source polygon’s diameter gives a sufficient cutoff, and a lower bound on polygon area does not give C4.", "id": "P004-E007", "location": "Section 4, p. 752 (PDF p. 9), the paragraph after C3; p. 753 (PDF p. 10), the sentence on polygon interventions after C4; footnote 25, p. 763 (PDF p. 20)", "pdf_page": 9, "printed_page": "752", "reaches": [ { "effect": "uses_only_what_survives", "effect_text": "relies only on the part that survives", "explanation": null, "package": null, "recorded_as": null, "to": "work", "url": "https://pmaronow.github.io/llm-audit/works/P008/", "where": null, "work_id": "P008" } ], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P004-T16", "P004-T17" ], "text": { "as_printed": "P.~752: ``For an interpretable sufficient condition, suppose that the intervention at every node has no influence on the outcome points more than $\\bar d$ away from it. Then C3 would be satisfied if the distance between the boundary of the radius-d circle around node i and the radius-d circle around node j is larger than $2\\bar d$. \\hl{In this case, $h(d)$ in C3 can be set to $2\\bar d+2d$.}''\n\nP.~753: ``For polygon-intervention experiments, one may require that the size of each polygon is larger than a threshold value \\hl{(thus ensuring adequate spacing between nonadjacent polygons)}.''\n\nFootnote 25, p.~763: ``To be precise, for AME at distance value d, we set the bandwidth at \\hl{$2\\tilde d+2d$}, as discussed in Section 4.''", "corrected": "P.~752: ``For an interpretable sufficient condition, suppose that the intervention at every node has no influence on the outcome points more than $\\bar d$ away from it. Then, \\hl{for point interventions}, C3 would be satisfied if the distance between the boundary of the radius-d circle around node i and the radius-d circle around node j is larger than $2\\bar d$. In this case, $h(d)$ in C3 can be set to $2\\bar d+2d$. \\hl{For polygon interventions, if every polygon has diameter at most $D<\\infty$, uniformly in the sample size, $h(d)$ in C3 can be set to $2\\bar d+2d+D$.}''\n\nStated in full, the cutoff rule for polygon interventions is as follows. Let $d_{ij}$ be the minimum distance between the points of node $i$ and the points of node $j$, as in footnote 3, and let the diameter of a node be the largest distance between two of its points. Suppose that the circle average (2) at distance value $d$ uses only outcome points within distance $d$ of its node, which holds for the circle, the donut and the disk. Suppose further that the intervention at every node has no influence on the outcome points more than $\\bar d$ away from it, and that every node has diameter at most $D<\\infty$, where $\\bar d$ and $D$ do not depend on the sample size. Then for all nodes $i\\ne j$,\n\\[\ns_{ij}(d)=1\\quad\\text{implies}\\quad d_{ij}\\le2\\bar d+2d+\\hl{D},\n\\]\nand C3 holds with $h(d)=2\\bar d+2d+\\hl{D}$. For point interventions every node has diameter $0$, and the rule is the printed $h(d)=2\\bar d+2d$.\n\nP.~753: ``For polygon-intervention experiments, \\hl{a lower bound on the area of each polygon does not ensure C4. In the Euclidean plane, if the polygons have pairwise disjoint interiors, area at least $a>0$ and diameter at most $D<\\infty$, uniformly in the sample size, then $|S_N(i;r)|\\le\\pi(r+2D)^2/a$ for all $i$ and all $r\\ge0$, and C4 holds with $b(r)=\\pi(r+2D)^2/a$.}''\n\nFootnote 25: ``To be precise, for AME at distance value d, we set the bandwidth at $2\\tilde d+2d$, \\hl{the rule of Section 4 for point interventions. For village polygons this bandwidth is a sensitivity choice; the premise of a local effect radius does not establish that it is sufficient.}''\n\nAlternatively, C3 and C4 may be assumed directly for polygon interventions, without the claim that a local effect radius and a lower bound on area establish them.", "evidence": "The evidence is two exact counterexamples and a derivation of the corrected bounds. First, let $A=[0,1]\\times[0,1]$, $B=[10,11]\\times[0,1]$ and $C=[0,11]\\times[2,3]$, with independent Bernoulli$(1/2)$ assignments, $\\bar d=2$, $d=0$ and Lebesgue measure. Let $Y_x(\\mathbf z)=z_C$ for $x\\in A\\cup B$ and $Y_x(\\mathbf z)=0$ elsewhere. Every affected point is within distance $2$ of $C$, and $A$ and $B$ affect nothing. The circle averages at $A$ and $B$ both equal $Z_C$. Then $s_{AB}(0)=1$ and their covariance is $1/4$, while $d_{AB}=9>2\\bar d+2d=4$. A variant with $d=\\bar d=1$ has $C=[0,11]\\times[3,4]$ and unit mass on the outcome points $(1/2,2)$ and $(21/2,2)$, where the outcome equals $z_C$. In it $d_{AB}=9>4$, the radius-$1$ contours of $A$ and $B$ are $7>2\\bar d=2$ apart, and again $s_{AB}(1)=1$ with covariance $1/4$.\n\nThe figure draws the first example to scale, and the table gives the distances in both examples. In each example the printed cutoff is $4$, and $A$ and $B$ are $9$ apart. The polygon $C$ has diameter $D=\\sqrt{11^2+1^2}=\\sqrt{122}\\approx11.05$, which is the largest of the three diameters. With this diameter, I obtain the corrected cutoff $2\\bar d+2d+D=4+\\sqrt{122}\\approx15.05$, which is larger than $d_{AB}=9$.\n\\begin{audittable}{lcc}{The two counterexamples to the printed cutoff rule.}\nQuantity & First example & Second example \\\\\nDistance value $d$ & $0$ & $1$ \\\\\nEffect radius $\\bar d$ & $2$ & $1$ \\\\\nPolygon $C$ & $[0,11]\\times[2,3]$ & $[0,11]\\times[3,4]$ \\\\\nPrinted cutoff $2\\bar d+2d$ & $4$ & $4$ \\\\\nDistance $d_{AB}$ & $9$ & $9$ \\\\\n$s_{AB}(d)$ & $1$ & $1$ \\\\\nCovariance of the circle averages at $A$ and $B$ & $1/4$ & $1/4$ \\\\\nLargest diameter $D$, that of $C$ & $\\sqrt{122}\\approx11.05$ & $\\sqrt{122}\\approx11.05$ \\\\\nCorrected cutoff $2\\bar d+2d+D$ & $4+\\sqrt{122}\\approx15.05$ & $4+\\sqrt{122}\\approx15.05$ \\\\\n\\end{audittable}\n\nSecond, for $n\\ge1$ let $P_0=[0,1]\\times[0,1]$ and $P_{k+1}=[2,2+n]\\times[2k/n,(2k+1)/n]$ for $0\\le k0$ and all sufficiently large $N$}, then\n\\[\n\\frac{\\sum_{i=1}^NV_i}{\\sqrt{\\operatorname{Var}\\big(\\sum_{i=1}^NV_i\\big)}}\\Rightarrow N(0,1).\\text{''}\n\\]\nHere a dependency neighborhood means that the variables indexed by two sets of units are independent whenever no unit of one set lies in the neighborhood of a unit of the other. In the proof of Proposition 2 the lemma is applied to the unnormalized summands $V_i$ such that $\\widehat\\tau_{\\mathrm{HT}}(d)-\\mathrm{AME}(d;\\eta)=N^{-1}\\sum_iV_i$, and in the proof of Proposition 3 to the summands of the linearized Hajek estimator. Equivalently, for the normalized summands $U_i$ of the printed proof, the conditions are $\\operatorname{Var}(\\sum_iU_i)=1$, $c_i(d)\\le\\bar c$ and $|U_i|\\le K/\\sqrt N$ for a constant $K$.", "evidence": "The evidence is an exact counterexample and a derivation of the corrected statement. For every $N$, let $U_1$ equal $+1$ or $-1$ with probability $1/2$ each, and let $U_2=\\cdots=U_N=0$. These variables are independent, bounded by $1$ and have mean zero, and $\\mathcal B(i;d)=\\{i\\}$ is a dependency neighborhood. Then $c_i(d)=1$ and $\\tilde c^2/N=1/N\\to0$. The sum equals $U_1$ and has variance $1$, and the standardized sum takes the values $\\pm1$ with probability $1/2$ each for every $N$. Its distribution function at $-1/2$ equals $1/2$, and $\\Phi(-1/2)<1/2$. A second example has every marginal variance equal to $1$. Let $N=2m+1$, let $R_0,\\ldots,R_m$ be independent signs, and let $(U_1,\\ldots,U_N)=(R_0,R_1,-R_1,\\ldots,R_m,-R_m)$. The neighborhoods have at most two elements, and the sum is again $R_0$.\n\nFor the corrected statement, let $\\bar c$ be a constant that bounds the sizes of the neighborhoods for all sample sizes. A joint cumulant of $r$ summands vanishes unless their indices are connected through the neighborhoods. There are at most $Nr^{r-2}\\bar c^{\\,r-1}$ connected $r$-tuples of indices, and each cumulant is at most $C_rM^r$ in absolute value, where $M$ bounds the summands and $C_r$ depends only on $r$. With $X_N$ the standardized sum and $v$ the constant in the lower bound on the variance, I obtain\n\\[\n|\\kappa_r(X_N)|\\le\\frac{r^{r-2}\\bar c^{\\,r-1}C_rM^r}{v^{r/2}}\\,N^{1-r/2}\\to0\\qquad\\text{for each }r\\ge3.\n\\]\nSince $\\kappa_1(X_N)=0$ and $\\kappa_2(X_N)=1$, this bound implies that the moments of $X_N$ converge to those of $N(0,1)$, and these moments determine the normal law. Note that I did not compare the lemma with Lemmas 1 and 2 of Ogburn et al.\\ (2020); nothing here concerns the original result.", "problem": "The lemma states a central limit theorem for bounded, centered summands under a single condition, on the sizes $c_i(d)$ of the dependency neighborhoods. Nothing in it keeps the variance of the sum from being small relative to $N$, and without such a condition the conclusion fails. A positive lower bound on the variance of the sum is not enough either: the standardized sum does not change when every summand is multiplied by a constant. The propositions that use the lemma assume that $N$ times the variance of the estimator is bounded below. The condition is available where the lemma is applied, but it is missing from the lemma.", "what_survives": "Propositions 2 and 3 keep their normal limits. In their proofs the unnormalized summands are bounded by a constant multiple of $b/\\min(p,1-p)$, conditions C3 and C4 bound the neighborhoods (C4A), and each proposition assumes that $N$ times the variance is bounded below. Proposition 4 uses these limits and assumes the same bound. The observational extension in A.6 also states its limit under a lower bound on $N$ times the variance. The corrected lemma covers only neighborhoods of bounded size: I do not claim a version for neighborhoods whose sizes grow while $\\tilde c^2/N\\to0$. No estimand, simulation or reported number changes." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E012", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of Proposition 2, 'Under C1-C4, we have', with the displayed bound on the variance of the linearized Hajek estimator. Red boxes mark, in the second sum, the bold 1 in mu_i(1; d) and the exponent 2 that follows the closing bracket of the expectation.", "caption": "Supplement A, p. 9, the statement of Proposition 2. In the control term, the marked argument $\\mathbf 1$ should be $\\mathbf 0$, and the marked outer exponent should be absent.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E013.png" }, { "alt": "Crop of four lines of a displayed computation. Red boxes mark the terms 'minus (E[(mu_i(1; d) minus mu-bar-1(d)) squared]) squared' and 'minus (E[(mu_i(0; d) minus mu-bar-0(d)) squared]) squared'. The last line is plus 2 E[mu_i(1; d) minus mu-bar-1(d)] times E[mu_i(0; d) minus mu-bar-0(d)].", "caption": "Supplement A, p. 9, the first lines of the proof of Proposition 2. The two marked terms subtract the squares of second moments; the squares of the means are required.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E013-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T21", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T21" }, { "id": "P004-T43", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T43" } ], "headline": "In the bound $\\bar{\\mathrm V}_{\\mathrm{HA}}(d)$ the control term is $\\mathrm E[(\\mu_i(\\mathbf 0;d)-\\bar\\mu^0(d))^2]/(1-p)$, with no outer square, and the proof subtracts squared means, not squared second moments.", "id": "P004-E013", "location": "Supplement A, A.4.2, Proposition 2, p. 9 (supplement PDF p. 9): the bound $\\bar{\\mathrm V}_{\\mathrm{HA}}(d)$ and the first lines of its proof", "pdf_page": 9, "printed_page": "9", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P004-T21", "P004-T43" ], "text": { "as_printed": "``PROPOSITION 2. Under C1-C4, we have,\n\\[\n\\begin{aligned}\n&\\operatorname{Var}\\big(\\hat\\tau^{\\mathrm L}_{\\mathrm{HA}}(d)\\big)\\le\\bar{\\mathrm V}_{\\mathrm{HA}}(d)\\\\\n&\\quad=\\frac1{N^2}\\sum_{i=1}^Nc_i(d)\\frac{\\mathrm E[(\\mu_i(\\mathbf 1;d)-\\bar\\mu^1(d))^2]}{p}+\\frac1{N^2}\\sum_{i=1}^Nc_i(d)\\frac{\\mathrm E[(\\mu_i(\\hl{\\mathbf 1};d)-\\bar\\mu^0(d))^2]\\hl{{}^2}}{1-p}.\\text{''}\n\\end{aligned}\n\\]\nIn the proof, the variance of the summand of node $i$ is printed as\n\\[\n\\begin{aligned}\n={}&\\frac{\\mathrm E[(\\mu_i(\\mathbf 1;d)-\\bar\\mu^1(d))^2]}{p}-\\hl{(\\mathrm E[(\\mu_i(\\mathbf 1;d)-\\bar\\mu^1(d))^2])^2}\\\\\n&+\\frac{\\mathrm E[(\\mu_i(\\mathbf 0;d)-\\bar\\mu^0(d))^2]}{1-p}-\\hl{(\\mathrm E[(\\mu_i(\\mathbf 0;d)-\\bar\\mu^0(d))^2])^2}\\\\\n&+2\\mathrm E[\\mu_i(\\mathbf 1;d)-\\bar\\mu^1(d)]\\times\\mathrm E[\\mu_i(\\mathbf 0;d)-\\bar\\mu^0(d)]\n\\end{aligned}\n\\]", "corrected": "``PROPOSITION 2. Under C1-C4, we have,\n\\[\n\\begin{aligned}\n&\\operatorname{Var}\\big(\\hat\\tau^{\\mathrm L}_{\\mathrm{HA}}(d)\\big)\\le\\bar{\\mathrm V}_{\\mathrm{HA}}(d)\\\\\n&\\quad=\\frac1{N^2}\\sum_{i=1}^Nc_i(d)\\frac{\\mathrm E[(\\mu_i(\\mathbf 1;d)-\\bar\\mu^1(d))^2]}{p}+\\frac1{N^2}\\sum_{i=1}^Nc_i(d)\\frac{\\mathrm E[(\\mu_i(\\hl{\\mathbf 0};d)-\\bar\\mu^0(d))^2]}{1-p}.\\text{''}\n\\end{aligned}\n\\]\nIn the proof:\n\\[\n\\begin{aligned}\n={}&\\frac{\\mathrm E[(\\mu_i(\\mathbf 1;d)-\\bar\\mu^1(d))^2]}{p}-\\hl{(\\mathrm E[\\mu_i(\\mathbf 1;d)-\\bar\\mu^1(d)])^2}\\\\\n&+\\frac{\\mathrm E[(\\mu_i(\\mathbf 0;d)-\\bar\\mu^0(d))^2]}{1-p}-\\hl{(\\mathrm E[\\mu_i(\\mathbf 0;d)-\\bar\\mu^0(d)])^2}\\\\\n&+2\\mathrm E[\\mu_i(\\mathbf 1;d)-\\bar\\mu^1(d)]\\times\\mathrm E[\\mu_i(\\mathbf 0;d)-\\bar\\mu^0(d)]\n\\end{aligned}\n\\]", "evidence": "The evidence is two exact counterexamples and a derivation. Let $N=2$ and $p=1/4$, with no interference, $c_i(d)=1$, treated outcomes $0$ and control outcomes $+1$ and $-1$. Both marginal means equal $0$. The printed bound equals $0$, the variance of the linearized estimator equals $1/6$, and the corrected bound equals $2/3$. For the proof step, let $p=1/4$ and consider a node such that $\\mu_i(\\mathbf 1;d)-\\bar\\mu^1(d)=1/2$ and $\\mu_i(\\mathbf 0;d)-\\bar\\mu^0(d)=0$. The variance of its summand is $3/4$, and the printed expression equals $15/16$.\n\nFor the derivation, I write $a_i=\\mu_i(\\mathbf 1;d)-\\bar\\mu^1(d)$ and $b_i=\\mu_i(\\mathbf 0;d)-\\bar\\mu^0(d)$. Let $A_i=Z_ia_i/p$ and $B_i=(1-Z_i)b_i/(1-p)$, as on p.~10. Since $A_iB_i=0$, I have\n\\[\n\\operatorname{Var}(A_i-B_i)=\\frac{\\mathrm E[a_i^2]}{p}+\\frac{\\mathrm E[b_i^2]}{1-p}-\\big(\\mathrm E[a_i]-\\mathrm E[b_i]\\big)^2\\le\\frac{\\mathrm E[a_i^2]}{p}+\\frac{\\mathrm E[b_i^2]}{1-p}.\n\\]\nThe covariance of the summands of $i$ and $j$ vanishes for $j\\notin\\mathcal B(i;d)$ and is otherwise at most half the sum of the two variances. Since $j\\in\\mathcal B(i;d)$ if and only if $i\\in\\mathcal B(j;d)$, the variance of the linearized estimator is at most $N^{-2}\\sum_ic_i(d)\\operatorname{Var}(A_i-B_i)$, which is at most the corrected bound.", "problem": "The term for the control arm is printed with the treated outcome $\\mu_i(\\mathbf 1;d)$ and with its expectation squared. As printed, the bound is false. The first lines of the proof compute the variance of the summand of node $i$ and subtract the squares of two second moments where the squares of the two means are required. The continuation of the proof on p.~10 has the correct identity and the correct bound, and the estimator (24) on p.~9, which is equation (20) of the article, uses the control residual.", "what_survives": "The bound as corrected, the SAH estimator (24) and the inference results that rest on the bound are unaffected. The correct expressions are printed on p.~10 of Supplement A." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E013", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of equations (39) and (40). Equation (39) reads: plus one over N squared times the sum over i and over j in B of i and d with j not equal to i, of the sums over a and b from 0 to 1, of E of the product of mu sub i of a sub i, b sub j and d minus O sub i prime beta sub a,N, and mu sub j of a sub i, b sub j and d minus O sub i prime beta sub b,N. Equation (40) reads: minus the same sums of minus 1 to the a plus b times E of mu sub i of a and d minus O sub i prime beta sub a,N, times E of mu sub j of b and d minus O sub i prime beta sub b,N. Red outlines surround the letter E that opens the expectation in (39) and the symbol O sub i prime in the second factor of each equation.", "caption": "Supplement A, equations (39) and (40), p. 15. The outline at the expectation sign of (39) marks the place where the factor $(-1)^{a+b}$ is missing; the two other outlines mark $O_i'$ in the residual of node $j$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E014.png" }, { "alt": "Crop of a displayed inequality under the words In addition under C2. It bounds V of tau-hat-L sub IPW of d by the two diagonal sums of (37) plus one over N squared times the sum over i and over j in B of i and d with j not equal to i, of the sums over a and b, of E of the product of mu sub i of a sub i, b sub j and d minus O sub i prime beta sub a,N, and mu sub j of a sub i, b sub j and d minus O sub i prime beta sub b,N. Red outlines surround the letter E that opens this expectation and the symbol O sub i prime in its second factor.", "caption": "Supplement A, the bound on the variance, p. 16. The outlines mark the same two places in the repeated term.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E014-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T22", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T22" } ], "headline": "The joint term (39) of the variance carries the factor $(-1)^{a+b}$, and in (39), in (40) and in the bound on p. 16 the residual of node $j$ is formed with $O_j$, not with $O_i$.", "id": "P004-E014", "location": "Supplement A, Appendix A.6, equations (39) and (40), p. 15, and the bound that repeats (39), p. 16", "pdf_page": 15, "printed_page": "15", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P004-T22" ], "text": { "as_printed": "\\emph{Equation (39), repeated in the bound on p.~16.}\n\\[\n+\\frac{1}{N^2}\\sum_{i;\\,j\\in\\mathcal B(i;d),\\,j\\ne i}\\ \\sum_{a=0}^{1}\\sum_{b=0}^{1}\\hl{\\mathrm E}\\Bigl[\\bigl(\\mu_i(\\mathbf a_i,\\mathbf b_j;d)-O_i'\\beta_{a,N}\\bigr)\\bigl(\\mu_j(\\mathbf a_i,\\mathbf b_j;d)-\\hl{O_i'}\\beta_{b,N}\\bigr)\\Bigr]\n\\]\n\\emph{Equation (40).}\n\\[\n-\\frac{1}{N^2}\\sum_{i;\\,j\\in\\mathcal B(i;d),\\,j\\ne i}\\ \\sum_{a=0}^{1}\\sum_{b=0}^{1}(-1)^{a+b}\\,\\mathrm E\\bigl[\\mu_i(\\mathbf a;d)-O_i'\\beta_{a,N}\\bigr]\\,E\\bigl[\\mu_j(\\mathbf b;d)-\\hl{O_i'}\\beta_{b,N}\\bigr].\n\\]", "corrected": "\\emph{Equation (39), and the same term in the bound on p.~16.}\n\\[\n+\\frac{1}{N^2}\\sum_{i;\\,j\\in\\mathcal B(i;d),\\,j\\ne i}\\ \\sum_{a=0}^{1}\\sum_{b=0}^{1}\\hl{(-1)^{a+b}}\\,\\mathrm E\\Bigl[\\bigl(\\mu_i(\\mathbf a_i,\\mathbf b_j;d)-O_i'\\beta_{a,N}\\bigr)\\bigl(\\mu_j(\\mathbf a_i,\\mathbf b_j;d)-\\hl{O_j'}\\beta_{b,N}\\bigr)\\Bigr]\n\\]\n\\emph{Equation (40).}\n\\[\n-\\frac{1}{N^2}\\sum_{i;\\,j\\in\\mathcal B(i;d),\\,j\\ne i}\\ \\sum_{a=0}^{1}\\sum_{b=0}^{1}(-1)^{a+b}\\,\\mathrm E\\bigl[\\mu_i(\\mathbf a;d)-O_i'\\beta_{a,N}\\bigr]\\,E\\bigl[\\mu_j(\\mathbf b;d)-\\hl{O_j'}\\beta_{b,N}\\bigr].\n\\]", "evidence": "The evidence is a derivation and an exact example. I write $r_{ia}=\\mu_i(\\mathbf Y;d)-O_i'\\beta_{a,N}$ and $S_i=\\sum_{a=0}^{1}(2a-1)\\mathbf 1\\{Z_i=a\\}\\,r_{ia}/\\Pr(Z_i=a)$. Then the printed linear form is a constant plus $N^{-1}\\sum_iS_i$. For $i\\ne j$, the assignments $Z_i$ and $Z_j$ are independent, their joint probability cancels the two denominators, and I have\n\\[\nE[S_iS_j]=\\sum_{a=0}^{1}\\sum_{b=0}^{1}(-1)^{a+b}E\\Bigl[\\bigl(\\mu_i(\\mathbf a_i,\\mathbf b_j;d)-O_i'\\beta_{a,N}\\bigr)\\bigl(\\mu_j(\\mathbf a_i,\\mathbf b_j;d)-O_j'\\beta_{b,N}\\bigr)\\Bigr].\n\\]\nThe example has three nodes with probability $1/2$ and $O_i=1$, and the circle averages are $\\mu_1(\\mathbf Y;d)=\\mu_2(\\mathbf Y;d)=Z_3-1/2$ and $\\mu_3(\\mathbf Y;d)=0$. All arm means and both coefficients vanish. For the pair of nodes 1 and 2, each of the four conditional products equals $1/4$, so that the signed sum is 0 and the sum without signs is 1. Enumerating the eight assignments, I find that the variance of the linear form is $2/9$. The expression (37)--(40) as printed equals $4/9$, and with the sign in (39) it equals $2/9$.", "problem": "Equations (37)--(40) expand the variance of a contrast between the treated and the control arm. In such an expansion the term for arm $a$ at node $i$ and arm $b$ at node $j$ has the sign $(-1)^{a+b}$, as in Lemma A.4 (Supplement A, p.~5), to which the proof refers, and as in (40). Equation (39) and its repetition in the bound on p.~16 omit the sign. In addition, the residual of node $j$ is written with $O_i'\\beta_{b,N}$ in (39), in (40) and in the bound. The fitted value of node $j$ is $O_j'\\beta_{b,N}$.", "what_survives": "Lemma A.4 has the signed form, and the sign in (40) is correct. With the two corrections, (37)--(40) is the variance of the printed linear form of Proposition 4 of A.6. That linear form is itself replaced in \\fid{P004-E001}, where the fitted values $O_i'\\beta_{a,N}$ of these displays become $f_{ia}$." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E014", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of the sentence 'A similar homophily condition is needed for the HAC variance estimation.' and of condition C 2, a displayed inequality: one over N times the sum over i of (tau_i minus (O_i' beta_1 minus O_i' beta_0)) times the sum over j in B(i; d) of (tau_j minus (O_i' beta_1 minus O_i' beta_0)) is at least 0. Red boxes mark the two O_i' in the second factor.", "caption": "Supplement A, p. 14, condition “C 2” of A.6. The two marked $O_i'$ stand in the factor that is summed over the neighbor $j$, where $O_j'$ is meant.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E015.png" }, { "alt": "Crop of the definition of the N by 2 matrix X, the display V-hat-obs-HAC(d) = (X'X) inverse times a double sum times (X'X) inverse, and the line that defines epsilon-hat_i. Red boxes mark the left-hand side V-hat-obs-HAC(d) and the symbol O_j in each of the two denominators p(O_j | theta-hat_MLE) and 1 minus p(O_j | theta-hat_MLE).", "caption": "Supplement A, p. 15, the HAC-type estimator of A.6 and the residual $\\widehat\\epsilon_i$. The marked left-hand side names a $2\\times2$ matrix as a variance; the two marked $O_j$ stand in the weights of the residual of node $i$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E015-b.png" }, { "alt": "Crop of the line 'Further more, for each alpha less than one half,' and of items (i) and (ii), each a limit of the probability that the absolute value of a studentized estimator lies between z_(alpha/2) and z_(1 minus alpha/2). Red boxes mark tau-hat_HA(d) in the numerator of each item.", "caption": "Supplement A, p. 16, items (i) and (ii) of Proposition 4 of A.6. The two marked numerators carry $\\widehat\\tau_{\\mathrm{HA}}(d)$; the estimator defined and analyzed in A.6 is $\\widehat\\tau_{\\mathrm{IPW}}(d)$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E015-c.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T23", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T23" } ], "headline": "In A.6 the neighbor factor of “C 2” takes $O_j$, the residual $\\widehat\\epsilon_i$ takes $p(O_i|\\widehat\\theta_{\\mathrm{MLE}})$, the scalar HAC variance is the $(2,2)$ entry of the displayed matrix, and the intervals are for $\\widehat\\tau_{\\mathrm{IPW}}$.", "id": "P004-E015", "location": "Supplement A, A.6: condition “C 2”, p. 14; the display of $\\widehat{\\mathrm V}^{\\mathrm{obs}}_{\\mathrm{HAC}}(d)$ and the residual $\\widehat\\epsilon_i$, p. 15; items (i)–(ii) of Proposition 4, p. 16 (supplement PDF pp. 14–16)", "pdf_page": 14, "printed_page": "14", "reaches": [ { "effect": "implementation_differs_from_paper", "effect_text": "the code implements something other than the printed method", "explanation": null, "package": "SpatialEffect", "recorded_as": null, "to": "package", "url": null, "where": null, "work_id": "P004" } ], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P004-T23" ], "text": { "as_printed": "P.~14: ``C 2. For all sample size $N$,\n\\[\n\\frac1N\\sum_{i=1}^N\\Big(\\tau_i(d;\\eta)-\\big(O_i'\\beta_{1,N}-O_i'\\beta_{0,N}\\big)\\Big)\\sum_{j\\in\\mathcal B(i;d)}\\Big(\\tau_j(d;\\eta)-\\big(\\hl{O_i'}\\beta_{1,N}-\\hl{O_i'}\\beta_{0,N}\\big)\\Big)\\ge0.\\text{''}\n\\]\nP.~15:\n\\[\n\\hl{\\widehat{\\mathrm V}^{\\mathrm{obs}}_{\\mathrm{HAC}}(d)}=(\\mathbf X'\\mathbf X)^{-1}\\left(\\sum_{i=1}^N\\sum_{j=1}^N\\mathbf X_i\\mathbf X_j'\\hat\\epsilon_i\\hat\\epsilon_j\\mathbf 1\\{j\\in\\mathcal B(i;d)\\}\\right)(\\mathbf X'\\mathbf X)^{-1},\n\\]\n``where $\\widehat\\epsilon_i=\\frac{N_1}{N}\\frac{Z_i}{p(\\hl{O_j}|\\widehat\\theta_{\\mathrm{MLE}})}\\big(\\mu_i(\\mathbf Y;d)-O_i'\\widehat\\beta_{1,N}\\big)+\\frac{N_0}{N}\\frac{1-Z_i}{1-p(\\hl{O_j}|\\widehat\\theta_{\\mathrm{MLE}})}\\big(\\mu_i(\\mathbf Y;d)-O_i'\\widehat\\beta_{0,N}\\big)$.''\n\nP.~16:\n\\[\n\\begin{aligned}\n&\\text{``(i) }\\lim_{N\\to\\infty}\\mathbf{Prob}\\Bigg(z_{\\frac\\alpha2}\\le\\Bigg|\\frac{\\hl{\\widehat\\tau_{\\mathrm{HA}}(d)}-\\mathrm{AME}(d;\\eta)}{\\sqrt{\\widehat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)}}\\Bigg|\\le z_{1-\\frac\\alpha2}\\Bigg)\\ge1-\\alpha;\\\\\n&\\text{(ii) additionally under C2, }\\lim_{N\\to\\infty}\\mathbf{Prob}\\Bigg(z_{\\frac\\alpha2}\\le\\Bigg|\\frac{\\hl{\\widehat\\tau_{\\mathrm{HA}}(d)}-\\mathrm{AME}(d;\\eta)}{\\sqrt{\\widehat V^{\\mathrm{obs}}_{\\mathrm{HAC}}(d)}}\\Bigg|\\le z_{1-\\frac\\alpha2}\\Bigg)\\ge1-\\alpha.\\text{''}\n\\end{aligned}\n\\]", "corrected": "P.~14: ``C 2. For all sample size $N$,\n\\[\n\\frac1N\\sum_{i=1}^N\\Big(\\tau_i(d;\\eta)-\\big(O_i'\\beta_{1,N}-O_i'\\beta_{0,N}\\big)\\Big)\\sum_{j\\in\\mathcal B(i;d)}\\Big(\\tau_j(d;\\eta)-\\big(\\hl{O_j'}\\beta_{1,N}-\\hl{O_j'}\\beta_{0,N}\\big)\\Big)\\ge0.\\text{''}\n\\]\nP.~15:\n\\[\n\\begin{aligned}\n{}\\hl{\\widehat\\Sigma^{\\mathrm{obs}}_{\\mathrm{HAC}}(d)}&=(\\mathbf X'\\mathbf X)^{-1}\\left(\\sum_{i=1}^N\\sum_{j=1}^N\\mathbf X_i\\mathbf X_j'\\hat\\epsilon_i\\hat\\epsilon_j\\mathbf 1\\{j\\in\\mathcal B(i;d)\\}\\right)(\\mathbf X'\\mathbf X)^{-1},\\\\\n{}\\hl{\\widehat{\\mathrm V}^{\\mathrm{obs}}_{\\mathrm{HAC}}(d)}&{}\\hl{{}=w'\\widehat\\Sigma^{\\mathrm{obs}}_{\\mathrm{HAC}}(d)\\,w,\\quad w=(0,1)',}\n\\end{aligned}\n\\]\nthat is, the $(2,2)$ entry of the matrix, ``where $\\widehat\\epsilon_i=\\frac{N_1}{N}\\frac{Z_i}{p(\\hl{O_i}|\\widehat\\theta_{\\mathrm{MLE}})}\\big(\\mu_i(\\mathbf Y;d)-O_i'\\widehat\\beta_{1,N}\\big)+\\frac{N_0}{N}\\frac{1-Z_i}{1-p(\\hl{O_i}|\\widehat\\theta_{\\mathrm{MLE}})}\\big(\\mu_i(\\mathbf Y;d)-O_i'\\widehat\\beta_{0,N}\\big)$.''\n\nP.~16:\n\\[\n\\begin{aligned}\n&\\text{``(i) }\\lim_{N\\to\\infty}\\mathbf{Prob}\\Bigg(z_{\\frac\\alpha2}\\le\\Bigg|\\frac{\\hl{\\widehat\\tau_{\\mathrm{IPW}}(d)}-\\mathrm{AME}(d;\\eta)}{\\sqrt{\\widehat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)}}\\Bigg|\\le z_{1-\\frac\\alpha2}\\Bigg)\\ge1-\\alpha;\\\\\n&\\text{(ii) additionally under C2, }\\lim_{N\\to\\infty}\\mathbf{Prob}\\Bigg(z_{\\frac\\alpha2}\\le\\Bigg|\\frac{\\hl{\\widehat\\tau_{\\mathrm{IPW}}(d)}-\\mathrm{AME}(d;\\eta)}{\\sqrt{\\widehat V^{\\mathrm{obs}}_{\\mathrm{HAC}}(d)}}\\Bigg|\\le z_{1-\\frac\\alpha2}\\Bigg)\\ge1-\\alpha.\\text{''}\n\\end{aligned}\n\\]", "evidence": "The evidence is a check of pp.~14--16 of Supplement A and a short derivation. With $h_i=\\tau_i(d;\\eta)-O_i'(\\beta_{1,N}-\\beta_{0,N})$, the homophily condition is a sum of products $h_ih_j$, and the second factor must contain $O_j$. The weight of the residual of node $i$ is the propensity of node $i$, as in the IPW estimator on p.~14. Since the matrix $\\mathbf X$ is $N$-by-$2$, the displayed matrix is $2\\times2$; the variance of the treatment contrast is $w'\\Sigma w=\\Sigma_{22}$ with $w=(0,1)'$, as on p.~755 of the article. The limit stated directly before item (i) is for $\\widehat\\tau_{\\mathrm{IPW}}$.", "problem": "Four displays of A.6 carry a wrong index or label. In ``C 2'' the factor summed over the neighbor $j$ repeats $O_i$ where the covariate of node $j$ is meant. In $\\widehat\\epsilon_i$ the propensity is written with $O_j$, although no $j$ is bound there and the residual belongs to node $i$. The display named $\\widehat{\\mathrm V}^{\\mathrm{obs}}_{\\mathrm{HAC}}(d)$ is a $2\\times2$ matrix, while items (i)--(ii) need the scalar variance of the treatment contrast. Items (i)--(ii) are written for $\\widehat\\tau_{\\mathrm{HA}}$, while A.6 defines and analyzes $\\widehat\\tau_{\\mathrm{IPW}}$. The absolute value in (i)--(ii) is harmless: since $z_{\\alpha/2}<0$, the lower inequality always holds, and the event is the usual symmetric interval.", "what_survives": "These corrections concern notation only, and nothing in the article's experimental results changes. They do not repair the linearization of Proposition 4 of A.6 (\\fid{P004-E001}), its covariance displays (\\fid{P004-E014}) or its SAH display (\\fid{P004-E002}). They also do not establish an interval for the Hajek estimator in the observational setting; such a claim would need its own argument." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E015", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of three lines of text and equation (44). The text reads: Define lambda equals w-prime times the inverse of X-prime X, times X, in R to the 1 by N; M equals I minus X-prime times the inverse of X-prime X, times X, in R to the N by N; and the weight matrix Omega in R to the N by N where Omega sub ij is the indicator that j is in B of i and d. Young (2015)'s effective degree of freedom adjustment in our setting is calculated as. Equation (44) defines mu as a trace. Red outlines surround the expression for lambda and the expression for M.", "caption": "Supplement A, Appendix A.7, p. 17. The outlines mark the two products with misplaced transposes.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E016.png" }, { "alt": "Crop of equation (45) and two lines of text. Equation (45) defines nu as 2 times the trace of the square of the matrix in (44). The text reads: We use the alpha over 2 quantile and the 1 minus alpha over 2 quantile of the t-distribution with 2 mu squared over v degree of freedom to construct the confidence interval. A red outline surrounds the letter v in the denominator.", "caption": "Supplement A, equation (45) and the last sentence of Appendix A.7, p. 17. The outline marks the denominator $v$, where the $\\nu$ of (45) is meant.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E016-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T24", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T24" } ], "headline": "In Appendix A.7 the transposes are on the wrong factors: $\\lambda=w'(X'X)^{-1}X'$ and $M=I-X(X'X)^{-1}X'$; the degrees of freedom are $2\\mu^2/\\nu$, with the $\\nu$ of (45).", "id": "P004-E016", "location": "Supplement A, Appendix A.7, the sentence “Define $\\lambda=\\dots$” before equation (44) and the last sentence of the section, p. 17", "pdf_page": 17, "printed_page": "17", "reaches": [ { "effect": "implementation_differs_from_paper", "effect_text": "the code implements something other than the printed method", "explanation": null, "package": "SpatialEffect", "recorded_as": null, "to": "package", "url": null, "where": null, "work_id": "P004" } ], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P004-T24" ], "text": { "as_printed": "Define $\\lambda=\\mathbf w'(\\mathbf X'\\mathbf X)^{-1}\\hl{\\mathbf X}\\in\\mathbb R^{1\\times N}$, $\\mathbf M=\\mathbf I-\\hl{\\mathbf X'}(\\mathbf X'\\mathbf X)^{-1}\\hl{\\mathbf X}\\in\\mathbb R^{N\\times N}$, and the weight matrix $\\Omega\\in\\mathbb R^{N\\times N}$ where $\\Omega_{ij}=\\mathbf 1\\{j\\in\\mathcal B(i;d)\\}$.\n\nWe use the $\\frac\\alpha2$-quantile and $\\bigl(1-\\frac\\alpha2\\bigr)$-quantile of the t-distribution with $\\frac{2\\mu^2}{\\hl{v}}$ degree of freedom to construct the confidence interval.", "corrected": "Define $\\lambda=\\mathbf w'(\\mathbf X'\\mathbf X)^{-1}\\hl{\\mathbf X'}\\in\\mathbb R^{1\\times N}$, $\\mathbf M=\\mathbf I_N-\\hl{\\mathbf X}(\\mathbf X'\\mathbf X)^{-1}\\hl{\\mathbf X'}\\in\\mathbb R^{N\\times N}$, and the weight matrix $\\Omega\\in\\mathbb R^{N\\times N}$ where $\\Omega_{ij}=\\mathbf 1\\{j\\in\\mathcal B(i;d)\\}$.\n\nWe use the $\\frac\\alpha2$-quantile and $\\bigl(1-\\frac\\alpha2\\bigr)$-quantile of the t-distribution with $\\frac{2\\mu^2}{\\hl{\\nu}}$ degree of freedom to construct the confidence interval.", "evidence": "The evidence is a check of the dimensions. With $X\\in\\mathbb R^{N\\times2}$, the row $w'(X'X)^{-1}X'$ has dimension $1\\times N$, and $I_N-X(X'X)^{-1}X'$ has dimension $N\\times N$, is symmetric and idempotent, and annihilates $X$. The package in the replication files uses these forms: \\texttt{z\\_vec <- c(0, 1) \\%*\\% XX\\_mat\\_inv \\%*\\% t(X\\_mat)} and \\texttt{M\\_mat <- diag(1, nz, nz) - X\\_mat \\%*\\% XX\\_mat\\_inv \\%*\\% t(X\\_mat)} (\\texttt{SpatialEffect/R/SpatialEffect.R}, lines 196--197).", "problem": "The matrix $X$ has dimension $N\\times2$ and $w=(0,1)'$. The product $w'(X'X)^{-1}$ is a row of length 2 and cannot multiply $X$, and $X'(X'X)^{-1}X$ is not a product of conforming matrices for general $N$. The dimensions stated in the sentence, $1\\times N$ and $N\\times N$, identify what is meant: the row of weights whose product with the outcomes is the coefficient of the treatment in the regression, and the matrix that maps the outcomes to the residuals. In the last sentence, $v$ stands for the $\\nu$ defined in (45).", "what_survives": "Equations (44) and (45) are as intended once $\\lambda$ and $M$ are read in the corrected form, and the code computes them from the corrected matrices. This correction concerns the two definitions and the symbol only. It does not bear on the accuracy of the adjustment in finite samples." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E016", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of page 26 of Supplement A with the definition of tau hat i N set in the text, the sentence Note E of tau hat i N equals 0, the statement of Lemma C.2 and equation (57). Red boxes mark the last term of the definition, minus tau i of d and eta, and the last term of equation (57), minus tau i of d.", "caption": "Supplement A, p. 26: the definition of $\\widehat\\tau_{i,N}(\\mu_1,\\mu_0;d)$ before Lemma C.2, and equation (57). The two marked terms lack the factor $1/N$ that every other term carries.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E017.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T25", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T25" } ], "headline": "The centring term of the summand $\\widehat\\tau_{i,N}(\\mu_1,\\mu_0;d)$ is $\\tau_i(d;\\eta)/N$ and not $\\tau_i(d;\\eta)$, both in the definition before Lemma C.2 and in equation (57).", "id": "P004-E017", "location": "Supplement A, Appendix C, definition of $\\widehat\\tau_{i,N}(\\mu_1,\\mu_0;d)$ before Lemma C.2 and equation (57), p. 26 (PDF p. 26 of the supplement)", "pdf_page": 26, "printed_page": "26", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P004-T25" ], "text": { "as_printed": "The following lemma states that provided $\\{\\mu_i(\\mathbf Y;d),i\\in\\mathcal S_N\\}$ is uniform $L_2$-NED with size $-4$, so is the transformation\n\\[\n\\widehat\\tau_{i,N}(\\mu_1,\\mu_0;d)=\\Big(\\frac{\\mathbf Z_i}{Np}-\\frac{(1-\\mathbf Z_i)}{N(1-p)}\\Big)\\mu_i(\\mathbf Y;d)+\\frac1N(\\mu_1-\\mu_0)-\\Big(\\frac{\\mathbf Z_i\\mu_1}{Np}-\\frac{(1-\\mathbf Z_i)\\mu_0}{N(1-p)}\\Big)\\hl{-\\tau_i(d;\\eta)}.\n\\]\nNote $\\mathrm E[\\widehat\\tau_{i,N}(\\mu_1,\\mu_0;d)]=0$.\n\nProof. Note $\\mathrm E[\\widehat\\tau_{i,N}(\\mu_1,\\mu_0;d)|\\mathcal F_i(s)]$ is equal to:\n\\[\n\\Big(\\frac{\\mathbf Z_i}{Np}-\\frac{(1-\\mathbf Z_i)}{N(1-p)}\\Big)\\mathrm E[\\mu_i(\\mathbf Y;d)|\\mathcal F_i(s)]+\\frac1N(\\mu_1-\\mu_0)-\\Big(\\frac{\\mathbf Z_i\\mu_1}{Np}-\\frac{(1-\\mathbf Z_i)\\mu_0}{N(1-p)}\\Big)\\hl{-\\tau_i(d)},\\qquad\\text{(57)}\n\\]", "corrected": "The following lemma states that provided $\\{\\mu_i(\\mathbf Y;d),i\\in\\mathcal S_N\\}$ is uniform $L_2$-NED with size $-4$, so is the transformation\n\\[\n\\widehat\\tau_{i,N}(\\mu_1,\\mu_0;d)=\\Big(\\frac{\\mathbf Z_i}{Np}-\\frac{(1-\\mathbf Z_i)}{N(1-p)}\\Big)\\mu_i(\\mathbf Y;d)+\\frac1N(\\mu_1-\\mu_0)-\\Big(\\frac{\\mathbf Z_i\\mu_1}{Np}-\\frac{(1-\\mathbf Z_i)\\mu_0}{N(1-p)}\\Big)-\\hl{\\frac1N}\\tau_i(d;\\eta).\n\\]\nNote $\\mathrm E[\\widehat\\tau_{i,N}(\\mu_1,\\mu_0;d)]=0$ under C1, for each pair $\\mu_1,\\mu_0\\in\\mathbb R$.\n\nProof. Note $\\mathrm E[\\widehat\\tau_{i,N}(\\mu_1,\\mu_0;d)|\\mathcal F_i(s)]$ is equal to:\n\\[\n\\Big(\\frac{\\mathbf Z_i}{Np}-\\frac{(1-\\mathbf Z_i)}{N(1-p)}\\Big)\\mathrm E[\\mu_i(\\mathbf Y;d)|\\mathcal F_i(s)]+\\frac1N(\\mu_1-\\mu_0)-\\Big(\\frac{\\mathbf Z_i\\mu_1}{Np}-\\frac{(1-\\mathbf Z_i)\\mu_0}{N(1-p)}\\Big)-\\hl{\\frac1N}\\tau_i(d),\\qquad\\text{(57)}\n\\]", "evidence": "The evidence is a derivation and an exact example. Let $m_{iz}=\\mathrm E[\\mu_i(\\mathbf Y;d)\\mid\\mathbf Z_i=z]$. Under the independent Bernoulli assignments of C1, I have $\\tau_i(d;\\eta)=m_{i1}-m_{i0}$ and\n\\[\n\\mathrm E\\Big[\\Big(\\frac{\\mathbf Z_i}{Np}-\\frac{1-\\mathbf Z_i}{N(1-p)}\\Big)\\mu_i(\\mathbf Y;d)\\Big]=\\frac{m_{i1}-m_{i0}}{N}=\\frac{\\tau_i(d;\\eta)}{N},\\qquad\n\\mathrm E\\Big[\\frac{\\mathbf Z_i\\mu_1}{Np}-\\frac{(1-\\mathbf Z_i)\\mu_0}{N(1-p)}\\Big]=\\frac{\\mu_1-\\mu_0}{N}.\n\\]\nThe second expectation cancels the constant $(\\mu_1-\\mu_0)/N$. The printed summand therefore has mean $\\tau_i(d;\\eta)/N-\\tau_i(d;\\eta)$, and the summand with $\\tau_i(d;\\eta)/N$ subtracted has mean zero. For the example, suppose that there is no interference and that $\\mu_i(\\mathbf Y;d)=\\mathbf Z_i$, with $N=8$, $p=1/2$ and $\\mu_1=\\mu_0=0$. Then $\\tau_i(d;\\eta)=1$, the printed summand has mean $-7/8$, and the corrected summand has mean $0$. By (56) and the definition (9) of the AME in the article, the corrected summands sum to $\\widehat\\tau(\\mu_1,\\mu_0;d)-\\mathrm{AME}(d;\\eta)$.", "problem": "The summand is meant to have mean zero and order $1/N$, and the sentence after the definition asserts that its mean is zero. Every other term of the definition carries the factor $1/N$, and the last term does not. As printed, the mean of the summand is $(1/N-1)\\tau_i(d;\\eta)$. The same term appears without the factor in (57). The proofs of Lemmas C.3 and C.7 use summands of mean zero and order $1/N$, and they need the corrected definition.", "what_survives": "Lemma C.2 and the bound (58)--(59) are unchanged, since the centring term is deterministic and cancels in the difference between $\\widehat\\tau_{i,N}$ and its conditional expectation. Covariances are unchanged for the same reason. The asymptotic arguments of Appendix C that use these summands are unaffected once the definition is corrected." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E017", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of assumption C 4 and the two sentences that introduce it. Red boxes mark s greater than or equal to 0 and the formula b of s equals C b times s squared.", "caption": "Supplement A, p. 27: assumption C4 of Appendix C. The marked range of radii includes $s=0$, where the marked bound $C_bs^2$ equals zero.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E018.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T26", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T26" }, { "id": "P004-T27", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T27" }, { "id": "P004-T28", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T28" }, { "id": "P004-T29", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T29" }, { "id": "P004-T44", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T44" } ], "headline": "Assumption C4 of Appendix C is a quadratic bound on the number of nodes within distance $s$ of a node and is meant for radii $s\\ge1$; at smaller radii, down to the printed $s\\ge0$, the ball still contains its own centre and the bound $C_bs^2$ cannot hold.", "id": "P004-E018", "location": "Supplement A, Appendix C, assumption C4, p. 27 (PDF p. 27 of the supplement); used in equations (69)–(73), p. 28", "pdf_page": 27, "printed_page": "27", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [ "P004-T26", "P004-T27", "P004-T28", "P004-T29", "P004-T44" ], "text": { "as_printed": "C 4. For each radius $\\hl{s\\ge0}$ and all large sample sizes $N$, there exists a $C_b>0$ such that $\\sup_{i\\in\\mathcal S_N}|\\mathcal B_N(i;s)|\\le b(s)$, with $\\hl{b(s)=C_bs^2}$.", "corrected": "C 4. \\hl{Let} $\\hl{\\mathcal B_N(i;s)=\\{j\\in\\mathcal S_N:d_{ij}\\le s\\}}$\\hl{.} There exists \\hl{one constant} $C_b>0$\\hl{, the same for all radii and all sample sizes,} such that for each radius $\\hl{s\\ge1}$ and all large sample sizes $N$, $\\sup_{i\\in\\mathcal S_N}|\\mathcal B_N(i;s)|\\le C_bs^2$. In the displays (69)--(73), $b(0)=0$ and $b(k)=C_bk^2$ for integers $k\\ge1$.", "intended_meaning": "The assumption is meant to bound the number of nodes within distance $s$ of any node by a constant times $s^2$, with one constant for all radii and all sample sizes. The counting arguments in (69)--(73) and (76)--(79) use nothing more, and the bound holds for nodes that are spaced apart: a square grid with unit spacing satisfies it with $C_b=9$. The printed range of radii is too wide. At $s=0$ the set $\\mathcal B_N(i;0)$ contains $i$, and its size is at least $1$, while $b(0)=0$. Two further points concern the notation. First, $\\mathcal B_N(i;s)$ here denotes the ball $\\{j\\in\\mathcal S_N:d_{ij}\\le s\\}$, which the article writes $\\mathcal S_N(i;s)$, while in equation (12) of the article the same symbol denotes a neighbourhood of radius $h(d)$. Second, note that the constant $C_b$ must not depend on $s$ or $N$. With the wording below, the weighted bound (69) holds with the formal values $b(0)=0$ and $b(k)=C_bk^2$ for integers $k\\ge1$. This follows by summation by parts applied to the number of nodes within distance $k+1$ of a node, which is at most $C_b(k+1)^2$." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E018", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "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 C.10 with display (123) and the concluding display. A red box marks the subscript j in B of i and b N under the sum in the second hypothesis.", "caption": "Supplement A, p. 35: the statement of Lemma C.10. The marked index set puts the sum over the nodes inside the ball $\\mathcal B(i,b_N)$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E019.png" }, { "alt": "Crop of the last two lines of the chain of displays in the proof of Lemma C.10. A red box marks the subscript j not in B of i and b tilde N under the final sum, which is followed by equals o of 1.", "caption": "Supplement A, p. 35: the last two lines of the proof of Lemma C.10. The marked index set puts the sum over the nodes outside the ball $\\mathcal B(i,\\tilde b_N)$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E019-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T26", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T26" } ], "headline": "The second hypothesis of Lemma C.10 should bound the sum of $\\phi_d(d_{ij})$ over the nodes $j$ outside the ball $\\mathcal B(i,b_N)$, as its proof and Remark 1 require, and not over the nodes inside it.", "id": "P004-E019", "location": "Supplement A, Appendix C, Lemma C.10, second hypothesis after (123), and the proof of the lemma, p. 35 (PDF p. 35 of the supplement)", "pdf_page": 35, "printed_page": "35", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P004-T26" ], "text": { "as_printed": "Lemma C.10. Under C2, C4 and C5, and suppose for all large $N$ and any pair $i$ and $j$ we have\n\\[\n|\\tau_i(d)-\\mathrm{AME}(d;\\eta)|\\times|\\tau_j(d)-\\mathrm{AME}(d;\\eta)|\\le\\phi_d(d_{ij}),\\qquad\\text{(123)}\n\\]\nand $\\lim_N\\sup_{i\\in\\mathcal S_N}\\sum_{\\hl{\\scriptstyle j\\in\\mathcal B(i,b_N)}}\\phi_d(d_{ij})\\to0$ for any $b_N\\to\\infty$. We have\n\\[\n\\liminf_N\\Big(\\frac1N\\sum_{i=1}^N\\sum_{j=1}^NK(\\frac{d_{ij}}{b_N})(\\tau_i(d)-\\mathrm{AME}(d;\\eta))(\\tau_j(d)-\\mathrm{AME}(d;\\eta))\\Big)=0.\n\\]\nProof. [\\ldots] Let $\\tilde b_N=o(b_N^{\\frac13})$. Then, for large $N$, [\\ldots]\n\\[\n\\lesssim\\lim_N\\frac{\\tilde b_N^3}{b_N}+\\lim_N\\sup_{i\\in\\mathcal S_N}\\sum_{j\\notin\\mathcal B(i,\\tilde b_N)}|(\\tau_i(d)-\\mathrm{AME}(d;\\eta))(\\tau_j(d)-\\mathrm{AME}(d;\\eta))|=o(1).\n\\]", "corrected": "Lemma C.10. Under C2, C4 and C5, and suppose for all large $N$ and any pair $i$ and $j$ we have\n\\[\n|\\tau_i(d)-\\mathrm{AME}(d;\\eta)|\\times|\\tau_j(d)-\\mathrm{AME}(d;\\eta)|\\le\\phi_d(d_{ij}),\\qquad\\text{(123)}\n\\]\nand $\\lim_N\\sup_{i\\in\\mathcal S_N}\\sum_{\\hl{\\scriptstyle j\\notin\\mathcal B(i,b_N)}}\\phi_d(d_{ij})\\to0$ for any $b_N\\to\\infty$. We have\n\\[\n\\liminf_N\\Big(\\frac1N\\sum_{i=1}^N\\sum_{j=1}^NK(\\frac{d_{ij}}{b_N})(\\tau_i(d)-\\mathrm{AME}(d;\\eta))(\\tau_j(d)-\\mathrm{AME}(d;\\eta))\\Big)=0.\n\\]\nProof. [\\ldots] Let $\\tilde b_N=\\hl{b_N^{\\frac16}}$\\hl{, so that} $\\hl{\\tilde b_N\\to\\infty}$ \\hl{and} $\\hl{\\tilde b_N^3/b_N\\to0}$. Then, for large $N$, [\\ldots]", "evidence": "The evidence is a check of the page and a derivation of both versions. On p.~35 the hypothesis is printed with $j\\in\\mathcal B(i,b_N)$ and the last line of the proof with $j\\notin\\mathcal B(i,\\tilde b_N)$. Let $a_i=\\tau_i(d)-\\mathrm{AME}(d;\\eta)$. For the printed version, recall that $K$ is supported on $[0,1]$ and bounded by $K_{\\max}$. By (123), I have\n\\[\n\\Big|\\frac1N\\sum_{i=1}^N\\sum_{j=1}^NK\\Big(\\frac{d_{ij}}{b_N}\\Big)a_ia_j\\Big|\\le K_{\\max}\\sup_{i\\in\\mathcal S_N}\\sum_{j\\in\\mathcal B(i,b_N)}\\phi_d(d_{ij}),\n\\]\nwhich tends to zero. For the corrected version, let $\\tilde b_N=b_N^{1/6}$. Since $\\sum_ia_i=0$, the double sum is unchanged when $K(d_{ij}/b_N)$ is replaced by $K(d_{ij}/b_N)-1$. For $d_{ij}\\le\\tilde b_N$ and large $N$, C5 implies that $|K(d_{ij}/b_N)-1|\\le C\\tilde b_N/b_N$, C2 bounds $|a_ia_j|$, and C4 bounds the number of such $j$ by $C_b\\tilde b_N^2$. This part is at most a constant times $\\tilde b_N^3/b_N=b_N^{-1/2}$. For $d_{ij}>\\tilde b_N$, I have $|K(d_{ij}/b_N)-1|\\le K_{\\max}+1$ and $|a_ia_j|\\le\\phi_d(d_{ij})$. This part is at most $(K_{\\max}+1)\\sup_i\\sum_{j\\notin\\mathcal B(i,\\tilde b_N)}\\phi_d(d_{ij})$, which tends to zero by the corrected hypothesis applied to the diverging sequence $\\tilde b_N$.", "problem": "As printed, the hypothesis bounds the sum of $\\phi_d(d_{ij})$ over the nodes inside the growing ball. That sum contains the term $j=i$, and (123) then implies that $\\sup_i(\\tau_i(d)-\\mathrm{AME}(d;\\eta))^2$ is at most $\\sup_i\\sum_{j\\in\\mathcal B(i,b_N)}\\phi_d(d_{ij})$, which tends to zero. With the single function $\\phi_d$ of the statement, the condition forces $\\phi_d(0)=0$ and hence $\\tau_i(d)=\\mathrm{AME}(d;\\eta)$ for every $i$ and all large $N$. The effects are then homogeneous. If $\\phi_d$ were allowed to vary with $N$, the effects would be homogeneous asymptotically, uniformly in $i$. The conclusion of the lemma holds in that case. The printed lemma is true, but it covers only this case. Remark 1 announces a condition on heterogeneity ``when nodes are far apart'', and the last line of the proof uses the sum over $j\\notin\\mathcal B(i,\\tilde b_N)$, which the printed hypothesis does not control. Note that the proof also sets $\\tilde b_N=o(b_N^{1/3})$ without requiring that $\\tilde b_N$ diverge, which the bound on the outside sum needs.", "what_survives": "The conclusion of Lemma C.10 holds under either hypothesis, and C6 and Proposition 6 are unaffected. The corrected hypothesis restores the case of heterogeneous effects that Remark 1 describes. Its proof uses C4 as worded in \\fid{P004-E018}." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E019", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the two sentences that introduce Proposition 6 and of its statement, with parts (i) to (v). Red boxes mark the phrase Under C1, C2, C3, C4, C5 and C6 and the symbol lim as N tends to infinity at the start of part (v).", "caption": "Supplement A, p. 35: Proposition 6. The marked list of assumptions contains no lower bound on $N\\times\\operatorname{AVar}(\\widehat\\tau_{\\mathrm{HA}}(d))$, and the marked conclusion of part (v) is a limit.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E020.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T27", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T27" }, { "id": "P004-T28", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T28" } ], "headline": "Part (v) of Proposition 6 needs the condition that $N\\times\\operatorname{AVar}(\\widehat\\tau_{\\mathrm{HA}}(d))$ stays bounded away from zero, which Proposition 5 imposes, and its conclusion should be stated with a limit inferior.", "id": "P004-E020", "location": "Supplement A, Appendix C, Proposition 6, the list of assumptions and part (v), p. 35 (PDF p. 35 of the supplement); compare Proposition 5, p. 29", "pdf_page": 35, "printed_page": "35", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P004-T27", "P004-T28" ], "text": { "as_printed": "Proposition 6. Let $b_N=o(N^{\\frac16})$ and $b_N\\to\\infty$. \\hl{Under C1, C2, C3, C4, C5 and C6,}\n\n(i) $N\\times\\big(\\widehat{\\mathrm V}_{\\mathrm{HAC}}(d;b_N)-\\widehat{\\mathrm V}_1(d,b_N)\\big)=o_p(1)$.\n\n(ii) $N\\times\\big(\\widehat{\\mathrm V}_1(d;b_N)-\\widehat{\\mathrm V}_2(d,b_N)\\big)=o_p(1)$.\n\n(iii) $N\\times\\big(\\widehat{\\mathrm V}_2(d;b_N)-\\mathrm E[\\widehat{\\mathrm V}_2(d;b_N)]\\big)=o_p(1)$.\n\n(iv) $\\liminf_N\\big(N\\times\\big(\\mathrm E[\\widehat{\\mathrm V}_2(d;b_N)]-\\operatorname{AVar}(\\widehat\\tau_{\\mathrm{HA}}(d))\\big)\\big)\\ge0$.\n\n(v) $\\hl{\\lim_{N\\to\\infty}}\\operatorname{Prob}\\Big(z_{\\frac\\alpha2}\\le\\dfrac{\\widehat\\tau_{\\mathrm{HA}}(d)-\\mathrm{AME}(d;\\eta)}{\\sqrt{\\widehat{\\mathrm V}_{\\mathrm{HAC}}(d,b_N)}}\\le z_{1-\\frac\\alpha2}\\Big)\\ge1-\\alpha$", "corrected": "Proposition 6. Let $b_N=o(N^{\\frac16})$ and $b_N\\to\\infty$. Under C1, C2, C3, C4, C5 and C6,\n\n(i) $N\\times\\big(\\widehat{\\mathrm V}_{\\mathrm{HAC}}(d;b_N)-\\widehat{\\mathrm V}_1(d,b_N)\\big)=o_p(1)$.\n\n(ii) $N\\times\\big(\\widehat{\\mathrm V}_1(d;b_N)-\\widehat{\\mathrm V}_2(d,b_N)\\big)=o_p(1)$.\n\n(iii) $N\\times\\big(\\widehat{\\mathrm V}_2(d;b_N)-\\mathrm E[\\widehat{\\mathrm V}_2(d;b_N)]\\big)=o_p(1)$.\n\n(iv) $\\liminf_N\\big(N\\times\\big(\\mathrm E[\\widehat{\\mathrm V}_2(d;b_N)]-\\operatorname{AVar}(\\widehat\\tau_{\\mathrm{HA}}(d))\\big)\\big)\\ge0$.\n\n(v) \\hl{If in addition} $\\hl{\\liminf_{N\\to\\infty}N\\times\\operatorname{AVar}(\\widehat\\tau_{\\mathrm{HA}}(d))>0}$\\hl{, then for each} $\\hl{0<\\alpha<1}$\\hl{,}\n\\[\n\\hl{\\liminf_{N\\to\\infty}}\\operatorname{Prob}\\Big(z_{\\frac\\alpha2}\\le\\frac{\\widehat\\tau_{\\mathrm{HA}}(d)-\\mathrm{AME}(d;\\eta)}{\\sqrt{\\widehat{\\mathrm V}_{\\mathrm{HAC}}(d,b_N)}}\\le z_{1-\\frac\\alpha2}\\Big)\\ge1-\\alpha.\n\\]\nHere C4 is read as worded in \\fid{P004-E018}.", "evidence": "The evidence is an exact counterexample to part (v), a second example for the limit, and a derivation of the corrected statement. Both examples satisfy C4 as worded in \\fid{P004-E018}. With C4 exactly as printed, no nonempty set of nodes satisfies the assumptions, and the failure concerns the proposition as it is meant.\n\nFor each $N\\ge3$, let the nodes be the points $(iN^2,0)$ for $1\\le i\\le N$, with independent assignments of probability $p=1/2$. Let the outcome points be the nodes themselves, with counting measure, and let the distance be $d=1$, with $\\Omega_d=[0,d]$, so that each disc contains its own node only. Let $Y_1(\\mathbf z)=1/4+(3/4)z_1$ and $Y_i(\\mathbf z)=0$ for $i>1$. Let $K(u)=\\mathbf 1\\{0\\le u\\le1\\}$ and $b_N=\\lfloor N^{1/12}\\rfloor$. Then C1, C2, C3 and C5 hold, a ball of radius $s\\ge1$ contains at most $2s+1\\le3s^2$ nodes, and the expression in C6 equals $9(N-1)/(16N^2)\\ge0$. A direct computation gives\n\\[\nN\\times\\operatorname{AVar}(\\widehat\\tau_{\\mathrm{HA}}(d))=\\frac{25(N-1)}{16N^2},\n\\]\nwhich tends to zero, and parts (i)--(iv) hold. Since the spacing $N^2$ exceeds $b_N$, the estimator $\\widehat{\\mathrm V}_{\\mathrm{HAC}}$ has no cross terms. Let $q_N$ be the number of nodes in the arm that contains node 1. Outside four assignments, of total probability $4/2^N$, I have\n\\[\n\\widehat\\tau_{\\mathrm{HA}}(d)=\\frac1{q_N},\\quad\\widehat{\\mathrm V}_{\\mathrm{HAC}}=\\frac{q_N-1}{q_N^3}\\quad\\text{if }Z_1=1,\\qquad\\widehat\\tau_{\\mathrm{HA}}(d)=-\\frac1{4q_N},\\quad\\widehat{\\mathrm V}_{\\mathrm{HAC}}=\\frac{q_N-1}{16q_N^3}\\quad\\text{if }Z_1=0.\n\\]\nHere $\\mathrm{AME}(d;\\eta)=3/(4N)$, and $q_N/N$ tends to $1/2$ in probability. The studentized statistic of part (v) then tends in probability to $5/8$ given $Z_1=1$ and to $-5/2$ given $Z_1=0$. Since $5/80$ for all large $N$. By (i)--(iii), I have $R_N=N(\\widehat{\\mathrm V}_{\\mathrm{HAC}}(d;b_N)-m_N)=o_p(1)$, and (iv) implies that $D_N=N(m_N-\\sigma_N^2)$ satisfies $\\liminf_ND_N\\ge0$. Fix $\\varepsilon\\in(0,1)$. For large $N$, I have $D_N\\ge-\\varepsilon c/2$, and $R_N\\ge-\\varepsilon c/2$ with probability tending to one. On this event,\n\\[\n\\frac{\\widehat{\\mathrm V}_{\\mathrm{HAC}}(d;b_N)}{\\sigma_N^2}=1+\\frac{D_N+R_N}{N\\sigma_N^2}\\ge1-\\varepsilon.\n\\]\nLet $T_N$ denote the studentized statistic of part (v) and let $a=z_{1-\\alpha/2}$. By Proposition 5, the ratio $(\\widehat\\tau_{\\mathrm{HA}}(d)-\\mathrm{AME}(d;\\eta))/\\sigma_N$ converges in distribution to $G$. It follows that $\\liminf_N\\operatorname{Prob}(|T_N|\\le a)\\ge\\operatorname{Prob}(|G|\\le a\\sqrt{1-\\varepsilon})$. Letting $\\varepsilon$ tend to zero, I obtain the bound $1-\\alpha$.", "problem": "Part (v) asserts that the Wald interval built on $\\widehat{\\mathrm V}_{\\mathrm{HAC}}$ has asymptotic coverage at least $1-\\alpha$ under C1--C6 alone. Its proof (``(v) follows similarly as in Proposition 1'', p.~39) rests on the normal limit of Proposition 5, and Proposition 5 assumes in addition that $N\\times\\operatorname{AVar}(\\widehat\\tau_{\\mathrm{HA}}(d))$ is uniformly bounded below for all large $N$. Proposition 6 does not state this condition on the variance, and C1--C6 do not imply it. Parts (i)--(iv) control the error of the variance estimator at the scale $1/N$ only. When $N\\times\\operatorname{AVar}(\\widehat\\tau_{\\mathrm{HA}}(d))$ tends to zero, they say nothing about the ratio of $\\widehat{\\mathrm V}_{\\mathrm{HAC}}$ to the variance, and the studentized statistic need not be approximately normal. The conclusion is also written as a limit, and the coverage probability need not converge even when the condition on the variance holds. Propositions 3 and 4 of the article, the second of which is its coverage result, state the condition on the variance.", "what_survives": "The estimator, the estimand, Proposition 5 with its stated premise and parts (i)--(iv) of Proposition 6 are not contradicted. Proposition 4 of the article already carries the condition on the variance. What changes is part (v), and with it the coverage claim of the extension announced in Section 6.5 (p.~759). The added condition excludes sequences with sparse effects such as the first example, and the conclusion is a lower bound on the limit inferior of the coverage and not a limit. No later result uses part (v). I did not recalculate the simulations or the applications, and no change to them is established." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E020", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of parts (i), (ii) and (iii) of Lemma C.9 with displays (108) and (109). A red box marks the condition s greater than 3 b N at the start of part (iii).", "caption": "Supplement A, p. 33: Lemma C.9, parts (i)–(iii). The marked condition of part (iii) is on $s$, while the bound (109) below it depends on $d_{ij}$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E021.png" }, { "alt": "Crop of displays (114) to (116), the expansion of the HAC variance estimator. A red box marks the factor Z i in the second sum of (115), the sum that is divided by N 0 squared.", "caption": "Supplement A, p. 34: equations (114)–(116). The marked factor in the second sum of (115), the diagonal term of the control arm, should be $1-Z_i$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E021-b.png" }, { "alt": "Crop of equation (120), the sentence Note V hat 1 equals the sum of v i, and equation (121), which defines V hat 2 as a sum of v i. Red boxes mark the symbol v i, printed without a hat, in both places.", "caption": "Supplement A, p. 34: the sentence after (120), and equation (121). The marked $v_i$ is the expectation of $\\widehat v_i$, and the sums should run over $\\widehat v_i$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E021-c.png" }, { "alt": "Crop of displays (135) and (136). Red boxes mark the denominator N squared times the square of 1 minus p, under the numerators (1 minus Z i) Z j and (1 minus Z j) Z i.", "caption": "Supplement A, p. 37: equations (135) and (136). The marked denominators should be $N^2p(1-p)$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E021-d.png" }, { "alt": "Crop of displays (146), (147) and (148). A red box marks the whole of (148): equals the double sum over i and j of K of d i j over b N, times tau i of d minus AME, times tau j of d minus AME.", "caption": "Supplement A, p. 38: equations (146)–(148). The marked right-hand side of (148) lacks the factor $N^{-2}$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E021-e.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T29", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T29" } ], "headline": "Five displays in the proof that the HAC variance estimator is consistent are misprinted: Lemma C.9(iii) requires $d_{ij}>3b_N$, the diagonal term of the control arm in (115) carries $1-Z_i$, the sums that define $\\widehat{\\mathrm V}_1$ and $\\widehat{\\mathrm V}_2$ run over the random $\\widehat v_{i,N}$, the mixed terms (135)–(136) have denominator $N^2p(1-p)$, and (148) carries the factor $N^{-2}$.", "id": "P004-E021", "location": "Supplement A, Appendix C: Lemma C.9(iii), p. 33; equation (115) and the definitions after (120) and in (121), p. 34; equations (135)–(136), p. 37; equation (148), p. 38 (PDF pages of the supplement with the same numbers)", "pdf_page": 33, "printed_page": "33", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P004-T29" ], "text": { "as_printed": "\\begin{enumerate}\n\\item Lemma C.9, p.~33: (iii) If $\\hl{s}>3b_N$,\n\\[\n\\begin{aligned}\n|\\operatorname{Cov}(\\widehat v_{i,N}(d,b_N),\\widehat v_{j,N}(d,b_N))|\\lesssim{}&\\frac{b_N^2}{N^4}\\times\\max\\Big\\{\\sum_{\\{k:d_{ik}\\le b_N\\}}\\psi_d(\\frac{d_{ij}}3-d_{ik}),\\\\\n&\\sum_{\\{l:d_{jl}\\le b_N\\}}\\psi_d(\\frac{d_{ij}}3-d_{jl})\\Big\\},\\qquad\\text{(109)}\n\\end{aligned}\n\\]\nwhere by using the symbol $\\lesssim$ we suppress a constant dependent of $B$ but independent of $N$.\n\\item Equations (114)--(116), p.~34:\n\\[\n\\begin{aligned}\n\\widehat{\\mathrm V}_{\\mathrm{HAC}}(d;b_N)={}&\\frac1{N_1^2}\\sum_{i=1}^NZ_i\\hat e_i^2(d)+\\frac1{N_0^2}\\sum_{i=1}^N\\hl{Z_i}\\hat e_i^2(d)+\\frac1{N_1^2}\\sum_{i=1}^N\\sum_{j\\ne i}Z_iZ_j\\hat e_i(d)\\hat e_j(d)K(\\frac{d_{ij}}{b_N})\\\\\n&-\\frac2{N_1N_0}\\sum_{i=1}^N\\sum_{j\\ne i}(1-Z_i)Z_j\\hat e_i(d)\\hat e_j(d)K(\\frac{d_{ij}}{b_N})\\\\\n&+\\frac1{N_0^2}\\sum_{i=1}^N\\sum_{j\\ne i}(1-Z_i)(1-Z_j)\\hat e_i(d)\\hat e_j(d)K(\\frac{d_{ij}}{b_N}).\n\\end{aligned}\n\\]\n\\item After (120), p.~34: Note $\\widehat{\\mathrm V}_1(d;b_N)=\\sum_{i=1}^N\\hl{v_i}(\\widehat{\\bar\\mu}_1(d),\\widehat{\\bar\\mu}_0(d);d,b_N)$. Also define\n\\[\n\\widehat{\\mathrm V}_2(d;b_N)=\\sum_{i=1}^N\\hl{v_i}(\\bar\\mu_1(d),\\bar\\mu_0(d);d,b_N)\\qquad\\text{(121)}\n\\]\n\\item Equations (135)--(136), p.~37, two of the terms of $u_{ij}$:\n\\[\n\\begin{aligned}\n&-\\frac{(1-Z_i)Z_j}{\\hl{N^2(1-p)^2}}\\Big(-\\mu_i(\\mathbf Y;d)\\big(\\widehat{\\bar\\mu}^1(d)-\\bar\\mu^1(d)\\big)-\\mu_j(\\mathbf Y;d)\\big(\\widehat{\\bar\\mu}^0(d)-\\bar\\mu^0(d)\\big)\\\\\n&\\qquad\\qquad\\qquad\\qquad+\\widehat{\\bar\\mu}^0(d)\\widehat{\\bar\\mu}^1(d)-\\bar\\mu^0(d)\\bar\\mu^1(d)\\Big)&&\\text{(135)}\\\\\n&-\\frac{(1-Z_j)Z_i}{\\hl{N^2(1-p)^2}}\\Big(-\\mu_i(\\mathbf Y;d)\\big(\\widehat{\\bar\\mu}^0(d)-\\bar\\mu^0(d)\\big)-\\mu_j(\\mathbf Y;d)\\big(\\widehat{\\bar\\mu}^1(d)-\\bar\\mu^1(d)\\big)\\\\\n&\\qquad\\qquad\\qquad\\qquad+\\widehat{\\bar\\mu}^0(d)\\widehat{\\bar\\mu}^1(d)-\\bar\\mu^0(d)\\bar\\mu^1(d)\\Big)&&\\text{(136)}\n\\end{aligned}\n\\]\n\\item Equations (147)--(148), p.~38:\n\\[\n\\begin{aligned}\n&=\\sum_{i=1}^N\\sum_{j=1}^NK\\Big(\\frac{d_{ij}}{b_N}\\Big)\\mathrm E[\\widehat\\epsilon_{i,N}(d)]\\mathrm E[\\widehat\\epsilon_{j,N}(d)]&&\\text{(147)}\\\\\n&=\\hl{\\sum_{i=1}^N\\sum_{j=1}^N}K\\Big(\\frac{d_{ij}}{b_N}\\Big)(\\tau_i(d)-\\mathrm{AME}(d;\\eta))(\\tau_j(d)-\\mathrm{AME}(d;\\eta)).&&\\text{(148)}\n\\end{aligned}\n\\]\n\\end{enumerate}", "corrected": "\\begin{enumerate}\n\\item Lemma C.9, p.~33: (iii) If $\\hl{d_{ij}}>3b_N$,\n\\[\n\\begin{aligned}\n|\\operatorname{Cov}(\\widehat v_{i,N}(d,b_N),\\widehat v_{j,N}(d,b_N))|\\lesssim{}&\\frac{b_N^2}{N^4}\\times\\max\\Big\\{\\sum_{\\{k:d_{ik}\\le b_N\\}}\\psi_d(\\frac{d_{ij}}3-d_{ik}),\\\\\n&\\sum_{\\{l:d_{jl}\\le b_N\\}}\\psi_d(\\frac{d_{ij}}3-d_{jl})\\Big\\},\\qquad\\text{(109)}\n\\end{aligned}\n\\]\nwhere by using the symbol $\\lesssim$ we suppress a constant dependent of $B$ but independent of $N$.\n\\item Equations (114)--(116), p.~34:\n\\[\n\\begin{aligned}\n\\widehat{\\mathrm V}_{\\mathrm{HAC}}(d;b_N)={}&\\frac1{N_1^2}\\sum_{i=1}^NZ_i\\hat e_i^2(d)+\\frac1{N_0^2}\\sum_{i=1}^N\\hl{(1-Z_i)}\\hat e_i^2(d)+\\frac1{N_1^2}\\sum_{i=1}^N\\sum_{j\\ne i}Z_iZ_j\\hat e_i(d)\\hat e_j(d)K(\\frac{d_{ij}}{b_N})\\\\\n&-\\frac2{N_1N_0}\\sum_{i=1}^N\\sum_{j\\ne i}(1-Z_i)Z_j\\hat e_i(d)\\hat e_j(d)K(\\frac{d_{ij}}{b_N})\\\\\n&+\\frac1{N_0^2}\\sum_{i=1}^N\\sum_{j\\ne i}(1-Z_i)(1-Z_j)\\hat e_i(d)\\hat e_j(d)K(\\frac{d_{ij}}{b_N}).\n\\end{aligned}\n\\]\n\\item After (120), p.~34: Note $\\widehat{\\mathrm V}_1(d;b_N)=\\sum_{i=1}^N\\hl{\\widehat v_{i,N}}(\\widehat{\\bar\\mu}_1(d),\\widehat{\\bar\\mu}_0(d);d,b_N)$. Also define\n\\[\n\\widehat{\\mathrm V}_2(d;b_N)=\\sum_{i=1}^N\\hl{\\widehat v_{i,N}}(\\bar\\mu_1(d),\\bar\\mu_0(d);d,b_N)\\qquad\\text{(121)}\n\\]\n\\item Equations (135)--(136), p.~37:\n\\[\n\\begin{aligned}\n&-\\frac{(1-Z_i)Z_j}{N^2\\hl{p(1-p)}}\\Big(-\\mu_i(\\mathbf Y;d)\\big(\\widehat{\\bar\\mu}^1(d)-\\bar\\mu^1(d)\\big)-\\mu_j(\\mathbf Y;d)\\big(\\widehat{\\bar\\mu}^0(d)-\\bar\\mu^0(d)\\big)\\\\\n&\\qquad\\qquad\\qquad\\qquad+\\widehat{\\bar\\mu}^0(d)\\widehat{\\bar\\mu}^1(d)-\\bar\\mu^0(d)\\bar\\mu^1(d)\\Big)&&\\text{(135)}\\\\\n&-\\frac{(1-Z_j)Z_i}{N^2\\hl{p(1-p)}}\\Big(-\\mu_i(\\mathbf Y;d)\\big(\\widehat{\\bar\\mu}^0(d)-\\bar\\mu^0(d)\\big)-\\mu_j(\\mathbf Y;d)\\big(\\widehat{\\bar\\mu}^1(d)-\\bar\\mu^1(d)\\big)\\\\\n&\\qquad\\qquad\\qquad\\qquad+\\widehat{\\bar\\mu}^0(d)\\widehat{\\bar\\mu}^1(d)-\\bar\\mu^0(d)\\bar\\mu^1(d)\\Big)&&\\text{(136)}\n\\end{aligned}\n\\]\n\\item Equations (147)--(148), p.~38:\n\\[\n\\begin{aligned}\n&=\\sum_{i=1}^N\\sum_{j=1}^NK\\Big(\\frac{d_{ij}}{b_N}\\Big)\\mathrm E[\\widehat\\epsilon_{i,N}(d)]\\mathrm E[\\widehat\\epsilon_{j,N}(d)]&&\\text{(147)}\\\\\n&=\\hl{\\frac1{N^2}}\\sum_{i=1}^N\\sum_{j=1}^NK\\Big(\\frac{d_{ij}}{b_N}\\Big)(\\tau_i(d)-\\mathrm{AME}(d;\\eta))(\\tau_j(d)-\\mathrm{AME}(d;\\eta)).