{ "claims": [ { "claim": "Under a general, non-parametric model with unrestricted interference and possibly misspecified exposures, the design-conditional targets (EPO, AEPO, EED, AEED) are well-defined whenever the design is known and individual positivity holds, without SUTVA.", "claim_id": "P061-H1", "related_corrections": [ "P061-N014", "P061-N015", "P061-N003", "P061-N004" ], "tied_to": "Abstract; Introduction (p. 2); Conclusion (p. 14); Equations (1)-(8); Condition 4.1" }, { "claim": "The defined contrasts coincide with the usual average treatment effects when SUTVA holds, and can then be interpreted beyond the support of the implemented design.", "claim_id": "P061-H2", "related_corrections": [ "P061-N016" ], "tied_to": "Introduction (p. 2); Section 4 (p. 6); Section 5.1 (p. 8); Condition 5.2; Equations (5)-(8); Table 3" }, { "claim": "NURVA is a weaker, design-dependent variant of SUTVA that requires stability only on the support of the design; for a given design the two are observationally equivalent; NURVA can hold where SUTVA fails, and the AEED then depends on the design.", "claim_id": "P061-H3", "related_corrections": [ "P061-N016", "P061-N018" ], "tied_to": "Abstract; Introduction (p. 2); Section 5 (pp. 8-10); Conditions 5.1 and 5.2; Tables 1-3" }, { "claim": "NURVA alone has practical limits: a contrast that is causal on the support of one design need not equal the contrast under another allocation, with hidden treatment versions, or under another sampling-treatment design; only SUTVA licenses off-support interpretation, and researchers should state the design, its support, the exposure mapping and the inferential scope.", "claim_id": "P061-H4", "related_corrections": [ "P061-N017", "P061-N008" ], "tied_to": "Abstract; Introduction (p. 2); Section 6 (pp. 11-13); Conclusion (p. 14); Tables 4 and 5; Section 6.3 and footnote 10" }, { "claim": "Horvitz-Thompson-type estimators are finite-sample unbiased for the AEPO and its contrasts even when neither SUTVA nor NURVA holds; under standard fixed-size designs they reduce to the sample mean and the difference in means.", "claim_id": "P061-H5", "related_corrections": [ "P061-N020", "P061-N021" ], "tied_to": "Section 7 (pp. 13-14); supplement pp. 1-2, Equations (1)-(4)" }, { "claim": "Under NURVA the traditional results carry over: an exact variance expression, a conservative variance estimator, and consistency under the stated regularity conditions.", "claim_id": "P061-H6", "related_corrections": [ "P061-N010", "P061-N011" ], "tied_to": "Section 7 (p. 13); Conclusion (p. 14); supplement pp. 3-6, Equations (5)-(9), Conditions 1.1-1.3, Proposition 1" }, { "claim": "The paper states that covariate-adjusted estimators can be finite-sample unbiased even without NURVA, under the displayed prediction moment condition. The authors present this result as potentially novel.", "claim_id": "P061-H7", "related_corrections": [], "tied_to": "Section 7 (p. 14); supplement p. 7, Equation (10) and the moment condition displayed before it" }, { "claim": "Inference on AEEDs is asymptotically equivalent to inference on the average direct effect when statistical dependence in $\\mathbf D$ and unmodeled interference are sufficiently local.", "claim_id": "P061-H8", "related_corrections": [ "P061-N019", "P061-N009" ], "tied_to": "Conclusion, second paragraph (p. 14); no numbered result; attributed to Sävje (2024); compare footnote 8" } ], "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 sentence that introduces the EPO and of Equation (5). A red box marks the words 'Given that relevant individual positivity assumptions hold,' at the start of the sentence.", "caption": "Section 4.1, p. 6 (PDF p. 6): the definition of the EPO, Equation (5). The marked words give the only condition under which the EPO is defined.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N014.png" }, { "alt": "Crop of two lines of text and Condition 4.1 (Individual positivity). A red box marks the inequality that says c is positive and at most pi_i(d) for every unit i.", "caption": "Section 4.1, p. 6 (PDF p. 6): Condition 4.1. The marked inequality is the whole of the condition.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N014-cond41.png" } ], "formal_release": null, "formal_statements": [ { "id": "P061-T5", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T5" }, { "id": "P061-T6", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T6" }, { "id": "P061-T7", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T7" }, { "id": "P061-T8", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T8" }, { "id": "P061-T11", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T11" }, { "id": "P061-T16", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T16" }, { "id": "P061-T20", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T20" } ], "headline": "Individual positivity alone does not make the EPO of Equation (5) a finite number when the design has infinitely many assignments; the definition also needs a finite first moment for each targeted unit and exposure, a condition that every design with finitely many assignments satisfies.", "id": "P061-N014", "location": "Section 4.1, p. 6 (PDF p. 6), Condition 4.1 and Equation (5); Equations (6)–(8), p. 7; statements on pp. 2 and 14", "pdf_page": 