&&\\text{(148)}\n\\end{aligned}\n\\]\n\\end{enumerate}", "evidence": "The evidence is a derivation from the adjacent definitions, with two exact checks. In the regression of the circle averages on the assignment, node $i$ enters the sandwich formula for the variance of the coefficient through $(Z_i/N_1-(1-Z_i)/N_0)\\hat e_i(d)$. Since $Z_i(1-Z_i)=0$, its square is $Z_i\\hat e_i^2(d)/N_1^2+(1-Z_i)\\hat e_i^2(d)/N_0^2$, which is the diagonal of the corrected (115). With $N=4$, assignments $(1,1,0,0)$ and residuals $(0,0,1,-1)$, which have mean zero in each arm, the diagonal terms sum to $0$ as printed and to $1/2$ as corrected. Multiplying the two terms of (102) for a control node $i$ and a treated node $j$, I obtain the factor $-(1-Z_i)Z_j/(N^2p(1-p))$ of (135), and exchanging $i$ and $j$ gives (136). At $p=1/4$ the printed factor $1/(1-p)^2$ equals $16/9$ and the correct factor $1/(p(1-p))$ equals $16/3$. Substituting footnote 22 into (147), I obtain (148) with the factor $N^{-2}$. Multiplying by $N$ as in (157) then gives the normalization $1/N$ of C6 in (122). The expansions (117)--(120) and (130)--(132) are sums of the random products in (107), that is, of the $\\widehat v_{i,N}$. The proof of Lemma C.9(iii) uses the separation $d_{ij}/3>b_N$.", "problem": "Each display disagrees with a definition or an identity next to it.\n\\begin{enumerate}\n\\item In Lemma C.9(iii) the radius $s$ plays no role in the bound (109), which depends on $d_{ij}$, and the proof of that part begins ``if $d_{ij}>3b_N$''.\n\\item The second sum of (115) is the diagonal term of the control arm and carries $1-Z_i$, as in (117). As printed, the residuals of control nodes do not enter the diagonal.\n\\item The line after (107) defines $v_{i,N}$ as the expectation of $\\widehat v_{i,N}$. The printed sums are then constants, while $\\widehat{\\mathrm V}_1$ and $\\widehat{\\mathrm V}_2$ are random quadratic forms, and the proof bounds the variance of $\\widehat{\\mathrm V}_2$ through $\\operatorname{Cov}(\\widehat v_{i,N},\\widehat v_{j,N})$ in (138).\n\\item The mixed terms multiply a factor $Z/(Np)$ for a treated node by a factor $(1-Z)/(N(1-p))$ for a control node. Their denominator is $N^2p(1-p)$, as in (119).\n\\item By footnote 22, $\\mathrm E[\\widehat\\epsilon_{i,N}(d)]=(\\tau_i(d)-\\mathrm{AME}(d;\\eta))/N$, and (147) equals $N^{-2}$ times the right-hand side of (148) as printed.\n\\end{enumerate}", "what_survives": "The variance estimator, the orders in probability asserted in the proof and the proof of Proposition 6(i)--(iv) are unchanged after these substitutions, for fixed $0\\frac94&&\\text{with the stored variance}.\n\\end{aligned}\n\\]\nThe interval of the caption does not cover the true value, and the interval that is plotted does.\n\n\\emph{Second example.} The eight nodes are the corners of two unit squares, and the cutoff is 1. In this order, the nodes are $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$ and $(10,0)$, $(11,0)$, $(11,1)$, $(10,1)$. Let\n\\[\nZ=(0,0,1,1,0,0,1,1),\\qquad Y=(3,-3,1,-1,3,-3,1,-1).\n\\]\n\\begin{audittable}{lrr}{Second example: the variances as defined and as stored by the simulation code.}\n & HAC, (19) & HAC\\_PD, (22) \\\\\nVariance as defined & $3/2$ & $2$ \\\\\nFactor $\\mu$ & $1/2$ & $3/4$ \\\\\nVariance divided by $\\mu$, as stored & $3$ & $8/3$ \\\\\n\\end{audittable}\nThe ratio of the HAC\\_PD variance to the HAC variance is $4/3$ as defined and $8/9$ as stored. The order of the two variances is reversed.\n\n\\emph{Direction for the uniform kernel.} For the uniform kernel the factor satisfies $0\\le\\mu<1$. On the draws where $\\mu>0$ and the variance of (19) is nonnegative, the plotted normal interval contains the interval built on (19), and the coverage of the latter cannot exceed that of the plotted curve on those draws.\n\n\\emph{What is not established.} The simulation outputs are not in the replication archive, and I did not regenerate the simulations. The size of the differences in Figures 7 and 8 is not known, and I do not claim that the intervals built on (22) are substantially longer than those built on (19) in the designs of the paper. The two examples show what the rescaling can do. They are not reconstructions of the simulations.", "problem": "Equations (19) and (22) define the two variance estimators without any rescaling. Appendix A.7 introduces a separate adjustment with two parts: the variance is divided by the factor $\\mu$ of (44), and the normal critical value is replaced by a quantile of a $t$ distribution. According to the captions, the curves HAC and HAC\\_PD use the variances of (19) and (22) with the critical value 1.96, and the curves marked ``(edof)'' carry the adjustment. The simulation code in the replication files computes something else. The function that returns the variances divides each of them by its own factor and returns $\\widehat V_{HAC}/\\mu_{HAC}$ and $\\widehat V_{PD}/\\mu_{PD}$. The simulation stores these two numbers, and the plotting code uses them for all four curves, with 1.96 for the curves HAC and HAC\\_PD, which I call the plain curves, and a $t$ quantile for the ``(edof)'' curves.\n\nThere are three consequences. The figures do not show how the intervals built on (19) and (22) behave without the adjustment. The difference between a plain curve and its ``(edof)'' curve shows the effect of the critical value only, not of the whole adjustment of A.7. Finally, the remarks that the positive semidefinite interval is ``not substantially longer'' (p.~761) and comes with ``little loss in efficiency'' (p.~756) compare two variances that are each divided by a different factor. Since\n\\[\n\\frac{\\widehat V_{PD}/\\mu_{PD}}{\\widehat V_{HAC}/\\mu_{HAC}}=\\frac{\\widehat V_{PD}}{\\widehat V_{HAC}}\\cdot\\frac{\\mu_{HAC}}{\\mu_{PD}},\n\\]\nthis is not the comparison of (22) with (19).", "what_survives": "This discrepancy does not refute the estimators (19) and (22), the positive semidefinite construction, the asymptotic results, the figures for the mean squared error or the randomization tests. The plotted curves describe the procedures that were computed, once they are relabelled. The captions of the two applications already name the positive semidefinite estimator with the adjustment; I have not established a change in their numerical results. The recommendation on p.~761 can stand as a statement about the adjusted procedures. What changes is the scope of the two observations: they hold for the rescaled variances, and the figures do not separate the whole adjustment of A.7 from the intervals built on (19) and (22). Whether the positive semidefinite variance may be divided by its factor, and the degrees of freedom used for HAC\\_PD (edof), are treated in \\fid{P004-N004} and \\fid{P004-E026}." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E025", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Five lines from the italic caption of Figure 7. Red boxes mark the sentence: HAC_PD (edof) refers to the CI with HAC variance estimator in (22) and empirical degree of freedom adjustment.", "caption": "Caption of Figure 7, p. 761 (PDF p. 18). The marked sentence defines the HAC_PD (edof) series as the interval with the positive semidefinite HAC variance estimator (22) and the degrees-of-freedom adjustment of Supplement A.7.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E026.png" }, { "alt": "Equation (45): nu equals 2 times the trace of the square of the matrix (N1 N0 / N) M (Omega circ lambda-prime lambda) M, with a red box around the factor 2 and the star after it. Below it, the sentence: We use the alpha/2-quantile and (1 - alpha/2)-quantile of the t-distribution with 2 mu squared over v degree of freedom to construct the confidence interval. A red box marks the fraction 2 mu squared over v.", "caption": "Supplement A, Section A.7, p. 17: equation (45) and the sentence after it. The marks show the factor 2 in $\\nu$ and the stated degrees of freedom $2\\mu^2/v$; the $v$ of that sentence is the $\\nu$ of (45).", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E026-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "The simulation code uses $\\mu_{\\mathrm{PD}}^2/\\nu_{\\mathrm{PD}}$ degrees of freedom for the HAC_PD (edof) interval, half of the $2\\mu^2/\\nu$ stated in Supplement A.7, so the coverage and half-lengths plotted for this interval in Figures 7 and 8 belong to an interval at least as wide as the one described; the code should use $2\\mu_{\\mathrm{PD}}^2/\\nu_{\\mathrm{PD}}$, and the series should be computed again.", "id": "P004-E026", "location": "Figure 7 (caption) and Figure 8, p. 761 (PDF p. 18); Supplement A, Section A.7, equation (45) and the sentence after it, p. 17; replication archive (Supplement B), simulation/scripts/6_functions3.R, lines 196 and 312, and simulation/scripts/5_functions_generate_figures.R, lines 78–79, 87 and 97", "pdf_page": 18, "printed_page": "761", "reaches": [ { "effect": "uses_only_what_survives", "effect_text": "relies only on the part that survives", "explanation": null, "package": "SpatialEffect", "recorded_as": null, "to": "package", "url": null, "where": null, "work_id": "P004" } ], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [], "text": { "as_printed": "Caption of Figure 7, p.~761: ``HAC\\_PD (edof) refers to the CI with HAC variance estimator in (22) and empirical degree of freedom adjustment.''\n\nSupplement A, Section A.7, p.~17:\n\\[\n\\nu = 2 * \\mathbf{Trace}\\left(\\left(\\frac{N_1N_0}{N}\\mathbf{M}\\left(\\Omega\\circ\\left(\\lambda'\\lambda\\right)\\right)\\mathbf{M}\\right)\\left(\\frac{N_1N_0}{N}\\mathbf{M}\\left(\\Omega\\circ\\left(\\lambda'\\lambda\\right)\\right)\\mathbf{M}\\right)\\right). \\tag{45}\n\\]\n``We use the $\\frac{\\alpha}{2}$-quantile and $\\left(1-\\frac{\\alpha}{2}\\right)$-quantile of the t-distribution with $\\frac{2\\mu^2}{v}$ degree of freedom to construct the confidence interval.''\n\n\\texttt{6\\_functions3.R}, line 196 (line 312 is identical), shown here on two lines:\n\n$\\texttt{return(c(Conley\\_var2/mu2,Conley\\_var2\\_pd/mu2\\_pd,SAHvar,}$\n\n$\\texttt{edof=2*mu2\\textasciicircum 2/v2,edof\\_pd=}\\hl{\\texttt{mu2\\_pd\\textasciicircum 2}}\\texttt{/v2\\_pd))}$\n\n\\texttt{5\\_functions\\_generate\\_figures.R}, lines 78--79:\n\n\\texttt{edof[which(edof<=1)]=1}\n\n\\texttt{edof\\_pd[which(edof\\_pd<=1)]=1}", "corrected": "In both return statements of \\texttt{6\\_functions3.R} (lines 196 and 312):\n\n$\\texttt{edof\\_pd=}\\hl{\\texttt{2*}}\\texttt{mu2\\_pd\\textasciicircum 2/v2\\_pd}$\n\nThen compute again the coverage and the half-length of the HAC\\_PD (edof) series in Figures 7 and 8 and in Figures 2, 3, 5, 6, 8 and 9 of Supplement A. In \\texttt{5\\_functions\\_generate\\_figures.R}, remove the reset of degrees of freedom below one (lines 78--79), or state the rule in A.7 if it is kept for the HAC series. After the factor is restored, the reset has no effect on HAC\\_PD whenever the quadratic form is nonzero. The effect of the correction on the published figures remains to be computed.", "evidence": "The evidence is an exact example, computed in rational arithmetic and reproduced by running the unchanged R and C++ routines in the archive. I take six nodes at $0,10,\\dots,50$ on a line, with $Z=(0,0,0,1,1,1)$, observed circle averages $Y=(0,0,3,1,1,1)$ and a kernel cutoff of $1$. Since the kernel matrix is the identity, the HAC and HAC\\_PD estimators coincide. Here $\\lambda=(-1,-1,-1,1,1,1)/3$, and the matrix in (45) is $\\mathbf{M}/6$, where $\\mathbf{M}$ is a projection of rank $4$. A direct computation gives $\\mu=2/3$ and $\\nu=2/9$. By A.7, I have $2\\mu^2/\\nu=4$ for both estimators. The routines return $4$ for HAC and $2$ for HAC\\_PD. The estimate is $0$, and both adjusted standard errors equal $1$. The half-length of the 95\\% interval is $2.776$ with $4$ degrees of freedom and $4.303$ with $2$. The coded interval contains the value $3$, and the stated interval does not.\n\nThe table adds two examples. In the second, the kernel is positive semidefinite with two clusters of three nodes, and the reset in the figure code restores the stated value. The third comes from a separate check with eight nodes, where the half-lengths are $2.234$ as stated and $2.905$ as coded.\n\nThe C++ routine stores $\\mathbf{M}A$ where A.7 has $\\mathbf{M}A\\mathbf{M}$, with $A$ the kernel-weighted matrix. Since $\\mathbf{M}^2=\\mathbf{M}$, the two matrices have the same trace, and their squares have the same trace. The routine computes the $\\mu$ and $\\nu$ of A.7, and no other normalization accounts for the missing factor.\n\n\\begin{audittable}{lrrrrrr}{Degrees of freedom in three exact examples: the value $2\\mu^2/\\nu$ of A.7, the values returned by the simulation routines, and the value used in the figure code. In each example the HAC and HAC\\_PD kernels are equal.}\nExample & $\\mu$ & $\\nu$ & A.7 & HAC, code & HAC\\_PD, code & HAC\\_PD, plotted \\\\\nSix nodes, identity kernel & $2/3$ & $2/9$ & $4$ & $4$ & $2$ & $2$ \\\\\nSix nodes, two clusters & $4/9$ & $32/81$ & $1$ & $1$ & $1/2$ & $1$ \\\\\nEight nodes, identity kernel & $3/4$ & $3/16$ & $6$ & $6$ & $3$ & $3$ \\\\\n\\end{audittable}", "problem": "Supplement A.7 chooses the degrees of freedom by matching two moments. Suppose that a quadratic form has mean $\\mu$ and variance $\\nu$. Divided by $\\mu$, it has mean $1$ and variance $\\nu/\\mu^2$. A chi-square variable with $f$ degrees of freedom, divided by $f$, has mean $1$ and variance $2/f$. Equating the two variances, I obtain $f=2\\mu^2/\\nu$, where $\\nu$ is twice the trace of the squared matrix, as in (45). The caption of Figure 7 applies this adjustment to the HAC interval and to the HAC\\_PD interval alike.\n\nThe two simulation routines, \\texttt{Regression\\_AME} and \\texttt{Regression\\_AME\\_obs}, return $2\\mu^2/\\nu$ for the HAC estimator and $\\mu_{\\mathrm{PD}}^2/\\nu_{\\mathrm{PD}}$ for the positive semidefinite one. One C++ routine computes $\\mu$ and $\\nu$ for both kernels, and its $\\nu$ already contains the factor 2 of (45). The value returned for HAC\\_PD is half of the stated one. The figure code takes this value as the degrees of freedom of the $t$ quantile, in the coverage and in the median half-length of the HAC\\_PD (edof) series. Since the positive upper $t$ critical value decreases as the degrees of freedom grow, the plotted interval is at least as wide as the interval that A.7 describes, and its coverage is at least as high.\n\nThe figure code also resets any degrees of freedom below one to one, a rule that A.7 does not state. Let $b_i\\ge 0$ be the eigenvalues of the matrix in (45) for a positive semidefinite kernel, not all zero. Then the stated value is $(\\sum_i b_i)^2/\\sum_i b_i^2\\ge 1$, which implies that the reset has no effect on HAC\\_PD once the factor is restored. With the factor missing, the reset raises a coded value below one back to one, and it then hides part of the error.\n\nThe affected series are HAC\\_PD (edof) in Figures 7 and 8 and in Figures 2, 3, 5, 6, 8 and 9 of Supplement A. These series are the simulation evidence for the interval that Section 7 recommends. The replication archive does not contain the simulation results needed to recompute these series, and I have not computed the series again.", "what_survives": "No formula in the paper changes. The moment-matching formula in A.7 is correct, and the HAC return expression already uses $2\\mu^2/\\nu$. Restoring the factor changes no variance estimate, and so it changes no series computed with a normal critical value and no SAH series. The point estimators, the asymptotic results, the mean squared errors and the randomization test do not involve the degrees of freedom. The application scripts compute their intervals by a separate path (\\fid{P004-N007}). What changes is the HAC\\_PD (edof) series: with the stated degrees of freedom its intervals are no wider, and its coverage no higher, than plotted. The size of the change is not known, and nothing here shows that the recommendation of Section 7 fails." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E026", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Equation (46): f_x(d) equals 3 alpha_x times (Gamma(d;1,1) minus Gamma(d;5,0.5)) times max((1 - d squared over 36), 0), with a red box around the max factor. Two lines below, a red box marks the words: is used to guarantee that there is no treatment effect beyond 6 units of distance.", "caption": "Supplement A, Appendix B.1, p. 18: equation (46) and the sentence that explains its last factor. The marks show the factor $\\max((1-d^2/36),0)$ and the statement that no effect extends beyond 6 units.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E027.png" } ], "formal_release": null, "formal_statements": [], "headline": "The generator of the additive-effect simulations multiplies by $1-d^2/36$ without the maximum with zero that equation (46) prescribes, and its effects do not vanish beyond distance 6; the code should use $\\max(1-d^2/36,0)$.", "id": "P004-E027", "location": "Supplement A, Appendix B.1, equation (46) and the sentence after it, p. 18; replication archive (Supplement B), SpatialEffect/src/effect_nonmo.cpp, line 14", "pdf_page": 18, "printed_page": "18", "reaches": [ { "effect": "implementation_wrong", "effect_text": "the package or replication code has the reported defect", "explanation": null, "package": "SpatialEffect", "recorded_as": null, "to": "package", "url": null, "where": null, "work_id": "P004" } ], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Supplement A, p.~18:\n\\[\nf_x(d)=3\\alpha_x\\times\\left(\\Gamma(d;1,1)-\\Gamma(d;5,0.5)\\right)\\times\\max\\left(\\left(1-\\frac{d^2}{36}\\right),0\\right), \\tag{46}\n\\]\n``The term $\\max\\left(\\left(1-\\frac{d^2}{36}\\right),0\\right)$ is used to guarantee that there is no treatment effect beyond 6 units of distance.''\n\n\\texttt{effect\\_nonmo.cpp}, line 14:\n\n\\texttt{return ( (Rcpp::dgamma(d, sh, 2*sc,0) - a*Rcpp::dgamma(d, 5*sh, sc,0))*\\hl{((-1)*d*d/36+1)});}", "corrected": "\\texttt{return ( (Rcpp::dgamma(d, sh, 2*sc,0) - a*Rcpp::dgamma(d, 5*sh, sc,0))*\\hl{Rcpp::pmax(1-d*d/36, 0.0)});}\n\nEquation (46) stays as printed. The additive simulation data, and the results computed from them, should be generated again before corrected figures are reported.", "evidence": "The evidence is an exact calculation and a compilation of the unchanged source. Fix $\\alpha_x=1$ and one treated node at $d=7$. Since $\\Gamma(d;1,1)=e^{-d}$ and $\\Gamma(d;5,0.5)=(4/3)d^4e^{-2d}$, the printed increment is $0$, and the coded increment is\n\\[\n3\\left(e^{-7}-\\frac{9604}{3}e^{-14}\\right)\\left(1-\\frac{49}{36}\\right)=\\frac{13}{12}\\left(\\frac{9604}{3}-e^{7}\\right)e^{-14},\n\\]\nwhich lies between $0.001895961$ and $0.001895963$. The compiled source returns $0.0018960$ at $d=7$, $0.00065$ at $d=8$ and $-0.000096$ at $d=10$, and it agrees with (46) for $d\\le 6$. For $d\\ge 13$ the coded function is negative. With $\\alpha_x=1$, a treated node at distance 14 then lowers every outcome on the circle of radius $1/2$ around another node. A copy of the source with the correction below agrees with (46) at every distance tested.", "problem": "Equation (46) is correct as printed. The code that generates the additive outcomes of (47) evaluates the effect function with the factor $1-d^2/36$, which is negative beyond $d=6$, where the printed factor is zero. The generator for the interactive case does truncate. In the additive simulations the effect of a treated node extends beyond 6 units. The effect there is small: about $0.0019\\,\\alpha_x$ at $d=7$, against about $1.7\\,\\alpha_x$ at $d=0.5$. The simulations set the HAC cutoff to $2\\cdot 6+2d$ because no effect is supposed to extend beyond 6 units. With the code as supplied this reason fails: two nodes at distance 14 have dependent circle averages at $d=1/2$, although $h(1/2)=13$.", "what_survives": "Equation (46), the theory, the interactive and null-effect generators and the two applications are unaffected by this omission. The supplied generator for the additive simulations illustrated in Figures 5, 6, 7 and 9 and Appendix B differs from (46) by this tail; I have not traced every published panel to its data. At $d=7$ the tail is about a thousandth of the effect at $d=0.5$. I have not computed the simulations again, and no conclusion is shown to change." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E027", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Listing of source code with line numbers, in four parts. In line 7 of Conley.cpp a red box marks the name ConleySE in a definition that returns Rcpp::List and takes eight parameters. In line 7 of Conley_debug.cpp a red box marks the name ConleySE in a definition that returns void and takes the same eight parameters. In line 11 of effect_nonmo.cpp and in line 11 of effect_interactive.cpp red boxes mark the name nonmonoeffect_c, each with the parameters NumericVector d, double sh, double sc, double a. The first body returns a single expression; the second builds a vector of weights and returns the product of an output vector and the weights.", "caption": "Source code of the package in the replication archive, not a page of the article: src/Conley.cpp and src/Conley_debug.cpp, lines 6–7, and src/effect_nonmo.cpp and src/effect_interactive.cpp, from line 11, typeset from the files with line numbers added. Each marked name is defined in two files with the same list of parameter types. The two helper functions have different bodies.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E028.png" }, { "alt": "Listing of source code with line numbers, in three parts. In line 15 of RcppExports.cpp a red box marks the declaration ConleySE(arma::colvec res, arma::mat W_meat, arma::mat dist, double cutoff, int kernel). In line 25 a red box marks the call ConleySE(res, W_meat, dist, cutoff, kernel). Line 44 registers _SpatialEffect_ConleySE with 5 arguments. Lines 4 to 6 of RcppExports.R define the R function ConleySE with the arguments res, W_meat, dist, cutoff, kernel. Line 7 of Conley.cpp defines ConleySE with eight parameters, including XXinv, trim and if_edof.", "caption": "The generated bindings in the same package: src/RcppExports.cpp, lines 14–16, 25 and 43–44, and R/RcppExports.R, lines 4–6, above the definition in src/Conley.cpp, line 7. The marked declaration and call have five arguments; the definition has eight.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E028-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "The source of the SpatialEffect package in the replication archive does not build as distributed: two C++ functions are each defined twice, and the generated bindings call a five-argument ConleySE that no file defines; the debug file should be left out, the two helper functions kept under distinct names, and the bindings and callers brought into agreement with one ConleySE interface.", "id": "P004-E028", "location": "Replication archive (Supplement B), package folder SpatialEffect/: src/Conley.cpp, line 7; src/Conley_debug.cpp, line 7; src/effect_nonmo.cpp and src/effect_interactive.cpp, line 11; src/RcppExports.cpp, lines 15, 25 and 44; R/RcppExports.R, lines 4–6. The code is not printed in the article; the archive is named on p. 766 (PDF p. 23).", "pdf_page": null, "printed_page": null, "reaches": [ { "effect": "result_fails", "effect_text": "a result or code path there relies on it and fails with it", "explanation": null, "package": null, "recorded_as": null, "to": "work", "url": "https://pmaronow.github.io/llm-audit/works/P008/", "where": null, "work_id": "P008" }, { "effect": "result_fails", "effect_text": "a result or code path there relies on it and fails with it", "explanation": null, "package": "SpatialEffect", "recorded_as": null, "to": "package", "url": null, "where": null, "work_id": "P004" } ], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\texttt{src/Conley.cpp}, line 7:\n\n\\texttt{Rcpp::List \\hl{ConleySE}(arma::colvec res, arma::mat W\\_meat, arma::mat dist, arma::mat XXinv, double cutoff, int kernel, int trim,int if\\_edof)\\{}\n\n\\texttt{src/Conley\\_debug.cpp}, line 7:\n\n\\texttt{void \\hl{ConleySE}(arma::colvec res, arma::mat W\\_meat, arma::mat dist, arma::mat XXinv, double cutoff, int kernel, int trim,int if\\_edof)\\{}\n\n\\texttt{src/effect\\_nonmo.cpp} and \\texttt{src/effect\\_interactive.cpp}, line 11 of each:\n\n\\texttt{NumericVector \\hl{nonmonoeffect\\_c}(NumericVector d, double sh, double sc, double a)\\{}\n\n\\texttt{src/RcppExports.cpp}, lines 15, 25 and 44:\n\n\\texttt{Rcpp::List ConleySE(\\hl{arma::colvec res, arma::mat W\\_meat, arma::mat dist, double cutoff, int kernel});}\n\n\\texttt{rcpp\\_result\\_gen = Rcpp::wrap(ConleySE(res, W\\_meat, dist, cutoff, kernel));}\n\n\\texttt{\\{\"\\_SpatialEffect\\_ConleySE\", (DL\\_FUNC) \\&\\_SpatialEffect\\_ConleySE, 5\\},}\n\n\\texttt{R/RcppExports.R}, line 4:\n\n\\texttt{ConleySE <- function(res, W\\_meat, dist, cutoff, kernel) \\{}", "corrected": "\\begin{itemize}\n\\item Leave \\texttt{\\hl{src/Conley\\_debug.cpp}} out of the package, and keep \\texttt{src/Conley.cpp}.\n\\item Keep both bodies of \\texttt{nonmonoeffect\\_c}, under \\hl{distinct names} or with internal linkage, and remove the export annotation above each where it is not needed.\n\\item Choose one interface for \\texttt{ConleySE}, and bring \\texttt{src/RcppExports.cpp}, \\texttt{R/RcppExports.R} and the callers into agreement with it. The callers in \\texttt{R/SpatialEffect.R} (lines 218--219) and \\texttt{R/SpatialEffect\\_internal.R} (lines 208--209) pass five arguments. An explicitly specified five-argument adapter is an alternative to changing every caller.\n\\item Check that the package installs from clean source and that representative calls of the replication run.\n\\end{itemize}", "evidence": "The evidence is a compilation of the unchanged source, with the binaries shipped in the folder removed. All eleven C++ files compile. Linking the resulting objects reports both duplicate symbols, and linking each pair of files alone reproduces each collision. The symbol tables show that the object built from \\texttt{RcppExports.cpp} requires a five-argument \\texttt{ConleySE} that no object defines; the entry point that the file registers, \\texttt{\\_SpatialEffect\\_ConleySE}, is itself defined. A library linked without the duplicates, for diagnosis only, fails to load for want of that function. A complete installation with \\texttt{R CMD INSTALL} stopped earlier, at a Fortran library missing from the machine I used. For that reason I showed the defects by linking the compiled objects directly.", "problem": "The ReadMe of the archive presents this folder as a standalone R package and says that installing it is essential for replicating the empirical results. When R builds a package, it compiles every \\texttt{.cpp} file in \\texttt{src/}, and neither \\texttt{Makevars} file excludes any of them. The files \\texttt{Conley.cpp} and \\texttt{Conley\\_debug.cpp} each define \\texttt{ConleySE} with eight parameters of the same types. Since a return type does not distinguish two C++ functions, the two definitions are the same external symbol, and the library cannot be linked. The same holds for \\texttt{nonmonoeffect\\_c}, which is defined in both effect files. Its two bodies differ: the one in \\texttt{effect\\_interactive.cpp} sets the effect to zero beyond distance 6, and the other does not. Apart from the duplicates, \\texttt{RcppExports.cpp} declares and calls a \\texttt{ConleySE} with five arguments, and the only definitions have eight. The library therefore needs a function that no source file provides.", "what_survives": "This finding shows no error in a theorem, estimand or printed number. The simulation scripts compile selected files one at a time with \\texttt{sourceCpp}, which generates its own bindings and avoids both collisions. Results produced earlier with a different installed binary are not shown to be wrong, and I did not compute them again. What changes is that the package cannot be installed from the archive without repair. I did not build a repaired package." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E028", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Listing of source code with line numbers. In line 178, inside a call to SpatialEffect, a red box marks the argument trim = trim. In line 330, at the end of a call to ConleySE, a red box marks a sixth argument, as.integer(1). In line 366, inside a call to LocalReg, a red box marks the arguments as.integer(1), edf = TRUE; they are followed by dist.metric and bias_correction = 0.", "caption": "Source code in the replication archive, not a page of the article: application/scripts/Jayachandran_etal_2017.R, lines 176–179, 329–330 and 364–366, typeset from the file with line numbers added; long lines continue on the next row. The marked arguments are the ones that the called functions do not accept.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E029.png" }, { "alt": "Listing of source code with line numbers, in three parts. Lines 1 to 6 of SpatialEffect.R give the arguments of SpatialEffect; a red box marks the end of the list, edf = FALSE, m = 2, lambda = 0.02, nPerms = 1000, and the list has no argument named trim and no three dots. In line 4 of RcppExports.R a red box marks the five arguments of ConleySE: res, W_meat, dist, cutoff, kernel. In line 93 of SpatialEffect_internal.R red boxes mark the end of the arguments of LocalReg, dist.metric = \"Euclidean\", bias_correction = TRUE; the list has no argument named edf.", "caption": "The interfaces that these calls meet, in the package folder of the same archive: R/SpatialEffect.R, lines 1–6; R/RcppExports.R, lines 4–6; R/SpatialEffect_internal.R, lines 92–93. The marks are on the end of each list of arguments: SpatialEffect has no trim and no ..., ConleySE has five arguments, and LocalReg has no edf.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E029-b.png" }, { "alt": "Listing of source code with line numbers. Lines 127 to 137 build the data frame Zdata with the columns treatment, cov1, cov2, cov3 and prob_treatment. Lines 153 and 154 are comments that would have named columns x_coord and y_coord. Red boxes mark line 158, x_coord_Z assigned \"x_coord\", and line 159, y_coord_Z assigned \"y_coord\". In line 238 a red box marks the arguments x_coord_Z = x_coord_Z, y_coord_Z = y_coord_Z, ras_Z = NULL in the first call to SpatialEffect.", "caption": "application/scripts/Ferraro_etal_2015.R, lines 127–137, 153–154, 158–159 and 238–241. The table Zdata receives the columns treatment, cov1, cov2, cov3 and prob_treatment; the two lines that would name coordinate columns are commented out. The marked lines request the columns x_coord and y_coord in a call with ras_Z = NULL.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E029-c.png" } ], "formal_release": null, "formal_statements": [], "headline": "Four calls in the two application scripts of the replication archive do not run against the package supplied with them: three calls in the Uganda script pass arguments that the functions do not have, and the first call in the Costa Rica script asks for coordinate columns that the script never creates.", "id": "P004-E029", "location": "Replication archive (Supplement B): application/scripts/Jayachandran_etal_2017.R, lines 176–179, 329–330 and 364–368; application/scripts/Ferraro_etal_2015.R, lines 127–137, 158–159 and 238–241; compared with SpatialEffect/R/SpatialEffect.R, lines 1–6, SpatialEffect/R/RcppExports.R, lines 4–6, and SpatialEffect/R/SpatialEffect_internal.R, lines 92–93. The code is not printed in the article; the applications are in Section 8, pp. 763–765 (PDF pp. 20–22).", "pdf_page": null, "printed_page": null, "reaches": [ { "effect": "result_fails", "effect_text": "a result or code path there relies on it and fails with it", "explanation": null, "package": null, "recorded_as": null, "to": "work", "url": "https://pmaronow.github.io/llm-audit/works/P008/", "where": null, "work_id": "P008" }, { "effect": "result_fails", "effect_text": "a result or code path there relies on it and fails with it", "explanation": null, "package": "SpatialEffect", "recorded_as": null, "to": "package", "url": null, "where": null, "work_id": "P004" } ], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\texttt{Jayachandran\\_etal\\_2017.R}, line 178, inside the call to \\texttt{SpatialEffect} on lines 176--179:\n\n\\texttt{per.se = per.se, conley.se = conley.se, cutoff = cutoff, alpha = alpha, edf = edf, \\hl{trim = trim}, bw = 10.996, bw\\_debias = 25.13,}\n\nLines 329--330:\n\n\\texttt{Conley\\_result <- ConleySE(as.vector(res\\_d), as.matrix(W\\_meat), as.matrix(dist),}\n\n\\texttt{as.double(c), as.integer(k), \\hl{as.integer(1)})}\n\nLines 364--366:\n\n\\texttt{wls\\_results <- LocalReg(dVec, Sdata, bw = 18.707, bw\\_debias = 25.13, dist = dist, Zup,}\n\n\\texttt{xevals = NULL, smooth.conley.se = 1, kernel, as.numeric(cutoff),}\n\n\\texttt{\\hl{as.integer(1), edf = TRUE}, dist.metric, bias\\_correction = 0)}\n\n\\texttt{Ferraro\\_etal\\_2015.R}, lines 158--159, and line 238, the first line of the call on lines 238--241:\n\n\\texttt{x\\_coord\\_Z <- \"x\\_coord\"}\n\n\\texttt{y\\_coord\\_Z <- \"y\\_coord\"}\n\n\\texttt{result.list <- SpatialEffect(ras = ras, Zdata = Zdata, \\hl{x\\_coord\\_Z = x\\_coord\\_Z, y\\_coord\\_Z = y\\_coord\\_Z, ras\\_Z = NULL},}", "corrected": "\\begin{itemize}\n\\item Bring the package and the two scripts to one interface, keeping the positive semidefinite trimming and the degrees-of-freedom calculation that the article describes. If the R interface of \\texttt{SpatialEffect} is kept as printed, remove \\texttt{\\hl{trim = trim}} from the call.\n\\item Give \\texttt{ConleySE} one definition, one generated binding and callers that agree with it (see \\fid{P004-E028}).\n\\item Call \\texttt{LocalReg} with named arguments that it has: omit \\texttt{\\hl{edf = TRUE}} and the extra \\texttt{\\hl{as.integer(1)}}, name \\texttt{dist.metric}, and keep \\texttt{bias\\_correction}. Supply the quantity \\texttt{edof} that line 368 uses.\n\\item In the Costa Rica script, skip the first call, whose result the later polygon calls overwrite, or pass it the polygons through \\texttt{\\hl{ras\\_Z = ras\\_Z}}. Coordinate columns are needed only if an analysis of point nodes is intended.\n\\item Run the repaired applications and compare them with the reported results.\n\\end{itemize}", "evidence": "The evidence is an exact check of the calls. I read each of the three Uganda calls from the script and evaluated it against the unchanged R functions of the package. The errors are ``unused argument (trim = trim)'', ``unused argument (as.integer(1))'' and ``unused arguments (edf = TRUE, dist.metric)''; they arise before any data are used. For Costa Rica, I ran the script's own construction of \\texttt{Zdata} and its first call on a small explicit input, a treatment raster of four cells and an outcome raster of nine cells equal to 1. The sampling points are empty, the four circle averages are \\texttt{NaN}, and the regression stops with ``0 (non-NA) cases''. With coordinate columns added the same call completes. The call on lines 364--366 also passes \\texttt{bias\\_correction = 0}; \\texttt{LocalReg} has that argument, and it is not part of this finding.", "problem": "The function \\texttt{SpatialEffect} of the supplied package has no argument \\texttt{trim} and no \\texttt{...}, and R rejects the first Uganda call with ``unused argument (trim = trim)''. The R function \\texttt{ConleySE} has five arguments and the script passes six. \\texttt{LocalReg} has no argument \\texttt{edf}; in the call on lines 364--366 the unnamed \\texttt{as.integer(1)} takes the place of \\texttt{dist.metric}, and both \\texttt{edf = TRUE} and the unnamed \\texttt{dist.metric} are left without a match. Removing \\texttt{edf} alone is therefore not enough. Line 368 then reads \\texttt{wls\\_results[[\"edof\"]]}, which \\texttt{LocalReg} never returns. In the Costa Rica script, \\texttt{Zdata} has the five columns \\texttt{treatment}, \\texttt{cov1}, \\texttt{cov2}, \\texttt{cov3} and \\texttt{prob\\_treatment}. Since the first call sets \\texttt{ras\\_Z = NULL}, the function treats the nodes as points and looks up the columns \\texttt{x\\_coord} and \\texttt{y\\_coord}, which do not exist. The two later calls pass the polygons, but they run only after the first call returns.", "what_survives": "No estimand, proposition or proof depends on these calls, and no reported estimate or interval is shown to be wrong. The maps and the plot of the propensity scores come before the failing calls. What changes is that the application estimates and intervals of Section 8 cannot be regenerated by running these scripts as supplied. I did not run a repaired application, and I did not test whether the output of each script matches Figures 11 and 13." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E029", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Listing of source code with line numbers. A red box marks line 1, rm(list=ls()). In line 20 a red box marks today_date inside paste0, which builds the name of the first output file. Lines 22 and 23 load the null results and read simulation_result. Lines 50 and 51 load the interactive results and read simulation_result. Line 77 builds the output name with nonmono in it. A red box marks line 79, temp=simulation_result[['output_rt_list']], which is not preceded by a load. Line 93 sets the title Rejection Probability (Additive Effect).", "caption": "Source code in the replication archive, not a page of the article: simulation/scripts/2_generate_figures_03.R, lines 1, 20–23, 48–51, 77–82 and 93, typeset from the file with line numbers added. Line 1 clears the workspace and line 20 uses today_date. The block for the additive panel reads simulation_result on line 79 without a load, unlike the blocks on lines 20–23 and 48–51.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-E030.png" } ], "formal_release": null, "formal_statements": [], "headline": "The script for the randomization-test panels of Figure 9 clears the workspace and then uses a date variable that it never defines, and it would draw the additive panel from the interactive results; two inserted lines repair both defects.", "id": "P004-E030", "location": "Replication archive (Supplement B): simulation/scripts/2_generate_figures_03.R, lines 1, 20, 48, 50 and 77–96. The code is not printed in the article; the script is associated with the three panels of Figure 9, p. 762 (PDF p. 19).", "pdf_page": null, "printed_page": null, "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\texttt{2\\_generate\\_figures\\_03.R}, lines 1, 20, 50, 77, 79 and 93:\n\n\\texttt{\\hl{rm(list=ls())}}\n\n\\texttt{figure\\_title\\_null =paste0('../graph\\_new/Randomization\\_test\\_null\\_', \\hl{today\\_date},'.png') \\#figure name}\n\n\\texttt{load('../data\\_new/Paired\\_randomization\\_test\\_interactive\\_10\\_Jun\\_12\\_points.RData')}\n\n\\texttt{figure\\_title\\_nonmono =paste0('../graph\\_new/Randomization\\_test\\_nonmono\\_', today\\_date,'.png') \\#figure name}\n\n\\texttt{\\hl{temp=simulation\\_result[['output\\_rt\\_list']]}}\n\n\\texttt{ggtitle('Rejection Probability (Additive Effect)')+}\n\nNo line of the file assigns \\texttt{today\\_date}, and no \\texttt{load} stands between line 50 and line 96.", "corrected": "Insert after line 1:\n\n\\texttt{\\hl{today\\_date = format(Sys.Date(), format=\"\\%b\\_\\%d\")}}\n\nInsert immediately before line 79:\n\n\\texttt{\\hl{load('../data\\_new/Paired\\_randomization\\_test\\_nonmono\\_10\\_Jun\\_12\\_points.RData')}}", "evidence": "The evidence is an exact counterexample, obtained by running lines 1--96 of the script itself; only line 9, which sets the working directory through RStudio, was left out. I replaced the three input files by small ones in which every rejection rate equals $0$ (null), $1/4$ (interactive) or $3/4$ (additive). As shipped, the script stops at line 20. With a date supplied after line 1, the panel titled ``Additive Effect'' shows $1/4$ at each of the 39 distances and three sample sizes, the same 117 values as the interactive panel. With the missing \\texttt{load} inserted it shows $3/4$. The name of the missing file appears in the neighbouring script \\texttt{2\\_generate\\_figures\\_02.R}.", "problem": "Line 1 clears the workspace, and nothing defines \\texttt{today\\_date} before line 20 needs it for the name of the first output file. Once the working directory on line 9 is set through RStudio, the script stops at line 20 with ``object 'today\\_date' not found''. Once a date is supplied, a second defect appears. The third block has the output name \\texttt{nonmono} and the title ``Rejection Probability (Additive Effect)'', but it reads \\texttt{simulation\\_result} on line 79 without loading the additive results. The object still holds the interactive results loaded on line 50, and the additive panel repeats the interactive one.", "what_survives": "The published Figure 9 has distinct additive and interactive curves, and this finding does not show it to be wrong. The simulation results that the script loads are not in the archive, and I could not regenerate its rejection probabilities. The null and interactive blocks load their own results. Later, the script summarises the test based on the supremum statistic and loads each scenario explicitly. The missing \\texttt{load} does not affect that summary, but the date error stops the script before it is reached." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-E030", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of four lines of text. A red box marks the upper limit n of the sum in 'the Hajek estimator replaces them with N_1 = sum from i = 1 to n of Z_i and N_0 = N minus N_1, respectively'.", "caption": "Section 5, p. 754 (PDF p. 11), the sentence before equation (16). The marked upper limit of the sum is $n$; the number of intervention nodes is $N$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N001.png" }, { "alt": "Crop of the line 'First, note that the HAC variance estimator can be expressed as', of equation (21) and of four lines of text. Red boxes mark 'e-hat_n)' in 'where e-hat = (e-hat_1, ..., e-hat_n)' and the words 'n-by-n' in 'K is a n-by-n symmetric matrix'. The next two lines write the eigenvalue decomposition of K and the matrix K^PD with sums from i = 1 to n.", "caption": "Section 5, p. 756 (PDF p. 13), equation (21) and the paragraph after it. The marked $\\widehat e_n$ and “$n$-by-$n$” use $n$ for the number of nodes, as do the three sums and index sets that follow.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N001-b.png" }, { "alt": "Crop of footnote 22, 'That is, Y_i(Z) = Y_i(0) for all Z.' A red box marks 'Y_i(Z) = Y_i(0)'.", "caption": "Footnote 22, p. 761 (PDF p. 18). The marked null hypothesis is indexed by the node index $i$; outcome points are indexed by $x$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N001-c.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T31", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T31" } ], "headline": "The number of intervention nodes is $N$, not $n$, in the treatment count and in the eigenvalue decomposition of $\\mathcal K$, and footnote 22 should index outcome points by $x$.", "id": "P004-N001", "location": "Section 5, p. 754 (PDF p. 11), the sentence before equation (16); p. 756 (PDF p. 13), the paragraph after equation (21); footnote 22, p. 761 (PDF p. 18)", "pdf_page": 11, "printed_page": "754", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P004-T31" ], "text": { "as_printed": "P.~754: ``the Hajek estimator replaces them with $N_1=\\textstyle\\sum_{i=1}^{\\hl{n}}Z_i$ and $N_0=N-N_1$, respectively,''.\n\nP.~756: ``where $\\widehat e=(\\widehat e_1,\\ldots,\\widehat e_{\\hl{n}})$, $\\mathcal K$ is a \\hl{$n$}-by-\\hl{$n$} symmetric matrix with $\\mathcal K_{ij}=K(\\frac{d_{ij}}{\\tilde d})$, and $\\circ$ denotes the pointwise (Hadamard) matrix product. Denote the eigenvalue decomposition of the matrix $\\mathcal K$ as $\\mathcal K=\\textstyle\\sum_{i=1}^{\\hl{n}}\\lambda_iv_iv_i'$, where $\\{\\lambda_i\\}_{i=1}^{\\hl{n}}$ are the eigenvalues and $\\{v_i\\}_{i=1}^{\\hl{n}}$ are the eigenvectors. We define $\\mathcal K^{\\mathrm{PD}}=\\textstyle\\sum_{i=1}^{\\hl{n}}\\max\\{\\lambda_i,0\\}v_iv_i'$''.\n\nFootnote 22, p.~761: ``That is, $Y_{\\hl{i}}(\\mathbf Z)=Y_{\\hl{i}}(0)$ for all $\\mathbf Z$.''", "corrected": "P.~754: ``the Hajek estimator replaces them with $N_1=\\textstyle\\sum_{i=1}^{\\hl{N}}Z_i$ and $N_0=N-N_1$, respectively,''.\n\nP.~756: ``where $\\widehat e=(\\widehat e_1,\\ldots,\\widehat e_{\\hl{N}})$, $\\mathcal K$ is a \\hl{$N$}-by-\\hl{$N$} symmetric matrix with $\\mathcal K_{ij}=K(\\frac{d_{ij}}{\\tilde d})$, and $\\circ$ denotes the pointwise (Hadamard) matrix product. Denote the eigenvalue decomposition of the matrix $\\mathcal K$ as $\\mathcal K=\\textstyle\\sum_{i=1}^{\\hl{N}}\\lambda_iv_iv_i'$, where $\\{\\lambda_i\\}_{i=1}^{\\hl{N}}$ are the eigenvalues and $\\{v_i\\}_{i=1}^{\\hl{N}}$ are the eigenvectors. We define $\\mathcal K^{\\mathrm{PD}}=\\textstyle\\sum_{i=1}^{\\hl{N}}\\max\\{\\lambda_i,0\\}v_iv_i'$''.\n\nFootnote 22: ``That is, $Y_{\\hl{x}}(\\mathbf Z)=Y_{\\hl{x}}(0)$ \\hl{for all $x\\in\\mathcal X$ and} all $\\mathbf Z$.''", "evidence": "The evidence is a check of pp.~751, 753--756, 758, 759 and 761 of the article.", "problem": "The article writes $N$ for the number of intervention nodes: the sums in (16) run to $N$, and the design matrix $\\mathbf X$ on p.~755 is $N$-by-$2$. The letter $n$ is introduced only in Section 7. In footnote 22 the sharp null hypothesis is written with the node index $i$, while outcomes are indexed by points $x\\in\\mathcal X$; Section 6.3 states the same hypothesis with $x$. The meaning is clear in each place. The letter $b$ denotes a constant in C2 and a function in C4; each is defined where it is used, and no correction is needed there.", "what_survives": "Nothing else changes. The estimators, the positive semidefinite variance estimator (22) and the randomization tests are as intended." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-N001", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of seven lines of text. They read: Figure 6 shows the mean squared errors of the Hajek estimator for both additive-effect (31) and interactive-effect cases (32), with intervention-node sample sizes of 64, 100, and 144. In both cases the MSEs decrease as sample sizes increase, as predicted by our theory. Figure 7 and Figure 8 report coverage rates and median half-lengths for the Hajek estimator with different confidence interval constructions in the additive-effect case and the interactive-effect case, respectively. For brevity, we only display the case with a sample size of 144. The results are illustrated for AMEs at different distance values. Red outlines surround the words the mean squared errors of the Hajek estimator and the words coverage rates.", "caption": "Section 7, p. 760 (PDF p. 17). The outlines mark the two quantities that are computed against the approximated AME.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N003.png" }, { "alt": "Crop of three lines of text, equation (50) and one more line. The text reads: For our simulation, we use a Bernoulli design where Z sub i equals 1 with probability 0.5 for all intervention nodes. We run the simulation for 2000 times. Denote the pth assignment of Z as Z sub p and the value of Z sub i under the assignment as Z sub pi. To obtain AME, we first calculate. Equation (50) gives tau sub ix of eta as approximately the ratio of the sum over p from 1 to 2000 of Z sub pi times Y sub x of Z sub p to the sum of Z sub pi, minus the same ratio with 1 minus Z sub pi. The last line reads: Then tau sub i of d and eta and the AME can be constructed following their definitions. One red outline surrounds the words To obtain AME, we first calculate, and another surrounds the equation.", "caption": "Supplement A, Appendix B.1, p. 18. The outlines mark the sentence that introduces the approximation and equation (50).", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N003-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "The mean squared errors and coverage rates of Figures 6 to 8 are computed against an AME that is itself estimated, by (50), from the same 2000 draws of the assignment that are being scored; the text does not say so.", "id": "P004-N003", "location": "Section 7, the two paragraphs on Figures 6 to 8, p. 760 (PDF p. 17); Supplement A, Appendix B.1, “Experimental Designs” and equation (50), p. 18", "pdf_page": 17, "printed_page": "760", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "\\emph{Main text, p.~760.} Figure 6 shows \\hl{the mean squared errors} of the Hajek estimator for both additive-effect (31) and interactive-effect cases (32), with intervention-node sample sizes of 64, 100, and 144. In both cases the MSEs decrease as sample sizes increase, as predicted by our theory.\n\nFigure 7 and Figure 8 report \\hl{coverage rates} and median half-lengths for the Hajek estimator with different confidence interval constructions in the additive-effect case and the interactive-effect case, respectively.\n\n\\emph{Supplement A, p.~18.} We run the simulation for 2000 times. Denote the $p$th assignment of $\\mathbf Z$ as $\\mathbf Z_p$ and the value of $Z_i$ under the assignment as $Z_{pi}$. \\hl{To obtain AME, we first calculate}\n\\[\n\\tau_{ix}(\\eta)\\approx\\frac{\\sum_{p=1}^{2000}Z_{pi}Y_x(\\mathbf Z_p)}{\\sum_{p=1}^{2000}Z_{pi}}-\\frac{\\sum_{p=1}^{2000}(1-Z_{pi})Y_x(\\mathbf Z_p)}{\\sum_{p=1}^{2000}(1-Z_{pi})}.\n\\]\nThen $\\tau_i(d;\\eta)$ and the AME can be constructed following their definitions.", "corrected": "Add after (50): \\hl{The mean squared errors and coverage rates reported below are computed relative to this approximation of the AME, which is obtained from the same 2000 draws. The error of the approximation is not assessed separately.}\n\nThe approximation can be avoided, since both designs have a closed form. Under (47), $\\tau_{ix}(\\eta)=f_x(d_{ix})$. Under (49) with assignment probability $1/2$,\n\\[\n\\tau_{ix}(\\eta)=f_x(d_{ix})+\\tfrac12\\,g_x(d_{ix})+\\tfrac12\\sum_{j:\\,\\mathcal N(j)=i}g_x(d_{jx}).\n\\]", "intended_meaning": "The figures give Monte Carlo approximations to the mean squared error and the coverage for the fixed, design-based AME. Equation (50) states that the AME is approximated from the simulated draws, and the formula is a correct approximation: under independent Bernoulli assignment, $\\tau_{ix}$ is the difference of the conditional means of $Y_x(\\mathbf Z)$ given $Z_i=1$ and given $Z_i=0$, and (50) is the difference of the corresponding sample means. Nothing printed is false.\n\nWhat the text leaves out is that the replication code computes this benchmark from the same outcomes and assignments on which the estimator is then evaluated. Let $B$ be the benchmark, $\\theta$ the AME, $\\delta=B-\\theta$, and $\\bar e$ the mean error of the estimates over the draws. Expanding the square, I find that the mean squared error computed against $B$ minus the one computed against $\\theta$ equals $\\delta^2-2\\delta\\bar e$ in every run. In a constructed example with 64 nodes and 2000 draws, the mean squared error is 4 against the AME and 3 against the benchmark, and an interval of half-width 1 covers the AME in no draw and the benchmark in three quarters of the draws. This example is not the design of the paper. The simulation outputs are not in the replication archive, and neither the size nor the sign of any effect on the published curves is known. The half-lengths of the intervals do not depend on the benchmark." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-N003", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Six lines of body text. A red box marks the sentence: we recommend researchers to use the positive semidefinite HAC variance estimator with the empirical degree of adjustment when applying our method.", "caption": "Section 7, p. 761 (PDF p. 18): the recommendation that closes the discussion of Figures 7 and 8.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N004.png" }, { "alt": "Equation (44): mu equals the trace of (N1 N0 / N) M (Omega circ lambda-prime lambda) M. Below it a red box marks the sentence: The estimated variance is inflated by 1 over mu, becoming w-prime Sigma-hat HAC w over mu. Equation (45) for nu follows.", "caption": "Supplement A, Section A.7, p. 17: equation (44) and the sentence that describes the rescaling of the variance. A.7 defines $\\mu$ with the uniform kernel matrix $\\Omega$ only.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N004-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T7", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T7" }, { "id": "P004-T8", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T8" } ], "headline": "The recommended interval divides the positive semidefinite HAC variance by a factor $\\mu$ computed from the trimmed kernel, and this factor can converge to a limit above one; under C1–C5 the adjusted variance can then stay below the variance of the estimator, and the interval can cover too rarely; the guarantee of Proposition 4 carries over once the factor is capped at one.", "id": "P004-N004", "location": "Section 7, p. 761 (PDF p. 18), end of the paragraph between Figures 7 and 8; Section 5, p. 756 (PDF p. 13), equation (22) and footnote 15; Supplement A, Section A.7, equation (44) and the sentence after it, p. 17; replication archive (Supplement B), simulation/scripts/6_functions3.R, line 196", "pdf_page": 18, "printed_page": "761", "reaches": [ { "effect": "uses_only_what_survives", "effect_text": "relies only on the part that survives", "explanation": null, "package": "SpatialEffect", "recorded_as": null, "to": "package", "url": null, "where": null, "work_id": "P004" } ], "related_entries": [], "revisions": [ { "classification_after": { "class": "error", "severity": "Moderate" }, "classification_before": { "class": "error", "severity": "Moderate" }, "note": "One sentence of what survives was qualified; the classification is unchanged (error, Moderate). How this came to light: a Lean formalization of Proposition 4 showed that, under its conditions, the coverage probabilities need not have limits, which the clarity note P004-N024 now records. Why it changed: the sentence said that Proposition 4 is correct as printed; it now refers to the proposition read with limits inferior, which is what the proof of the proposition supports and what this entry relies on. P. M. Aronow decided to add the clarity note; this note was written by the model session that runs the audit." } ], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P004-T7", "P004-T8" ], "text": { "as_printed": "Section 7, p.~761: ``Based on these observations, we recommend researchers to use the positive semidefinite HAC variance estimator \\hl{with the empirical degree of adjustment} when applying our method.''\n\nSection 5, p.~756: ``Nonetheless, it is possible to design a biased-upward positive definite estimator.'' After (22): ``In addition, $\\widehat{\\Sigma}^{\\mathrm{PD}}_{\\mathrm{HAC}}(d)\\geq\\widehat{\\Sigma}_{\\mathrm{HAC}}(d)$.''\n\nSupplement A, Section A.7, p.~17: ``The estimated variance is inflated by $\\frac{1}{\\mu}$, becoming $\\frac{\\mathbf{w}'\\widehat{\\Sigma}_{\\mathrm{HAC}}\\mathbf{w}}{\\mu}$.''\n\n\\texttt{6\\_functions3.R}, line 196, second returned value: \\texttt{Conley\\_var2\\_pd/\\hl{mu2\\_pd}}", "corrected": "Recommendation of Section 7, with the adjustment of A.7 written out for the positive semidefinite estimator. Let $K^{\\mathrm{PD}}$ be the positive part of the kernel matrix $K$ of (21), let $\\widehat{V}_{\\mathrm{PD}}(d)$ be the $(2,2)$ entry of $\\widehat{\\Sigma}^{\\mathrm{PD}}_{\\mathrm{HAC}}(d)$ in (22), and let $\\mu_{\\mathrm{PD}}$ and $\\nu_{\\mathrm{PD}}$ be the quantities (44) and (45) computed with $K^{\\mathrm{PD}}$ in place of $\\Omega$. The recommended variance estimate is\n\\[\n\\widehat{V}^{\\mathrm{adj}}_{\\mathrm{PD}}(d)=\\frac{\\widehat{V}_{\\mathrm{PD}}(d)}{\\hl{\\min\\{\\mu_{\\mathrm{PD}},1\\}}}\\quad\\text{if }\\mu_{\\mathrm{PD}}>0,\\qquad \\widehat{V}^{\\mathrm{adj}}_{\\mathrm{PD}}(d)=\\widehat{V}_{\\mathrm{PD}}(d)\\quad\\text{otherwise.}\n\\]\nIn the first case the critical value $c$ is the $(1-\\alpha/2)$-quantile of the $t$ distribution with $2\\mu_{\\mathrm{PD}}^2/\\nu_{\\mathrm{PD}}$ degrees of freedom, and in the second it is the normal quantile $z_{1-\\alpha/2}$. The interval is $\\widehat{\\tau}_{\\mathrm{HA}}(d)\\pm c\\,\\widehat{V}^{\\mathrm{adj}}_{\\mathrm{PD}}(d)^{1/2}$.\n\nLet $K(\\cdot)$ in (19) be the uniform kernel with cutoff $\\tilde d=h(d)$. Under C1--C5, and if $N\\times\\operatorname{AVar}(\\widehat{\\tau}_{\\mathrm{HA}}(d))$ is uniformly bounded away from zero for all large $N$, the limit inferior of this interval's coverage probability for $\\operatorname{AME}(d;\\eta)$ is at least $1-\\alpha$, for each $0<\\alpha<1$. Indeed, $\\widehat{V}^{\\mathrm{adj}}_{\\mathrm{PD}}(d)\\ge\\widehat{V}_{\\mathrm{PD}}(d)\\ge\\widehat{V}_{\\mathrm{HAC}}(d)$ and $c\\ge z_{1-\\alpha/2}$, so the interval contains the interval of Proposition 4(ii) whenever $\\widehat{V}_{\\mathrm{HAC}}(d)\\ge 0$.\n\nI have not assessed the finite-sample coverage of the capped interval. If the uncapped division is kept, Section 7 should present the interval as a finite-sample device supported by the simulations, and it should say that Proposition 4 does not cover it.", "evidence": "The evidence is a pair of exact examples that satisfy C1--C5, with the limits derived and the finite matrices computed in rational arithmetic.\n\n\\emph{First example.} For each $m\\ge 1$, the design has $N=5m$ nodes in $m$ blocks. Block $b$ has a centre at $(10b,0)$ and four leaves at $(10b\\pm 1,0)$ and $(10b,\\pm 1)$. The assignments are independent with probability $1/2$. The circle averages at $d=0$ are $y=(0,1,1,-1,-1)$ in every block under every assignment, so that no interference is present and every treatment effect is zero. Conditions C1--C5 hold, and $h(0)=6/5$ is a valid cutoff in C3. With this cutoff the uniform kernel links the centre to each leaf and links no two leaves. In each block the kernel has eigenvalues $-1,1,1,1,3$. After the eigenvalue $-1$ is trimmed, the matrix $K^{\\mathrm{PD}}$ has diagonal $(3/2,9/8,9/8,9/8,9/8)$ and trace $6$.\n\nI write $s_i=2Z_i-1$. Since $\\sum_i y_i=0$ in each block, I have $\\widehat{\\tau}_{\\mathrm{HA}}=2N^{-1}\\sum_i s_iy_i+O_p(N^{-3/2})$, and hence $N\\operatorname{Var}(\\widehat{\\tau}_{\\mathrm{HA}})\\to (4/N)\\sum_i y_i^2=16/5$. For a kernel $Q$ made of identical blocks, $N\\widehat{V}_Q$ equals $4/N$ times the sum over blocks of $t'Qt$, up to a term that tends to zero in probability, where $t_i=s_iy_i$. Using the independence of the signs, I obtain $\\mathbb{E}[t'Qt]=\\sum_i Q_{ii}y_i^2$. This sum is $4$ for the uniform kernel and $4\\cdot 9/8=9/2$ for $K^{\\mathrm{PD}}$. By the law of large numbers, I find that $N\\widehat{V}_{\\mathrm{HAC}}\\to 16/5$ and $N\\widehat{V}_{\\mathrm{PD}}\\to 18/5$. Finally, $\\mu_Q$ equals $N^{-1}\\operatorname{Trace}(Q)$ up to a term that tends to zero, which implies that $\\mu\\to 1$ for the uniform kernel and $\\mu_{\\mathrm{PD}}\\to 6/5$. Combining these limits, I obtain\n\\[\nN\\,\\frac{\\widehat{V}_{\\mathrm{PD}}}{\\mu_{\\mathrm{PD}}}\\to\\frac{18/5}{6/5}=3<\\frac{16}{5},\n\\]\nso the adjusted variance converges to $15/16$ of the variance of the estimator. The degrees of freedom $2\\mu_{\\mathrm{PD}}^2/\\nu_{\\mathrm{PD}}$ tend to infinity, and the coverage of the nominal 95\\% interval tends to $2\\Phi(z_{0.975}\\sqrt{15/16})-1=0.9423$.\n\n\\emph{Second example.} The blocks now have four nodes at the corners of a unit square, $(10b,0)$, $(10b+1,0)$, $(10b+1,1)$ and $(10b,1)$. I use disk averages with counting measure. Each node $i$ has outcome $2Z_j-1$, where $j$ is its horizontal neighbour; at the reflection of $j$ across $i$, the outcome is $1-2Z_i$. The two effects of $Z_i$ are both at distance one from $i$ and cancel in every disk centred on $i$, so every marginal effect is zero. Here the cutoff $h(0)=1$ equals the range of the dependence. The trimmed kernel has diagonal $5/4$. Both unadjusted estimators are asymptotically exact, and the adjusted one converges to $4/5$ of the variance of the estimator.\n\n\\begin{audittable}{lrrrrrr}{Limits in the two examples as $N\\to\\infty$. The last column is the limiting coverage of the nominal 95\\% interval with variance $\\widehat{V}_{\\mathrm{PD}}/\\mu_{\\mathrm{PD}}$.}\nExample & $\\mu_{\\mathrm{PD}}$ & $N\\operatorname{Var}(\\widehat{\\tau}_{\\mathrm{HA}})$ & $N\\widehat{V}_{\\mathrm{PD}}$ & $N\\widehat{V}_{\\mathrm{PD}}/\\mu_{\\mathrm{PD}}$ & Ratio & Coverage \\\\\nFive-node blocks, no interference & $6/5$ & $16/5$ & $18/5$ & $3$ & $15/16$ & $0.9423$ \\\\\nFour-node blocks, paired interference & $5/4$ & $8$ & $8$ & $32/5$ & $4/5$ & $0.9204$ \\\\\n\\end{audittable}\n\nThe ordering also fails in finite samples. In the first example with $N=20$ and a balanced assignment, $\\widehat{V}_{\\mathrm{PD}}=\\widehat{V}_{\\mathrm{HAC}}=4/25$, $\\mu=41/50$ and $\\mu_{\\mathrm{PD}}=101/100$, and the adjusted values are $16/101<8/41$. In the second with $N=16$, both variances equal $3/4$, $\\mu=13/16$ and $\\mu_{\\mathrm{PD}}=33/32$, and the adjusted values are $8/11<12/13$. The unchanged R and C++ functions in the archive reproduce the values of the second case. The missing factor of two in the coded degrees of freedom (\\fid{P004-E026}) plays no part here, because the stated and the coded degrees of freedom both tend to infinity.", "problem": "Section 5 shows that the positive semidefinite estimator is at least as large as the HAC estimator, and under the conditions of Proposition 4(ii) the unadjusted HAC interval has asymptotic coverage of at least $1-\\alpha$. The interval with the positive semidefinite variance $\\widehat{V}_{\\mathrm{PD}}$ and a normal critical value inherits this guarantee. The recommended interval is a different one. Section A.7 defines the adjustment for the uniform kernel matrix $\\Omega$ only. The replication code extends it to the positive semidefinite estimator: it computes $\\mu_{\\mathrm{PD}}$ and $\\nu_{\\mathrm{PD}}$ from the trimmed kernel $K^{\\mathrm{PD}}$ of Section 5 in place of $\\Omega$, and it returns $\\widehat{V}_{\\mathrm{PD}}/\\mu_{\\mathrm{PD}}$.\n\nThe diagonal entries of $\\Omega$ equal one, and in the examples below $\\mu$ tends to one. Trimming the negative eigenvalues adds a positive semidefinite matrix to the kernel, which raises its trace. In the examples below, $\\mu_{\\mathrm{PD}}$ tends to the average diagonal entry of $K^{\\mathrm{PD}}$, which is $6/5$ in one and $5/4$ in the other. Dividing by a factor above one does not inflate the variance. It deflates it, and the upward bias of $\\widehat{V}_{\\mathrm{PD}}$ need not make up the difference. The ordering $\\widehat{V}_{\\mathrm{PD}}/\\mu_{\\mathrm{PD}}\\ge\\widehat{V}_{\\mathrm{HAC}}/\\mu$ can then fail, and $N\\widehat{V}_{\\mathrm{PD}}/\\mu_{\\mathrm{PD}}$ can converge to a limit below the limit of $N\\operatorname{Var}(\\widehat{\\tau}_{\\mathrm{HA}})$. In the examples the $t$ critical value does not compensate, because the degrees of freedom tend to infinity.\n\nThe paper states no theorem for the adjusted estimator; this failure does not contradict Proposition 4 or the printed unadjusted inequality. What fails is the recommendation when it is read as a procedure valid under C1--C5. Figures 7 and 8 report this interval, and the captions of Figures 11 and 13 describe the intervals of the two applications in the same terms.", "what_survives": "Proposition 4, read with limits inferior of its coverage probabilities as in \\fid{P004-N024}, the inequality $\\widehat{\\Sigma}^{\\mathrm{PD}}_{\\mathrm{HAC}}(d)\\geq\\widehat{\\Sigma}_{\\mathrm{HAC}}(d)$ and footnote 15 are correct; they concern the unadjusted estimators. The definition and identification of the AME, the point estimators with their limit theorems, and the randomization test are unaffected. The examples do not show that the coverage plotted in Figures 7 and 8 is wrong for those designs, or that an interval in Figure 11 or 13 is too short, and they do not show that $\\mu_{\\mathrm{PD}}$ exceeds one in every design. What changes is the recommendation. As implemented, the adjusted positive semidefinite interval has no asymptotic guarantee under C1--C5. The capped version below has one, and it is wider wherever $\\mu_{\\mathrm{PD}}>1$." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-N004", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Five lines of body text. A red box marks the words: cutoff values (d-tilde = 2 km, 5 km and 10 km). A footnote mark 25 follows the parenthesis.", "caption": "Section 8.1, p. 763 (PDF p. 20): the values 2, 5 and 10 km are introduced as cutoff values and written $\\tilde{d}$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N005.png" }, { "alt": "Footnotes 24 and 25. In footnote 25 a red box marks the words: bandwidth at 2 d-tilde + 2d.", "caption": "Footnote 25, p. 763 (PDF p. 20): the bandwidth actually used is $2\\tilde{d}+2d$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N005-b.png" }, { "alt": "Three lines of body text. A red box marks the sentence: d-tilde is a cutoff value, and our setup suggests setting it at d-tilde = h(d).", "caption": "Section 5, p. 755 (PDF p. 12), after equation (19): $\\tilde{d}$ is defined as the cutoff of the kernel, with the suggested value $h(d)$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N005-c.png" } ], "formal_release": null, "formal_statements": [], "headline": "In Section 8 and in the legends of Figures 11 and 13, $\\tilde{d}$ and “cutoff” denote the assumed range of interference, not the cutoff of the HAC kernel that Section 5 calls $\\tilde{d}$; the kernel cutoff used at distance $d$ is twice the stated value plus $2d$.", "id": "P004-N005", "location": "Section 8.1, p. 763 (PDF p. 20), and footnote 25; Section 5, p. 755 (PDF p. 12), after equation (19); legends of Figures 11 and 13, pp. 764–765", "pdf_page": 20, "printed_page": "763", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "Section 5, p.~755: ``$\\tilde{d}$ is a cutoff value, and our setup suggests setting it at $\\tilde{d}=h(d)$.''\n\nSection 8.1, p.~763: ``we assess the robustness of inferences by considering a range of \\hl{cutoff values ($\\tilde{d}$ = 2 km, 5 km and 10 km)}.''\n\nFootnote 25: ``To be precise, for AME at distance value d, we set the \\hl{bandwidth at $2\\tilde{d}+2d$}, as discussed in Section 4.''\n\nLegends of Figures 11 and 13: cutoff=2km, cutoff=5km, cutoff=10km.", "corrected": "Section 8.1: ``we assess the robustness of inferences by considering a range of \\hl{assumed ranges of interference ($\\bar{d}$ = 2 km, 5 km and 10 km)}.''\n\nFootnote 25: ``To be precise, for AME at distance value $d$, we set the \\hl{kernel cutoff at $\\tilde{d}=2\\bar{d}+2d$}, as discussed in Section 4.''\n\nIn the legends and captions of Figures 11 and 13, and in the sentence of Section 8.2 that sets the cutoff value, write ``assumed interference range'' in place of ``cutoff''.", "intended_meaning": "Footnote 25 settles the meaning. The values 2, 5 and 10 km are assumed ranges of interference, the quantity written $\\bar{d}$ in Section 4, where $h(d)$ is set to $2\\bar{d}+2d$. The kernel cutoff of (19) at distance $d$ is then $4+2d$, $10+2d$ or $20+2d$ km. The application script in the replication archive computes its kernel cutoff in the same way, which supports this reading. No bandwidth or interval is wrong because of the notation." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-N005", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "recalculation_from_source", "exhibits": [ { "alt": "Four lines of body text and two footnotes. Red boxes mark the value -0.034, the words 10,000 random draws, and the p-value is around 0.065, and in footnote 24 the words a triangular kernel with a bandwidth of five km.", "caption": "Section 8.1, last paragraph of p. 763 (PDF p. 20), with footnotes 24 and 25. The marks show the reported statistic, the number of draws with the p-value, and the smoothing bandwidth.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N006.png" }, { "alt": "Paragraph headed Replication archive, with the DOI 10.1214/24-AOAS1986SUPPB. Two red boxes mark the words: All code and data to replicate simulation and empirical results are available in the Supplementary Material.", "caption": "Supplementary Material, p. 766 (PDF p. 23): the statement that the archive contains all code and data to replicate the empirical results.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N006-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "The supplied Uganda results and script do not reproduce the published cumulative statistic $-0.034$: the stored results give $-0.052$ over 0–5 km and $-0.039$ over 1–5 km, and the script differs from the article in the range of the statistic, the number of draws, the smoothing bandwidth and the cutoffs; the computation behind the published values needs to be identified and supplied.", "id": "P004-N006", "location": "Section 8.1, last paragraph of p. 763 (PDF p. 20), continuing on p. 764, and footnote 24; Supplementary Material statement, p. 766 (PDF p. 23); replication archive (Supplement B), Jayachandran_etal_2017.R, lines 116, 178, 193, 291 and 364, and forestResult_reproj.RData", "pdf_page": 20, "printed_page": "763", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Section 8.1, p.~763: ``The test statistic is the sum of the point Hajek estimates between zero and five km. The estimated value is \\hl{$-0.034$} (a 3.4 percentage point decline in deforested area). For the randomization test, we use \\hl{10,000 random draws}, and the p-value is around \\hl{0.065}.''\n\nFootnote 24: ``We use a triangular kernel with a bandwidth of \\hl{five km} to construct the smoothed Hajek estimator.''\n\nSupplementary Material, p.~766: ``Replication archive (DOI: 10.1214/24-AOAS1986SUPPB; .zip). All code and data to replicate simulation and \\hl{empirical results are available in the Supplementary Material} as well as on the following github repository: \\url{https://github.com/YeWang1576/SpatialEffectPackage}''\n\n\\texttt{Jayachandran\\_etal\\_2017.R}, line 116 and the call on line 193:\n\n\\texttt{nPerms <- \\hl{500}}\n\n\\texttt{test.result <- SpatialEffectTest(result.list, test.range = \\hl{c(1, 5)}, smooth = 0, alpha = 0.1)}", "corrected": "No replacement value is proposed. For the results of Section 8.1, the archive should contain:\n\\begin{itemize}\n\\item the processing of the outcome raster, the geometry and projection, the distance grid, the matrix of circle averages, the estimates and the saved result object that give \\hl{$-0.034$};\n\\item the randomization design, the tail convention, the seed and the package versions that give the reported p-value of \\hl{around $0.065$};\n\\item a script whose settings agree with the article: range \\hl{zero to five km}, \\hl{10,000} draws, smoothing bandwidth \\hl{five km}, cutoff values \\hl{2, 5 and 10 km}.\n\\end{itemize}\nThe published numbers should not be replaced by the stored ones before the final computation is identified and checked.", "evidence": "The evidence is a recalculation from the archive in exact arithmetic. I read the stored circle averages and the six stored estimates as binary fractions and computed (16) again from the 726 circle averages. Each recomputed estimate agrees with the stored one to within $10^{-16}$.\n\n\\begin{audittable}{rrrr}{Stored results of the Uganda application: mean circle average in control and treated villages, and the Hajek estimate (16).}\nDistance (km) & Control & Treated & Estimate \\\\\n0 & $0.02608$ & $0.01263$ & $-0.01344$ \\\\\n1 & $0.02353$ & $0.01566$ & $-0.00786$ \\\\\n2 & $0.02283$ & $0.01472$ & $-0.00811$ \\\\\n3 & $0.02352$ & $0.01389$ & $-0.00963$ \\\\\n4 & $0.02181$ & $0.01541$ & $-0.00640$ \\\\\n5 & $0.02308$ & $0.01610$ & $-0.00698$ \\\\\nSum, 0--5 km & & & $-0.05242$ \\\\\nSum, 1--5 km & & & $-0.03898$ \\\\\nPublished & & & $-0.034$ \\\\\n\\end{audittable}\n\n\\begin{audittable}{lll}{Settings of Section 8.1 in the article and in the script of the archive.}\nSetting & Article & Script \\\\\nRange of the test statistic & zero to five km & \\texttt{test.range = c(1, 5)} \\\\\nRandom draws & 10,000 & \\texttt{nPerms <- 500} \\\\\nSmoothing bandwidth & five km & \\texttt{bw = 10.996} and \\texttt{bw = 18.707} \\\\\nCutoff values & 2, 5 and 10 km & \\texttt{seq(2, 10, 2)} \\\\\n\\end{audittable}\n\nAs a further check, I held the stored outcomes fixed and drew 10,000 assignments of 60 treated villages among the 121. The absolute value of the sum over 0--5 km is at least the observed one in 626 draws, and over 1--5 km in 857. These are counts for one Monte Carlo sample.\n\nSeparately, I recalculated the circle averages from the supplied outcome raster using the authors' functions, with changes to the supplied script. The estimates over 0--5 km sum to $-0.0427592$, with a two-sided Monte Carlo fraction of $0.0928$ from 10,000 draws (Monte Carlo standard error $0.00290$). This calculation uses newly computed circle averages, rather than the stored result object above. All eight selected unsmoothed and smoothed point estimates agree with Figure 11 within the stated graph-reading tolerances; selected interval endpoints were also compared. At 0 km the lower endpoint using the 5 km cutoff and the degrees of freedom of A.7 is $-0.027219$, against approximately $-0.023$ in the figure, a difference just beyond the graph-reading tolerance of $0.004$. These comparisons do not establish an exact discrepancy in an endpoint that the figure does not print numerically.", "problem": "The archive contains the result object that the Uganda script saves and then loads. It holds the circle averages of 121 villages, 60 treated and 61 control, at distances $0,1,\\dots,30$ km, and the Hajek estimates computed from them. The six stored estimates for 0 to 5 km sum to $-0.0524$. Over 1 to 5 km, the range that the script passes to the test, they sum to $-0.0390$. Both sums are below $-0.0345$, so neither rounds to $-0.034$. The script also differs from the article in four settings, listed in the second table. These stored results and script settings do not reproduce the published calculation.\n\nWhich computation produced $-0.034$ and the p-value around $0.065$ is not identified. I do not conclude that the published values are wrong, because nothing ties the stored object to the final computation.", "what_survives": "Equation (16), the estimator and the procedure of the randomization test are not in question. The statistic from the stored results is still negative, and the Monte Carlo fraction for 0--5 km is $0.0626$, below $0.10$. This calculation gives no reversal of the reported rejection at the 10\\% level. The published statistic $-0.034$ and p-value around $0.065$ are not shown to be wrong. They are not reproduced by the supplied Uganda results and script. The simulations and the Costa Rica application are not affected by this finding. The interpretation of the statistic is a separate matter (\\fid{P004-E008})." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-N006", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Five lines of body text and two footnotes. Red boxes mark the sentence: For the randomization test, we use 10,000 random draws, and the p-value is around 0.065.", "caption": "Section 8.1, last paragraph of p. 763 (PDF p. 20): the randomization test and its p-value.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N007.png" }, { "alt": "Two lines of body text and footnote 3. Red boxes mark the sentence: Under polygon intervention, d_ij = min over x in i and x-prime in j of gamma(x, x-prime).", "caption": "Footnote 3, p. 751 (PDF p. 8): the distance between two polygons is the minimal distance between their points.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N007-b.png" }, { "alt": "Equation (45): nu equals 2 times the trace of the square of the matrix (N1 N0 / N) M (Omega circ lambda-prime lambda) M, with a red box around the factor 2 and the star after it. Below it, the sentence that sets the degrees of freedom of the t-distribution, with a red box around the fraction 2 mu squared over v.", "caption": "Supplement A, Section A.7, p. 17: equation (45) and the stated degrees of freedom, $2\\mu^2/v$ with $v$ the $\\nu$ of (45).", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N007-c.png" } ], "formal_release": null, "formal_statements": [], "headline": "The Uganda script in the replication archive draws its randomization distribution by permuting treatment with a fixed count and no blocks, which is not the lottery within subcounties among covariate-balanced lists that assigned the villages; the same script measures HAC distances between centroids where footnote 3 prescribes minimal distances between polygons, and it uses twice the degrees of freedom of Supplement A.7.", "id": "P004-N007", "location": "Section 8.1, last paragraph of p. 763 (PDF p. 20), and Figure 11, p. 764; footnote 3, p. 751 (PDF p. 8); Supplement A, Section A.7, equation (45) and the sentence after it, p. 17; replication archive (Supplement B), application/scripts/Jayachandran_etal_2017.R, lines 111, 170–173, 286–289 and 339–342", "pdf_page": 20, "printed_page": "763", "reaches": [ { "effect": "result_needs_same_repair", "effect_text": "a result or code path there needs the same repair", "explanation": null, "package": null, "recorded_as": "P008-N012", "to": "work", "url": "https://pmaronow.github.io/llm-audit/works/P008/", "where": null, "work_id": "P008" }, { "effect": "implementation_wrong", "effect_text": "the package or replication code has the reported defect", "explanation": null, "package": "SpatialEffect", "recorded_as": null, "to": "package", "url": null, "where": null, "work_id": "P004" } ], "related_entries": [], "revisions": [], "same_defect_as": [ { "id": "P008-N012", "url": "https://pmaronow.github.io/llm-audit/works/P008/#P008-N012", "work_id": "P008" } ], "severity": "Moderate", "subject": "paper", "target_ids": [], "text": { "as_printed": "Section 8.1, p.~763: ``For the randomization test, we use 10,000 random draws, and the p-value is around 0.065.''\n\nFootnote 3, p.~751: ``As before, under point intervention, $d_{ij}=\\gamma(x(i),x(j))$. Under polygon intervention, $d_{ij}=\\min_{x\\in i,x'\\in j}\\gamma(x,x')$.''\n\nSupplement A, Section A.7, p.~17: equation (45),\n\\[\n\\nu = 2 * \\mathbf{Trace}\\left(\\left(\\frac{N_1N_0}{N}\\mathbf{M}\\left(\\Omega\\circ(\\lambda'\\lambda)\\right)\\mathbf{M}\\right)\\left(\\frac{N_1N_0}{N}\\mathbf{M}\\left(\\Omega\\circ(\\lambda'\\lambda)\\right)\\mathbf{M}\\right)\\right).\n\\]\n``We use the $\\frac{\\alpha}{2}$-quantile and $\\left(1-\\frac{\\alpha}{2}\\right)$-quantile of the t-distribution with $\\frac{2\\mu^2}{v}$ degree of freedom to construct the confidence interval.''\n\n\\texttt{Jayachandran\\_etal\\_2017.R}, lines 111, 170, 286 and 341--342:\n\n\\texttt{blockvar <- \\hl{NULL}}\n\n\\texttt{centr <- st\\_coordinates(\\hl{st\\_centroid}(map))}\n\n\\texttt{Zmat <- Zdata[, c(1:2)]}\n\n$\\texttt{nu <- (nz/((nz-2)*XX\\_mat\\_inv[2, 2]))\\textasciicircum 2*sum(diag(dof\\_mat \\%*\\% dof\\_mat))}$\n\n$\\texttt{cvs[d] <- (}\\hl{\\texttt{2*}}\\texttt{dofs[d]\\textasciicircum 2) / nu}$", "corrected": "\\begin{itemize}\n\\item Draw reference assignments from the trial's lottery: in each of the seven subcounties, \\hl{choose with equal probability between its two original balanced lists}, conditioning on those paired lists where available; otherwise reproduce their original selection procedure before the lottery. Setting \\texttt{blockvar} to the subcounty alone does not reproduce this lottery.\n\\item In the loop over cutoffs, take the distances between villages as \\hl{minimal distances between polygons}, as in footnote 3, for the HAC kernel and for the degrees of freedom.\n\\item Replace line 342 by $\\texttt{cvs[d] <- }\\hl{\\texttt{dofs[d]\\textasciicircum 2}}\\texttt{ / nu}$.\n\\item Compute the intervals and the p-value again before any published number is revised.\n\\end{itemize}", "evidence": "For the reference distribution, the evidence is a check of the trial's published design against the code. The article of Jayachandran et al.\\ (2017) also states that the randomization was stratified by subcounty.\n\nFor the other two parts, the evidence is a pair of exact examples and a recalculation from the data in the archive. I take six nodes with $Z=(0,1,0,1,0,1)$ and $Y=(0,2,1,1,2,0)$ and an identity kernel. By A.7, I have $4$ degrees of freedom, and the expressions in the script return $8$. Next, I keep this assignment and these outcomes, place six rectangles of width 3 and height 1 on a line, with centres at $0,4,12,20,28,36$, and take a cutoff of 2. The first two rectangles are at distance $1$ as polygons and $4$ as centroids, and no other pair is within the cutoff. The unadjusted HAC variance is $2/3$ with polygon distances and $4/9$ with centroid distances.\n\nFor the Uganda data at $d=0$, the script's calculation, done again from the data, reproduces the five stored pairs of interval endpoints to within $10^{-12}$. The first table compares the script's degrees of freedom with those of A.7 and lists the half-widths at the stored standard errors. At the largest cutoff the upper endpoint of the stored interval moves from $-0.00037$ to $+0.00051$, so that this interval contains zero after the change. The second table counts the pairs of villages within the kernel cutoff. The cutoffs of the first table are those of the script, not those of Figure 11.\n\n\\begin{audittable}{rrrrrr}{Uganda data, $d=0$, stored estimate $-0.01344$: degrees of freedom and half-widths of the 90\\% interval at the stored standard errors.}\nNominal cutoff (km) & df, script & df, A.7 & Half-width, script & Half-width, A.7 & Ratio \\\\\n2 & $77.48$ & $38.74$ & $0.01144$ & $0.01158$ & $1.012$ \\\\\n4 & $39.28$ & $19.64$ & $0.01480$ & $0.01516$ & $1.025$ \\\\\n6 & $26.51$ & $13.26$ & $0.01555$ & $0.01613$ & $1.038$ \\\\\n8 & $18.05$ & $9.03$ & $0.01443$ & $0.01526$ & $1.057$ \\\\\n10 & $15.58$ & $7.79$ & $0.01307$ & $0.01395$ & $1.067$ \\\\\n\\end{audittable}\n\n\\begin{audittable}{rrr}{Pairs of distinct villages within the kernel cutoff at $d=0$, out of 7,260 pairs, by the script's ellipsoidal centroid distances and by minimal spherical distances between polygons.}\nKernel cutoff (km) & By centroid distance & By polygon distance \\\\\n4 & 309 & 837 \\\\\n10 & 1,480 & 2,154 \\\\\n20 & 3,378 & 3,902 \\\\\n\\end{audittable}", "problem": "I find three discrepancies in the script that the archive offers for Section 8.1.\n\n\\emph{The reference distribution.} The test of Section 6.3 compares the observed statistic with its distribution over assignments drawn from the design. The script draws assignments uniformly among those with 60 treated villages out of 121, with no blocks. Note that fixing the number treated is not in itself an error: under Bernoulli assignment with a common probability, the assignment is uniform given its count. The trial was not assigned in that way. Its supplementary materials (Jayachandran et al., 2017, \\href{https://seemajayachandran.com/supplementary_material_pes.pdf}{Section 2.2}) describe a public lottery in each of seven subcounties. The villages of a subcounty were divided into two lists of nearly equal size, and only divisions balanced on four covariates were admitted: ``For each subcounty, among the balanced permutations, we randomly chose one to use in the public lottery.'' The lottery then selected the list to be treated. Permutations without blocks ignore both the subcounties and the balance restriction. The reference distribution of the script is not that of the trial, and the p-value it yields is not shown to be valid for the trial's design.\n\n\\emph{The distances.} Footnote 3 defines the distance between two polygons as the minimal distance between their points. The script computes these distances (lines 85--88) and then rebuilds the table of nodes \\texttt{Zdata} from the centroids of the polygons (lines 170--173). The loop that computes the HAC intervals takes its distances from these centroids (lines 286--289). In the Uganda data, hundreds of pairs of villages that are within the kernel cutoff by polygon distance are beyond it by centroid distance.\n\n\\emph{The degrees of freedom.} Section A.7 prescribes $2\\mu^2/\\nu$ with $\\nu$ equal to twice the trace of the squared matrix, that is, the squared trace divided by the trace of the square. The script computes \\texttt{nu} without the factor 2 and then sets \\texttt{cvs} to $2\\,\\texttt{dofs}^2/\\texttt{nu}$. The scalar normalization cancels in this ratio, and the result is twice the stated value, with or without the positive semidefinite trimming. More degrees of freedom give a smaller $t$ critical value and a narrower interval.\n\nIt is not established that this script produced Figure 11 or the p-value $0.065$. The stored results in the archive differ from the published ones (\\fid{P004-N006}).", "what_survives": "The polygon-distance rule in footnote 3 and the degrees-of-freedom formula in Section A.7 are correct; the estimand, the estimators and the theory under its stated assumptions are unaffected. Correcting the distances and the degrees of freedom changes neither the test statistic nor its reference distribution, and correcting the sampler does not change the intervals. The published p-value and the conclusion of no substantial leakage are not shown to be wrong: the computation behind the published numbers is not identified, and the direction of any change in those results is unknown. The theory in the paper assumes Bernoulli assignment. I have not examined how it extends to the restricted design of the trial." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-N007", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Six lines of body text. Red boxes mark the two sentences: To construct the circle averages, we generate buffers around each of the village polygons. The distance range for estimating the AME is set to run from zero km to 15 km.", "caption": "Section 8.1, p. 763 (PDF p. 20): the buffers around the village polygons and the range of distances, in km.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N009.png" }, { "alt": "Figure 11 with its caption. Two panels, Hajek Estimator on the left and Smoothed Hajek Estimator (Triangular Kernel, bw=5km) on the right, each showing an effect curve near zero with upper and lower interval curves for cutoffs of 2, 5 and 10 km, over distances from 0 to about 14 km. Red boxes mark the label Distance (km) under each horizontal axis.", "caption": "Figure 11, p. 764 (PDF p. 21). Both panels plot the estimates against distance in km; the marks show the two axis labels.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N009-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "The Uganda script in the replication archive projects the raster and the village polygons with a UTM zone of the central United States and then sends projected coordinates to a geodesic routine; its buffers of one unit are not buffers of one kilometre on the ground, and a projection for the study area with Euclidean distances, or geographic coordinates with geodesic distances, is needed.", "id": "P004-N009", "location": "Section 8.1, p. 763 (PDF p. 20), and Figure 11, p. 764 (PDF p. 21); replication archive (Supplement B), Jayachandran_etal_2017.R, lines 117 and 166–168, and SpatialEffect.R, lines 23–41 and 168–176", "pdf_page": 20, "printed_page": "763", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [], "text": { "as_printed": "Section 8.1, p.~763: ``To construct the circle averages, we generate buffers around each of the village polygons. The distance range for estimating the AME is set to run from zero km to 15 km.'' Axes of Figure 11, p.~764: ``Distance (km)''.\n\n\\texttt{Jayachandran\\_etal\\_2017.R}, lines 117 and 166--168:\n\n\\texttt{dist.metric <- \\hl{\"Geodesic\"}}\n\n\\texttt{sr <- \"+proj=utm \\hl{+zone=15 +ellps=GRS80 +datum=NAD83} +units=km +no\\_defs\"}\n\n\\texttt{ras <- projectRaster(ras, crs = sr)}\n\n\\texttt{ras\\_Z <- spTransform(ras\\_Z, crs(ras))}", "corrected": "\\texttt{sr <- \"+proj=utm \\hl{+zone=36 +datum=WGS84} +units=km +no\\_defs\"}\n\n\\texttt{dist.metric <- \\hl{\"Euclidean\"}}\n\nTransform the raster and the polygons with the same projection, using transformations appropriate to their classes, and compute distances and buffers in the projected kilometres. \\hl{This is a local approximation to ground distances.} The alternative is to keep geographic coordinates with geodesic distances and to bring the buffer widths (in metres), the distances and the cutoffs to the same unit. The separate script/package interface errors also need repair (\\fid{P004-E029}). Any result obtained from the buffers of the supplied script should be computed again.", "evidence": "I checked the projection's returned coordinates by exact comparisons, with an analytic upper bound for distances on the ground, after bypassing the separate execution errors; I did not reproduce the estimates. The first centroid is sent to the routine as about $(8019, 19696)$, and all 121 fail its bounds of 360 and 90. The table lists the two comparisons of distances. The first uses the points at longitudes $31.2^\\circ$ and $31.21^\\circ$ E on the parallel $1.2^\\circ$ N. With a projection for UTM zone 36 the same two points are $1.113$ units apart, and their geodesic distance is $1.113$ km. The second uses a boundary vertex of the first village and the point $0.006^\\circ$ west of it on the same parallel.\n\n\\begin{audittable}{lll}{Distances on the ground and in the projected coordinates of the script.}\nPair & On the ground & In the script's projection \\\\\nTwo points $0.01^\\circ$ of longitude apart & less than $1.135$ km & more than $1.99$ units \\\\\nA point and the first village & less than $0.681$ km & more than $1$ unit (outside the buffer) \\\\\n\\end{audittable}", "problem": "The villages lie near $31^\\circ$ E and $1^\\circ$ N, in UTM zone 36. The script projects the outcome raster and the polygons to UTM zone 15, which covers the central United States, with kilometre units. So far from its own zone the projection distorts lengths, and the package draws its buffers in the projected coordinates. A buffer of width one is then not a buffer of one km on the ground. Two points of the study area that are less than $1.135$ km apart on the ground are more than $1.99$ units apart after projection, a ratio above $7/4$. A point less than $0.681$ km from the boundary of the first village lies outside that village's buffer of width one.\n\nThe script also selects geodesic distances. The package then takes the centroids of the projected polygons and passes their coordinates, in projected kilometres, to a routine that expects longitude and latitude in degrees. All 121 centroids have a first coordinate above 360, and the routine stops with the error \\texttt{longitude > 360}. This path cannot produce HAC distances. The later block of the script, which computes the intervals for the varying cutoffs, uses the original geographic coordinates and valid geodesic distances, and this defect does not reach it.\n\nNo dependence of Figure 11 on this path is established. If the published analysis used these buffers, every estimate at a nonzero distance and the cumulative statistic refer to regions other than those stated.", "what_survives": "These code defects do not refute the definitions of distance or the statistical theory. The region at distance zero is the village itself and needs no positive buffer width; I have not verified its published numerical estimate. Figure 11, the cumulative statistic, its p-value and the conclusion about leakage are not shown to depend on this path: the script as supplied does not run against the package supplied with it (\\fid{P004-E029}), and the code that produced Figure 11 is not identified. What changes is that the archive cannot confirm the distance scale of Figure 11." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-N009", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of five lines of text that begin: study. The distance range is set to run from zero km and 20 km. A red box marks the word and, between zero km and 20 km.", "caption": "Section 8.2, p. 765 (PDF p. 22), the first lines of text below the caption of Figure 12. The marked word should be “to”.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N011.png" }, { "alt": "Crop of Figure 13 with its caption. Two panels, titled Hajek Estimator and Smoothed Hajek Estimator (Triangular Kernel, bw=5km), plot an effect curve and interval curves against distance in km. Red boxes mark the horizontal axis of each panel, with the tick labels 0, 5 and 10 and the label Distance (km). The curves stop a little before the right edge of each panel, near 14 km.", "caption": "Figure 13, p. 765 (PDF p. 22). The marked horizontal axes carry ticks at 0, 5 and 10 km. The curves in both panels end about four units beyond the tick at 10 km, short of the 20 km stated in the text.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N011-b.png" }, { "alt": "Crop of Figure 11 with its caption. Two panels, titled Hajek Estimator and Smoothed Hajek Estimator (Triangular Kernel, bw=5km), plot an effect curve and interval curves against distance in km. Red boxes mark the horizontal axis of each panel, with the tick labels 0, 5 and 10 and the label Distance (km). The curves stop a little before the right edge of each panel, near 14 km.", "caption": "Figure 11, p. 764 (PDF p. 21). The marked horizontal axes carry ticks at 0, 5 and 10 km. The curves in both panels end about four units beyond the tick at 10 km, short of the 15 km stated in Section 8.1.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N011-c.png" } ], "formal_release": null, "formal_statements": [], "headline": "In Section 8.2, “from zero km and 20 km” should read “from zero km to 20 km”, and Figures 11 and 13 display estimates only up to about 14 km, a shorter range than the 15 km and 20 km stated in the text.", "id": "P004-N011", "location": "Section 8.2, p. 765 (PDF p. 22), first line of text; Figure 13, p. 765; Figure 11, p. 764 (PDF p. 21); compare Section 8.1, p. 763 (PDF p. 20)", "pdf_page": 22, "printed_page": "765", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Section 8.2, p.~765: ``The distance range is set to run from zero km \\hl{and} 20 km.''\n\nSection 8.1, p.~763: ``The distance range for estimating the AME is set to run from zero km to 15 km.''\n\nFigures 11 and 13: the horizontal axes carry ticks at 0, 5 and 10 km, and the curves end at about 14 km.", "corrected": "Section 8.2, p.~765: ``The distance range is set to run from zero km \\hl{to} 20 km.''\n\nCaptions of Figures 11 and 13, added sentence: ``\\hl{Displayed estimates cover distances from 0 to about 14 km.}''", "evidence": "The evidence is a check of pp.~763--765. In each panel the curves end about four units beyond the tick at 10 km, and no label is printed at the endpoint. The Costa Rica script in the replication files sets the distances to run from 0 to 20 km in steps of 500 m, which agrees with the text of Section 8.2. The Uganda script in the same files sets \\texttt{len=30}, which does not confirm the range of 15 km stated in Section 8.1. I recomputed no estimate.", "problem": "The word ``and'' is a slip for ``to''. The two sentences give the range of distances at which the AME is estimated, 15 km in Section 8.1 and 20 km in Section 8.2. Both figures show curves that end near 14 km, and neither caption says that the display is shorter than the range given in the text.", "what_survives": "The stated ranges of estimation are not shown to be wrong, and nothing displayed in the two figures changes. The shorter display does not by itself establish an error in the statements about nearby areas and about leakage." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-N011", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Three displayed lines of a proof. The first is an upper bound whose first term has the coefficient 1 over N squared p. After the words We also have, the next display repeats the expression with a third term subtracted; a red box marks the coefficient of its first term, 1 over N p, and a second red box marks the right-hand side, O(1). The text continues: since C2 implies that all the moments are bounded.", "caption": "Supplement A, p. 3, proof of Lemma A.2: the bound that ends the expansion of the diagonal terms and the display after “We also have:”. The marked coefficient reads $1/(Np)$ where the line above has $1/(N^2p)$, and the marked rate reads $O(1)$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N013.png" }, { "alt": "A displayed difference of two sums over pairs i and j with j in the neighbourhood B(i;d) and j different from i, each divided by N squared. A red box marks the right-hand side, O(1). The text continues: since the size of B(i;d) is bounded by (C4a). Other covariance terms have similar forms.", "caption": "Supplement A, p. 3, end of the proof of Lemma A.2: the display after “Moreover,”. The marked rate reads $O(1)$.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N013-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T32", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T32" } ], "headline": "In the proof of Lemma A.2, the first term of the display after “We also have:” should carry the factor $1/(N^2p)$, and that display and the one after “Moreover,” should conclude $O(1/N)$, the rate that the lemma states.", "id": "P004-N013", "location": "Supplement A, proof of Lemma A.2, p. 3, the displays after “We also have:” and after “Moreover,”", "pdf_page": 3, "printed_page": "3", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P004-T32" ], "text": { "as_printed": "We also have:\n\\[\n\\frac{1}{\\hl{Np}}\\sum_{i=1}^N \\mathrm{E}\\left[\\mu_i(\\mathbf{1};d)^2\\right]+\\frac{1}{N^2(1-p)}\\sum_{i=1}^N \\mathrm{E}\\left[\\mu_i(\\mathbf{0};d)^2\\right]-\\frac{1}{N^2}\\sum_{i=1}^N\\left(\\mathrm{E}\\left[\\mu_i(\\mathbf{1};d)-\\mu_i(\\mathbf{0};d)\\right]\\right)^2=\\hl{O(1)},\n\\]\nsince C2 implies that all the moments are bounded. [\\ldots] Moreover,\n\\[\n\\frac{1}{N^2}\\sum_{i;j\\in\\mathcal{B}(i;d),j\\ne i}\\mathrm{E}\\left[\\mu_i(\\mathbf{1}_{\\mathbf{i}},\\mathbf{1}_{\\mathbf{j}};d)\\mu_j(\\mathbf{1}_{\\mathbf{i}},\\mathbf{1}_{\\mathbf{j}};d)\\right]-\\frac{1}{N^2}\\sum_{i;j\\in\\mathcal{B}(i;d),j\\ne i}\\mathrm{E}\\left[\\mu_i(\\mathbf{1};d)\\right]\\mathrm{E}\\left[\\mu_j(\\mathbf{1};d)\\right]=\\hl{O(1)},\n\\]\nsince $|\\mathcal{B}(i;d)|$ is bounded by (C4a).", "corrected": "We also have:\n\\[\n\\frac{1}{\\hl{N^2p}}\\sum_{i=1}^N \\mathrm{E}\\left[\\mu_i(\\mathbf{1};d)^2\\right]+\\frac{1}{N^2(1-p)}\\sum_{i=1}^N \\mathrm{E}\\left[\\mu_i(\\mathbf{0};d)^2\\right]-\\frac{1}{N^2}\\sum_{i=1}^N\\left(\\mathrm{E}\\left[\\mu_i(\\mathbf{1};d)-\\mu_i(\\mathbf{0};d)\\right]\\right)^2=\\hl{O\\left(\\frac{1}{N}\\right)},\n\\]\nsince C2 implies that all the moments are bounded. [\\ldots] Moreover,\n\\[\n\\frac{1}{N^2}\\sum_{i;j\\in\\mathcal{B}(i;d),j\\ne i}\\mathrm{E}\\left[\\mu_i(\\mathbf{1}_{\\mathbf{i}},\\mathbf{1}_{\\mathbf{j}};d)\\mu_j(\\mathbf{1}_{\\mathbf{i}},\\mathbf{1}_{\\mathbf{j}};d)\\right]-\\frac{1}{N^2}\\sum_{i;j\\in\\mathcal{B}(i;d),j\\ne i}\\mathrm{E}\\left[\\mu_i(\\mathbf{1};d)\\right]\\mathrm{E}\\left[\\mu_j(\\mathbf{1};d)\\right]=\\hl{O\\left(\\frac{1}{N}\\right)},\n\\]\nsince $|\\mathcal{B}(i;d)|$ is bounded by (C4a).", "evidence": "The evidence is a derivation under the conditions of the lemma. By C2, there exists a constant $b$ such that $|\\mu_i(\\cdot\\,;d)|\\le b$. The corrected first expression is a sum of $N$ variances divided by $N^2$. It is nonnegative, and since its last term is subtracted I have\n\\[\n0\\le\\frac{1}{N^2p}\\sum_{i=1}^N \\mathrm{E}\\left[\\mu_i(\\mathbf{1};d)^2\\right]+\\frac{1}{N^2(1-p)}\\sum_{i=1}^N \\mathrm{E}\\left[\\mu_i(\\mathbf{0};d)^2\\right]-\\frac{1}{N^2}\\sum_{i=1}^N\\left(\\mathrm{E}\\left[\\mu_i(\\mathbf{1};d)-\\mu_i(\\mathbf{0};d)\\right]\\right)^2\\le\\frac{b^2}{Np}+\\frac{b^2}{N(1-p)}.\n\\]\nBy C4A, there exists a constant $c$ such that $|\\mathcal{B}(i;d)|\\le c$. Each sum in the second display then has at most $N(c-1)$ terms, and each term is bounded by $b^2$ in absolute value. The difference of the two sums, divided by $N^2$, is at most $2b^2(c-1)/N$ in absolute value. Both quantities are $O(1/N)$.", "problem": "Lemma A.2 asserts that $\\operatorname{Var}(\\widehat\\tau_{\\mathrm{HT}}(d))=O(1/N)$ under C1--C4. The proof expands the diagonal part of the variance, and both the exact expression and the bound printed just above ``We also have:'' carry $1/(N^2p)$ on the treated sum. The next display repeats the expression with $1/(Np)$. Neither printed statement is false. With the factor $1/(Np)$ the first expression is of order one and no smaller: if $\\mu_i(\\mathbf{1};d)=\\mu_i(\\mathbf{0};d)=1$ for every $i$, it equals $1/p+1/(N(1-p))$. But it is then not the diagonal part of the variance, and a bound of order one on each part does not imply the rate that the lemma states. Once the factor is restored, both displays are $O(1/N)$. Replacing only the first $O(1)$ by $O(1/N)$ would make that display false.", "what_survives": "Lemma A.2, its variance bound and its conclusion $O(1/N)$ under C1--C4 are not affected, and neither is any later use of the lemma. The exact expansion printed above the first display is correct. Only the two displays that justify the rate change." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-N013", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "The last display of a proof, a double sum that is shown to be at least zero, followed by the sentence: by C5. Hence the term combining expressions in (2) and (5) is non-positive in the limit under C5, which proves the lemma. A red box marks (2).", "caption": "Supplement A, p. 5, last lines of the proof of Lemma A.4. The marked reference reads (2); the display above it combines lines (3) and (5).", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N014.png" }, { "alt": "A paragraph of text. A red box marks the words Proposition A.4 in the sentence: The Hajek estimator's asymptotic distribution can be then obtained via the linear expansion in Proposition A.4 and a similar application of the CLT result.", "caption": "Supplement A, p. 6, Section A.3, first paragraph. The marked reference reads “Proposition A.4”; the linear expansion is Lemma A.3.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N014-b.png" }, { "alt": "A paragraph of text. Red boxes mark the words Lemma, at the end of one line, and A.3, at the start of the next, in the sentence: we show that the HAC variance estimator is consistent for the variance bound defined in Lemma A.3 and lines (6)-(7).", "caption": "Supplement A, p. 6, Section A.4.1, first paragraph. The marked reference reads “Lemma A.3”; lines (6)–(7) belong to Lemma A.4.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N014-c.png" }, { "alt": "Three lines of text. A red box marks the words Proposition A.4 in the sentence: The characterization of the variance is similar to that in Proposition A.4.", "caption": "Supplement A, p. 12, first line of the proof of Proposition 3. The marked reference reads “Proposition A.4”.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N014-d.png" }, { "alt": "Text and one display. A red box marks (6) in the sentence: We can similarly define a HAC type estimator as in (6). The display defines a HAC estimator as a sandwich of the inverse of X prime X around a double sum of X_i X_j prime times two residuals and a neighbourhood indicator. A second red box marks (6) in the sentence: The formula is similar to (6), except now that the residuals are weighted with the inverse propensity scores.", "caption": "Supplement A, p. 15, the observational HAC estimator. Both marked references read (6); the estimator displayed between them has the form of equation (19) of the article.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N014-e.png" }, { "alt": "Five lines of text. A red box marks (5) in the sentence: Recall the regression interpretation of the Hajek estimator discussed in (5) and the HAC variance estimator in (19).", "caption": "Supplement A, p. 17, Section A.7, second paragraph. The marked reference reads (5); the regression form of the Hajek estimator is equation (18) of the article.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N014-f.png" } ], "formal_release": null, "formal_statements": [], "headline": "Seven cross-references in Supplement A name the wrong item; each should name the lemma or equation listed in the correction.", "id": "P004-N014", "location": "Supplement A: end of the proof of Lemma A.4, p. 5; first paragraphs of Sections A.3 and A.4.1, p. 6; proof of Proposition 3, p. 12; the text around the observational HAC estimator, p. 15; Section A.7, p. 17.", "pdf_page": 5, "printed_page": "5", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\begin{itemize}\n\\item p.~5: ``Hence the term combining expressions in \\hl{(2)} and (5) is non-positive in the limit under C5''\n\\item p.~6: ``via the linear expansion in \\hl{Proposition A.4}''\n\\item p.~6: ``the variance bound defined in \\hl{Lemma A.3} and lines (6)-(7)''\n\\item p.~12: ``The characterization of the variance is similar to that in \\hl{Proposition A.4}.''\n\\item p.~15: ``We can similarly define a HAC type estimator as in \\hl{(6)}.'' and ``The formula is similar to \\hl{(6)}''\n\\item p.~17: ``Recall the regression interpretation of the Hajek estimator discussed in \\hl{(5)}''\n\\end{itemize}", "corrected": "\\begin{audittable}{lll}{The seven references of Supplement A, as printed and corrected.}\nPlace & As printed & Corrected \\\\\np.~5, end of the proof of Lemma A.4 & (2) and (5) & \\hl{(3)} and (5) \\\\\np.~6, Section A.3 & the linear expansion in Proposition A.4 & the linear expansion in \\hl{Lemma A.3} \\\\\np.~6, Section A.4.1 & Lemma A.3 and lines (6)-(7) & \\hl{Lemma A.4} and lines (6)-(7) \\\\\np.~12, proof of Proposition 3 & similar to that in Proposition A.4 & similar to that in \\hl{Lemma A.4} \\\\\np.~15, first sentence & as in (6) & as in \\hl{(19) of the article} \\\\\np.~15, after the display & similar to (6) & similar to \\hl{(19) of the article} \\\\\np.~17, Section A.7 & discussed in (5) & discussed in \\hl{(18) of the article} \\\\\n\\end{audittable}", "evidence": "The evidence is a check of the pages. I compared each reference with the item that it names and with the item that matches the description in the sentence, on pp.~4--6, 12, 15 and 17 of Supplement A and on pp.~754--755 of the article.", "problem": "On p.~5 the proof combines lines (3) and (5) a few lines earlier; line (2) is the positive sum of second moments on the diagonal. The linear expansion of the Hajek estimator is Lemma A.3, its variance and the bound in lines (6)--(7) are Lemma A.4, and the supplement has no Proposition A.4. The spatial HAC estimator is equation (19) of the article. In the supplement, (6) is the variance bound of Lemma A.4. The regression form of the Hajek estimator is equation (18) of the article; equation (5) there defines the marginalised circle average. Each intended item can be identified from its content.", "what_survives": "Replacing these labels leaves the arguments and formulas unchanged." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-N014", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the variance bound of Lemma A.4, lines (6) and (7), with the sentence that introduces it and the first two lines of the proof. Red boxes mark the fragment '; d)' that follows mu_i(0; d) in the control term of line (6), and the fragment '; d)' that follows mu_j(a_i, b_j; d) in the second factor of line (7).", "caption": "Supplement A, p. 5, lines (6)–(7) of Lemma A.4. The two marked fragments “$;d)$” repeat the distance argument after the outcome argument is already closed.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N015.png" }, { "alt": "Crop of the line 'Proof. We first prove the case for the HT estimator. Define U_i as' and the display that defines U_i. A red box marks the first 'mu_i(' in the numerator (1 - Z_i) mu_i( mu_i(0; d) of the second fraction.", "caption": "Supplement A, p. 6, the definition of $U_i$ in the proof of Propositions 2 and 3. The marked “$\\mu_i($” is the stray fragment in the control term.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N015-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T33", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T33" } ], "headline": "Lines (6) and (7) each carry a stray “$;d)$”, and the control term of $U_i$ carries a stray “$\\mu_i($”.", "id": "P004-N015", "location": "Supplement A, Lemma A.4, p. 5 (supplement PDF p. 5), lines (6)–(7); A.3.1, p. 6 (supplement PDF p. 6), the definition of $U_i$", "pdf_page": 5, "printed_page": "5", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P004-T33" ], "text": { "as_printed": "P.~5: ``Under conditions C1-C5, we have the following variance bound for $\\operatorname{Var}\\big(\\widehat\\tau^{\\mathrm L}_{\\mathrm{HA}}(d)\\big)$:\n\\[\n\\begin{aligned}\n\\tilde{\\mathrm V}_{\\mathrm{HA}}={}&\\frac{1}{N^2p}\\sum_{i=1}^N\\mathrm E\\Big[\\big(\\mu_i(\\mathbf 1;d)-\\bar\\mu^1(d)\\big)^2\\Big]+\\frac{1}{N^2(1-p)}\\sum_{i=1}^N\\mathrm E\\Big[\\big(\\mu_i(\\mathbf 0;d)\\hl{;d)}-\\bar\\mu^0(d)\\big)^2\\Big]\\quad\\text{(6)}\\\\\n&+\\frac{1}{N^2}\\sum_{i;j\\in\\mathcal B(i;d),j\\neq i}\\ \\sum_{a,b=0}^{1}(-1)^{a+b}\\mathrm E\\big[(\\mu_i(\\mathbf{a_i},\\mathbf{b_j};d)-\\bar\\mu^a(d))\\big(\\mu_j(\\mathbf{a_i},\\mathbf{b_j};d)\\hl{;d)}-\\bar\\mu^b(d)\\big)\\big].\\quad\\text{(7)''}\n\\end{aligned}\n\\]\nP.~6: the control term inside the parentheses of $U_i$ is printed as\n\\[\n-\\frac{(1-Z_i)\\hl{\\mu_i(}\\mu_i(\\mathbf 0;d)}{N(1-p)}.\n\\]", "corrected": "P.~5: ``Under conditions C1-C5, we have the following variance bound for $\\operatorname{Var}\\big(\\widehat\\tau^{\\mathrm L}_{\\mathrm{HA}}(d)\\big)$:\n\\[\n\\begin{aligned}\n\\tilde{\\mathrm V}_{\\mathrm{HA}}={}&\\frac{1}{N^2p}\\sum_{i=1}^N\\mathrm E\\Big[\\big(\\mu_i(\\mathbf 1;d)-\\bar\\mu^1(d)\\big)^2\\Big]+\\frac{1}{N^2(1-p)}\\sum_{i=1}^N\\mathrm E\\Big[\\big(\\hl{\\mu_i(\\mathbf 0;d)}-\\bar\\mu^0(d)\\big)^2\\Big]\\quad\\text{(6)}\\\\\n&+\\frac{1}{N^2}\\sum_{i;j\\in\\mathcal B(i;d),j\\neq i}\\ \\sum_{a,b=0}^{1}(-1)^{a+b}\\mathrm E\\big[(\\mu_i(\\mathbf{a_i},\\mathbf{b_j};d)-\\bar\\mu^a(d))\\big(\\hl{\\mu_j(\\mathbf{a_i},\\mathbf{b_j};d)}-\\bar\\mu^b(d)\\big)\\big].\\quad\\text{(7)''}\n\\end{aligned}\n\\]\nP.~6:\n\\[\nU_i=\\frac{1}{\\sqrt{\\operatorname{Var}(\\widehat\\tau_{\\mathrm{HT}}(d))}}\\left(\\frac{Z_i\\mu_i(\\mathbf 1;d)}{Np}-\\frac{(1-Z_i)\\hl{\\mu_i(\\mathbf 0;d)}}{N(1-p)}-\\frac{\\mathrm E[\\mu_i(\\mathbf 1;d)]-\\mathrm E[\\mu_i(\\mathbf 0;d)]}{N}\\right).\n\\]", "evidence": "The evidence is a check of pp.~5--6 of Supplement A against lines (2) and (4) of Lemma A.4 and equation (13) of the article. The identity stated directly below the definition of $U_i$ fixes the summand that is meant.", "problem": "The three fragments are typesetting slips. In (6) and (7) the argument of the outcome is closed before the second ``$;d)$''. Line (2) of the same lemma has the control term without the fragment, and line (4) has the second-node term with one distance argument. In $U_i$ the function $\\mu_i$ does not take another $\\mu_i$ as its argument: the Horvitz--Thompson estimator of equation (13), p.~754, has one control outcome in that term.", "what_survives": "The variance bound of Lemma A.4 and the centered sum used for asymptotic normality are unaffected. Only the three displays change." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-N015", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the heading 'A.6. Theoretical Results on Observational Studies', its introductory paragraph, and the line 'C 1. The following properties hold for all sample size N and for all i in S_N:' followed by item (i). A red box marks the label 'C 1.'", "caption": "Supplement A, p. 13, the start of A.6. The marked label “C 1.” names a new block of conditions on the logistic assignment model; C1 of the article is the Bernoulli design.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N016.png" }, { "alt": "Crop of the line 'Under C2, C3, C4, C6, and C1,' and of equation (36), which states that the linearized and the fitted IPW estimators differ by o_p(N to the power minus one half). Red boxes mark 'C2' and 'C1'.", "caption": "Supplement A, p. 15, Proposition 4 of A.6, the hypothesis of (36). The marked “C2” is the article’s bounded-outcomes condition and the marked “C1” is the logistic block of p. 13.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N016-b.png" }, { "alt": "Crop of the line 'In addition under C2,' and the first line of the displayed upper bound on the variance of the linearized IPW estimator. A red box marks 'C2'.", "caption": "Supplement A, p. 16, the continuation of Proposition 4. The marked “C2” is the homophily condition of p. 14, not the condition of the same name on p. 15.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N016-c.png" }, { "alt": "Crop of the sentence 'The following assumption is similar as C5.', the line 'C 6. Given a kernel function K and a sequence of bandwidth b_N,' and equation (122), a liminf condition on a kernel-weighted sum of products of centered node effects. Red boxes mark 'C5' and the label 'C 6.'", "caption": "Supplement A, p. 34, Appendix C. The marked “C5” refers to the homophily condition of the article; the marked “C 6.” is a new condition local to Appendix C.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N016-d.png" } ], "formal_release": null, "formal_statements": [], "headline": "Supplement A reuses the labels C1–C6 for different assumptions: in A.6, “C 1” and “C 2” are a logistic-model block and a homophily condition, and in Appendix C, “C 3”–“C 6” are four new conditions.", "id": "P004-N016", "location": "Supplement A, A.6, pp. 13–16, and Appendix C, pp. 26–35 (the same supplement PDF pages)", "pdf_page": 13, "printed_page": "13–16", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "A.6, p.~13: ``\\hl{C 1.} The following properties hold for all sample size $N$ and for all $i\\in\\mathcal S_N$: \\ldots''; p.~14: ``\\hl{C 2.} For all sample size $N$, \\ldots''; Proposition 4, pp.~15--16: ``Under \\hl{C2}, C3, C4, C6, and \\hl{C1}, \\ldots'' and ``In addition under \\hl{C2}, \\ldots''.\n\nAppendix C, p.~26: ``We shall inherit C1 and C2. We relax C3 with the following weak-dependence assumption. \\hl{C 3.} \\ldots''; p.~27: ``\\hl{C 4.} \\ldots''; p.~32: ``\\hl{C 5.} The kernel function \\ldots''; p.~34: ``The following assumption is similar as \\hl{C5}. \\hl{C 6.} \\ldots'', with footnote 20: ``Note if $b_N$ is fixed and $K$ is the uniform kernel, this reduces to \\hl{C5}.''; Proposition 6, p.~35: ``Under C1, C2, C3, C4, C5 and C6''.", "corrected": "Give the two conditions of A.6 the labels \\hl{O1} (logistic model) and \\hl{O2} (homophily), and state Proposition 4 as ``Under C2, C3, C4, C6, and \\hl{O1}, \\ldots\\ In addition under \\hl{O2}, \\ldots''. Give the conditions of Appendix C the labels \\hl{W3}--\\hl{W6}, and state its results under ``C1, C2, \\hl{W3, W4, W5 and W6}''. On p.~34, write ``similar to C5 \\hl{of the article}'' and ``this reduces to C5 \\hl{of the article}''.", "intended_meaning": "In A.6, C6 is the article's assignment condition and ``C 1'' is the new block of conditions on the logistic model. In Proposition 4 the first ``C2'' is the article's bounded outcomes: the proof on p.~16 uses it, with bounded confounders, to control remainders. The ``C2'' of ``In addition under C2'' and of item (ii) is the homophily condition of p.~14. Appendix C states that it keeps C1 and C2 of the article and replaces C3. Its ``C 3''--``C 6'' are local conditions (near-epoch dependence, a quadratic bound on neighborhood counts, kernel conditions and a homophily limit), and Proposition 6 uses that local set. The two comparisons on p.~34, ``similar as C5'' and ``this reduces to C5'', refer to the homophily condition C5 of the article, not to the kernel condition labeled ``C 5'' on p.~32. With these readings the hypotheses of the supplement's propositions are recoverable, and no result depends on the labels." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-N016", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of displays (127), (128) and (129) and the sentence that follows them. Red boxes mark the factor N squared over N 1 squared minus 1 over p squared, set in parentheses, at the start of (128) and at the start of (129).", "caption": "Supplement A, p. 36: equations (127)–(129). The marked factor is printed without absolute value bars, while (127) and the second factor of (128) are inside them.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N018.png" } ], "formal_release": null, "formal_statements": [ { "id": "P004-T34", "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T34" } ], "headline": "The leading factor $N^2/N_1^2-1/p^2$ of the bounds (128) and (129) should appear in absolute value.", "id": "P004-N018", "location": "Supplement A, Appendix C, proof of Proposition 6(i), equations (128)–(129), p. 36 (PDF p. 36 of the supplement)", "pdf_page": 36, "printed_page": "36", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P004-T34" ], "text": { "as_printed": "\\[\n\\begin{aligned}\n&\\Big|N\\times(\\frac1{N_1^2}\\sum_{i=1}^N\\sum_{j\\ne i}Z_iZ_j\\hat e_i(d)\\hat e_j(d)K(\\frac{d_{ij}}{b_N})-\\frac1{N^2p^2}\\sum_{i=1}^N\\sum_{j\\ne i}Z_iZ_j\\hat e_i(d)\\hat e_j(d)K(\\frac{d_{ij}}{b_N}))\\Big|&&\\text{(127)}\\\\\n&\\le\\hl{\\Big(\\frac{N^2}{N_1^2}-\\frac1{p^2}\\Big)}\\Big|\\frac1N\\sum_{i=1}^N\\sum_{j\\ne i}Z_iZ_j\\hat e_i(d)\\hat e_j(d)K(\\frac{d_{ij}}{b_N})\\Big|&&\\text{(128)}\\\\\n&\\lesssim\\hl{\\Big(\\frac{N^2}{N_1^2}-\\frac1{p^2}\\Big)}\\big(b_N^2\\times K_{max}\\big)=O_p(N^{-\\frac12}b_N^2)=o_p(1)&&\\text{(129)}\n\\end{aligned}\n\\]", "corrected": "\\[\n\\begin{aligned}\n&\\Big|N\\times(\\frac1{N_1^2}\\sum_{i=1}^N\\sum_{j\\ne i}Z_iZ_j\\hat e_i(d)\\hat e_j(d)K(\\frac{d_{ij}}{b_N})-\\frac1{N^2p^2}\\sum_{i=1}^N\\sum_{j\\ne i}Z_iZ_j\\hat e_i(d)\\hat e_j(d)K(\\frac{d_{ij}}{b_N}))\\Big|&&\\text{(127)}\\\\\n&\\le\\hl{\\Big|\\frac{N^2}{N_1^2}-\\frac1{p^2}\\Big|}\\,\\Big|\\frac1N\\sum_{i=1}^N\\sum_{j\\ne i}Z_iZ_j\\hat e_i(d)\\hat e_j(d)K(\\frac{d_{ij}}{b_N})\\Big|&&\\text{(128)}\\\\\n&\\lesssim\\hl{\\Big|\\frac{N^2}{N_1^2}-\\frac1{p^2}\\Big|}\\big(b_N^2\\times K_{max}\\big)=O_p(N^{-\\frac12}b_N^2)=o_p(1)&&\\text{(129)}\n\\end{aligned}\n\\]", "evidence": "The evidence is an exact counterexample and a derivation of the corrected bound. Let $N=8$, $p=1/2$ and $N_1=5$, with residuals $(1,-1,0,0,0)$ for the treated nodes and zero for the control nodes, and with $K(d_{ij}/b_N)=1$ for all pairs. The residuals have mean zero in each arm. Then $A=-36/25$ and $B=-1/4$. The quantity (127) equals $9/25$, and the right-hand side of (128), as printed, equals $-9/25$. For the corrected bound, note that $|A|=O_p(N^{-1/2})$ by (124). The residuals are bounded by C2, and by C4 and C5 each node has at most a constant times $b_N^2$ neighbours with nonzero kernel weight, each weight being at most $K_{\\max}$. I obtain $|A|\\,|B|\\lesssim|A|\\,b_N^2K_{\\max}=O_p(N^{-1/2}b_N^2)$, which is $o_p(1)$ because $b_N=o(N^{1/6})$.", "problem": "Let $A=N^2/N_1^2-1/p^2$ and $B=N^{-1}\\sum_{i}\\sum_{j\\ne i}Z_iZ_j\\hat e_i(d)\\hat e_j(d)K(d_{ij}/b_N)$. The quantity (127) is an absolute value and equals $|A|\\,|B|$, while the printed bound (128) is $A|B|$. The factor $A$ is negative whenever $N_1/N>p$, which happens on ordinary assignments, and the right-hand side as printed is then negative when $B\\ne0$. The same factor appears without bars in (129).", "what_survives": "The rate $O_p(N^{-1/2}b_N^2)$ and Proposition 6(i) are unchanged. Only the two displayed bounds change." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-N018", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the notation table with the columns Notations, Definitions and First Appear in, from the row for S to the row for tau i of d and eta. Red boxes mark the symbol Y x of z i and eta in the row Marginalized potential outcome at point x, and the entry Page 6 in the row Marginalized circle averages.", "caption": "Supplement A, p. 39: the first rows of the notation table of Appendix D. The marked symbol lacks the index $i$, and the marked entry should be Page 7.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N019.png" } ], "formal_release": null, "formal_statements": [], "headline": "In the notation table, the marginalized potential outcome is $Y_{ix}(z_i;\\eta)$, with the index $i$ of the node, and the marginalized circle averages first appear on page 7 of the article and not on page 6.", "id": "P004-N019", "location": "Supplement A, Appendix D, notation table, p. 39 (PDF p. 39 of the supplement); compare equations (4) and (5) of the article, pp. 749–750 (PDF pp. 6–7)", "pdf_page": 39, "printed_page": "39", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\begin{audittable}{lll}{Three rows of the notation table, as printed.}\nNotations & Definitions & First Appear in \\\\\n$\\hl{Y_x}(z_i;\\eta)$ & Marginalized potential outcome at point x & Page 6 \\\\\n$\\mu_i(z_i;d,\\eta)$ & Marginalized circle averages & Page \\hl{6} \\\\\n$\\tau_{ix}(\\eta)$ & Individual marginalized effect & Page 7 \\\\\n\\end{audittable}", "corrected": "\\begin{audittable}{lll}{The same rows, corrected.}\nNotations & Definitions & First Appear in \\\\\n$\\hl{Y_{ix}}(z_i;\\eta)$ & Marginalized potential outcome at point x & Page 6 \\\\\n$\\mu_i(z_i;d,\\eta)$ & Marginalized circle averages & Page \\hl{7} \\\\\n$\\tau_{ix}(\\eta)$ & Individual marginalized effect & Page 7 \\\\\n\\end{audittable}", "evidence": "The evidence is a check of p.~39 of the supplement against pp.~749--750 of the article.", "problem": "Equation (4) of the article defines the marginalized potential outcome $Y_{ix}(z_i;\\eta)$ at the point $x$ for the node $i$ whose assignment is held fixed. The table drops the index $i$ from the symbol and keeps it in $z_i$. The table refers to pages of the article's PDF: equation (4) is on PDF p.~6 (p.~749). The marginalized circle average $\\mu_i(z_i;d,\\eta)$ first appears in equation (5), on PDF p.~7 (p.~750). The row for $\\tau_{ix}(\\eta)$ is correct as printed.", "what_survives": "The definitions (4)--(8) of the article are unaffected, and nothing depends on the table." }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-N019", "work": { "arxiv_id": "2010.13599", "doi": "10.1214/24-AOAS1986", "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "url": "https://pmaronow.github.io/llm-audit/works/P004/" }, "work_id": "P004" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Listing of lines 21 to 34 of Conley.cpp with line numbers. In the branch metric == 1, line 24 reads dist(j) = sum(pow(point1 - point2, 2)). In the branch metric == 2, lines 26 to 28 call distGeo from the package geosphere, and a red box marks line 29, dist(j) = geo_distance[0]. In line 33 a red box marks the test sqrt(dist) < cutoff, whose result is assigned to dist_u.", "caption": "Source code of the package in the GitHub repository, not a page of the article: SpatialEffect/src/Conley.cpp, lines 21–34, typeset from the file with line numbers added. Line 24 stores a squared Euclidean distance. The marked line 29 stores the geodesic distance itself, and the marked test on line 33 takes the square root in both cases.", "url": "https://pmaronow.github.io/llm-audit/works/P004/exhibits/P004-N020.png" } ], "formal_release": null, "formal_statements": [], "headline": "In the older source of the package, the spatial HAC routine compares the square root of a geodesic distance with the cutoff, so that a pair of nodes is weighted when its distance is below the square of the cutoff; the geodesic branch should store the squared distance.", "id": "P004-N020", "location": "R package SpatialEffect in the GitHub repository named on p. 766 (commit 7c873da): SpatialEffect/src/Conley.cpp at the top level of the repository, lines 23–33. The code is not printed in the article; the estimator that it implements is equation (19), p. 755 (PDF p. 12).", "pdf_page": null, "printed_page": null, "reaches": [ { "effect": "implementation_wrong", "effect_text": "the package or replication code has the reported defect", "explanation": null, "package": null, "recorded_as": null, "to": "work", "url": "https://pmaronow.github.io/llm-audit/works/P008/", "where": null, "work_id": "P008" }, { "effect": "implementation_wrong", "effect_text": "the package or replication code has the reported defect", "explanation": null, "package": "SpatialEffect", "recorded_as": null, "to": "package", "url": null, "where": null, "work_id": "P004" } ], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "\\texttt{SpatialEffect/src/Conley.cpp}, lines 24, 29 and 33:\n\n\\texttt{dist(j) = sum(pow(point1 - point2, 2));}\n\n\\texttt{dist(j) = \\hl{geo\\_distance[0]};}\n\n\\texttt{arma::uvec dist\\_u = (\\hl{sqrt(dist)} < cutoff);}\n\nLine 24 belongs to the branch \\texttt{metric == 1} (Euclidean) and line 29 to the branch \\texttt{metric == 2} (geodesic).", "corrected": "Line 29:\n\n\\texttt{dist(j) = \\hl{pow(geo\\_distance[0], 2)};}\n\nBoth branches then store squared distances, and line 33 compares $d_{ij}$ with the cutoff under either metric. The kernel of footnote 12 includes the endpoint $d_{ij}=\\tilde d$; to agree with it there as well, line 33 would read \\texttt{sqrt(dist) <= cutoff}. That is a separate choice.", "evidence": "The evidence is an exact counterexample. Let four nodes lie on the equator at arc offsets $0$, $2500$, $50{,}000$ and $52{,}500$ metres, and let $c=100$, $Z=(0,0,1,1)$ and $Y=(-1,1,0,2)$. The regression of $Y$ on an intercept and $Z$ has coefficients $(0,1)$ and residuals $e=(-1,1,-1,1)$, and\n\\[\n\\mathbf{X}'\\mathbf{X}=\\begin{pmatrix}4&2\\\\2&2\\end{pmatrix},\\qquad (\\mathbf{X}'\\mathbf{X})^{-1}=\\begin{pmatrix}1/2&-1/2\\\\-1/2&1\\end{pmatrix}.\n\\]\nNo two nodes are within 100 metres, and the kernel matrix of equation (19) is the identity. Since every $e_i^2$ equals 1, the double sum in (19) equals $\\mathbf{X}'\\mathbf{X}$, and I obtain the estimate $(\\mathbf{X}'\\mathbf{X})^{-1}$, in which the treatment coefficient has variance $1$. The code also weights the pairs $(1,2)$ and $(3,4)$, since $\\sqrt{2500}=50<100$. In each pair the rows of $\\mathbf{X}$ agree and the residuals sum to zero, so the double sum vanishes and the variance is $0$. When I compile the unchanged file and run it on these coordinates, it returns the zero matrix; with the corrected line it returns the variance $1$.\n\\begin{audittable}{lll}{The four-node example at a cutoff of 100 metres.}\nQuantity & Equation (19) & Code as written \\\\\nPairs with weight one, off the diagonal & none & $(1,2)$ and $(3,4)$ \\\\\nDistances given weight one & $d_{ij}\\le 100$ m & $d_{ij}<10{,}000$ m \\\\\nVariance of the treatment coefficient & $1$ & $0$ \\\\\n\\end{audittable}", "problem": "Equation (19) weights the pair $(i,j)$ by $K(d_{ij}/\\tilde d)$, and for the uniform kernel of footnote 12, $K(x)=\\mathbf{1}\\{x\\in[0,1]\\}$, a pair counts when $d_{ij}\\le\\tilde d$. In the Euclidean branch the routine stores $d_{ij}^2$, and the square root on line 33 recovers $d_{ij}$. In the geodesic branch it stores $d_{ij}$ itself, in metres. The test on line 33 is then $\\sqrt{d_{ij}}Supplement A pp. 13-17, Proposition 4, eqs. (36), (42)-(43). The printed claim is false: the balanced three-type spatial design satisfies every stated hypothesis, with the displayed exact information, coefficients, effects and variance constants. Its independent blocks have the stated exact second and fourth moments and the positive uniform tail bound. An actual constrained logistic MLE has the corrected order-N⁻¹ expansion, so the printed root-N remainder fails to converge to zero. The fitted estimator's Gaussian variance is 3/625, its printed-variance standardization has limiting variance 3600/1753, and intervals consistent for the printed variance have the stated strict coverage deficit for every 0<α<1.