6, "printed_page": "6", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P061-T11", "P061-T16", "P061-T20", "P061-T5", "P061-T6", "P061-T7", "P061-T8" ], "text": { "as_printed": "``\\textbf{Condition 4.1} (\\textbf{Individual positivity}). An exposure $d$ satisfies \\emph{individual positivity} if there exists a $c$ such that $00$. Then $\\operatorname{E}[|Y_i|\\mid D_i=d]=M_{i,d}/\\pi_i(d)<\\infty$. The numerator of Equation (5) converges absolutely, and $\\overline{y}_i(d)$ is the finite conditional expectation. Since $N$ is finite, Equations (6)--(8) are finite sums and differences of finite numbers. For the estimator, let $T_i=Y_i\\mathbf{1}\\{D_i=d\\}/\\pi_i(d)$. I have $\\operatorname{E}|T_i|=M_{i,d}/\\pi_i(d)<\\infty$ and $\\operatorname{E}[T_i]=\\overline{y}_i(d)$, and linearity of expectation gives $\\operatorname{E}[\\widehat{\\overline{y}}(d)]=\\overline{y}(d)$. The condition $M_{i,d}<\\infty$ holds whenever the support is finite. It also holds under NURVA at a targeted exposure: there $y_i$ takes a single value $a$ on the assignments in the support with exposure $d$, and $M_{i,d}=|a|\\pi_i(d)$. For the bounded outcomes $y_1(k)=(-1)^{k+1}$ on the same design, I find $M_{1,1}=1/2$, and the EPO at $d=1$ and the expectation of the Horvitz--Thompson estimator both equal $1/3$.", "problem": "Section 3 fixes a finite population, real-valued outcomes and a probability measure that is a sum of point masses. It does not restrict the design to finitely many assignments, and it does not bound the outcomes. If the support of the design is countably infinite, then the sum in the numerator of Equation (5) need not converge absolutely, so a finite conditional mean is not guaranteed. Individual positivity controls only the denominator. In the generality of Section 3, positivity does not ensure that the EPO, the AEPO, the EED and the AEED exist, or that the Horvitz--Thompson estimator of supplement Equation (1) has an expectation. Every design in the article has finitely many assignments, and for those designs the targets are well-defined as stated.", "what_survives": "Equations (5)--(8) are unchanged, and the Horvitz--Thompson estimator is unbiased for the AEPO whenever the targeted first moments are finite. The survey, experiment and sampling-treatment examples of the article and Tables 1--5 have finitely many assignments and are unaffected, as are the distinction between NURVA and SUTVA and the variance formulas under NURVA. Proposition 1 of the supplement assumes NURVA and the bound of Condition 1.1, which imply that the targeted first moments are finite. The added condition is needed in the statements on pp.~2, 6--7 and~14 that the targets are well-defined, in the unbiasedness derivation of supplement Equation (3), and for the AEPO and the Horvitz--Thompson expectation used in supplement Equation (10). The proof of supplement Equation (10) also needs the expectations of the predictions in it to be finite." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N014", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of five lines of text from page 6. A red box marks the word always in the sentence: in practice, researchers can always trim the population to a subset where the assumption is valid.", "caption": "Section 4.1, p. 6 (PDF p. 6), paragraph after Condition 4.1. The marked word “always” omits the case in which no unit is eligible for every exposure in the target.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N015.png" } ], "formal_release": null, "formal_statements": [], "headline": "Trimming recovers positivity only when some unit has positive probability of every exposure that the target requires, and the word “always” omits this proviso.", "id": "P061-N015", "location": "Section 4.1, paragraph after Condition 4.1, p. 6 (PDF p. 6), sentence beginning “While this may seem like a stringent assumption”", "pdf_page": 6, "printed_page": "6", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "In our setting, each individual that we conduct inference over must have a non-zero probability of being in the relevant exposure condition. While this may seem like a stringent assumption, in practice, researchers can \\hl{always} trim the population to a subset where the assumption is valid to arrive at interpretable functions of EPOs---although this changes the target population and therefore the estimand.", "corrected": "While this may seem like a stringent assumption, in practice, \\hl{when a nonempty subset of units has positive exposure probabilities for every exposure required by the target,} researchers can trim the \\hl{target} population to \\hl{that subset} to arrive at interpretable functions of EPOs---although this changes the target population and therefore the estimand.", "intended_meaning": "The advice is to restrict the target population to the units that are eligible for the target, and it is correct whenever such units exist. For a finite set $T$ of target exposures, let $U_T=\\{i:\\pi_i(d)>0\\text{ for every }d\\in T\\}$. If $U_T$ is nonempty, then $c=\\min_{i\\in U_T,\\,d\\in T}\\pi_i(d)$ is positive because $U_T$ is finite. Then Condition 4.1 holds on $U_T$, and the targets are averages over $U_T$ with its size as denominator. If $U_T$ is empty, no trimmed population exists. For example, let there be two units with equally likely assignments $(0,0)$ and $(1,1)$, and let $g_1(\\mathbf z)=z_1$ and $g_2(\\mathbf z)=1+z_2$. Then each of the exposures $0$ and $2$ has probability $1/2$ for one unit and probability $0$ for the other. No definition or result in the paper uses the remark." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N015", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of Condition 5.2 on page 8 with two lines of text above it and two below. A red box marks the last equality in the display, equals y bar i of g i of z, which stands before the quantifier for all z and z prime in the design space.", "caption": "Condition 5.2 (SUTVA), p. 8 (PDF p. 8). The marked equality identifies the common raw outcome with the EPO $\\overline y_i(g_i(\\mathbf z))$, which Equation (5) defines only when the exposure has positive probability under the design.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N016.png" } ], "formal_release": null, "formal_statements": [ { "id": "P061-T12", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T12" }, { "id": "P061-T13", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T13" } ], "headline": "Condition 5.2 should require stability of the raw potential outcomes on all of $\\mathcal Z$; its last equality refers to an EPO that Equation (5) leaves undefined at exposures the design never produces.", "id": "P061-N016", "location": "Condition 5.2 (SUTVA), display, p. 8 (PDF p. 8); Equation (5), p. 6", "pdf_page": 8, "printed_page": "8", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [ "P061-T12", "P061-T13" ], "text": { "as_printed": "Condition 5.2 (SUTVA).\n\\[\ng_i(\\mathbf z)=g_i(\\mathbf z')\\implies y_i(\\mathbf z)=y_i(\\mathbf z')\\hl{{}=\\overline y_i(g_i(\\mathbf z))},\\quad\\forall\\mathbf z,\\mathbf z'\\in\\mathcal Z.\n\\]", "corrected": "Condition 5.2 (SUTVA). For every $i\\in\\{1,\\ldots,N\\}$ and every $\\mathbf z,\\mathbf z'\\in\\mathcal Z$,\n\\[\ng_i(\\mathbf z)=g_i(\\mathbf z')\\implies y_i(\\mathbf z)=y_i(\\mathbf z').\n\\]\nThis defines a common raw outcome $h_i(d)$ for each $d\\in g_i(\\mathcal Z)$. \\hl{When $\\pi_i(d)>0$,} Equation (5) implies $\\overline y_i(d)=h_i(d)$.", "intended_meaning": "The intended condition is that, for every unit and every pair of feasible assignments, equal exposures imply equal raw potential outcomes. Sections 5 and 6 use this condition, and it is correct. The EPO of Equation (5) is a conditional mean under the implemented design, defined only when $\\pi_i(d)>0$, while Condition 5.2 ranges over all of $\\mathcal Z$. For example, let $N=2$ and $\\mathcal Z=\\{0,1\\}^2$, let the assignments $(0,0)$, $(0,1)$, $(1,0)$, $(1,1)$ have probabilities $1/2$, $1/4$, $1/4$, $0$, and let $g_i(\\mathbf z)=z_1+z_2$ and $y_i(\\mathbf z)=i(z_1+z_2)$. The raw potential outcomes satisfy the intended condition on $\\mathcal Z$, and the exposures $0$ and $1$ each have probability $1/2$. Since $\\pi_i(2)=0$, the last term of the display is $0/0$ at $\\mathbf z=\\mathbf z'=(1,1)$. No such gap occurs in Condition 5.1, and no supported EPO, table or later argument changes." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N016", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" }, { "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 the Introduction. A red box marks the clause 'but only SUTVA permits off-support interpretations.' at the end of the paragraph that introduces NURVA.", "caption": "Introduction, p. 2 (PDF p. 2): the end of the paragraph that introduces NURVA. The marked clause is the statement about off-support interpretations.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N017.png" }, { "alt": "Crop of six lines of the Conclusion. Red boxes mark the sentence 'SUTVA is required for those off-support interpretations.', which runs from the end of one line to the start of the next.", "caption": "Conclusion, p. 14 (PDF p. 14), first paragraph. The marked sentence repeats the statement.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N017-p14.png" } ], "formal_release": null, "formal_statements": [], "headline": "The statements that “only SUTVA permits off-support interpretations” (p. 2) and that SUTVA “is required” for them (p. 14) mean that NURVA alone does not guarantee such an interpretation and that SUTVA suffices for one; a weaker restriction can suffice for a particular target, as in the partial-interference example on p. 9.", "id": "P061-N017", "location": "Introduction, p. 2 (PDF p. 2), last sentence of the paragraph that introduces NURVA; Conclusion, p. 14 (PDF p. 14), first paragraph", "pdf_page": 2, "printed_page": "2", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "Introduction, p.~2: ``When NURVA holds but SUTVA does not, contrasts remain causal under the design; but \\hl{only SUTVA permits} off-support interpretations.''\n\nConclusion, p.~14: ``What NURVA buys in such cases is a within-support causal interpretation; what it does not buy is extrapolation to alternative allocations, protocols or populations. \\hl{SUTVA is required} for those off-support interpretations.''", "corrected": "Introduction, p.