", "headline": "Supplement A, Appendix A.6, Proposition 4, equation (36): the linearization of the IPW estimator with estimated logistic propensity scores. A design satisfying every stated hypothesis in which N(τ^IPWL(d)−τ^IPW(d))\\sqrt N(\\hat\\tau^{\\mathrm L}_{\\mathrm{IPW}}(d)-\\hat\\tau_{\\mathrm{IPW}}(d))N​(τ^IPWL​(d)−τ^IPW​(d)) does not tend to zero in probability", "location": "Supplement A p. 15 (Proposition 4, eq. (36)); pp. 16-17 (proof: eqs. (42)-(43) and the display 'Hence (42) becomes'); definitions on Supplement A pp. 13-14; advertised in the article, Section 6.4, p. 759 (PDF p. 16)" }, "id": "P004-T1", "text": { "body": "Supplement A pp. 13-17, Proposition 4, eqs. (36), (42)-(43). The printed claim is false: the balanced three-type spatial design satisfies every stated hypothesis, with the displayed exact information, coefficients, effects and variance constants. Its independent blocks have the stated exact second and fourth moments and the positive uniform tail bound. An actual constrained logistic MLE has the corrected order-N⁻¹ expansion, so the printed root-N remainder fails to converge to zero. The fitted estimator's Gaussian variance is 3/625, its printed-variance standardization has limiting variance 3600/1753, and intervals consistent for the printed variance have the stated strict coverage deficit for every 0<α<1.", "headline": "Supplement A, Appendix A.6, Proposition 4, equation (36): the linearization of the IPW estimator with estimated logistic propensity scores. A design satisfying every stated hypothesis in which $\\sqrt N(\\hat\\tau^{\\mathrm L}_{\\mathrm{IPW}}(d)-\\hat\\tau_{\\mathrm{IPW}}(d))$ does not tend to zero in probability", "location": "Supplement A p. 15 (Proposition 4, eq. (36)); pp. 16-17 (proof: eqs. (42)-(43) and the display 'Hence (42) becomes'); definitions on Supplement A pp. 13-14; advertised in the article, Section 6.4, p. 759 (PDF p. 16)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T1" }, { "declarations": [ { "claim_revision_id": "catalog:3ec33acd458528e5a80968ee3624f3e15d25d7a3ec641f2f5db7ad43f3a1edc2", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.corrected_fits", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L425", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:3ec33acd458528e5a80968ee3624f3e15d25d7a3ec641f2f5db7ad43f3a1edc2", "conditional_on": [], "credit_reason": "The summand identity is used inside corrected_ipw_finite, which supplies the exact estimator representation.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.corrected_score_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L437", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:3ec33acd458528e5a80968ee3624f3e15d25d7a3ec641f2f5db7ad43f3a1edc2", "conditional_on": [], "credit_reason": "These finite identities remain valid as algebraic statements in singular or empty dimensions; this adds cases and does not exclude the nonsingular logistic specialization the claim needs.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.corrected_ipw_finite", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L908", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:3ec33acd458528e5a80968ee3624f3e15d25d7a3ec641f2f5db7ad43f3a1edc2", "conditional_on": [], "credit_reason": "The corrected residual mean is already supplied by corrected_ipw_finite.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.residualContrast_mean", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L516", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:3ec33acd458528e5a80968ee3624f3e15d25d7a3ec641f2f5db7ad43f3a1edc2", "conditional_on": [], "credit_reason": "Unbiasedness is already supplied by corrected_ipw_finite; this helper is not separately invoked.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.augmented_unbiased", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L546", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:3ec33acd458528e5a80968ee3624f3e15d25d7a3ec641f2f5db7ad43f3a1edc2", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.augmented_variance", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_VarianceCompletion.lean#L88", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:3ec33acd458528e5a80968ee3624f3e15d25d7a3ec641f2f5db7ad43f3a1edc2", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.corrected_ipw_oracle", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_VarianceCompletion.lean#L177", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:3ec33acd458528e5a80968ee3624f3e15d25d7a3ec641f2f5db7ad43f3a1edc2", "conditional_on": [], "credit_reason": "The credited component is the corrected T2(a,e), whose text already requires interior truth, corrected score fits, liminf coverage, and HAC truncation; these are not extra restrictions relative to that component. No conclusion is hidden in a hypothesis.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.corrected_ipw_asymptotics_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GLP0042B1.lean#L2750", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:3ec33acd458528e5a80968ee3624f3e15d25d7a3ec641f2f5db7ad43f3a1edc2", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T2", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T2" }, "html": { "body": "

Supplement A pp. 14-17 (Appendix A.6: 'C 2', Proposition 4, eqs. (36)-(43)); article Section 6.4, p. 759 (PDF p. 16). The corrected IPW estimator has the expansion, exact representations, unbiasedness, variance formula, oracle bounds, normal limit and conservative intervals in clauses (a)-(e). The usual measure-theoretic conditions are assumed.

", "headline": "Supplement A, Appendix A.6, Proposition 4, corrected: first-order expansion of the IPW estimator with an estimated logistic propensity score, its variance, and the HAC and SAH variance bounds built on the corrected residuals", "location": "Supplement A pp. 14-17 (Appendix A.6: 'C 2', Proposition 4, eqs. (36)-(43)); article Section 6.4, p. 759 (PDF p. 16)" }, "id": "P004-T2", "text": { "body": "Supplement A pp. 14-17 (Appendix A.6: 'C 2', Proposition 4, eqs. (36)-(43)); article Section 6.4, p. 759 (PDF p. 16). The corrected IPW estimator has the expansion, exact representations, unbiasedness, variance formula, oracle bounds, normal limit and conservative intervals in clauses (a)-(e). The usual measure-theoretic conditions are assumed.", "headline": "Supplement A, Appendix A.6, Proposition 4, corrected: first-order expansion of the IPW estimator with an estimated logistic propensity score, its variance, and the HAC and SAH variance bounds built on the corrected residuals", "location": "Supplement A pp. 14-17 (Appendix A.6: 'C 2', Proposition 4, eqs. (36)-(43)); article Section 6.4, p. 759 (PDF p. 16)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T2" }, { "declarations": [ { "claim_revision_id": "catalog:3b53dd889163a3a84210092374597fb9a1ec7a14c495186d02c83285c5f58c55", "conditional_on": [], "credit_reason": "The row covers the actual-MLE variance-ratio witness, not the target's additional finite tables, normal limit, or all-alpha coverage formula. The alternative realization satisfies every hypothesis of the printed reading, so it remains an exact counterexample component.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.observational_sah_counterexample_proof", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GLP0042B3.lean#L460", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:3b53dd889163a3a84210092374597fb9a1ec7a14c495186d02c83285c5f58c55", "conditional_on": [], "credit_reason": "Entered rational comparisons do not identify the actual estimator expectations or variances.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.sah_witness_constants", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1669", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:3b53dd889163a3a84210092374597fb9a1ec7a14c495186d02c83285c5f58c55", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T3", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T3" }, "html": { "body": "