~2: ``When NURVA holds but SUTVA does not, contrasts remain causal under the design; but \\hl{NURVA alone does not guarantee} off-support interpretations. \\hl{Global exposure stability (Condition 5.2) is sufficient for invariance across designs on the same population, with a fixed exposure mapping and positive probabilities for the relevant exposures, while weaker restrictions linking the particular target assignments can suffice, including the partial-interference case discussed on p.~9.}''\n\nConclusion, p.~14: ``What NURVA buys in such cases is a within-support causal interpretation; what it does not buy is extrapolation to alternative allocations, protocols or populations. \\hl{Such off-support interpretations require additional restrictions; full SUTVA is sufficient for exposure stability on the specified feasible set, but target-specific restrictions may suffice.}''", "intended_meaning": "On p.~8 the article says what SUTVA adds to NURVA. Under Condition 5.2, the outcome of a unit depends on the assignment only through the exposure of that unit, over the whole design space $\\mathcal Z$. The EED can then be interpreted without the qualifiers about the design. NURVA requires this stability only on the support of the design. In that sense both sentences are correct: NURVA alone does not guarantee an off-support interpretation, and the interpretation without qualifiers holds exactly when Condition 5.2 does.\n\nRead as statements about every off-support target, the words ``only'' and ``required'' say too much, and Section 5.2 (p.~9) contains an exception. I consider four units in the clusters $\\{1,2\\}$ and $\\{3,4\\}$, with the design space $\\{0,1\\}^4$, the exposure mapping $g_i(\\mathbf z)=z_i$ and the outcomes $y_i(\\mathbf z)=z_i+z_{m(i)}$, where $m(i)$ is the other unit in the cluster of $i$. Let the design put probability $1/2$ on each of $(1,1,0,0)$ and $(0,0,1,1)$. Individual positivity and NURVA hold. SUTVA fails, since $y_1(1,0,0,0)=1$ and $y_1(1,1,0,0)=2$ at the same exposure. The AEED is $\\tau(1,0)=2$, and it equals the average effect of the assignment $(1,1,1,1)$ against $(0,0,0,0)$, two assignments outside the support. The equality holds for all outcomes of the form $y_i(\\mathbf z)=F_i(z_i,z_{m(i)})$, because under this design $\\overline{y}_i(d)=F_i(d,d)$. NURVA alone does not guarantee this equality. If the outcomes at $(1,1,1,1)$ are changed to $0$, then the distribution of $(\\mathbf Z,\\mathbf Y)$ under the design and the AEED are unchanged, and the average effect of $(1,1,1,1)$ against $(0,0,0,0)$ becomes $0$. No calculation that I inspected uses the stronger reading as a premise." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N017", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of five lines of text from the conclusion on page 14. Red boxes mark the two terms of the formula for the average direct effect: y i of d and bold D minus i at the end of one line, and y i of d prime and bold D minus i at the start of the next line.", "caption": "Section 8 (Conclusion), p. 14 (PDF p. 14), second paragraph. In the two marked terms, the arguments of the raw potential outcome $y_i$, which Equation (1) defines on assignment vectors, are an exposure $d$ or $d'$ and the other units' exposures $\\mathbf D_{-i}$.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N019.png" } ], "formal_release": null, "formal_statements": [ { "id": "P061-T1", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T1" }, { "id": "P061-T2", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T2" }, { "id": "P061-T3", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T3" } ], "headline": "The “average direct effect” of the conclusion is defined for the individualistic exposure mapping $g_i(\\mathbf z)=z_i$, where it equals the AEED under independent assignments; for the general exposure mappings that the paper allows, the expression $y_i(d;\\mathbf D_{-i})$ need not be defined, and the AEED of a coarsened exposure can differ from the average direct effect at every population size.", "id": "P061-N019", "location": "Section 8 (Conclusion), second paragraph, second sentence, p. 14 (PDF p. 14): the formula for the “average direct effect”", "pdf_page": 14, "printed_page": "14", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P061-T1", "P061-T2", "P061-T3" ], "text": { "as_printed": "There is additional hope for further refining inference on targets such as AEEDs from Sävje (2024); inference on AEEDs is asymptotically equivalent to inference on the ``average direct effect'' $N^{-1}\\sum_{i=1}^{N}\\bigl(\\mathrm E[y_i(d;\\hl{\\mathbf D_{-i}})-y_i(d';\\hl{\\mathbf D_{-i}})]\\bigr)$ if both statistical dependence in $\\mathbf D$ and unmodeled interference are sufficiently local.", "corrected": "There is additional hope for further refining inference on targets such as AEEDs from Sävje (2024). \\hl{For the individualistic exposure mapping $g_i(\\mathbf z)=z_i$,} define the ``average direct effect'' as\n\\[\nN^{-1}\\sum_{i=1}^{N}\\mathrm E\\bigl[y_i(d;\\hl{\\mathbf Z_{-i}})-y_i(d';\\hl{\\mathbf Z_{-i}})\\bigr],\n\\]\nwhere $y_i(d;\\mathbf z_{-i})$ is the raw potential outcome at the assignment with $i$th coordinate $d$ and remaining coordinates $\\mathbf z_{-i}$, as in footnote 8, and both assignments are assumed to lie in $\\mathcal Z$. Suppose that, for every $i\\in\\{1,\\ldots,N\\}$, $\\pi_i(d)>0$ and $\\pi_i(d')>0$, the assignment \\hl{$Z_i$ is independent of $\\mathbf Z_{-i}$}, and $y_i(\\mathbf Z)$ is integrable. Then the average direct effect \\hl{equals the AEED} $\\tau(d,d')$. For dependent assignments the two targets can differ, and inference on one is inference on the other only under further conditions.