Supplement A pp. 15-16, observational residual, SAH variance and Proposition 4(i). The printed claim is false: the specified N=8r two-group disk field and actual clipped logistic MLE satisfy the stated assumptions, with singleton neighborhoods and zero AME. The exact fitted estimator, coefficients, residual sum and double-weighted variance have the stated formulas. The N=32 and N=64 variances and expectations give the stated strict finite underestimation. The root-N Gaussian variance and scaled actual variance tend to 1/4, the pooled treatment fraction N₁/N and normalized residual sum R/N tend in probability to 1/5 and 1/8, and the printed scaled variance and variance ratio tend to 2/25 and 8/25. Every 0<α<1/2 has the stated exact limiting coverage strictly below 1-α.

", "headline": "Supplement A, Appendix A.6: the observational SAH variance estimator V^SAHobs(d)\\hat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)V^SAHobs​(d) and the coverage statement (i) of Proposition 4 built on it. A design in which NV^SAHobsN\\hat V^{\\mathrm{obs}}_{\\mathrm{SAH}}NV^SAHobs​ converges to 825\\frac8{25}258​ of N Var(τ^IPW)N\\,\\mathrm{Var}(\\hat\\tau_{\\mathrm{IPW}})NVar(τ^IPW​)", "location": "Supplement A p. 15 (definitions of ϵ^i\\hat\\epsilon_iϵ^i​ and V^SAHobs(d)\\hat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)V^SAHobs​(d)); p. 16 (Proposition 4, item (i))" }, "id": "P004-T3", "text": { "body": "Supplement A pp. 15-16, observational residual, SAH variance and Proposition 4(i). The printed claim is false: the specified N=8r two-group disk field and actual clipped logistic MLE satisfy the stated assumptions, with singleton neighborhoods and zero AME. The exact fitted estimator, coefficients, residual sum and double-weighted variance have the stated formulas. The N=32 and N=64 variances and expectations give the stated strict finite underestimation. The root-N Gaussian variance and scaled actual variance tend to 1/4, the pooled treatment fraction N₁/N and normalized residual sum R/N tend in probability to 1/5 and 1/8, and the printed scaled variance and variance ratio tend to 2/25 and 8/25. Every 0<α<1/2 has the stated exact limiting coverage strictly below 1-α.", "headline": "Supplement A, Appendix A.6: the observational SAH variance estimator $\\hat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)$ and the coverage statement (i) of Proposition 4 built on it. A design in which $N\\hat V^{\\mathrm{obs}}_{\\mathrm{SAH}}$ converges to $\\frac8{25}$ of $N\\,\\mathrm{Var}(\\hat\\tau_{\\mathrm{IPW}})$", "location": "Supplement A p. 15 (definitions of $\\hat\\epsilon_i$ and $\\hat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)$); p. 16 (Proposition 4, item (i))" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T3" }, { "declarations": [ { "claim_revision_id": "catalog:cabd2613c2131d97ba5cf25fd2764aeceb75333098398260e0417a5e739a12c3", "conditional_on": [], "credit_reason": "The modelling-choice sentence about a finite variance bound and symmetry describes the accompanying T4(b) result; this row itself is only the pointwise T4(a) identity and needs no neighborhood restrictions.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.sah_weighting_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L867", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:cabd2613c2131d97ba5cf25fd2764aeceb75333098398260e0417a5e739a12c3", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.raw_sah_bound", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L842", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:cabd2613c2131d97ba5cf25fd2764aeceb75333098398260e0417a5e739a12c3", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T4", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T4" }, "html": { "body": "

Supplement A p. 15, definitions of the observational residual and SAH variance. Use raw fitted residuals with squared inverse propensities. With both arms nonempty this has the stated equivalent arm-count form; the printed weighted residual instead produces fourth powers. Under independent assignment, symmetric self-containing neighborhoods and zero covariance outside them, the variance of the residual contrast mean is bounded by the stated conditional second moments, exactly the expectation of the raw SAH estimator. In the T3 design its N-scaled limit is 1/2, compared with actual variance limit 1/4, and its N=32 expectation is exactly 493725989/38038425600, above 18239/1505280. General fitted-IPW intervals use the residuals of the corrected linearization as specified in T2.

", "headline": "Supplement A, Appendix A.6: the observational SAH variance estimator, corrected so that each residual is weighted by the inverse propensity once", "location": "Supplement A p. 15 (definitions of ϵ^i\\hat\\epsilon_iϵ^i​ and V^SAHobs(d)\\hat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)V^SAHobs​(d))" }, "id": "P004-T4", "text": { "body": "Supplement A p. 15, definitions of the observational residual and SAH variance. Use raw fitted residuals with squared inverse propensities. With both arms nonempty this has the stated equivalent arm-count form; the printed weighted residual instead produces fourth powers. Under independent assignment, symmetric self-containing neighborhoods and zero covariance outside them, the variance of the residual contrast mean is bounded by the stated conditional second moments, exactly the expectation of the raw SAH estimator. In the T3 design its N-scaled limit is 1/2, compared with actual variance limit 1/4, and its N=32 expectation is exactly 493725989/38038425600, above 18239/1505280. General fitted-IPW intervals use the residuals of the corrected linearization as specified in T2.", "headline": "Supplement A, Appendix A.6: the observational SAH variance estimator, corrected so that each residual is weighted by the inverse propensity once", "location": "Supplement A p. 15 (definitions of $\\hat\\epsilon_i$ and $\\hat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)$)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T4" }, { "declarations": [ { "claim_revision_id": "catalog:3f1a86ed00aeb5cb38a502b6f8a3efa6cc745a7d157467eb4f27f035ef837fb0", "conditional_on": [], "credit_reason": "This duplicate projection is contained in the stronger all-assignment spatial Case B row.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.spatial_area_sum_counterexample_proof", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GLP0042B4.lean#L14", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:3f1a86ed00aeb5cb38a502b6f8a3efa6cc745a7d157467eb4f27f035ef837fb0", "conditional_on": [], "credit_reason": "This projection is contained in the stronger all-assignment spatial Case B row.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.area_sum_counterexample_proof", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GP0042B2b.lean#L17", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:3f1a86ed00aeb5cb38a502b6f8a3efa6cc745a7d157467eb4f27f035ef837fb0", "conditional_on": [], "credit_reason": "Finite-table arithmetic is subsumed by the actual spatial Case B row.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.area_witness_realized", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1036", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:3f1a86ed00aeb5cb38a502b6f8a3efa6cc745a7d157467eb4f27f035ef837fb0", "conditional_on": [], "credit_reason": "The all-assignment spatial row already supplies the required design witness; assignment enumeration is not separately invoked.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.assignment_two_univ", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1004", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:3f1a86ed00aeb5cb38a502b6f8a3efa6cc745a7d157467eb4f27f035ef837fb0", "conditional_on": [], "credit_reason": "The statement exports the all-assignment equality and exact partition, as well as the area-integral sign reversal; selecting Case B is the stated component, not an undisclosed restriction on a universal positive theorem.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.area_sum_spatial_caseB", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/AreaSum.lean#L23", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:3f1a86ed00aeb5cb38a502b6f8a3efa6cc745a7d157467eb4f27f035ef837fb0", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T5", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T5" }, "html": { "body": "

pp. 763-764 (PDF pp. 20-21), interpretation of the sum of point Hajek estimates; eqs. (2), (8)-(9), (16). These examples refute the reading that an unweighted sum of point Hajek contrasts over a partition equals the change in the normalized deforested area share of the union. Three specified binary area fields on the two radius-one disk nodes realize Cases A, B and C. All six bin areas, positive finite averages, all-assignment shares and whole-region weighted averages have the stated values. At assignment (1,0), each bin Hajek contrast is its AME and each six-bin sum is -17/500; the actual whole-region contrasts and AMEs are -17/3000, 121/18000 and -7/1800. Case B reverses the sign and Case C's sum is below every bin effect. Refining the same Case A field into eleven bins changes the sum to -187/3000 while preserving its whole-region effect -17/3000. No corrected empirical area effect is asserted for the application.

", "headline": "Section 8.1, the reading of the summed Hajek estimates between zero and five km as 'a 3.4 percentage point decline in deforested area': binary-outcome designs with the same sum −0.034-0.034−0.034 and area-average effects −173000-\\frac{17}{3000}−300017​, +12118000+\\frac{121}{18000}+18000121​ and −71800-\\frac7{1800}−18007​", "location": "p. 763 (PDF p. 20), last paragraph, continued on p. 764 (PDF p. 21); definitions: eq. (2), p. 747; eqs. (8)-(9), p. 750; eq. (16), p. 754" }, "id": "P004-T5", "text": { "body": "pp. 763-764 (PDF pp. 20-21), interpretation of the sum of point Hajek estimates; eqs. (2), (8)-(9), (16). These examples refute the reading that an unweighted sum of point Hajek contrasts over a partition equals the change in the normalized deforested area share of the union. Three specified binary area fields on the two radius-one disk nodes realize Cases A, B and C. All six bin areas, positive finite averages, all-assignment shares and whole-region weighted averages have the stated values. At assignment (1,0), each bin Hajek contrast is its AME and each six-bin sum is -17/500; the actual whole-region contrasts and AMEs are -17/3000, 121/18000 and -7/1800. Case B reverses the sign and Case C's sum is below every bin effect. Refining the same Case A field into eleven bins changes the sum to -187/3000 while preserving its whole-region effect -17/3000. No corrected empirical area effect is asserted for the application.", "headline": "Section 8.1, the reading of the summed Hajek estimates between zero and five km as 'a 3.4 percentage point decline in deforested area': binary-outcome designs with the same sum $-0.034$ and area-average effects $-\\frac{17}{3000}$, $+\\frac{121}{18000}$ and $-\\frac7{1800}$", "location": "p. 763 (PDF p. 20), last paragraph, continued on p. 764 (PDF p. 21); definitions: eq. (2), p. 747; eqs. (8)-(9), p. 750; eq. (16), p. 754" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T5" }, { "declarations": [ { "claim_revision_id": "catalog:887d2aef785be811698914eebef203211a96db27ffccbfbd2eee00591f65f26b", "conditional_on": [], "credit_reason": "Auxiliary realized-average identity; item 02 supplies the effect aggregation. The regularity assumptions are the ordinary meaning of the finite-mass averages in the claim, rather than a separate statistical restriction.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.area_partition_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_SpatialAlgebraCompletion.lean#L58", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:887d2aef785be811698914eebef203211a96db27ffccbfbd2eee00591f65f26b", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.weighted_effect_bounds", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L951", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:887d2aef785be811698914eebef203211a96db27ffccbfbd2eee00591f65f26b", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P004_Spatial.lean:2582 states that the bin-mass average commutes with each arm-specific marginal mean, for arbitrary node-specific weights. 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p. 763 (PDF p. 20), last paragraph, continued on p. 764 (PDF p. 21). Actual area averages and their marginalized effects aggregate with bin masses, lie within bin bounds, and positive scaling preserves randomization p-values.

", "headline": "Section 8.1, corrected: the summed distance-specific estimates are a summed contrast; the area-average effect is the area-weighted mean of the bin effects; the randomization p-value does not depend on the scaling of the statistic", "location": "p. 763 (PDF p. 20), last paragraph, continued on p. 764 (PDF p. 21)" }, "id": "P004-T6", "text": { "body": "p. 763 (PDF p. 20), last paragraph, continued on p. 764 (PDF p. 21). Actual area averages and their marginalized effects aggregate with bin masses, lie within bin bounds, and positive scaling preserves randomization p-values.", "headline": "Section 8.1, corrected: the summed distance-specific estimates are a summed contrast; the area-average effect is the area-weighted mean of the bin effects; the randomization p-value does not depend on the scaling of the statistic", "location": "p. 763 (PDF p. 20), last paragraph, continued on p. 764 (PDF p. 21)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T6" }, { "declarations": [ { "claim_revision_id": "catalog:1203558cd7c4a168cffb976cbde8383afa56776c31b81db358c9279c8a433bd3", "conditional_on": [], "credit_reason": "The finite row covers Witness 2 only. The original modelling paragraph mentions Student density/quantiles in the wider witness setup, but those occur in the separate asymptotic row, not this finite conclusion. The recorded review note about an unproved source and sorryAx is stale relative to this task's stipulated standard-axiom proofs; proof status is outside this fidelity audit.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.adjusted_pd_finite_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_ConfidenceCompletion.lean#L107", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:1203558cd7c4a168cffb976cbde8383afa56776c31b81db358c9279c8a433bd3", "conditional_on": [], "credit_reason": "The target fixes the adjusted-procedure interpretation and the 95-percent repeated-star witness. Direct Student-interval undercoverage is exported, not merely an underestimated scalar variance; the concrete Q=K+vv^T/8 is the genuine positive part since Kv=-v and ||v||^2=8.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.adjusted_pd_asymptotic_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GLP0042B5.lean#L2013", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:1203558cd7c4a168cffb976cbde8383afa56776c31b81db358c9279c8a433bd3", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.adjusted_pd_constants", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1937", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:1203558cd7c4a168cffb976cbde8383afa56776c31b81db358c9279c8a433bd3", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T7", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T7" }, "html": { "body": "

p. 761 (PDF p. 18), Section 7; p. 756 (PDF p. 13), eqs. (21)-(22) and footnotes 14-15; Supplement A p. 17 (A.7, eqs. (44)-(45)). The repeated-star construction gives a finite adjustment-order reversal and a guarded asymptotic coverage upper bound strictly below 95 percent.

", "headline": "Section 7 recommendation ('the positive semidefinite HAC variance estimator with the empirical degree of adjustment') read with Supplement A.7: a design satisfying C1-C5 in which the adjusted positive-semidefinite variance converges to 1516\\frac{15}{16}1615​ of the asymptotic variance, and an assignment in which the adjusted ordering is reversed", "location": "p. 761 (PDF p. 18), Section 7; p. 756 (PDF p. 13), eqs. (21)-(22) and footnotes 14-15; Supplement A p. 17 (A.7, eqs. (44)-(45))" }, "id": "P004-T7", "text": { "body": "p. 761 (PDF p. 18), Section 7; p. 756 (PDF p. 13), eqs. (21)-(22) and footnotes 14-15; Supplement A p. 17 (A.7, eqs. (44)-(45)). The repeated-star construction gives a finite adjustment-order reversal and a guarded asymptotic coverage upper bound strictly below 95 percent.", "headline": "Section 7 recommendation ('the positive semidefinite HAC variance estimator with the empirical degree of adjustment') read with Supplement A.7: a design satisfying C1-C5 in which the adjusted positive-semidefinite variance converges to $\\frac{15}{16}$ of the asymptotic variance, and an assignment in which the adjusted ordering is reversed", "location": "p. 761 (PDF p. 18), Section 7; p. 756 (PDF p. 13), eqs. (21)-(22) and footnotes 14-15; Supplement A p. 17 (A.7, eqs. (44)-(45))" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T7" }, { "declarations": [ { "claim_revision_id": "catalog:8ac73303c7f213b87f76b0bf04e17cfda3785b472415691f7c5d6c22a4cd4985", "conditional_on": [], "credit_reason": "The scalar cap inequality is incorporated in capped_pd_interval.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.capped_variance_ge", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L675", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8ac73303c7f213b87f76b0bf04e17cfda3785b472415691f7c5d6c22a4cd4985", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.capped_pd_interval", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_MatrixCompletion.lean#L150", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8ac73303c7f213b87f76b0bf04e17cfda3785b472415691f7c5d6c22a4cd4985", "conditional_on": [], "credit_reason": "This is exactly the corrected liminf claim, including arbitrary random critical values satisfying the stated lower bound. The hypotheses do not assume a CLT, variance consistency, positive adjusted trace, or the desired coverage conclusion.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.capped_pd_coverage_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GP0042B2b.lean#L840", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:8ac73303c7f213b87f76b0bf04e17cfda3785b472415691f7c5d6c22a4cd4985", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T8", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T8" }, "html": { "body": "

p. 761 (PDF p. 18), Section 7; p. 756 (PDF p. 13), eqs. (21)-(22), footnote 15; Supplement A p. 17 (A.7). Capping the positive-semidefinite HAC adjustment preserves variance ordering and interval inclusion for every symmetric kernel. Every positive-degree Student upper quantile dominates the normal quantile. Under C1-C5 and the variance floor, the capped intervals have asymptotic coverage at least one minus alpha, including arbitrary assignment-dependent positive Student degrees of freedom. Coverage counts only draws with both arms nonempty. The repeated-star design has well-defined uncapped intervals with probability tending to one and shows that the uncapped rule has no such universal guarantee.

", "headline": "Section 7 recommendation and Supplement A.7, corrected: an adjustment of the positive-semidefinite HAC variance that cannot reduce it, and the guarantee it inherits from Proposition 4(ii)", "location": "p. 761 (PDF p. 18), Section 7; p. 756 (PDF p. 13), eqs. (21)-(22), footnote 15; Supplement A p. 17 (A.7)" }, "id": "P004-T8", "text": { "body": "p. 761 (PDF p. 18), Section 7; p. 756 (PDF p. 13), eqs. (21)-(22), footnote 15; Supplement A p. 17 (A.7). Capping the positive-semidefinite HAC adjustment preserves variance ordering and interval inclusion for every symmetric kernel. Every positive-degree Student upper quantile dominates the normal quantile. Under C1-C5 and the variance floor, the capped intervals have asymptotic coverage at least one minus alpha, including arbitrary assignment-dependent positive Student degrees of freedom. Coverage counts only draws with both arms nonempty. The repeated-star design has well-defined uncapped intervals with probability tending to one and shows that the uncapped rule has no such universal guarantee.", "headline": "Section 7 recommendation and Supplement A.7, corrected: an adjustment of the positive-semidefinite HAC variance that cannot reduce it, and the guarantee it inherits from Proposition 4(ii)", "location": "p. 761 (PDF p. 18), Section 7; p. 756 (PDF p. 13), eqs. (21)-(22), footnote 15; Supplement A p. 17 (A.7)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T8" }, { "declarations": [ { "claim_revision_id": "catalog:418a0d01b4adb49f76affbf83743549c94e4c27eb562684f46f68b625fdb429f", "conditional_on": [], "credit_reason": "The target metadata asymptotic=false conflicts with its explicit non-o_p conclusion; the row correctly derives asymptotic failure from a finite tail bound uniform over N>0. The surviving consistency and root-N rate are not refuted.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.boundary_mle_counterexample_proof", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GLP0042B6.lean#L431", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:418a0d01b4adb49f76affbf83743549c94e4c27eb562684f46f68b625fdb429f", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T9", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T9" }, "html": { "body": "

Supplement A pp. 13-14, C 1(iv),(vi) and Lemma A.6. The boundary-constrained one-dimensional logistic MLE over [0,1], with constant confounder one and true coefficient zero, satisfies all stated premises. Its likelihood has the stated derivatives and unique clipped-logit maximizer; its information, score remainder, exact moments and explicit finite tail lower bounds have the displayed values. The score remainder fails to be smaller than root-N. The scaled MLE converges to twice the positive part of a standard normal, while the leading score term converges to twice a standard normal. Consistency and the root-N rate hold. Thus the printed universal score expansion is false.

", "headline": "Supplement A, Appendix A.6, Lemma A.6 (score expansion of the logistic MLE over a compact set) with the true coefficient on the boundary of Θ0\\Theta_0Θ0​, as 'C 1'(iv) allows", "location": "Supplement A pp. 13-14 ('C 1' items (iv) and (vi)); p. 14 (Lemma A.6)" }, "id": "P004-T9", "text": { "body": "Supplement A pp. 13-14, C 1(iv),(vi) and Lemma A.6. The boundary-constrained one-dimensional logistic MLE over [0,1], with constant confounder one and true coefficient zero, satisfies all stated premises. Its likelihood has the stated derivatives and unique clipped-logit maximizer; its information, score remainder, exact moments and explicit finite tail lower bounds have the displayed values. The score remainder fails to be smaller than root-N. The scaled MLE converges to twice the positive part of a standard normal, while the leading score term converges to twice a standard normal. Consistency and the root-N rate hold. Thus the printed universal score expansion is false.", "headline": "Supplement A, Appendix A.6, Lemma A.6 (score expansion of the logistic MLE over a compact set) with the true coefficient on the boundary of $\\Theta_0$, as 'C 1'(iv) allows", "location": "Supplement A pp. 13-14 ('C 1' items (iv) and (vi)); p. 14 (Lemma A.6)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T9" }, { "declarations": [ { "claim_revision_id": "catalog:3c55f3cade4c3c3b12ae07a18f520263ada1777d7a011ea69971bc8e99ae3aea", "conditional_on": [], "credit_reason": "Interiority is required only for the score expansion, exactly as corrected T10 requests. Compact membership alone supports consistency and the root-N rate; no interior or asymptotic conclusion is hidden in LogisticConditions.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.corrected_mle_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GLP0041B1.lean#L653", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:3c55f3cade4c3c3b12ae07a18f520263ada1777d7a011ea69971bc8e99ae3aea", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T10", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T10" }, "html": { "body": "

Supplement A pp. 13-14 ('C 1' item (iv)); p. 14 (Lemma A.6). Compact membership gives logistic-MLE consistency and its root-N rate, while interior truth gives the order-N^-1 and smaller-than-root-N score remainders.

", "headline": "Supplement A, Appendix A.6, 'C 1'(iv) and Lemma A.6, corrected: the true logistic coefficient is an interior point of the compact parameter set", "location": "Supplement A pp. 13-14 ('C 1' item (iv)); p. 14 (Lemma A.6)" }, "id": "P004-T10", "text": { "body": "Supplement A pp. 13-14 ('C 1' item (iv)); p. 14 (Lemma A.6). Compact membership gives logistic-MLE consistency and its root-N rate, while interior truth gives the order-N^-1 and smaller-than-root-N score remainders.", "headline": "Supplement A, Appendix A.6, 'C 1'(iv) and Lemma A.6, corrected: the true logistic coefficient is an interior point of the compact parameter set", "location": "Supplement A pp. 13-14 ('C 1' item (iv)); p. 14 (Lemma A.6)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T10" }, { "declarations": [ { "claim_revision_id": "catalog:e319924ec85ad37062dd629c4c5da597320c39d081de63564c10b5b05eb8f92a", "conditional_on": [], "credit_reason": "The positivity hypothesis makes the defined conditional means equal the paper's marginalization over the other assignments. For a fixed finite N, no extra uniform-overlap restriction is needed for exact unbiasedness.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.ht_unbiased", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L287", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e319924ec85ad37062dd629c4c5da597320c39d081de63564c10b5b05eb8f92a", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.printed_obs_ht_expectation", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1705", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:e319924ec85ad37062dd629c4c5da597320c39d081de63564c10b5b05eb8f92a", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T11", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T11" }, "html": { "body": "

p. 759 (PDF p. 16), display defining τ^HTobs(d)\\hat\\tau^{\\mathrm{obs}}_{\\mathrm{HT}}(d)τ^HTobs​(d); Supplement A p. 4 (A.1.3, last line of the display). Corrected observational HT is unbiased, the printed estimator has the stated expectation and exact equality criterion, and the constant-outcome example has printed expectation minus two.

", "headline": "Section 6.4, the observational Horvitz-Thompson estimator τ^HTobs(d)\\hat\\tau^{\\mathrm{obs}}_{\\mathrm{HT}}(d)τ^HTobs​(d) and its unbiasedness under C6, with the control weight corrected to 1/(1−p(Oi))1/(1-p(O_i))1/(1−p(Oi​)) (also the last line of the proof in Supplement A.1.3)", "location": "p. 759 (PDF p. 16), display defining τ^HTobs(d)\\hat\\tau^{\\mathrm{obs}}_{\\mathrm{HT}}(d)τ^HTobs​(d); Supplement A p. 4 (A.1.3, last line of the display)" }, "id": "P004-T11", "text": { "body": "p. 759 (PDF p. 16), display defining $\\hat\\tau^{\\mathrm{obs}}_{\\mathrm{HT}}(d)$; Supplement A p. 4 (A.1.3, last line of the display). Corrected observational HT is unbiased, the printed estimator has the stated expectation and exact equality criterion, and the constant-outcome example has printed expectation minus two.", "headline": "Section 6.4, the observational Horvitz-Thompson estimator $\\hat\\tau^{\\mathrm{obs}}_{\\mathrm{HT}}(d)$ and its unbiasedness under C6, with the control weight corrected to $1/(1-p(O_i))$ (also the last line of the proof in Supplement A.1.3)", "location": "p. 759 (PDF p. 16), display defining $\\hat\\tau^{\\mathrm{obs}}_{\\mathrm{HT}}(d)$; Supplement A p. 4 (A.1.3, last line of the display)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T11" }, { "declarations": [ { "claim_revision_id": "catalog:b37a088ca20a6a9c6291d10e352b31c425ccc0acbbab7761e2101116512a9a74", "conditional_on": [], "credit_reason": "The source comment calling eightNodeResampled one-treated is wrong: its definition treats seven nodes, giving j=127 exactly as the target requires. This is a source-comment issue, not an error in the credited original row description.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.mc_signed_spatial_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GP0041B1.lean#L126", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b37a088ca20a6a9c6291d10e352b31c425ccc0acbbab7761e2101116512a9a74", "conditional_on": [], "credit_reason": "This proves the explicitly credited Witness B at P=100. It does not refute the assertion that a fixed finite design may become conservative for all sufficiently large P, nor does it prove Witness C at P=10000.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.mc_level_spatial_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GLP0041B2.lean#L358", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b37a088ca20a6a9c6291d10e352b31c425ccc0acbbab7761e2101116512a9a74", "conditional_on": [], "credit_reason": "The retained explicit draw-array witness already demonstrates the signed-comparison defect; the separate universal assertion is not included.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.negative_observed_comparison", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1203", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:b37a088ca20a6a9c6291d10e352b31c425ccc0acbbab7761e2101116512a9a74", "conditional_on": [], "credit_reason": "The actual spatial rejection-size row already includes this exact size inequality.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.mc_size_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1218", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:b37a088ca20a6a9c6291d10e352b31c425ccc0acbbab7761e2101116512a9a74", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T12", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T12" }, "html": { "body": "

p. 758 (PDF p. 15), Section 6.3, second paragraph. The eight-node sharp-null design witnesses the signed-comparison defect and excess size of the uncorrected absolute-statistic Monte Carlo test.

", "headline": "Section 6.3, the Monte Carlo rejection rule and 'gives an α\\alphaα-level exact test': (A) with a negative observed statistic the printed rule never rejects; (B) the rule with absolute values on both sides, at P=100P=100P=100 and α=120\\alpha=\\frac1{20}α=201​, has size 0.0555>0.050.0555>0.050.0555>0.05 in a sharp-null Bernoulli design with eight nodes", "location": "p. 758 (PDF p. 15), Section 6.3, second paragraph" }, "id": "P004-T12", "text": { "body": "p. 758 (PDF p. 15), Section 6.3, second paragraph. The eight-node sharp-null design witnesses the signed-comparison defect and excess size of the uncorrected absolute-statistic Monte Carlo test.", "headline": "Section 6.3, the Monte Carlo rejection rule and 'gives an $\\alpha$-level exact test': (A) with a negative observed statistic the printed rule never rejects; (B) the rule with absolute values on both sides, at $P=100$ and $\\alpha=\\frac1{20}$, has size $0.0555>0.05$ in a sharp-null Bernoulli design with eight nodes", "location": "p. 758 (PDF p. 15), Section 6.3, second paragraph" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T12" }, { "declarations": [ { "claim_revision_id": "catalog:ad659299657db459b7eee6dab0293f563cf0d5457fc9fbacba9b35df758d78b1", "conditional_on": [], "credit_reason": "The current statement allows an arbitrary whole-assignment law w, so no independent-coordinate design restriction remains. Fixed sharp-null outcomes are absorbed in the arbitrary assignment statistic T. Historical repair and binding-assertion details do not change its mathematical scope.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.mc_exact_level", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/FidP004.lean#L15", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:ad659299657db459b7eee6dab0293f563cf0d5457fc9fbacba9b35df758d78b1", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.mc_plus_one_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1184", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:ad659299657db459b7eee6dab0293f563cf0d5457fc9fbacba9b35df758d78b1", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T13", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T13" }, "html": { "body": "

p. 758 (PDF p. 15), Section 6.3, second paragraph. The plus-one absolute-statistic Monte Carlo test has finite-sample level, the raw rule has the stated bound, and the correction changes its p-value by the stated amount.

", "headline": "Section 6.3, the Monte Carlo randomization test, corrected: absolute value of the observed statistic and the observed assignment counted among the draws", "location": "p. 758 (PDF p. 15), Section 6.3, second paragraph" }, "id": "P004-T13", "text": { "body": "p. 758 (PDF p. 15), Section 6.3, second paragraph. The plus-one absolute-statistic Monte Carlo test has finite-sample level, the raw rule has the stated bound, and the correction changes its p-value by the stated amount.", "headline": "Section 6.3, the Monte Carlo randomization test, corrected: absolute value of the observed statistic and the observed assignment counted among the draws", "location": "p. 758 (PDF p. 15), Section 6.3, second paragraph" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T13" }, { "declarations": [ { "claim_revision_id": "catalog:f610c0c1345a33b0dabb0eff1f12984658c3e060a1dea4db1ab04c01117fbc6a", "conditional_on": [], "credit_reason": "The record's review note says the source remains unproved, conflicting with the supplied counterexample_proved status and the task's compilation premise; this is stale proof-status metadata, not a statement defect.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.spatial_shell_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L2378", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f610c0c1345a33b0dabb0eff1f12984658c3e060a1dea4db1ab04c01117fbc6a", "conditional_on": [], "credit_reason": "The two names have the same exact spatial Witness 2 statement; only spatial_shell_counterexample is invoked.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.shell_counterexample", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L2447", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:f610c0c1345a33b0dabb0eff1f12984658c3e060a1dea4db1ab04c01117fbc6a", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T14", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T14" }, "html": { "body": "

p. 757 (PDF p. 14), sentence after eq. (26); eq. (2), p. 747 (PDF p. 4), and eqs. (4)-(9), pp. 749-750. All three explicit Euclidean witnesses refute the printed uniform-radius disk/donut identity. The one-node atomic witness has radius effects zero and one, uniform-radius mean one half, and band effects two thirds, with shell weights one third and two thirds. The two-node witness has zero circle AMEs, band AMEs one sixth, the stated individual effects and unequal shell weights, and defeats every common pair of weights. For area on the unit disk the continuous conditional circle-effect version is radius itself; radial area has density 2r/(b squared minus a squared). Disk averages are two thirds versus one half, and annulus averages seven ninths versus three quarters. Every atomic and positive-area band average exists; singleton area circles are conditional means.

", "headline": "Section 6.2, the sentence after eq. (26): 'The donut and disk AMEs, defined in Section 3, are special cases with (properly-normalized) uniform kernels.' Finite designs in which the disk and donut AMEs differ from every common-kernel smoothing of the circle AMEs", "location": "p. 757 (PDF p. 14), sentence after eq. (26); definitions: eq. (2), p. 747 (PDF p. 4); eqs. (4)-(9), pp. 749-750" }, "id": "P004-T14", "text": { "body": "p. 757 (PDF p. 14), sentence after eq. (26); eq. (2), p. 747 (PDF p. 4), and eqs. (4)-(9), pp. 749-750. All three explicit Euclidean witnesses refute the printed uniform-radius disk/donut identity. The one-node atomic witness has radius effects zero and one, uniform-radius mean one half, and band effects two thirds, with shell weights one third and two thirds. The two-node witness has zero circle AMEs, band AMEs one sixth, the stated individual effects and unequal shell weights, and defeats every common pair of weights. For area on the unit disk the continuous conditional circle-effect version is radius itself; radial area has density 2r/(b squared minus a squared). Disk averages are two thirds versus one half, and annulus averages seven ninths versus three quarters. Every atomic and positive-area band average exists; singleton area circles are conditional means.", "headline": "Section 6.2, the sentence after eq. (26): 'The donut and disk AMEs, defined in Section 3, are special cases with (properly-normalized) uniform kernels.' Finite designs in which the disk and donut AMEs differ from every common-kernel smoothing of the circle AMEs", "location": "p. 757 (PDF p. 14), sentence after eq. (26); definitions: eq. (2), p. 747 (PDF p. 4); eqs. (4)-(9), pp. 749-750" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T14" }, { "declarations": [ { "claim_revision_id": "catalog:66b31eeb803f19ba49c1d09a3f0478e8de10d2ebdb79f455ed09623be4baf84b", "conditional_on": [], "credit_reason": "The required component is T15(a); neither a common radial kernel nor planar area weighting is assumed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.band_effect_identity", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_SpatialAlgebraCompletion.lean#L111", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:66b31eeb803f19ba49c1d09a3f0478e8de10d2ebdb79f455ed09623be4baf84b", "conditional_on": [], "credit_reason": "No claim-stage verdict change. This duplicates R032-04 under a different name; arbitrary node-specific band totals and supports are retained and the common weights are only required for clause (c).", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.supported_shell_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/FidP004.lean#L102", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:66b31eeb803f19ba49c1d09a3f0478e8de10d2ebdb79f455ed09623be4baf84b", "conditional_on": [], "credit_reason": "This is a duplicate of supported_shell_identity with the same node-specific contract.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.shell_mass_identity", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/FidP004.lean#L128", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:66b31eeb803f19ba49c1d09a3f0478e8de10d2ebdb79f455ed09623be4baf84b", "conditional_on": [], "credit_reason": "This row covers T15(d), including a=0; the correction to the printed uniform density is the target itself, not a fidelity difference.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.annulus_radial_measure_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GP0041B1.lean#L262", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:66b31eeb803f19ba49c1d09a3f0478e8de10d2ebdb79f455ed09623be4baf84b", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T15", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T15" }, "html": { "body": "

p. 757 (PDF p. 14), sentence after eq. (26); eqs. (2), (7)-(9). Band effects average marginalized point effects using node-specific shell masses. Continuous circle effects are signed pushforward densities, with the actual shell means at positive atoms. Finite shell decompositions and their node mean hold without common weights; common weights give a weighted sum of circle AMEs. A common kernel works for every bounded measurable profile if and only if all normalized node measures equal it. Circle AMEs alone do not determine the band AME. Full planar annuli have density 2r/(b²-a²), including disks at a=0, and their radial measure differs from the uniform radial measure. The usual measure-theoretic conditions are assumed.

", "headline": "Section 6.2, the sentence on donut and disk AMEs, corrected: they are shell-mass-weighted, node-specific averages of the circle effects", "location": "p. 757 (PDF p. 14), sentence after eq. (26)" }, "id": "P004-T15", "text": { "body": "p. 757 (PDF p. 14), sentence after eq. (26); eqs. (2), (7)-(9). Band effects average marginalized point effects using node-specific shell masses. Continuous circle effects are signed pushforward densities, with the actual shell means at positive atoms. Finite shell decompositions and their node mean hold without common weights; common weights give a weighted sum of circle AMEs. A common kernel works for every bounded measurable profile if and only if all normalized node measures equal it. Circle AMEs alone do not determine the band AME. Full planar annuli have density 2r/(b²-a²), including disks at a=0, and their radial measure differs from the uniform radial measure. The usual measure-theoretic conditions are assumed.", "headline": "Section 6.2, the sentence on donut and disk AMEs, corrected: they are shell-mass-weighted, node-specific averages of the circle effects", "location": "p. 757 (PDF p. 14), sentence after eq. (26)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T15" }, { "declarations": [ { "claim_revision_id": "catalog:924026e015ba9a37cc4f1cc16ea2c37c14ca228f30dda5e7769b7e454419b3c5", "conditional_on": [], "credit_reason": "This row is a witness for the sufficient-condition failure, not a refutation of propositions assuming C3. The positive-distance alternative witness is unnecessary for this refutation.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.polygon_range_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GP0041B1.lean#L504", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:924026e015ba9a37cc4f1cc16ea2c37c14ca228f30dda5e7769b7e454419b3c5", "conditional_on": [], "credit_reason": "This is the full arbitrary-h witness family for A4, not a claim that a fixed finite population has no possible cutoff. Status proved is consistent with proving a counterexample conjunction.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.no_polygon_cutoff_from_range_alone_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GLP0041B2.lean#L634", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:924026e015ba9a37cc4f1cc16ea2c37c14ca228f30dda5e7769b7e454419b3c5", "conditional_on": [], "credit_reason": "The printed positive area threshold can be any fixed value below 1. The weaker exported distance estimate still refutes C4, so it loses none of the intended counterexample conclusion.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.polygon_spacing_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GP0041B2.lean#L167", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:924026e015ba9a37cc4f1cc16ea2c37c14ca228f30dda5e7769b7e454419b3c5", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T16", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T16" }, "html": { "body": "

p. 752 (PDF p. 9), paragraph 'For an interpretable sufficient condition ...'; p. 753 (PDF p. 10), polygon-intervention spacing sentence; footnote 25, p. 763. The zero-radius rectangle witness, every prescribed separation L>5, and the positive-radius two-atom witness refute the range-only cutoff, with exact distances, averages, interference indicators and covariance. Disjoint area-one thin polygons have the stated exact squared distances and an unbounded radius-two neighborhood count, refuting area-only spacing.

", "headline": "Section 4, the two sufficient conditions offered for C3 and C4, for polygon intervention nodes: (A) effects confined to within dˉ\\bar ddˉ of their source do not give h(d)=2dˉ+2dh(d)=2\\bar d+2dh(d)=2dˉ+2d; (B) a lower bound on polygon area does not give C4", "location": "p. 752 (PDF p. 9), paragraph 'For an interpretable sufficient condition ...'; p. 753 (PDF p. 10), sentence on polygon-intervention experiments after C4; footnote 25, p. 763 (PDF p. 20)" }, "id": "P004-T16", "text": { "body": "p. 752 (PDF p. 9), paragraph 'For an interpretable sufficient condition ...'; p. 753 (PDF p. 10), polygon-intervention spacing sentence; footnote 25, p. 763. The zero-radius rectangle witness, every prescribed separation L>5, and the positive-radius two-atom witness refute the range-only cutoff, with exact distances, averages, interference indicators and covariance. Disjoint area-one thin polygons have the stated exact squared distances and an unbounded radius-two neighborhood count, refuting area-only spacing.", "headline": "Section 4, the two sufficient conditions offered for C3 and C4, for polygon intervention nodes: (A) effects confined to within $\\bar d$ of their source do not give $h(d)=2\\bar d+2d$; (B) a lower bound on polygon area does not give C4", "location": "p. 752 (PDF p. 9), paragraph 'For an interpretable sufficient condition ...'; p. 753 (PDF p. 10), sentence on polygon-intervention experiments after C4; footnote 25, p. 763 (PDF p. 20)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T16" }, { "declarations": [ { "claim_revision_id": "catalog:8073b14267f80ca11375c80b690c096230ef70d44f85557d47f3f43bb559b16c", "conditional_on": [], "credit_reason": "Only corrected T17(a) is credited; the added D is the claimed correction, not a weakening of that corrected statement.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.diameter_interference_cutoff_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GP0041B2.lean#L241", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8073b14267f80ca11375c80b690c096230ef70d44f85557d47f3f43bb559b16c", "conditional_on": [], "credit_reason": "An auxiliary inequality, accurately described as the five-link step; it is not credited as the entire cutoff theorem.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.diameter_chain", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1418", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8073b14267f80ca11375c80b690c096230ef70d44f85557d47f3f43bb559b16c", "conditional_on": [], "credit_reason": "The declaration is also credited in R032-07 to T16(A4); its role here is corrected T17(b).", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.no_polygon_cutoff_from_range_alone_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GLP0041B2.lean#L634", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8073b14267f80ca11375c80b690c096230ef70d44f85557d47f3f43bb559b16c", "conditional_on": [], "credit_reason": "The polygon representation introduces no excluded ordinary closed polygon cases. The parameters a,D are arbitrary, so a common choice independent of N gives the stated C4 bound.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.polygon_count_bound_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GP0041B2.lean#L342", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:8073b14267f80ca11375c80b690c096230ef70d44f85557d47f3f43bb559b16c", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T17", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T17" }, "html": { "body": "

p. 752 (PDF p. 9), paragraph 'For an interpretable sufficient condition ...'; p. 753 (PDF p. 10), sentence on polygon-intervention experiments after C4; footnote 25, p. 763. A diameter bound gives the corrected local-interference cutoff and, with positive polygon area and disjoint interiors, the stated neighborhood count bound, while the two missing-bound counterexamples remain available.

", "headline": "Section 4, the sufficient conditions for C3 and C4, corrected: h(d)=2dˉ+2dh(d)=2\\bar d+2dh(d)=2dˉ+2d for point nodes, 2dˉ+2d+D2\\bar d+2d+D2dˉ+2d+D for nodes of diameter at most DDD; and a neighbourhood-count bound from disjoint interiors, an area lower bound and a diameter upper bound", "location": "p. 752 (PDF p. 9), paragraph 'For an interpretable sufficient condition ...'; p. 753 (PDF p. 10), sentence on polygon-intervention experiments after C4; footnote 25, p. 763" }, "id": "P004-T17", "text": { "body": "p. 752 (PDF p. 9), paragraph 'For an interpretable sufficient condition ...'; p. 753 (PDF p. 10), sentence on polygon-intervention experiments after C4; footnote 25, p. 763. A diameter bound gives the corrected local-interference cutoff and, with positive polygon area and disjoint interiors, the stated neighborhood count bound, while the two missing-bound counterexamples remain available.", "headline": "Section 4, the sufficient conditions for C3 and C4, corrected: $h(d)=2\\bar d+2d$ for point nodes, $2\\bar d+2d+D$ for nodes of diameter at most $D$; and a neighbourhood-count bound from disjoint interiors, an area lower bound and a diameter upper bound", "location": "p. 752 (PDF p. 9), paragraph 'For an interpretable sufficient condition ...'; p. 753 (PDF p. 10), sentence on polygon-intervention experiments after C4; footnote 25, p. 763" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T17" }, { "declarations": [ { "claim_revision_id": "catalog:04c66c2e48d75405904c9c9d7f2f38a60d193663e66817bb1e1dae8e2d5f6e96", "conditional_on": [], "credit_reason": "Both root-N negligibility and O_p(N^-1) appear as separate conclusions. Circle-average regularity is explicit eventually, as required to interpret the target.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.hajek_linearization_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GP0041B3.lean#L414", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:04c66c2e48d75405904c9c9d7f2f38a60d193663e66817bb1e1dae8e2d5f6e96", "conditional_on": [], "credit_reason": "This row is only the finite algebraic ingredient. Its modelling-choices sentence about both asymptotic rates refers to the accompanying full linearization result R032-14; those rates are not conclusions of this declaration. The record's claim that the source remains unproved conflicts with status proved and is stale proof-status metadata.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.hajek_ratio_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1947", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:04c66c2e48d75405904c9c9d7f2f38a60d193663e66817bb1e1dae8e2d5f6e96", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T18", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T18" }, "html": { "body": "

Supplement A p. 4 (Lemma A.3), p. 5 (Lemma A.4), and eq. (27), p. 11. Under C1-C4 the corrected displayed Hajek linearization has root-N negligible error and O_p(N^-1) remainder, with zero as each empty arm's mean. It equals the augmented class member. The balanced fixed-outcome witness at p=1/4 has the exact printed-minus-corrected formula and scaled variance 4/3, and the printed expansion fails even along even sample sizes. The printed and corrected linearizations coincide at p=1/2.

", "headline": "Supplement A, Lemma A.3 (asymptotic linear expansion of the Hajek estimator), with the control denominator corrected from NpNpNp to N(1−p)N(1-p)N(1−p)", "location": "Supplement A p. 4 (Lemma A.3 and its proof); used in Lemma A.4 (p. 5), eq. (27) (p. 11) and the proofs of Propositions 3 and 4 of the article" }, "id": "P004-T18", "text": { "body": "Supplement A p. 4 (Lemma A.3), p. 5 (Lemma A.4), and eq. (27), p. 11. Under C1-C4 the corrected displayed Hajek linearization has root-N negligible error and O_p(N^-1) remainder, with zero as each empty arm's mean. It equals the augmented class member. The balanced fixed-outcome witness at p=1/4 has the exact printed-minus-corrected formula and scaled variance 4/3, and the printed expansion fails even along even sample sizes. The printed and corrected linearizations coincide at p=1/2.", "headline": "Supplement A, Lemma A.3 (asymptotic linear expansion of the Hajek estimator), with the control denominator corrected from $Np$ to $N(1-p)$", "location": "Supplement A p. 4 (Lemma A.3 and its proof); used in Lemma A.4 (p. 5), eq. (27) (p. 11) and the proofs of Propositions 3 and 4 of the article" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T18" }, { "declarations": [ { "claim_revision_id": "catalog:4b751397332452e493e20509ce534316e2fc07fa7a859072bd01175f28d80460", "conditional_on": [], "credit_reason": "The counterexample uses positive sum variance and full tuple-event independence. Metadata inconsistency: the target marks asymptotic=false although its assertion is failure of an asymptotic distributional limit; this does not obscure the claim being refuted.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.dependency_clt_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GP0041B3.lean#L582", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:4b751397332452e493e20509ce534316e2fc07fa7a859072bd01175f28d80460", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T19", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T19" }, "html": { "body": "

Supplement A p. 6 (Lemma A.5). Both exact bounded dependency-neighborhood counterexamples satisfy the printed hypotheses and fail the normal limit. The first has one fair sign and zero remaining variables; the second has one unpaired fair sign and cancelling independent sign pairs, with every marginal variance one and closed neighborhoods of size at most two. Their sums and standardized sums are exactly the unpaired fair sign.

", "headline": "Supplement A, Lemma A.5 (central limit theorem for variables with dependency neighbourhoods), as stated without a condition on the variance of the sum", "location": "Supplement A p. 6 (Lemma A.5)" }, "id": "P004-T19", "text": { "body": "Supplement A p. 6 (Lemma A.5). Both exact bounded dependency-neighborhood counterexamples satisfy the printed hypotheses and fail the normal limit. The first has one fair sign and zero remaining variables; the second has one unpaired fair sign and cancelling independent sign pairs, with every marginal variance one and closed neighborhoods of size at most two. Their sums and standardized sums are exactly the unpaired fair sign.", "headline": "Supplement A, Lemma A.5 (central limit theorem for variables with dependency neighbourhoods), as stated without a condition on the variance of the sum", "location": "Supplement A p. 6 (Lemma A.5)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T19" }, { "declarations": [ { "claim_revision_id": "catalog:718153e21ee65fe788ea46a361453e1586a96e3dc543277bbdd795aa1e8d7ec4", "conditional_on": [], "credit_reason": "This establishes the general corrected CLT and its equivalent normalized form by rescaling. The estimator-specific applications in the target are not exported by this row; see targets.json.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.corrected_dependency_clt", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_DependencyCompletion.lean#L18", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:718153e21ee65fe788ea46a361453e1586a96e3dc543277bbdd795aa1e8d7ec4", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T20", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T20" }, "html": { "body": "

Supplement A p. 6 (Lemma A.5), and its use in Propositions 2 and 3. Uniformly bounded centered arrays with bounded-degree dependency graphs and a linear sum-variance lower bound satisfy the corrected normal limit. The unit-sum-variance normalization also holds. The specified HT and linearized Hajek contrasts have their exact centered identities, bounds 2b/min(p,1-p) and 4b/min(p,1-p), the shared-source dependency graph of eventually bounded degree, and the two variance-conditional CLTs.

", "headline": "Supplement A, Lemma A.5, corrected: a central limit theorem for uniformly bounded variables with a dependency graph of bounded degree and variance of the sum of order NNN", "location": "Supplement A p. 6 (Lemma A.5 and its use in the proofs of Propositions 2 and 3 of the article)" }, "id": "P004-T20", "text": { "body": "Supplement A p. 6 (Lemma A.5), and its use in Propositions 2 and 3. Uniformly bounded centered arrays with bounded-degree dependency graphs and a linear sum-variance lower bound satisfy the corrected normal limit. The unit-sum-variance normalization also holds. The specified HT and linearized Hajek contrasts have their exact centered identities, bounds 2b/min(p,1-p) and 4b/min(p,1-p), the shared-source dependency graph of eventually bounded degree, and the two variance-conditional CLTs.", "headline": "Supplement A, Lemma A.5, corrected: a central limit theorem for uniformly bounded variables with a dependency graph of bounded degree and variance of the sum of order $N$", "location": "Supplement A p. 6 (Lemma A.5 and its use in the proofs of Propositions 2 and 3 of the article)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T20" }, { "declarations": [ { "claim_revision_id": "catalog:c3c37f09a4ffcb266be882deb55f5f18ea63e21e161c469a3573db4ac7f51b2f", "conditional_on": [], "credit_reason": "The generalized identity needs no graph assumptions; the corrected Hajek specialization is obtained by a direct substitution.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.node_variance_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1302", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:c3c37f09a4ffcb266be882deb55f5f18ea63e21e161c469a3573db4ac7f51b2f", "conditional_on": [], "credit_reason": "The graph assumptions exactly include the C3 neighborhoods; the bound concerns Var(linearHajek), as claimed, rather than finite-sample Var(hajek).", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.hajek_sah_bound", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_VarianceCompletion.lean#L133", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:c3c37f09a4ffcb266be882deb55f5f18ea63e21e161c469a3573db4ac7f51b2f", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T21", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T21" }, "html": { "body": "

Supplement A pp. 9-10 (A.4.2, Proposition 2 and its first proof identity); eq. (24), p. 9, and article eq. (20), p. 755 (PDF p. 12). The corrected node variance equals both the expanded squares-of-first-means expression and its compact residual-mean-difference form. The corrected finite SAH bound holds, including its spatial C3 specialization. The two explicit two-node schedules give variance 1/6 versus printed bound zero (corrected bound 2/3), and node variance 3/4 versus printed expression 15/16. The excluded feasible-estimator consistency claim is not asserted.

", "headline": "Supplement A, A.4.2, Proposition 2 (the variance bound VˉHA(d)\\bar{\\mathrm V}_{\\mathrm{HA}}(d)VˉHA​(d) behind the SAH estimator) and the first identity of its proof, with the control term and the squared means corrected", "location": "Supplement A p. 9 (A.4.2, Proposition 2 and the first lines of its proof); the corrected form appears on p. 10; estimator: eq. (24), p. 9, and eq. (20) of the article, p. 755 (PDF p. 12)" }, "id": "P004-T21", "text": { "body": "Supplement A pp. 9-10 (A.4.2, Proposition 2 and its first proof identity); eq. (24), p. 9, and article eq. (20), p. 755 (PDF p. 12). The corrected node variance equals both the expanded squares-of-first-means expression and its compact residual-mean-difference form. The corrected finite SAH bound holds, including its spatial C3 specialization. The two explicit two-node schedules give variance 1/6 versus printed bound zero (corrected bound 2/3), and node variance 3/4 versus printed expression 15/16. The excluded feasible-estimator consistency claim is not asserted.", "headline": "Supplement A, A.4.2, Proposition 2 (the variance bound $\\bar{\\mathrm V}_{\\mathrm{HA}}(d)$ behind the SAH estimator) and the first identity of its proof, with the control term and the squared means corrected", "location": "Supplement A p. 9 (A.4.2, Proposition 2 and the first lines of its proof); the corrected form appears on p. 10; estimator: eq. (24), p. 9, and eq. (20) of the article, p. 755 (PDF p. 12)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T21" }, { "declarations": [ { "claim_revision_id": "catalog:38a50ada739f97de89c43f628bd94b1528c4d93081f048e88f0793f62701be63", "conditional_on": [], "credit_reason": "The exact variance identity retains both pair signs and the entire residual-effect quadratic form for arbitrary deterministic fits, a generalization of the printed projection-fit estimator.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.augmented_variance", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_VarianceCompletion.lean#L88", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:38a50ada739f97de89c43f628bd94b1528c4d93081f048e88f0793f62701be63", "conditional_on": [], "credit_reason": "The variance bound is valid for every deterministic node-and-arm fit and every self-containing dependency neighborhood, including the printed projection fits and spatial neighborhoods.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.augmented_homophily_bound", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_VarianceCompletion.lean#L144", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:38a50ada739f97de89c43f628bd94b1528c4d93081f048e88f0793f62701be63", "conditional_on": [], "credit_reason": "The auxiliary residual-contrast expectation identity holds for arbitrary deterministic fits under heterogeneous independent Bernoulli assignment and specializes to the claimed residual effects.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.residualContrast_mean", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L516", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:38a50ada739f97de89c43f628bd94b1528c4d93081f048e88f0793f62701be63", "conditional_on": [], "credit_reason": "Unbiasedness is a helper, not an additional variance claim.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.augmented_unbiased", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L546", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:38a50ada739f97de89c43f628bd94b1528c4d93081f048e88f0793f62701be63", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T22", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T22" }, "html": { "body": "

Supplement A pp. 15–16, equations (37)–(40) and the bound under C 2. The corrected five-term variance formula has both pair signs and the neighbour's own fit. Its third and fifth terms form the full negative residual-effect quadratic, yielding the corrected homophily bound. This includes the population projection-fit linear estimator and the three-node witness: true and corrected variance 2/9, printed sum 4/9.

", "headline": "Supplement A, Appendix A.6, eqs. (37)-(40) and the variance bound that follows them, with the sign (−1)a+b(-1)^{a+b}(−1)a+b restored in (39) and the second node's fitted value written with OjO_jOj​", "location": "Supplement A p. 15 (eqs. (37)-(40)); p. 16 (the bound 'In addition under C2')" }, "id": "P004-T22", "text": { "body": "Supplement A pp. 15–16, equations (37)–(40) and the bound under C 2. The corrected five-term variance formula has both pair signs and the neighbour's own fit. Its third and fifth terms form the full negative residual-effect quadratic, yielding the corrected homophily bound. This includes the population projection-fit linear estimator and the three-node witness: true and corrected variance 2/9, printed sum 4/9.", "headline": "Supplement A, Appendix A.6, eqs. (37)-(40) and the variance bound that follows them, with the sign $(-1)^{a+b}$ restored in (39) and the second node's fitted value written with $O_j$", "location": "Supplement A p. 15 (eqs. (37)-(40)); p. 16 (the bound 'In addition under C2')" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T22" }, { "declarations": [ { "claim_revision_id": "catalog:44dce085599829ec81e9e2aaca7bab5d1aba7b56d372b80a483e257c5b80bd63", "conditional_on": [], "credit_reason": "The weighted-residual cancellation and treatment sandwich identity hold pointwise for arbitrary positive propensity and fit arrays, so the fitted logistic quantities are included.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.observational_sandwich_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_RegressionCompletion.lean#L227", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:44dce085599829ec81e9e2aaca7bab5d1aba7b56d372b80a483e257c5b80bd63", "conditional_on": [], "credit_reason": "Only a scalar statement under an assumed sign; it does not establish the Gaussian quantile assertions.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.symmetric_critical_event", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1230", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:44dce085599829ec81e9e2aaca7bab5d1aba7b56d372b80a483e257c5b80bd63", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T23", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T23" }, "html": { "body": "

Supplement A pp. 14–16, C 2, observational HAC, and Proposition 4(i)–(ii). The corrected homophily uses O_j at the neighbour, and the residual uses its own fitted propensity. The actual inverse-Gram treatment row yields the scalar treatment sandwich entry and the inverse-propensity residual quadratic. The numerator is the fitted IPW estimator. Genuine normal quantiles have opposite signs and give all three equivalent event presentations, for both SAH and HAC. These are exact identities, without a coverage claim.

", "headline": "Supplement A, Appendix A.6, index and label corrections: 'C 2' with OjO_jOj​ in the neighbour's factor; ϵ^i\\hat\\epsilon_iϵ^i​ with p(Oi∣θ^MLE)p(O_i|\\hat\\theta_{\\mathrm{MLE}})p(Oi​∣θ^MLE​); the scalar observational HAC variance as the (2,2) entry of the displayed sandwich; items (i)-(ii) of Proposition 4 written for τ^IPW\\hat\\tau_{\\mathrm{IPW}}τ^IPW​", "location": "Supplement A p. 14 ('C 2'); p. 15 (V^HACobs(d)\\hat{\\mathrm V}^{\\mathrm{obs}}_{\\mathrm{HAC}}(d)V^HACobs​(d) and ϵ^i\\hat\\epsilon_iϵ^i​); p. 16 (Proposition 4, items (i)-(ii))" }, "id": "P004-T23", "text": { "body": "Supplement A pp. 14–16, C 2, observational HAC, and Proposition 4(i)–(ii). The corrected homophily uses O_j at the neighbour, and the residual uses its own fitted propensity. The actual inverse-Gram treatment row yields the scalar treatment sandwich entry and the inverse-propensity residual quadratic. The numerator is the fitted IPW estimator. Genuine normal quantiles have opposite signs and give all three equivalent event presentations, for both SAH and HAC. These are exact identities, without a coverage claim.", "headline": "Supplement A, Appendix A.6, index and label corrections: 'C 2' with $O_j$ in the neighbour's factor; $\\hat\\epsilon_i$ with $p(O_i|\\hat\\theta_{\\mathrm{MLE}})$; the scalar observational HAC variance as the (2,2) entry of the displayed sandwich; items (i)-(ii) of Proposition 4 written for $\\hat\\tau_{\\mathrm{IPW}}$", "location": "Supplement A p. 14 ('C 2'); p. 15 ($\\hat{\\mathrm V}^{\\mathrm{obs}}_{\\mathrm{HAC}}(d)$ and $\\hat\\epsilon_i$); p. 16 (Proposition 4, items (i)-(ii))" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T23" }, { "declarations": [ { "claim_revision_id": "catalog:c8793be3c1eb57a8975933501c36ed566d8c278093020b699ae91c96a498eb75", "conditional_on": [], "credit_reason": "The row accurately limits its credit to T24(a-b). The separate degrees-of-freedom spelling correction and the invalid printed matrix products are not conclusions of this declaration; see targets.json.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.regression_matrix_identities", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_RegressionCompletion.lean#L211", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:c8793be3c1eb57a8975933501c36ed566d8c278093020b699ae91c96a498eb75", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T24", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T24" }, "html": { "body": "

Supplement A p. 17, Appendix A.7 and equations (44)–(45). The corrected leverage and residual-maker matrices satisfy all projection, treatment-row, norm, reciprocal normalization and sandwich identities. The uniform kernel gives the stated residual quadratic. The trace adjustment uses degrees of freedom 2 mu²/nu on its domain nu ≠ 0. Each printed defective product has compatible dimensions exactly when N=2.

", "headline": "Supplement A, A.7 (effective degrees of freedom), the definitions of λ\\lambdaλ and M\\mathbf MM with the transposes corrected, and the degrees of freedom written with ν\\nuν", "location": "Supplement A p. 17 (A.7, sentence 'Define λ=…\\lambda=\\ldotsλ=…' before eq. (44), and the last sentence)" }, "id": "P004-T24", "text": { "body": "Supplement A p. 17, Appendix A.7 and equations (44)–(45). The corrected leverage and residual-maker matrices satisfy all projection, treatment-row, norm, reciprocal normalization and sandwich identities. The uniform kernel gives the stated residual quadratic. The trace adjustment uses degrees of freedom 2 mu²/nu on its domain nu ≠ 0. Each printed defective product has compatible dimensions exactly when N=2.", "headline": "Supplement A, A.7 (effective degrees of freedom), the definitions of $\\lambda$ and $\\mathbf M$ with the transposes corrected, and the degrees of freedom written with $\\nu$", "location": "Supplement A p. 17 (A.7, sentence 'Define $\\lambda=\\ldots$' before eq. (44), and the last sentence)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T24" }, { "declarations": [ { "claim_revision_id": "catalog:80d63fe5861c0006bc0bbc95597229b8eba3fb1fd9672aa1489e00c01b9ecec6", "conditional_on": [], "credit_reason": "All three stated centering identities hold with arbitrary heterogeneous positive Bernoulli propensities, encompassing the common-p corrected statement.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.centered_summand_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1268", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:80d63fe5861c0006bc0bbc95597229b8eba3fb1fd9672aa1489e00c01b9ecec6", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T25", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T25" }, "html": { "body": "

Supplement A p. 26, the summands preceding Lemma C.2 and equation (57). Subtracting tau_i/N gives mean zero, the exact augmented-estimator sum and its HT specialization. Conditioning on a ball containing the own coin gives corrected (57). The printed summand has mean (1/N-1)tau_i; the eight-node example gives -7/8 instead of zero. The own-coin affine transformation also preserves uniform L2-NED of size -4.

", "headline": "Supplement A, Appendix C, the summand τ^i,N(μ1,μ0;d)\\hat\\tau_{i,N}(\\mu_1,\\mu_0;d)τ^i,N​(μ1​,μ0​;d) before Lemma C.2 and eq. (57), with the centring term corrected to τi(d;η)/N\\tau_i(d;\\eta)/Nτi​(d;η)/N", "location": "Supplement A p. 26 (text before Lemma C.2; eq. (57) in the proof of Lemma C.2)" }, "id": "P004-T25", "text": { "body": "Supplement A p. 26, the summands preceding Lemma C.2 and equation (57). Subtracting tau_i/N gives mean zero, the exact augmented-estimator sum and its HT specialization. Conditioning on a ball containing the own coin gives corrected (57). The printed summand has mean (1/N-1)tau_i; the eight-node example gives -7/8 instead of zero. The own-coin affine transformation also preserves uniform L2-NED of size -4.", "headline": "Supplement A, Appendix C, the summand $\\hat\\tau_{i,N}(\\mu_1,\\mu_0;d)$ before Lemma C.2 and eq. (57), with the centring term corrected to $\\tau_i(d;\\eta)/N$", "location": "Supplement A p. 26 (text before Lemma C.2; eq. (57) in the proof of Lemma C.2)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T25" }, { "declarations": [ { "claim_revision_id": "catalog:c3e9af8f1e47ac11c1d4e7c5757c74e736ba27c5e093b4eb11435cddbe28358d", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"Wave1/Gap/GP0042B1.lean:120 assumes the corrected exterior-tail envelope and concludes convergence to zero of the actual centered-effect kernel form, exactly as corrected T26 requires. QuadraticSpacing and KernelConditions (P004_Spatial.lean:1690 and :1700) implement the target's uniform C_b(1+s^2) bound and kernel conditions; MetricNodeDistances implements the standing Appendix C metric premise. Requiring well-defined averages and positive bandwidth at every N rather than eventually concerns only a finite prefix, which can be replaced without changing any hypothesis or conclusion at infinity. These mathematically immaterial prefix conventions do not warrant a major narrowing verdict.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.tail_homophily_correction_proof", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GP0042B1.lean#L119", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:c3e9af8f1e47ac11c1d4e7c5757c74e736ba27c5e093b4eb11435cddbe28358d", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T26", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T26" }, "html": { "body": "

Supplement A pp. 34–35, Lemma C.10, equation (122) and Remark 1. The exterior-tail envelope, bounded outcomes, quadratic spacing and kernel conditions give convergence of the centered-effect kernel form to zero, liminf zero and C 6. The explicit radius b_N^(1/6) diverges and its cube divided by b_N tends to zero. The printed inside-ball premise also implies the conclusion, bounds the diagonal heterogeneity and the kernel form, and forces eventual exact homogeneity for fixed phi. Effects use S3 marginalization, including Bernoulli endpoints; regularity is only eventual.

", "headline": "Supplement A, Appendix C, Lemma C.10 (a sufficient condition for 'C 6'), with the second hypothesis corrected to a sum over the nodes outside the ball", "location": "Supplement A p. 35 (Lemma C.10 and its proof); Remark 1, p. 34" }, "id": "P004-T26", "text": { "body": "Supplement A pp. 34–35, Lemma C.10, equation (122) and Remark 1. The exterior-tail envelope, bounded outcomes, quadratic spacing and kernel conditions give convergence of the centered-effect kernel form to zero, liminf zero and C 6. The explicit radius b_N^(1/6) diverges and its cube divided by b_N tends to zero. The printed inside-ball premise also implies the conclusion, bounds the diagonal heterogeneity and the kernel form, and forces eventual exact homogeneity for fixed phi. Effects use S3 marginalization, including Bernoulli endpoints; regularity is only eventual.", "headline": "Supplement A, Appendix C, Lemma C.10 (a sufficient condition for 'C 6'), with the second hypothesis corrected to a sum over the nodes outside the ball", "location": "Supplement A p. 35 (Lemma C.10 and its proof); Remark 1, p. 34" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T26" }, { "declarations": [ { "claim_revision_id": "catalog:b8d76821795eb0c8262b2f51a37d32996456675438e6e0589f06738be020e3ad", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"Wave1/Gap/GLP0042B2.lean:821 states coverage tending to 1/2 for the explicit sparsePopulation with critical values +/-2, together with the corrected spacing, NED, kernel, bandwidth and homophily conditions and N times AVar tending to zero. T27 expressly permits this +/-2 alternative and its nominal coverage of at least 3/4, so omission of the separate 95-percent quantile calculation is not a fidelity defect. The concrete singleton-node distances are N^2 times the absolute index difference and hence are a metric; at any fixed nonnegative band d, only node 1's assignment can affect a nonzero disk mean and the envelope 1{s <= d} gives the required NED. Thus the unexported metric and other-band NED facts hold for the actual witness, and this refutes precisely the target's nonvacuous reading.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.ned_coverage_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GLP0042B2.lean#L821", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:b8d76821795eb0c8262b2f51a37d32996456675438e6e0589f06738be020e3ad", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T27", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T27" }, "html": { "body": "

Supplement A p. 35 (Proposition 6(v)); compare Proposition 5, p. 29; assumptions on pp. 26, 27, 32, 34. Under the reading of Proposition 6(v) with uniform spacing bound C_b(1+s^2), the sparse Bernoulli witness is a counterexample to its coverage assertion: it satisfies the NED, kernel, bandwidth and homophily assumptions, but its valid Wald intervals have coverage tending to one half, both at nominal 95 percent and with critical values plus/minus two. The statement includes the all-distance NED assertion, exact variance, conditional binomial count law, exceptional probability, estimator and studentized formulas, conditional limits, and the variance rates and bias.

", "headline": "Supplement A, Appendix C, Proposition 6(v) (asymptotic coverage of the Wald interval with the HAC variance under near-epoch dependence), stated without a lower bound on N×AVar(τ^HA(d))N\\times\\mathrm{AVar}(\\hat\\tau_{\\mathrm{HA}}(d))N×AVar(τ^HA​(d)): a design in which the coverage of the nominal 95% interval tends to 12\\frac1221​", "location": "Supplement A p. 35 (Proposition 6, item (v)); compare Proposition 5, p. 29; assumptions on pp. 26, 27, 32, 34" }, "id": "P004-T27", "text": { "body": "Supplement A p. 35 (Proposition 6(v)); compare Proposition 5, p. 29; assumptions on pp. 26, 27, 32, 34. Under the reading of Proposition 6(v) with uniform spacing bound C_b(1+s^2), the sparse Bernoulli witness is a counterexample to its coverage assertion: it satisfies the NED, kernel, bandwidth and homophily assumptions, but its valid Wald intervals have coverage tending to one half, both at nominal 95 percent and with critical values plus/minus two. The statement includes the all-distance NED assertion, exact variance, conditional binomial count law, exceptional probability, estimator and studentized formulas, conditional limits, and the variance rates and bias.", "headline": "Supplement A, Appendix C, Proposition 6(v) (asymptotic coverage of the Wald interval with the HAC variance under near-epoch dependence), stated without a lower bound on $N\\times\\mathrm{AVar}(\\hat\\tau_{\\mathrm{HA}}(d))$: a design in which the coverage of the nominal 95% interval tends to $\\frac12$", "location": "Supplement A p. 35 (Proposition 6, item (v)); compare Proposition 5, p. 29; assumptions on pp. 26, 27, 32, 34" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T27" }, { "declarations": [ { "claim_revision_id": "catalog:f0f988908b693940e585d069fc0f4ee4e801f8140805b1e750bc23e21bdd0221", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'Wave1/Gap/TailBNedHac.lean:1233 has exactly the type of ned_hac_correction at P004_Spatial.lean:1755. That statement includes all three convergence-in-probability approximations, the limiting bias lower bound, and conservative liminf coverage with Nondegenerate required only for coverage, matching corrected T28 rather than the uncorrected printing. Its quadratic spacing and metric premises match the corrected claim and standing Appendix C assumptions. All-N well-definedness and bandwidth positivity affect only a finite prefix, while totalized empty-arm and nonpositive-variance events have vanishing probability under the coverage assumptions, so none of these notes removes a material asymptotic case.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.TailB_ned_hac_correction", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/TailBNedHac.lean#L1205", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:f0f988908b693940e585d069fc0f4ee4e801f8140805b1e750bc23e21bdd0221", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T28", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T28" }, "html": { "body": "

Supplement A p. 35 (Proposition 6); proof pp. 35--39. Under the corrected uniform quadratic spacing condition and the stated NED, kernel, bandwidth and homophily assumptions, the independent five-term V1 is the random row sum, all three raw scaled approximation errors vanish in probability, and the asymptotic bias is nonnegative. With the claimed variance lower bound, defined Wald intervals have conservative liminf coverage for every alpha in (0,1). Gaussian quantile existence is a conclusion.

", "headline": "Supplement A, Appendix C, Proposition 6, with item (v) corrected: the lower bound on N×AVar(τ^HA(d))N\\times\\mathrm{AVar}(\\hat\\tau_{\\mathrm{HA}}(d))N×AVar(τ^HA​(d)) added and the coverage limit stated as a liminf", "location": "Supplement A p. 35 (Proposition 6); proof pp. 35-39" }, "id": "P004-T28", "text": { "body": "Supplement A p. 35 (Proposition 6); proof pp. 35--39. Under the corrected uniform quadratic spacing condition and the stated NED, kernel, bandwidth and homophily assumptions, the independent five-term V1 is the random row sum, all three raw scaled approximation errors vanish in probability, and the asymptotic bias is nonnegative. With the claimed variance lower bound, defined Wald intervals have conservative liminf coverage for every alpha in (0,1). Gaussian quantile existence is a conclusion.", "headline": "Supplement A, Appendix C, Proposition 6, with item (v) corrected: the lower bound on $N\\times\\mathrm{AVar}(\\hat\\tau_{\\mathrm{HA}}(d))$ added and the coverage limit stated as a liminf", "location": "Supplement A p. 35 (Proposition 6); proof pp. 35-39" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T28" }, { "declarations": [ { "claim_revision_id": "catalog:a91b80cba910b26f7321a3069b5297b1a5faf853a37d92e4a5f1dd905235e674", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'P004_HACCompletion.lean:68 exports the corrected five-term HAC expansion, so T29(2) is covered. Its second conjunct at line 79 only repeats the row-sum definition of nedV1 from P004_Spatial.lean:1725. T29(3) instead starts with Vhat_1 defined by the independent five-term expression (117)-(120) and asserts equality to the sum of random vhat rows. Neither that expression nor an equality connecting it to the row sum is in this declaration, so the claimed T29(3) component is missing; the correct random-row definition of Vhat_2 does not supply it.', 'suggested_fix': 'Add a conjunct equating the five terms (117)-(120), with denominators N^2 p^2, N^2(1-p)^2, and N^2 p(1-p), to the displayed nedVRow sum (or define Vhat_1 by those five terms and prove the row-sum equality). Retain the existing HAC expansion.', 'second_review': \"P004_HACCompletion.lean:73 exports T29(2)'s corrected HAC expansion, but line 79 merely repeats the row-sum definition of nedV1 at P004_Spatial.lean:1725. T29(3) explicitly requires equality between the independent five-term expression (117)-(120) and that row sum, and this equality is absent. The missing equality is routine finite algebra: expand the two residual contrasts defined at P004_Spatial.lean:503 and :1713, combine the symmetric cross terms, and split the diagonal using the existing kernel hypotheses, so the statement addition is minor.\", 'severity': 'minor'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P004.hac_expansion", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_HACCompletion.lean#L68", "status": "proved", "support_scope": "limited", "used_in_solution": true }, { "claim_revision_id": "catalog:a91b80cba910b26f7321a3069b5297b1a5faf853a37d92e4a5f1dd905235e674", "conditional_on": [], "credit_reason": "The exact corrected mixed-arm product expansion is proved for arbitrary nonzero real N and nondegenerate real p, including every admissible sample-size and probability instance.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.epsilon_product_difference", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1751", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:a91b80cba910b26f7321a3069b5297b1a5faf853a37d92e4a5f1dd905235e674", "conditional_on": [], "credit_reason": "The corrected epsilon means and N-to-the-minus-two mean-product identity hold for every real matrix Q, extending the claimed distance-kernel specialization.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.ned_epsilon_means", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1567", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:a91b80cba910b26f7321a3069b5297b1a5faf853a37d92e4a5f1dd905235e674", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'Wave1/NedFarCovariance.lean:22 assumes QuadraticSpacing, which P004_Spatial.lean:1690 defines using one constant Cb and one eventual sample-size threshold simultaneously for every nonnegative radius. T29(1) corrects only the far-pair condition to d_ij > 3 b_N; the quoted Appendix C C4 instead puts each radius before the large-sample qualification and the existential constant. The simultaneous bound is needed at the moving radius b_N and is not supplied by that radiuswise assumption, so it excludes substantive cases. Replacing s^2 by 1+s^2 resolves the separate zero-radius defect in printed C4, but does not justify exchanging its quantifiers; calling the uniform repair approved in the row description cannot insert it into the corrected claim.', 'suggested_fix': \"Export T29(1) under the target's stated spacing assumption, retaining actual neighborhood-count factors if the b_N^2 order cannot be obtained. If uniform quadratic spacing is the intended substantive correction, explicitly add 'there exist Cb > 0 and N0 such that for every N >= N0, i, and s >= 0, |B_N(i;s)| <= Cb(1+s^2)' to the corrected target before assigning this theorem that credit.\", 'second_review': \"Wave1/NedFarCovariance.lean:22 assumes QuadraticSpacing, whose definition at P004_Spatial.lean:1690 requires one eventual sample-size threshold valid simultaneously for every radius. T29(1) changes the far-pair condition to d_ij > 3 b_N, while the inherited Appendix C C4 says 'For each radius s' before the large-sample qualification; it supplies no simultaneous bound at a moving bandwidth. Even allowing the separate correction from s^2 to 1+s^2, this additional uniformity excludes substantive cases and must be added to the corrected target or removed from the theorem's hypotheses.\", 'severity': 'major'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P004.Wave1.ned_far_covariance_under_uniform_spacing", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/NedFarCovariance.lean#L18", "status": "proved", "support_scope": "limited", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:a91b80cba910b26f7321a3069b5297b1a5faf853a37d92e4a5f1dd905235e674", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T29", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T29" }, "html": { "body": "

Supplement A p. 33 (Lemma C.9(iii)); p. 34 (eqs. (114)-(121)); pp. 36-37 (eqs. (130)-(136)); pp. 37-38 (eqs. (144)-(148) and footnote 22). The exported far-pair covariance bound and corrected HAC and epsilon identities hold under their stated regularity conditions.

", "headline": "Supplement A, Appendix C, five displays of the HAC-consistency argument, corrected: Lemma C.9(iii); eq. (115); the definitions of V^1\\hat{\\mathrm V}_1V^1​ and V^2\\hat{\\mathrm V}_2V^2​ (after (120) and in (121)); eqs. (135)-(136); eq. (148)", "location": "Supplement A p. 33 (Lemma C.9(iii)); p. 34 (eqs. (114)-(121)); pp. 36-37 (eqs. (130)-(136)); pp. 37-38 (eqs. (144)-(148) and footnote 22)" }, "id": "P004-T29", "text": { "body": "Supplement A p. 33 (Lemma C.9(iii)); p. 34 (eqs. (114)-(121)); pp. 36-37 (eqs. (130)-(136)); pp. 37-38 (eqs. (144)-(148) and footnote 22). The exported far-pair covariance bound and corrected HAC and epsilon identities hold under their stated regularity conditions.", "headline": "Supplement A, Appendix C, five displays of the HAC-consistency argument, corrected: Lemma C.9(iii); eq. (115); the definitions of $\\hat{\\mathrm V}_1$ and $\\hat{\\mathrm V}_2$ (after (120) and in (121)); eqs. (135)-(136); eq. (148)", "location": "Supplement A p. 33 (Lemma C.9(iii)); p. 34 (eqs. (114)-(121)); pp. 36-37 (eqs. (130)-(136)); pp. 37-38 (eqs. (144)-(148) and footnote 22)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T29" }, { "declarations": [ { "claim_revision_id": "catalog:a49fbc19ef856d25e387f063e64d9642f292d9bace0134aa0ea2b8a0105f1670", "conditional_on": [], "credit_reason": "The finite cardinality ratio states the full uniform leave-one-out conditional law with exactly N1-a treatments and the count bounds ensure nonzero conditioning mass.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.complete_randomization_conditioning", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_ConditioningCompletion.lean#L38", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:a49fbc19ef856d25e387f063e64d9642f292d9bace0134aa0ea2b8a0105f1670", "conditional_on": [], "credit_reason": "The supporting population-arm-mean identity is an exact algebraic equality even for arbitrary real propensity arrays and the empty population.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.barMu_difference", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L539", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:a49fbc19ef856d25e387f063e64d9642f292d9bace0134aa0ea2b8a0105f1670", "conditional_on": [], "credit_reason": "The supporting independence and marginalization identity allows heterogeneous Bernoulli propensities and specializes to the common-p Bernoulli comparison in the target. The row text says T39 but is credited to T30 here; the formula genuinely supports the latter's Bernoulli comparison, so this is only a cross-target label.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.marginal_is_marginalization", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L362", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:a49fbc19ef856d25e387f063e64d9642f292d9bace0134aa0ea2b8a0105f1670", "conditional_on": [], "credit_reason": "{'source': 'dispute', 'reason': 'P004/P004_Spatial.lean:1312 asserts equality of the two assignment-indexed functions, with no additional hypotheses. Reading augmented at P004/P004_Spatial.lean:111, barMu at :83, and linearHajek at :117 shows that, for positive N and constant interior p, both sides expand to barMu₁−barMu₀ plus the treated residual sum divided by Np minus the control residual sum divided by N(1−p). This is precisely the corrected assignmentwise identity credited in T30(2), and the distinct arm means agree with the estimator-class formula in [Supplement A.5 of the preprint](https://arxiv.org/pdf/2010.13599#page=64). The proof by rfl is legitimate here: the definition gives the actual corrected formula, rather than assuming an unproved identification or building a statistical conclusion into a premise. This row need not independently establish the asymptotic linearization, the other T30 component, or the error of the equal-arm substitution.', 'defensible_statement': 'For positive N and common 0p. 751 (PDF p. 8), end of Section 3; Supplement A p. 13, eq. (33), with eq. (32) on p. 12. Complete randomization gives uniform conditional remainder laws with N1-1 and N1 treated nodes, positive conditioning events, and disjoint supports. Under Bernoulli assignment the two remainder laws coincide. The corrected arm centers give the explicit linearized Hajek estimator and the same variance; the wrong equal-center fit agrees identically exactly when the centers agree. The approximation in (33) has the exact nonnegative quadratic remainder in the interaction correction Delta, with the constant independent of both fits.

", "headline": "Two misprinted formulas among the local slips of finding P004-E022, corrected: the numbers of treated nodes over which one marginalizes under complete randomization (end of Section 3), and the second argument in eq. (33) of Supplement A.5", "location": "p. 751 (PDF p. 8), first paragraph (end of Section 3); Supplement A p. 13, eq. (33)" }, "id": "P004-T30", "text": { "body": "p. 751 (PDF p. 8), end of Section 3; Supplement A p. 13, eq. (33), with eq. (32) on p. 12. Complete randomization gives uniform conditional remainder laws with N1-1 and N1 treated nodes, positive conditioning events, and disjoint supports. Under Bernoulli assignment the two remainder laws coincide. The corrected arm centers give the explicit linearized Hajek estimator and the same variance; the wrong equal-center fit agrees identically exactly when the centers agree. The approximation in (33) has the exact nonnegative quadratic remainder in the interaction correction Delta, with the constant independent of both fits.", "headline": "Two misprinted formulas among the local slips of finding P004-E022, corrected: the numbers of treated nodes over which one marginalizes under complete randomization (end of Section 3), and the second argument in eq. (33) of Supplement A.5", "location": "p. 751 (PDF p. 8), first paragraph (end of Section 3); Supplement A p. 13, eq. (33)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T30" }, { "declarations": [ { "claim_revision_id": "catalog:75e1a4611f585c1b1e163a380d3608ad66f9b5bbfbd8929d5df7f5023f4fbb54", "conditional_on": [], "credit_reason": "Only T31(a) is assigned to this row; coverage of the entire target is assessed separately.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.hajek_ols", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_RegressionCompletion.lean#L257", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:75e1a4611f585c1b1e163a380d3608ad66f9b5bbfbd8929d5df7f5023f4fbb54", "conditional_on": [], "credit_reason": "The generalization preserves the ordinary invertible-design specialization. No conditional published theorem is assumed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.sandwich_sum_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_MatrixCompletion.lean#L108", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:75e1a4611f585c1b1e163a380d3608ad66f9b5bbfbd8929d5df7f5023f4fbb54", "conditional_on": [], "credit_reason": "All PSD-order conclusions of the assigned component are exported, including the scalar maximum bound.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.spectral_psd_order", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_MatrixCompletion.lean#L127", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:75e1a4611f585c1b1e163a380d3608ad66f9b5bbfbd8929d5df7f5023f4fbb54", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T31", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T31" }, "html": { "body": "

pp. 754, 756 and 761, eqs. (16), (18), (19), (21)-(22), footnotes 14-15 and 22. The corrected count identities hold for every assignment; with both arms nonempty the OLS fit is unique, the two sandwich formulas agree, and eigenvalue truncation gives the stated semidefinite ordering and treatment-entry bound; the sharp null indexes outcomes by x.