\n\nFor a general exposure mapping the same equality holds under an added assumption. Suppose that, for every $i$, there exists a function $H_i$ of the exposure vector such that $H_i(\\mathbf D)$ is integrable and $y_i(\\mathbf z)=H_i(g_1(\\mathbf z),\\ldots,g_N(\\mathbf z))$ for every $\\mathbf z\\in\\operatorname{Supp}(\\mathbf Z)$, that $\\pi_i(d)>0$ and $\\pi_i(d')>0$, and that $D_i$ is independent of $\\mathbf D_{-i}$. Then\n\\[\n\\tau(d,d')=N^{-1}\\sum_{i=1}^{N}\\mathrm E\\bigl[H_i(d,\\mathbf D_{-i})-H_i(d',\\mathbf D_{-i})\\bigr].\n\\]\nNeither statement asserts that inference procedures are equivalent under weaker conditions on the dependence.", "evidence": "The evidence is two exact examples and a derivation. In the first example, let $N=2$ and $\\mathcal Z=\\{0,1\\}^2$, let each assignment have probability $1/4$, and let the two units be adjacent in the network. The mapping of Equation (3) is then $g_1(\\mathbf z)=z_1+2z_2$ and $g_2(\\mathbf z)=z_2+2z_1$. Let $y_i(\\mathbf z)=z_i$.\n\\begin{audittable}{cccc}{The exposure mapping of Equation (3) on two adjacent units.}\nAssignment $\\mathbf z$ & Exposure vector $(D_1,D_2)$ & Outcomes $(y_1,y_2)$ & Probability \\\\\n$(0,0)$ & $(0,0)$ & $(0,0)$ & $1/4$ \\\\\n$(0,1)$ & $(2,1)$ & $(0,1)$ & $1/4$ \\\\\n$(1,0)$ & $(1,2)$ & $(1,0)$ & $1/4$ \\\\\n$(1,1)$ & $(3,3)$ & $(1,1)$ & $1/4$ \\\\\n\\end{audittable}\nEach exposure has probability $1/4$ for each unit, and the AEED $\\tau(1,0)$ equals $1$. Each value of $D_2$ occurs with exactly one value of $D_1$. For no assignment are both $(1,D_2)$ and $(0,D_2)$ exposure vectors, and the contrast for unit $1$ lacks a term in all four cases. The same holds for unit $2$. If $(d,D_2)$ is read as an assignment instead, it lies in $\\{0,1\\}^2$ only when $D_2\\in\\{0,1\\}$, which has probability $1/2$. The same failure occurs for every even $N$ in a population of independent copies of this pair, although each exposure depends on one other exposure only and no outcome depends on another unit's assignment.\n\nIn the second example, let $N\\ge1$, let $Z_1,\\ldots,Z_N$ be independent and uniform on $\\{0,1,2\\}$, and let $g_i(\\mathbf z)=\\mathbf 1\\{z_i>0\\}$, as in Section 6.2, and $y_i(\\mathbf z)=z_i$. Then $\\pi_i(1)=2/3$ and $\\pi_i(0)=1/3$. By Equation (5), I have $\\overline y_i(1)=(1+2)/2=3/2$ and $\\overline y_i(0)=0$. Using Equation (8), I obtain $\\tau(1,0)=3/2$ for every $N$. With $d=1$ and $d'=0$ read as assignment coordinates, the contrast $y_i(1;\\cdot)-y_i(0;\\cdot)$ equals $1$ for every unit and every draw, and the average direct effect is $1$. The two targets differ by $1/2$ at every $N$. The Horvitz--Thompson estimator of the AEED is $(3/(2N))\\sum_{i=1}^NZ_i$, with expectation $3/2$ and variance $3/(2N)$, and it converges in probability to $3/2$. The exposures do not determine a unique outcome either. For $N=2$, the assignments $(1,0)$ and $(2,0)$ have the same exposure vector $(1,0)$, and the outcomes of unit $1$ are $1$ and $2$.\n\\begin{audittable}{ccccc}{Exact values in the second example. The Horvitz--Thompson estimator targets the AEED at every $N$.}\n$N$ & Expectation of the estimator & Variance of the estimator & AEED $\\tau(1,0)$ & Average direct effect \\\\\n$1$ & $3/2$ & $3/2$ & $3/2$ & $1$ \\\\\n$2$ & $3/2$ & $3/4$ & $3/2$ & $1$ \\\\\n$4$ & $3/2$ & $3/8$ & $3/2$ & $1$ \\\\\n$6$ & $3/2$ & $1/4$ & $3/2$ & $1$ \\\\\n\\end{audittable}\nFor the derivation, suppose that $g_i(\\mathbf z)=z_i$, that $\\pi_i(d)>0$ and $\\pi_i(d')>0$, that $Z_i$ is independent of $\\mathbf Z_{-i}$, and that $y_i(\\mathbf Z)$ is integrable. For each $\\mathbf w$ such that $\\Pr[\\mathbf Z_{-i}=\\mathbf w]>0$, the assignment $(d;\\mathbf w)$ has probability $\\Pr[Z_i=d]\\Pr[\\mathbf Z_{-i}=\\mathbf w]>0$, and hence it lies in the support of the design. By Equation (5), I find\n\\[\n\\overline y_i(d)=\\sum_{\\mathbf w}y_i(d;\\mathbf w)\\Pr[\\mathbf Z_{-i}=\\mathbf w\\mid Z_i=d]=\\sum_{\\mathbf w}y_i(d;\\mathbf w)\\Pr[\\mathbf Z_{-i}=\\mathbf w]=\\mathrm E[y_i(d;\\mathbf Z_{-i})].\n\\]\nIn the second equality, I used that $Z_i$ is independent of $\\mathbf Z_{-i}$. The same identity holds for $d'$. Subtracting the two identities and averaging over $i$, I obtain\n\\[\n\\tau(d,d')=\\frac1N\\sum_{i=1}^N\\mathrm E\\bigl[y_i(d;\\mathbf Z_{-i})-y_i(d';\\mathbf Z_{-i})\\bigr].\n\\]\nThe identity uses neither SUTVA nor the absence of interference. In an example with three units, independent binary assignments with probabilities $1/3$, $1/2$, $2/3$ and outcomes $y_i(\\mathbf z)=z_i+2z_{i+1}+3z_iz_{i+1}$ with cyclic indices, both sides equal $5/2$. The second statement in the correction follows from the same computation with $\\mathbf D$ in place of $\\mathbf Z$ and $H_i$ in place of $y_i$.", "problem": "Equation (1) defines the raw potential outcome $y_i$ on assignment vectors $\\mathbf z\\in\\mathcal Z$, and Equation (2) defines the exposure vector $\\mathbf D=(g_1(\\mathbf Z),\\ldots,g_N(\\mathbf Z))$ through exposure mappings that the researcher chooses. In the marked expression, the arguments of $y_i$ are an exposure $d$ for unit $i$ and the exposures $\\mathbf D_{-i}$ of the other units. The paper does not define $y_i$ on such arguments. For the individualistic exposure mapping $g_i(\\mathbf z)=z_i$, I have $\\mathbf D=\\mathbf Z$, and the expression is the average of the contrast in footnote 8 (p.