", "headline": "Section 5 formulas written with nnn for NNN (the treated count N1N_1N1​ before eq. (16); the eigen-decomposition of the kernel matrix after eq. (21)) and footnote 22 (the sharp null written with the node index), corrected; with the ordering Σ^HACPD(d)≥Σ^HAC(d)\\hat\\Sigma^{\\mathrm{PD}}_{\\mathrm{HAC}}(d)\\ge\\hat\\Sigma_{\\mathrm{HAC}}(d)Σ^HACPD​(d)≥Σ^HAC​(d) that the corrected eigen-decomposition gives", "location": "p. 754 (PDF p. 11), text before eq. (16); p. 756 (PDF p. 13), eqs. (21)-(22) and footnotes 14-15; p. 761 (PDF p. 18), footnote 22" }, "id": "P004-T31", "text": { "body": "pp. 754, 756 and 761, eqs. (16), (18), (19), (21)-(22), footnotes 14-15 and 22. The corrected count identities hold for every assignment; with both arms nonempty the OLS fit is unique, the two sandwich formulas agree, and eigenvalue truncation gives the stated semidefinite ordering and treatment-entry bound; the sharp null indexes outcomes by x.", "headline": "Section 5 formulas written with $n$ for $N$ (the treated count $N_1$ before eq. (16); the eigen-decomposition of the kernel matrix after eq. (21)) and footnote 22 (the sharp null written with the node index), corrected; with the ordering $\\hat\\Sigma^{\\mathrm{PD}}_{\\mathrm{HAC}}(d)\\ge\\hat\\Sigma_{\\mathrm{HAC}}(d)$ that the corrected eigen-decomposition gives", "location": "p. 754 (PDF p. 11), text before eq. (16); p. 756 (PDF p. 13), eqs. (21)-(22) and footnotes 14-15; p. 761 (PDF p. 18), footnote 22" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T31" }, { "declarations": [ { "claim_revision_id": "catalog:bd83c6f61bb9a47642e19fc8fba69942bde615b52225a2aa81cf7417fd21f6c9", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P004_VarianceComponentsCompletion.lean:187 states both T32(a)'s exact HT variance and T32(c)'s explicit finite-N upper bound. Expanding secondMomentBound and homophilyQ (P004_Spatial.lean:569 and :579) restores the diagonal squared-effect subtraction and every signed local pair mean-product term, so the regrouping omits nothing. The self-containing symmetric neighborhoods and absence of shared sources outside them are implied by the claim's geometric neighborhoods and C3. Although the conjunction requests a strict outcome bound even for its exact-identity clause, every finite assignment table admits such a bound, so the free parameter b excludes no finite table from that identity; the numerical bound uses exactly the claim's C2 assumption.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.ht_variance_bound", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_VarianceComponentsCompletion.lean#L187", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:bd83c6f61bb9a47642e19fc8fba69942bde615b52225a2aa81cf7417fd21f6c9", "conditional_on": [], "credit_reason": "The weaker hypotheses enlarge the domain while preserving all four arm-pair bounds and the paper's b and c_N meanings.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.ht_variance_components", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_VarianceComponentsCompletion.lean#L138", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:bd83c6f61bb9a47642e19fc8fba69942bde615b52225a2aa81cf7417fd21f6c9", "conditional_on": [], "credit_reason": "WellDefined makes the meaning of a suitable spatial average explicit; it does not add a moment assumption beyond the integrals defining the claim. The C4 quantifiers concern fixed radii, which suffice for this fixed-d O(1/N) conclusion.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.ht_variance_order", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_VarianceOrderCompletion.lean#L70", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:bd83c6f61bb9a47642e19fc8fba69942bde615b52225a2aa81cf7417fd21f6c9", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T32", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T32" }, "html": { "body": "

Supplement A pp. 2-3, Lemma A.2 and its corrected proof displays. The HT variance has the exact local expansion, the corrected component and total bounds, and order 1/N under the stated boundedness and neighborhood conditions.

", "headline": "Supplement A, Lemma A.2 (variance of the Horvitz-Thompson estimator and its O(1/N)O(1/N)O(1/N) rate), with the two displays of its proof corrected (normalization 1N2p\\frac1{N^2p}N2p1​ and orders O(1N)O(\\frac1N)O(N1​))", "location": "Supplement A p. 2 (Lemma A.2); p. 3 (proof, displays after 'We also have:' and after 'Moreover,')" }, "id": "P004-T32", "text": { "body": "Supplement A pp. 2-3, Lemma A.2 and its corrected proof displays. The HT variance has the exact local expansion, the corrected component and total bounds, and order 1/N under the stated boundedness and neighborhood conditions.", "headline": "Supplement A, Lemma A.2 (variance of the Horvitz-Thompson estimator and its $O(1/N)$ rate), with the two displays of its proof corrected (normalization $\\frac1{N^2p}$ and orders $O(\\frac1N)$)", "location": "Supplement A p. 2 (Lemma A.2); p. 3 (proof, displays after 'We also have:' and after 'Moreover,')" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T32" }, { "declarations": [ { "claim_revision_id": "catalog:e261752c567bbe4f825fd8836c936a044281d45d0ec3084f3b1974f0d80b0ead", "conditional_on": [], "credit_reason": "This is the variance of the linearized estimator as specified by the target and standing assumption S6.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.hajek_variance_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_VarianceCompletion.lean#L125", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e261752c567bbe4f825fd8836c936a044281d45d0ec3084f3b1974f0d80b0ead", "conditional_on": [], "credit_reason": "Setting p_i=p and g_i(a)=barMu(a) gives h_i=tau_i-AME, exactly the C5 expression; no consistency of an estimated HAC variance is claimed by this component.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.hajek_homophily_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_VarianceCompletion.lean#L105", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e261752c567bbe4f825fd8836c936a044281d45d0ec3084f3b1974f0d80b0ead", "conditional_on": [], "credit_reason": "Positive variance is in T33(d) itself, so the nonzero standardization premise excludes no claimed case.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.ht_standardized_summands", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_VarianceCompletion.lean#L153", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:e261752c567bbe4f825fd8836c936a044281d45d0ec3084f3b1974f0d80b0ead", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T33", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T33" }, "html": { "body": "

Supplement A pp. 5-6, Lemma A.4 and A.3.1. The corrected linearized Hajek variance has the stated expansion and homophily identity, its second-moment bound holds exactly when C5 holds, and the corrected positive-variance HT summands have the stated sum, zero means and unit sum variance.

", "headline": "Supplement A, Lemma A.4 (variance of the linearized Hajek estimator and the bound V~HA\\tilde{\\mathrm V}_{\\mathrm{HA}}V~HA​, lines (6)-(7)) and the definition of UiU_iUi​ in A.3.1, with the malformed terms corrected", "location": "Supplement A p. 5 (Lemma A.4, lines (1)-(7), and its proof); p. 6 (A.3.1, definition of UiU_iUi​)" }, "id": "P004-T33", "text": { "body": "Supplement A pp. 5-6, Lemma A.4 and A.3.1. The corrected linearized Hajek variance has the stated expansion and homophily identity, its second-moment bound holds exactly when C5 holds, and the corrected positive-variance HT summands have the stated sum, zero means and unit sum variance.", "headline": "Supplement A, Lemma A.4 (variance of the linearized Hajek estimator and the bound $\\tilde{\\mathrm V}_{\\mathrm{HA}}$, lines (6)-(7)) and the definition of $U_i$ in A.3.1, with the malformed terms corrected", "location": "Supplement A p. 5 (Lemma A.4, lines (1)-(7), and its proof); p. 6 (A.3.1, definition of $U_i$)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T33" }, { "declarations": [ { "claim_revision_id": "source-claim-import-2026-11-08:P004-T34:1", "conditional_on": [], "credit_reason": "The sign of p^2-(N1/N)^2 is reversed in the factored numerator, which is immaterial inside the absolute value.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.denominator_absolute_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1239", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "source-claim-import-2026-11-08:P004-T34:1", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P004_Spatial.lean:1269 states the exact scalar values +9/25 and -9/25 and their failed ordering for N = 8, N1 = 5, p = 1/2 and S = -1/4, matching the arithmetic component of T34 and the row's own description. The row expressly covers this arithmetic rather than the entire residual-and-kernel construction, so absence of that construction from this declaration is not an omitted conclusion within its component. The target's residual vector is consistent with the supplied S: its treated off-diagonal product sum is -2 and division by N gives -1/4.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.missing_absolute_witness", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1255", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "source-claim-import-2026-11-08:P004-T34:1", "conditional_on": [], "credit_reason": "The kernel and distance generalizations are valid for the paper's positive bandwidths and nonnegative geographic distances. No stochastic order is stated in this row.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.kernel_cross_sum_bound", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_KernelCompletion.lean#L39", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "source-claim-import-2026-11-08:P004-T34:1", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'Wave1/Gap/GP0042B1.lean:332 states exactly the O_p(N^(-1/2)) order of |N^2/N1^2 - 1/p^2| under independent Bernoulli(p) assignment with 0 < p < 1, the component assigned by T34(c). BigOp (P004_Spatial.lean:191) is the usual eventual tightness definition at the displayed inverse-square-root rate. Lean assigns the expression the finite value 1/p^2 when N1 = 0, whereas the paper leaves it undefined there, but that event has probability (1-p)^N tending to zero. Any finite extension on that vanishing-probability event gives the same stochastic order, so the totalization is immaterial.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.bernoulli_denominator_rate_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GP0042B1.lean#L328", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "source-claim-import-2026-11-08:P004-T34:1", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T34", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T34" }, "html": { "body": "

Supplement A p. 36, eqs. (127)-(129), and p. 35, eq. (124). The corrected absolute-value identities, finite kernel cross-sum bound and Bernoulli denominator order hold, together with the scalar witness arithmetic. Restricted: omit the composed bandwidth conclusion O_p(N^(-1/2)b_N^2)=o_p(1) and the realization of the eight-node assignment, residuals and kernel (including the certified value S=-1/4); only its scalar arithmetic is retained. Require positive sample size and bandwidth for the finite kernel bound, and use zero denominator error on empty treated arms, so no reciprocal of a zero count is asserted there.

", "headline": "Supplement A, Appendix C, eqs. (127)-(129) in the proof of Proposition 6(i), with the absolute value restored on the factor N2N12−1p2\\frac{N^2}{N_1^2}-\\frac1{p^2}N12​N2​−p21​", "location": "Supplement A p. 36 (eqs. (127)-(129)); eq. (124), p. 35" }, "id": "P004-T34", "text": { "body": "Supplement A p. 36, eqs. (127)-(129), and p. 35, eq. (124). The corrected absolute-value identities, finite kernel cross-sum bound and Bernoulli denominator order hold, together with the scalar witness arithmetic. Restricted: omit the composed bandwidth conclusion O_p(N^(-1/2)b_N^2)=o_p(1) and the realization of the eight-node assignment, residuals and kernel (including the certified value S=-1/4); only its scalar arithmetic is retained. Require positive sample size and bandwidth for the finite kernel bound, and use zero denominator error on empty treated arms, so no reciprocal of a zero count is asserted there.", "headline": "Supplement A, Appendix C, eqs. (127)-(129) in the proof of Proposition 6(i), with the absolute value restored on the factor $\\frac{N^2}{N_1^2}-\\frac1{p^2}$", "location": "Supplement A p. 36 (eqs. (127)-(129)); eq. (124), p. 35" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T34" }, { "declarations": [ { "claim_revision_id": "catalog:69d8c483aa86271bc7d602f74bc45ac362ca5f13a65164e43245fb59651a6f6f", "conditional_on": [], "credit_reason": "Both usable narrower rows negate conjunctions universally quantified over arbitrary metric measurable spatial spaces. Neither exported type restricts the counterexample to the canonical two-dimensional spatial domain. Their weaker larger-domain negations do not refute the printed proposition, and no positive component of that proposition is established by these rows.", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P004.Wave1.printed_ipw_proposition_refuted", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/IpwProposition.lean#L30", "status": "counterexample_proved", "support_scope": "limited", "used_in_solution": null }, { "claim_revision_id": "catalog:69d8c483aa86271bc7d602f74bc45ac362ca5f13a65164e43245fb59651a6f6f", "conditional_on": [], "credit_reason": "Both usable narrower rows negate conjunctions universally quantified over arbitrary metric measurable spatial spaces. Neither exported type restricts the counterexample to the canonical two-dimensional spatial domain. Their weaker larger-domain negations do not refute the printed proposition, and no positive component of that proposition is established by these rows.", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P004.Wave1.printed_ipw_bound_and_literal_coverage_refuted", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/IpwBoundCoverage.lean#L30", "status": "counterexample_proved", "support_scope": "limited", "used_in_solution": null } ], "formalization": { "claim_revision_id": "catalog:69d8c483aa86271bc7d602f74bc45ac362ca5f13a65164e43245fb59651a6f6f", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T35", "decision": "omitted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T35" }, "html": { "body": "

No current theorem is supplied for this target.

", "headline": "Supplement A, Appendix A.6, Proposition 4 (linearization, variance, normal limit and coverage for the IPW estimator with an estimated logistic propensity score), as printed", "location": "Supplement A pp. 15-16 (Proposition 4, eqs. (36)-(40) and items (i)-(ii)); proof pp. 16-17; advertised in the article, Section 6.4, p. 759 (PDF p. 16)" }, "id": "P004-T35", "text": { "body": "No current theorem is supplied for this target.", "headline": "Supplement A, Appendix A.6, Proposition 4 (linearization, variance, normal limit and coverage for the IPW estimator with an estimated logistic propensity score), as printed", "location": "Supplement A pp. 15-16 (Proposition 4, eqs. (36)-(40) and items (i)-(ii)); proof pp. 16-17; advertised in the article, Section 6.4, p. 759 (PDF p. 16)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T35" }, { "declarations": [ { "claim_revision_id": "CT-01:P004-T36:accepted", "conditional_on": [], "credit_reason": "The only row has final status wrong_credit: its numerator is fitted IPW, whereas the corrected canonical as-printed assertion retains the distinct Hajek numerator. It proves neither that coverage assertion nor its refutation.", "credit_status": "wrong_credit", "fidelity_status_baseline": "wrong_credit", "lean_name": "AronowAudit.P004.Wave1.printed_observational_coverage_refuted", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/ObservationalCoverage.lean#L29", "status": "counterexample_proved", "support_scope": "withdrawn", "used_in_solution": null } ], "formalization": { "claim_revision_id": "CT-01:P004-T36:accepted", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T36", "decision": "omitted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T36" }, "html": { "body": "

No current theorem is supplied for this target.

", "headline": "Supplement A, Appendix A.6: the inverse-propensity-weighted residual ϵ^i\\hat\\epsilon_iϵ^i​, the observational SAH variance estimator V^SAHobs(d)\\hat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)V^SAHobs​(d) and item (i) of Proposition 4, as printed", "location": "Supplement A p. 15 (definitions of ϵ^i\\hat\\epsilon_iϵ^i​ and V^SAHobs(d)\\hat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)V^SAHobs​(d)); p. 16 (Proposition 4, item (i))" }, "id": "P004-T36", "text": { "body": "No current theorem is supplied for this target.", "headline": "Supplement A, Appendix A.6: the inverse-propensity-weighted residual $\\hat\\epsilon_i$, the observational SAH variance estimator $\\hat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)$ and item (i) of Proposition 4, as printed", "location": "Supplement A p. 15 (definitions of $\\hat\\epsilon_i$ and $\\hat V^{\\mathrm{obs}}_{\\mathrm{SAH}}(d)$); p. 16 (Proposition 4, item (i))" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T36" }, { "declarations": [ { "claim_revision_id": "catalog:9b6bfbcdfc0a54bf8975faf77c24763ae2ad607645fe03d0a784dca5ffd6b21c", "conditional_on": [], "credit_reason": "The narrower row negates an equality for arbitrary real circle tables and positive bin weights. It exports no binary spatial population, common outcome field, ordered distance bins or regional partition satisfying the canonical domain. The empirical estimate and p-value are not mathematical targets.", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P004.ConfidenceFollowup02.printed_area_sum_refuted", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_ConfidenceCompletion.lean#L15", "status": "counterexample_proved", "support_scope": "limited", "used_in_solution": null } ], "formalization": { "claim_revision_id": "catalog:9b6bfbcdfc0a54bf8975faf77c24763ae2ad607645fe03d0a784dca5ffd6b21c", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T37", "decision": "omitted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T37" }, "html": { "body": "

No current theorem is supplied for this target.

", "headline": "Section 8.1, the sentence giving the summed Hajek estimates between zero and five km and its reading as a decline in deforested area, as printed", "location": "p. 763 (PDF p. 20), last paragraph; continued on p. 764 (PDF p. 21)" }, "id": "P004-T37", "text": { "body": "No current theorem is supplied for this target.", "headline": "Section 8.1, the sentence giving the summed Hajek estimates between zero and five km and its reading as a decline in deforested area, as printed", "location": "p. 763 (PDF p. 20), last paragraph; continued on p. 764 (PDF p. 21)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T37" }, { "declarations": [ { "claim_revision_id": "catalog:f47abf5c2f29ce17ae9ec312db0ac2637fb0cdd8331ba5b709026444fefbed6b", "conditional_on": [], "credit_reason": "Accepted generalization; its specialization loses none of the cases of the component credited to this row.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.ht_unbiased", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L287", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f47abf5c2f29ce17ae9ec312db0ac2637fb0cdd8331ba5b709026444fefbed6b", "conditional_on": [], "credit_reason": "Auxiliary component already subsumed by the used main declaration; 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p. 754, Proposition 1, eq. (14). For positive sample size and an interior common Bernoulli probability, the HT estimator is unbiased for the AME defined by forced-coordinate marginalization, without a restriction on interference.

", "headline": "Proposition 1 (Unbiasedness)", "location": "p. 754 (PDF p. 11), Proposition 1, eq. (14); proof: Supplement A pp. 1-2 (Lemma A.1 and A.1.1)" }, "id": "P004-T39", "text": { "body": "p. 754, Proposition 1, eq. (14). For positive sample size and an interior common Bernoulli probability, the HT estimator is unbiased for the AME defined by forced-coordinate marginalization, without a restriction on interference.", "headline": "Proposition 1 (Unbiasedness)", "location": "p. 754 (PDF p. 11), Proposition 1, eq. (14); proof: Supplement A pp. 1-2 (Lemma A.1 and A.1.1)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T39" }, { "declarations": [ { "claim_revision_id": "catalog:f8a49b46e11e0ae8d6aa8096f7fce97c651cce0d585f98edd0070bf3bdc9aa8c", "conditional_on": [], "credit_reason": "Auxiliary component already subsumed by the used main declaration; no separate helper assertion is added to the Challenge.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.additive_potential_shift", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1790", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f8a49b46e11e0ae8d6aa8096f7fce97c651cce0d585f98edd0070bf3bdc9aa8c", "conditional_on": [], "credit_reason": "Auxiliary component already subsumed by the used main declaration; no separate helper assertion is added to the Challenge.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.marginal_law_shift", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1813", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f8a49b46e11e0ae8d6aa8096f7fce97c651cce0d585f98edd0070bf3bdc9aa8c", "conditional_on": [], "credit_reason": "Final status wrong_credit: the finite-outcome version omits the canonical general spatial-integral cases; the credited integral theorem is used instead.", "credit_status": "wrong_credit", "fidelity_status_baseline": "wrong_credit", "lean_name": "AronowAudit.P004.additive_design_invariance", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1829", "status": "proved", "support_scope": "withdrawn", "used_in_solution": false }, { "claim_revision_id": "catalog:f8a49b46e11e0ae8d6aa8096f7fce97c651cce0d585f98edd0070bf3bdc9aa8c", "conditional_on": [], "credit_reason": "This row supplies the full general spatial result and repairs the coverage gap left by the finite-support row; arbitrary marginal laws are represented without a Bernoulli or overlap restriction.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.additive_spatial_design_invariance", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L2156", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:f8a49b46e11e0ae8d6aa8096f7fce97c651cce0d585f98edd0070bf3bdc9aa8c", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T40", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T40" }, "html": { "body": "

pp. 756-757, eqs. (23)-(25). For additive potential outcomes and genuine spatial averages, every marginal design law gives the point effect g_i(x), its spatial average as the node effect, and their node mean as the AME, independently of the assignment law.

", "headline": "Section 6.1 (Structural interpretation of the AME), eqs. (23)-(25): under additive effects the marginalized effects, and hence the AME, do not depend on the assignment design", "location": "pp. 756-757 (PDF pp. 13-14), eqs. (23), (24), (25)" }, "id": "P004-T40", "text": { "body": "pp. 756-757, eqs. (23)-(25). For additive potential outcomes and genuine spatial averages, every marginal design law gives the point effect g_i(x), its spatial average as the node effect, and their node mean as the AME, independently of the assignment law.", "headline": "Section 6.1 (Structural interpretation of the AME), eqs. (23)-(25): under additive effects the marginalized effects, and hence the AME, do not depend on the assignment design", "location": "pp. 756-757 (PDF pp. 13-14), eqs. (23), (24), (25)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T40" }, { "declarations": [ { "claim_revision_id": "catalog:bad97a19efb70dd63eb911b2bc57e22cbf5518deb68243caf538ec8bf38ef7ca", "conditional_on": [], "credit_reason": "Accepted generalization; its specialization loses none of the cases of the component credited to this row. WellDefined is read as the existence requirements of the paper's suitable-measure average, not an added outcome-moment or metric-node-distance assumption.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.ht_central_limit", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_SpatialCLTCompletion.lean#L238", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:bad97a19efb70dd63eb911b2bc57e22cbf5518deb68243caf538ec8bf38ef7ca", "conditional_on": [], "credit_reason": "Accepted generalization; its specialization loses none of the cases of the component credited to this row. WellDefined is read as the existence requirements of the paper's suitable-measure average, not an added outcome-moment or metric-node-distance assumption.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.ht_variance_order", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_VarianceOrderCompletion.lean#L70", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:bad97a19efb70dd63eb911b2bc57e22cbf5518deb68243caf538ec8bf38ef7ca", "conditional_on": [], "credit_reason": "Auxiliary component already subsumed by the used main declaration; no separate helper assertion is added to the Challenge.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.design_dependency_graph", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_FiniteGraphCompletion.lean#L201", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:bad97a19efb70dd63eb911b2bc57e22cbf5518deb68243caf538ec8bf38ef7ca", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T41", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T41" }, "html": { "body": "

p. 754, Proposition 2, eq. (15), and Supplement A Lemma A.2. Under C1-C4 the HT variance is O(1/N), and under the eventual positive N-times-variance lower bound its AME-centred, actual-variance-standardized statistic tends to N(0,1).

", "headline": "Proposition 2 (Asymptotic Distribution for the Horvitz-Thompson estimator)", "location": "p. 754 (PDF p. 11), Proposition 2, eq. (15), footnote 7; proof: Supplement A p. 6 (A.3 and A.3.1), with Lemmas A.2 (p. 2) and A.5 (p. 6)" }, "id": "P004-T41", "text": { "body": "p. 754, Proposition 2, eq. (15), and Supplement A Lemma A.2. Under C1-C4 the HT variance is O(1/N), and under the eventual positive N-times-variance lower bound its AME-centred, actual-variance-standardized statistic tends to N(0,1).", "headline": "Proposition 2 (Asymptotic Distribution for the Horvitz-Thompson estimator)", "location": "p. 754 (PDF p. 11), Proposition 2, eq. (15), footnote 7; proof: Supplement A p. 6 (A.3 and A.3.1), with Lemmas A.2 (p. 2) and A.5 (p. 6)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T41" }, { "declarations": [ { "claim_revision_id": "catalog:8effbe633c8c01e3afa031da448ca08af610d764a1990616f6ac28571a47a005", "conditional_on": [], "credit_reason": "Accepted generalization; its specialization loses none of the cases of the component credited to this row. WellDefined is read as the existence requirements of the paper's suitable-measure average, not an added outcome-moment or metric-node-distance assumption.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.Wave1.hajek_central_limit_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/GP0042B1.lean#L927", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8effbe633c8c01e3afa031da448ca08af610d764a1990616f6ac28571a47a005", "conditional_on": [], "credit_reason": "Auxiliary component already subsumed by the used main declaration; no separate helper assertion is added to the Challenge.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.hajek_ratio_identity", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L1947", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:8effbe633c8c01e3afa031da448ca08af610d764a1990616f6ac28571a47a005", "conditional_on": [], "credit_reason": "Auxiliary component already subsumed by the used main declaration; no separate helper assertion is added to the Challenge.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.design_dependency_graph", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_FiniteGraphCompletion.lean#L201", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:8effbe633c8c01e3afa031da448ca08af610d764a1990616f6ac28571a47a005", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T42", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T42" }, "html": { "body": "

p. 754, Proposition 3, eq. (17), and Supplement A pp. 4-6. Under C1-C4 and the eventual positive N-times-AVar lower bound, the Hajek statistic centred at the AME and standardized by the variance of the corrected linearization tends to N(0,1), with zero means assigned to empty arms.

", "headline": "Proposition 3 (Asymptotic Distribution for the Hajek estimator)", "location": "p. 754 (PDF p. 11), Proposition 3, eq. (17), footnote 9; proof: Supplement A p. 6 (A.3.1), with Lemmas A.3 (p. 4), A.4 (p. 5) and A.5 (p. 6)" }, "id": "P004-T42", "text": { "body": "p. 754, Proposition 3, eq. (17), and Supplement A pp. 4-6. Under C1-C4 and the eventual positive N-times-AVar lower bound, the Hajek statistic centred at the AME and standardized by the variance of the corrected linearization tends to N(0,1), with zero means assigned to empty arms.", "headline": "Proposition 3 (Asymptotic Distribution for the Hajek estimator)", "location": "p. 754 (PDF p. 11), Proposition 3, eq. (17), footnote 9; proof: Supplement A p. 6 (A.3.1), with Lemmas A.3 (p. 4), A.4 (p. 5) and A.5 (p. 6)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T42" }, { "declarations": [ { "claim_revision_id": "catalog:6fd8864971dc2fce662570304e573cf9508f4de1b89be6052cbc91b46a472df3", "conditional_on": [], "credit_reason": "The narrower negative row ranges over arbitrary metric measurable spaces and does not export a refutation in the canonical two-dimensional spatial domain. The other two rows prove finite oracle variance bounds; those are auxiliaries and establish neither printed ordinary-limit coverage conclusion. A package of these helper facts would not state the canonical coverage claim.", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P004.Wave1.experimental_wald_coverage_printed_false", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/Gap/DecideD.lean#L718", "status": "counterexample_proved", "support_scope": "limited", "used_in_solution": null }, { "claim_revision_id": "catalog:6fd8864971dc2fce662570304e573cf9508f4de1b89be6052cbc91b46a472df3", "conditional_on": [], "credit_reason": "Finite oracle variance identity or bound only; it does not establish the canonical coverage limit.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.hajek_homophily_identity", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_VarianceCompletion.lean#L105", "status": "proved", "support_scope": "historical declaration", "used_in_solution": null }, { "claim_revision_id": "catalog:6fd8864971dc2fce662570304e573cf9508f4de1b89be6052cbc91b46a472df3", "conditional_on": [], "credit_reason": "Finite oracle variance identity or bound only; it does not establish the canonical coverage limit.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.hajek_sah_bound", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_VarianceCompletion.lean#L133", "status": "proved", "support_scope": "historical declaration", "used_in_solution": null } ], "formalization": { "claim_revision_id": "catalog:6fd8864971dc2fce662570304e573cf9508f4de1b89be6052cbc91b46a472df3", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T43", "decision": "omitted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T43" }, "html": { "body": "

No current theorem is supplied for this target.

", "headline": "Proposition 4 (asymptotically conservative Wald intervals for the Hajek estimator with the SAH variance estimator, and with the spatial HAC variance estimator under C5)", "location": "p. 755 (PDF p. 12), eqs. (19)-(20), C5, Proposition 4, footnotes 10-12; proof: Supplement A pp. 6-11 (A.4.1 Proposition 1, A.4.2 Proposition 2, A.4.3)" }, "id": "P004-T43", "text": { "body": "No current theorem is supplied for this target.", "headline": "Proposition 4 (asymptotically conservative Wald intervals for the Hajek estimator with the SAH variance estimator, and with the spatial HAC variance estimator under C5)", "location": "p. 755 (PDF p. 12), eqs. (19)-(20), C5, Proposition 4, footnotes 10-12; proof: Supplement A pp. 6-11 (A.4.1 Proposition 1, A.4.2 Proposition 2, A.4.3)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T43" }, { "declarations": [ { "claim_revision_id": "catalog:0e005ac3af8bfa3127129b780f602dac605ac432eb4fcffd98a48a0eb21b28c7", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': \"AronowAudit/P004/Wave1/NedHeadline.lean:25 assumes QuadraticSpacing, defined at AronowAudit/P004/P004_Spatial.lean:1690 as one positive Cb and one eventual sample-size threshold valid for every radius simultaneously. Appendix C's C4 instead says 'For each radius s >= 0 and all large sample sizes N'; even allowing the recorded small-radius correction to Cb*(1+s^2), radiuswise eventual bounds do not imply this uniform threshold. For example, N nodes with all distinct-node distances a_N = N^(1/4) have singleton balls eventually at every fixed radius, but at s = a_N the ratio of ball size to 1+s^2 is N/(1+sqrt(N)), which is unbounded. Such metrics can be realized by point nodes at the leaves of a planar star with the permitted path metric, and bounded own-node outcomes can satisfy NED, so the excluded cases are substantive. The replacement of Cb*s^2 at radius zero repairs its impossible literal center bound, but does not justify the additional uniformity.\", 'suggested_fix': \"Replace the spacing hypothesis by the radiuswise version: exists Cb > 0, for every s >= 0, eventually in N, for every i, card(B_N(i;s)) <= Cb*(1+s^2), and establish the conclusions under that hypothesis. Keep the positive small-radius bound explicit as the audit's correction; if the conclusions require uniformity, credit the current theorem to a corrected target that explicitly adds the stronger uniform assumption.\", 'second_review': \"Wave1/NedHeadline.lean:25 assumes QuadraticSpacing, defined at P004_Spatial.lean:1690 with one constant and one eventual sample-size threshold valid for all radii simultaneously. T44 instead inherits Appendix C C4's 'For each radius s' followed by 'all large sample sizes N', which permits radius-dependent thresholds. Even after correcting the impossible zero-radius bound to Cb*(1+s^2), equilateral node metrics with separation N^(1/4) satisfy a fixed-radius eventual bound but violate the simultaneous quadratic bound at the moving separation. The stronger uniform hypothesis therefore materially narrows the credited headline theorem and requires an explicit target correction or a weaker theorem premise.\", 'severity': 'major'}", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P004.Wave1.ned_headline_under_uniform_spacing", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/Wave1/NedHeadline.lean#L20", "status": "proved", "support_scope": "limited", "used_in_solution": true }, { "claim_revision_id": "catalog:0e005ac3af8bfa3127129b780f602dac605ac432eb4fcffd98a48a0eb21b28c7", "conditional_on": [], "credit_reason": "Auxiliary component already subsumed by the used main declaration; no separate helper assertion is added to the Challenge.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.ht_unbiased", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L287", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:0e005ac3af8bfa3127129b780f602dac605ac432eb4fcffd98a48a0eb21b28c7", "conditional_on": [], "credit_reason": "Auxiliary component already subsumed by the used main declaration; no separate helper assertion is added to the Challenge.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P004.ned_exact_linear_variance", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P004/P004_Spatial.lean#L2050", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:0e005ac3af8bfa3127129b780f602dac605ac432eb4fcffd98a48a0eb21b28c7", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P004-T44", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P004-T44" }, "html": { "body": "

Supplement A pp. 28-29, Lemma C.4 and Proposition 5, eq. (80). The HT estimator is unbiased with variance O(1/N), the root-N linearization error vanishes in probability, and the nondegenerate Hajek statistic tends to N(0,1). Restricted: replace the printed radiuswise-eventual C_b*s^2 spacing condition by C_b*(1+s^2), with one eventual sample-size threshold valid for all radii at once; the weaker radiuswise condition is not claimed to suffice. Finite unbiasedness is asserted for positive sample sizes with genuine spatial averages. The eventual regularity and the metric node-distance premise are explicit. No headline conclusion is dropped under these premises.

", "headline": "Supplement A, Appendix C: Lemma C.4 (root-NNN rate of the Horvitz-Thompson estimator and linearization of the Hajek estimator) and Proposition 5 (asymptotic normality of the Hajek estimator) under near-epoch dependence", "location": "Supplement A p. 28 (Lemma C.4); p. 29 (Proposition 5, eq. (80)); assumptions pp. 25-27; proof pp. 28-30; announced in the article, Section 6.5, p. 759 (PDF p. 16)" }, "id": "P004-T44", "text": { "body": "Supplement A pp. 28-29, Lemma C.4 and Proposition 5, eq. (80). The HT estimator is unbiased with variance O(1/N), the root-N linearization error vanishes in probability, and the nondegenerate Hajek statistic tends to N(0,1). Restricted: replace the printed radiuswise-eventual C_b*s^2 spacing condition by C_b*(1+s^2), with one eventual sample-size threshold valid for all radii at once; the weaker radiuswise condition is not claimed to suffice. Finite unbiasedness is asserted for positive sample sizes with genuine spatial averages. The eventual regularity and the metric node-distance premise are explicit. No headline conclusion is dropped under these premises.", "headline": "Supplement A, Appendix C: Lemma C.4 (root-$N$ rate of the Horvitz-Thompson estimator and linearization of the Hajek estimator) and Proposition 5 (asymptotic normality of the Hajek estimator) under near-epoch dependence", "location": "Supplement A p. 28 (Lemma C.4); p. 29 (Proposition 5, eq. (80)); assumptions pp. 25-27; proof pp. 28-30; announced in the article, Section 6.5, p. 759 (PDF p. 16)" }, "url": "https://pmaronow.github.io/llm-audit/works/P004/#P004-T44" } ] }, "missing_sources": [], "reproduction": { "agreement": "agrees_in_part", "status": "partial", "targets": [ { "agreement": "not_compared", "location": "p. 760 (PDF p. 17)", "method": "I inspected the curve-generation code; I did not compare numerical curves because the required simulation caches are absent.", "status": "blocked", "target": "Figure 5" }, { "agreement": "not_compared", "location": "p. 760 (PDF p. 17)", "method": "I inspected the MSE simulation code; I did not compare numerical MSE curves because the required simulation caches are absent.", "status": "blocked", "target": "Figure 6" }, { "agreement": "not_compared", "location": "p. 761 (PDF p. 18)", "method": "I inspected the additive-simulation interval code; I did not compare numerical coverage or length curves because the required simulation caches are absent.", "status": "blocked", "target": "Figure 7" }, { "agreement": "not_compared", "location": "p. 761 (PDF p. 18)", "method": "I inspected the interactive-simulation interval code; I did not compare numerical coverage or length curves because the required simulation caches are absent.", "status": "blocked", "target": "Figure 8" }, { "agreement": "not_compared", "location": "p. 762 (PDF p. 19), Section 7", "method": "I inspected the randomization-test code and compared archived images with Figure 9. I did not recalculate the rejection rates because the required simulation caches are absent.", "status": "blocked", "target": "Figure 9 and global-test rejection rates" }, { "agreement": "agrees_in_part", "location": "p. 764 (PDF p. 21), Section 8.1 pp. 763-764", "method": "I used the authors' code with changes to compare eight selected point estimates and selected interval endpoints with readings of Figure 11. The points agree at graph resolution; one lower endpoint differs just beyond the graph-reading tolerance. I did not reproduce the full interval curves.", "status": "recalculated", "target": "Figure 11" }, { "agreement": "disagrees", "location": "p. 763 (PDF p. 20), final paragraph of Section 8.1", "method": "I used the authors' functions with changes to recalculate circle averages from the supplied raster. The 0–5 km sum is -0.0427592 and the two-sided Monte Carlo fraction is 0.0928 from 10,000 draws, compared with the published -0.034 and approximately 0.065 (P004-N006).", "status": "recalculated", "target": "Uganda cumulative effect and randomization p-value" }, { "agreement": "agrees_in_part", "location": "pp. 764-765 (PDF pp. 21-22), Section 8.2, Figures 12-13", "method": "I used the authors' code with changes to reproduce the 233 treated and 522 untreated node counts and compare selected point estimates and intervals with Figure 13. All four selected smoothed estimates agree at graph resolution; three selected unsmoothed estimates and selected 2 km intervals differ beyond the reading tolerances. I recalculated the full point grid and selected intervals.", "status": "recalculated", "target": "Figure 13 and Costa Rica intervention counts" } ] }, "summary": "The paper studies spatial experiments in which treatment at one location may affect outcomes elsewhere in unknown ways. It defines an average marginalized effect by distance from an intervention node, gives asymptotically normal estimators under local interference, and reanalyses two studies of forest conservation. 43 entries are classified as errors (10 Moderate, 33 Minor), and 6 are clarity notes. The main finding is \\fid{P004-E001}: with an estimated logistic propensity score, the supplement's expansion of the weighted estimator and the variance estimators built on it can be wrong when propensities vary.", "what_was_checked": "I checked the following results, subject to the corrections and limits noted below.\n\\begin{itemize}\n\\item Proposition 1 and equations (4)--(9): the marginalisation, the circle averages and the definition of the AME are consistent, and the Horvitz--Thompson estimator is unbiased under Bernoulli assignment. I followed the proof and enumerated every assignment of two small designs with interference, a ring of seven nodes and a design of four nodes with interactions.\n\\item Supplement A, Lemma A.1 and Lemma A.2: the variance expansion and the bound $O(1/N)$ under C1--C4, apart from the displays corrected in \\fid{P004-N013}.\n\\item Lemma A.3 with the denominator corrected in \\fid{P004-E011}, and Lemma A.4: lines (2)--(5) equal the exact variance of the linearised Hajek estimator in an enumerated design, and lines (3) and (5) together equal $-N^{-2}\\sum_i(\\tau_i-\\mathrm{AME})\\sum_{j\\in\\mathcal{B}(i)}(\\tau_j-\\mathrm{AME})$.\n\\item Propositions 2 and 3: both include the lower bound on the scaled variance that a central limit theorem with a nondegenerate limit needs, and the argument through dependency neighbourhoods is sound under their premises. The restated Lemma A.5 is the subject of \\fid{P004-E012}.\n\\item Proposition 4 and Supplement A.4.1--A.4.3: the $(2,2)$ entry of the sandwich, the limit of the HAC estimator, the bias term under homophily, the bound behind the alternative variance estimator and the rescaling argument for conservative coverage with limits inferior as in \\fid{P004-N024}, apart from the other corrections recorded separately.\n\\item Equations (21)--(22) and footnotes 14--15: the truncated kernel is positive semidefinite and dominates the original kernel, and the positive semidefinite estimator is weakly larger.\n\\item Section 6.1, equations (23)--(25): under additivity the AME equals the average additive effect and does not depend on the design.\n\\item Supplement A.5: equation (32), equations (34)--(35), the comparison of the two estimators and the identity $p\\Delta(1)=(1-p)\\Delta(0)$ of its footnote 5.\n\\item Supplement A.6: the expansion of the maximum likelihood estimator of the logistic model in Lemma A.6, subject to \\fid{P004-E003}, and the Taylor expansion (42). The step that follows is the subject of \\fid{P004-E001}.\n\\item Supplement A.7, equations (44)--(45): the trace moments, and the degrees of freedom $2\\mu^2/\\nu$, which agree with the effective degrees of freedom of Satterthwaite and Young.\n\\item Randomization inference under the sharp null is valid for any fixed statistic when draws follow the experiment's assignment law. The simulation code implements a two-sided test against 1,000 redraws, pointwise and for the supremum over the grid of footnote 23, and the images in the archive match Figure 9. The printed rejection rule is the subject of \\fid{P004-E005}.\n\\item Supplement A, Appendix C: Lemmas C.1 and C.3, the counting bounds (69)--(74), Lemmas C.5--C.9, and Proposition 6 (i)--(iv) with the rate $b_N=o(N^{1/6})$ and the bound (150)--(155), after the corrections of printed expressions that are recorded separately. The hypothesis of Lemma C.10 is the subject of \\fid{P004-E019}.\n\\item Figure 3: the assignments listed in the text match the panels.\n\\item The simulation code agrees with Supplement B.1 in the numbers and layouts of nodes, the gamma densities, the interactive term, the 2,000 Bernoulli$(0.5)$ draws, the Monte Carlo AME of equation (50), the distance grid and the nominal HAC cutoff. The additive effect in the supplied code extends beyond its stated range, so that the justification for that cutoff fails (\\fid{P004-E027}).\n\\item Uganda application: the stored data have 121 villages, 60 of them treated; the stored Hajek estimates equal differences in means recomputed from the stored circle averages; the kernel cutoff in the script matches footnote 25; and the intervals stored in the saved result object follow the normal 90\\% formula. A separate recalculation from the supplied raster gives a 0--5 km sum of $-0.0427592$ and a two-sided Monte Carlo fraction of $0.0928$, compared with the published $-0.034$ and approximately $0.065$ (\\fid{P004-N006}). Eight selected point estimates agree with Figure 11 at graph resolution; selected interval endpoints were compared, with the small lower-endpoint discrepancy described in \\fid{P004-N006}.\n\\item Costa Rica application: I reproduced the counts of 233 treated and 522 untreated tiles from the supplied data. All four selected smoothed point estimates agree with Figure 13 at graph resolution; three selected unsmoothed estimates and selected 2 km intervals exceed the graph-reading tolerances. I recalculated the full point-estimate grid but only selected interval endpoints.\n\\item The notation table of Supplement D agrees with the definitions of the article, apart from \\fid{P004-N019}. Among the 51 figures and five simulation images of the archive, no discrepancy was established beyond \\fid{P004-N022} and \\fid{P004-N023}.\n\\end{itemize}\nMy checks had the following limits. The stored simulation results are not in the replication archive, so Figures 5--9, the simulation figures of the supplement and the rejection rate 0.056 could not be recomputed; I compared them only with the code that would produce them. The application scripts do not run as supplied (\\fid{P004-E029}); the recalculations for the two applications used the authors' functions on the supplied data, covered full point-estimate grids and selected interval endpoints, and were compared with readings of the published plots, which print no numerical values. I searched older variants of the simulation scripts and the package vignette for the relevant code. I consulted the original Uganda and Costa Rica studies for the assignment procedure and outcome date. The restated results in Lemma A.5 and the appeal to Theorem 2 of Jenish and Prucha (2012) were not compared with the originals. The article does not state the versions of R and of the spatial packages. The absence of further corrections does not establish that the rest of the paper is correct.", "work": { "arxiv_id": "2010.13599", "audit_complete": true, "authors": [ "Ye Wang", "Cyrus Samii", "Haoge Chang", "P. M. Aronow" ], "category": "Articles - Theory and Methodology", "citation": "Ye Wang, Cyrus Samii, Haoge Chang, and P. M. Aronow (2025). Design-Based Inference for Spatial Experiments under Unknown Interference. Annals of Applied Statistics 19(1):744–768.", "doi": "10.1214/24-AOAS1986", "edition": { "pages_in_audited_file": 25, "sha256": "decbf0e0589295a10ab2e2c596d0b20e1dbbfd0a51d71181854195aa8b8e833f", "source_url": "https://doi.org/10.1214/24-AOAS1986", "venue": "Annals of Applied Statistics", "year": "2025" }, "id": "P004", "later_materials": null, "title": "Design-Based Inference for Spatial Experiments under Unknown Interference", "type": "Journal article", "url": "https://pmaronow.github.io/llm-audit/works/P004/", "venue": "Annals of Applied Statistics", "year": "2025" } }