~7), which holds fixed the treatments assigned to the other units. The sentence does not state this restriction, and the paragraph before it speaks of arbitrary exposure misspecification. For a general exposure mapping the comparison fails in two ways. First, the vector $(d,\\mathbf D_{-i})$ need not lie in $\\mathcal Z$ and need not be the exposure vector of any assignment. Further, two assignments with the same exposure vector can have different outcomes, so the exposures need not determine a unique outcome. Second, when $d$ and $d'$ are read as assignment coordinates and the expression has a value, the AEED of a coarsened exposure can differ from it by the same amount at every $N$, even if the assignments are independent and no outcome depends on another unit's assignment. The comparison is correct for the individualistic exposure mapping under independent assignments, where the two targets are equal.", "what_survives": "The assignment space, the definitions of the EPO, AEPO, EED and AEED, the interpretation under NURVA and SUTVA, the examples and the arguments on external validity do not use this formula. The results on estimation and inference in the supplement do not use it either. No later result, table or figure depends on the sentence. Only the closing remark changes. The comparison with the average direct effect is restricted to the individualistic exposure mapping, or to exposure mappings for which the outcome is a function of the exposure vector. For dependent assignments a further condition is needed, which is the subject of \\fid{P061-N009}." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N019", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of three lines of the derivation in Equation (3) of the supplement. A red box marks the second term of the numerator in the fourth line, the conditional expectation of Y_i 1{D_i = d} given D_i not equal to d, multiplied by Pr[D_i not equal to d].", "caption": "Supplementary Appendix, p. 2: the third, fourth and fifth lines of Equation (3). The marked term conditions on the event $D_i\\neq d$.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N020.png" }, { "alt": "Crop of note 1 of the supplement. A red box marks the term E[Y_i 1{D_i = d} | D_i not equal to d] in the second line of the note.", "caption": "Supplementary Appendix, p. 10: note 1. The marked term is the same conditional expectation.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N020-note1.png" } ], "formal_release": null, "formal_statements": [ { "id": "P061-T16", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T16" } ], "headline": "The fourth line of supplement Equation (3), and note 1, condition on the event $D_i\\neq d$, which has probability zero when $\\pi_i(d)=1$; the term is meant to contribute zero, and the derivation holds without it.", "id": "P061-N020", "location": "Supplementary Appendix, p. 2 (PDF p. 2 of the supplement), Equation (3), fourth line; note 1, p. 10 (PDF p. 10 of the supplement)", "pdf_page": 2, "printed_page": "2", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [ "P061-T16" ], "text": { "as_printed": "Equation (3), fourth line:\n\\[\n=\\frac{1}{N}\\sum_{i=1}^{N}\\frac{\\operatorname{E}[Y_i\\mathbf{1}\\{D_i=d\\}\\mid D_i=d]\\Pr[D_i=d]+\\hl{\\operatorname{E}[Y_i\\mathbf{1}\\{D_i=d\\}\\mid D_i\\neq d]\\Pr[D_i\\neq d]}}{\\pi_i(d)}\n\\]\n\nNote 1, p.~10: ``The term involving $D_i\\neq d$ of Equation 3 demonstrates why the coding of the potential outcome for unsurveyed groups is irrelevant, because the term \\hl{$\\operatorname{E}[Y_i\\mathbf{1}\\{D_i=d\\}\\mid D_i\\neq d]$} will always be equal to zero.''", "corrected": "Equation (3), with the third line followed directly by the fifth:\n\\[\n\\frac{1}{N}\\sum_{i=1}^{N}\\frac{\\operatorname{E}[Y_i\\mathbf{1}\\{D_i=d\\}]}{\\pi_i(d)}=\\frac{1}{N}\\sum_{i=1}^{N}\\frac{\\operatorname{E}[Y_i\\mid D_i=d]\\Pr[D_i=d]}{\\pi_i(d)},\n\\]\n\\hl{where $\\operatorname{E}[Y_i\\mathbf{1}\\{D_i=d\\}]=\\pi_i(d)\\operatorname{E}[Y_i\\mid D_i=d]$, including when $\\pi_i(d)=1$.}\n\nNote 1: ``\\hl{The coding of the potential outcome for unsurveyed groups is irrelevant, because $Y_i\\mathbf{1}\\{D_i=d\\}$ is zero whenever $D_i\\neq d$.}''", "intended_meaning": "Condition 4.1 requires $\\pi_i(d)\\ge c>0$ and allows $\\pi_i(d)=1$, for instance in a census. Then the event $D_i\\neq d$ has probability zero, and the expectation conditional on it is not defined as a ratio. The term is meant to contribute zero, because $Y_i\\mathbf{1}\\{D_i=d\\}$ vanishes whenever $D_i\\neq d$. The derivation does not need the split. Summing over the assignments with $g_i(\\mathbf z)=d$, I have\n\\[\n\\operatorname{E}[Y_i\\mathbf{1}\\{D_i=d\\}]=\\sum_{\\mathbf z\\in\\operatorname{Supp}(\\mathbf Z):g_i(\\mathbf z)=d}y_i(\\mathbf z)\\Pr[\\mathbf Z=\\mathbf z]=\\pi_i(d)\\operatorname{E}[Y_i\\mid D_i=d],\n\\]\nand this identity holds when $\\pi_i(d)=1$. It uses only the first moments that the AEPO already requires (see \\fid{P061-N014}). Finite-$N$ unbiasedness and the remark that the coding of unsurveyed groups is irrelevant are unchanged." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N020", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the foot of page 2 of the supplement. A red box marks the words 'we remove all implicit probabilistic dependence of Y_i on Z:' in the sentence beginning 'Under NURVA'. Equation (4) follows below it.", "caption": "Supplementary Appendix, p. 2: the sentence that introduces Equation (4), and the equation. The marked words describe what NURVA removes.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N021.png" } ], "formal_release": null, "formal_statements": [ { "id": "P061-T18", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T18" } ], "headline": "Under NURVA the outcome does not vary within an exposure on the support of the design; it still depends on the assignment through the exposure, and Equation (4) uses only the identity within an exposure.", "id": "P061-N021", "location": "Supplementary Appendix, p. 2 (PDF p. 2 of the supplement), sentence before Equation (4)", "pdf_page": 2, "printed_page": "2", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [ "P061-T18" ], "text": { "as_printed": "``Under NURVA, we \\hl{remove all implicit probabilistic dependence of $Y_i$ on $\\mathbf Z$}:''\n\\[\n\\widehat{\\overline{y}}(d)=\\frac{1}{N}\\sum_{i=1}^{N}\\frac{Y_i\\mathbf{1}\\{D_i=d\\}}{\\pi_i(d)}=\\frac{1}{N}\\sum_{i=1}^{N}\\frac{\\overline{y}_i(d)\\mathbf{1}\\{D_i=d\\}}{\\Pr[D_i=d]}.\\qquad(4)\n\\]", "corrected": "``Under NURVA, \\hl{there is no residual outcome variation within exposure $d$ on the support of the design, so that $Y_i\\mathbf{1}\\{D_i=d\\}=\\overline{y}_i(d)\\mathbf{1}\\{D_i=d\\}$}:''\n\\[\n\\widehat{\\overline{y}}(d)=\\frac{1}{N}\\sum_{i=1}^{N}\\frac{Y_i\\mathbf{1}\\{D_i=d\\}}{\\pi_i(d)}=\\frac{1}{N}\\sum_{i=1}^{N}\\frac{\\overline{y}_i(d)\\mathbf{1}\\{D_i=d\\}}{\\Pr[D_i=d]}.\\qquad(4)\n\\]\nEquation (4) is unchanged.", "intended_meaning": "Under NURVA (Condition 5.1), the outcome of unit $i$ is the same at all assignments in the support of the design that have the same exposure. Hence $Y_i\\mathbf{1}\\{D_i=d\\}=\\overline{y}_i(d)\\mathbf{1}\\{D_i=d\\}$ with probability one. Equation (4) uses this identity and no more, and it is correct. NURVA does not make $Y_i$ independent of $\\mathbf Z$. With one unit, a fair binary assignment, $g_1(\\mathbf z)=z_1$ and $y_1(\\mathbf z)=z_1$, both NURVA and SUTVA hold and $Y_1=Z_1$, so that $\\Pr[Y_1=1\\mid Z_1=1]=1$ and $\\Pr[Y_1=1]=1/2$. In each summand of Equation (4), the remaining assignment randomness is carried by the indicator $\\mathbf{1}\\{D_i=d\\}$." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N021", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of six lines of text from page 9. A red box marks the phrase single exposure level in the sentence: NURVA will hold with respect to that outcome on all designs where all units in a cluster are assigned to a single exposure level.", "caption": "Section 5.2, p. 9 (PDF p. 9), the partial-interference illustration. The marked phrase is correct for a binary treatment with exposure equal to own treatment, and too broad for a coarse exposure label.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N018.png" } ], "formal_release": null, "formal_statements": [], "headline": "The partial-interference illustration holds for a binary treatment with the individualistic exposure mapping $g_i(\\mathbf z)=z_i$ and clusters assigned wholly to treatment or wholly to control; a common but coarser exposure label does not suffice.", "id": "P061-N018", "location": "Section 5.2, p. 9 (PDF p. 9), paragraph beginning “Whether each stability assumption holds”, sentence beginning “In such a setting, NURVA will hold”; footnote 9", "pdf_page": 9, "printed_page": "9", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "For example, consider a setting where there is partial interference on a given outcome. That is, there may be arbitrary interference within, but not across mutually exclusive clusters of units. In such a setting, NURVA will hold with respect to that outcome on all designs where all units in a cluster are assigned to a \\hl{single exposure level}, and the AEED will be equivalent to the global average treatment effect (the effect of treating all subjects relative to no subjects). However, NURVA will not necessarily hold for designs that result in within-cluster variation in treatment assignment.$^{9}$", "corrected": "In such a setting, \\hl{for a binary treatment with the individualistic exposure mapping $g_i(\\mathbf z)=z_i$,} partial interference implies NURVA with respect to that outcome on any design that assigns each cluster \\hl{wholly to treatment or wholly to control}. \\hl{Under the individual positivity required in Condition 4.1, with all-treated and all-control assignments feasible,} the AEED $\\tau(1,0)$ equals the global average treatment effect (the effect of treating all subjects relative to no subjects).", "intended_meaning": "The setting meant is the conventional one, as the contrast with within-cluster variation and footnote 9 indicate: a binary treatment, the individualistic exposure mapping $g_i(\\mathbf z)=z_i$, and a design that assigns each cluster wholly to treatment or wholly to control. In that setting, with all-treated and all-control assignments feasible, the claim is correct. Two assignments in the support under which unit $i$ has the same exposure $d$ agree on the cluster of $i$, and partial interference implies that $y_i$ takes the value $y_i(d\\mathbf 1_N)$ at both. Under the positivity of Condition 4.1, I have $\\overline y_i(d)=y_i(d\\mathbf 1_N)$ by Equation (5), and hence $\\tau(1,0)=N^{-1}\\sum_{i=1}^N[y_i(\\mathbf 1_N)-y_i(\\mathbf 0_N)]$. The wording is too broad only if ``a single exposure level'' may be a coarse label. For example, let the clusters $\\{1,2\\}$ and $\\{3,4\\}$ be assigned raw levels $0$, $1$ or $2$ independently with probability $1/3$ each, let $g_i(\\mathbf z)=\\mathbf 1\\{z_i>0\\}$, and let $y_i(\\mathbf z)$ be the sum of the raw levels of unit $i$ and of the other unit in its cluster. Under the assignments $(1,1,0,0)$ and $(2,2,0,0)$, unit $1$ has the exposure $1$ and the outcomes $2$ and $4$, and NURVA fails. The AEED is $3$, while treating every unit at raw level $1$ or at raw level $2$ has effect $2$ or $4$. No result in the paper uses the sentence." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N018", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" }, { "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 the Introduction. A red box marks the citation '(Harshaw, Sävje, and Wang 2022)'.", "caption": "Introduction, p. 2 (PDF p. 2). The marked citation gives the order Harshaw, Sävje, Wang.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N002.png" }, { "alt": "Crop of three entries of the reference list. A red box marks the authors and year 'Harshaw, C., Y. Wang, and F. Sävje. 2022.' at the start of the second entry.", "caption": "Reference list, p. 15 (PDF p. 15). The marked entry gives the order Harshaw, Wang, Sävje.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N002-p15.png" } ], "formal_release": null, "formal_statements": [], "headline": "The citation on p. 2 and the entry in the reference list on p. 15 give the authors of Harshaw et al. (2022) in different orders, and one of the two must be changed to agree with the cited work.", "id": "P061-N002", "location": "Introduction, p. 2 (PDF p. 2), paragraph on recent advances; reference list, p. 15 (PDF p. 15)", "pdf_page": 2, "printed_page": "2", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Introduction, p.~2: ``More general methods for randomization tests under interference (Basse, Feller, and Toulis 2019) and Riesz estimators (\\hl{Harshaw, Sävje, and Wang} 2022) have also been proposed.''\n\nReference list, p.~15: \\hl{Harshaw, C., Y. Wang, and F. Sävje.} 2022. ``A Design-Based Riesz Representation Framework for Randomized Experiments.'' In \\textit{NeurIPS 2022 Workshop on Causality for Real-World Impact}. \\url{https://openreview.net/forum?id=GM16YCLkEmz}.", "corrected": "The order of the authors must be the same in both places, after a check against the cited work. If the reference list is right, the citation on p.~2 reads ``(Harshaw, \\hl{Wang, and Sävje} 2022)''. If the citation is right, the entry on p.~15 begins ``Harshaw, C., \\hl{F. Sävje, and Y. Wang}. 2022.''", "evidence": "I compared the author orders on the two pages. I did not compare either order with the cited work, so I have not established the correct order.", "problem": "The citation places Sävje before Wang, and the entry in the reference list places Wang before Sävje. At least one of the two is misprinted. The article does not determine which order is that of the cited work.", "what_survives": "The cited work can be identified from either order. No argument in the article depends on the citation." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N002", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of the heading '3.1. Design' and the first three lines of the section. A red box marks the word 'bijective' in 'where Z is a bijective random vector Z: Omega to R^N'.", "caption": "Section 3.1, p. 3 (PDF p. 3), first sentence. The marked word describes the map $\\mathbf Z$.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N003.png" } ], "formal_release": null, "formal_statements": [], "headline": "The word “bijective” means injective here: $\\mathbf Z$ is a bijection from $\\Omega$ onto the design space $\\mathcal Z$, not onto $\\mathbb R^N$.", "id": "P061-N003", "location": "Section 3.1, p. 3 (PDF p. 3), first sentence", "pdf_page": 3, "printed_page": "3", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "``We define the design space $\\mathcal Z=\\mathbf Z(\\Omega)$, where $\\mathbf Z$ is a \\hl{bijective} random vector $\\mathbf Z:\\Omega\\to\\mathbb R^N$. The \\emph{design} is defined as the joint p.m.f. of $\\mathbf Z$, $f:\\mathbb R^N\\to[0,1]$.''", "corrected": "``We define the design space $\\mathcal Z=\\mathbf Z(\\Omega)$, where $\\mathbf Z$ is an \\hl{injective} random vector $\\mathbf Z:\\Omega\\to\\mathbb R^N$ \\hl{(a bijection from $\\Omega$ onto $\\mathcal Z$)}. The \\emph{design} is defined as the joint p.m.f. of $\\mathbf Z$, $f:\\mathbb R^N\\to[0,1]$.''", "intended_meaning": "A bijection from $\\Omega$ onto $\\mathbb R^N$ would force $\\mathcal Z=\\mathbf Z(\\Omega)=\\mathbb R^N$. This excludes the finite design spaces that the article uses, such as $\\mathcal Z\\subseteq\\{0,1,2\\}^N$ in Section 6.2 (p.~11). The article needs $\\mathbf Z$ to be one-to-one, so that $\\mathbf Z^{-1}(\\mathbf z)$ is defined for every $\\mathbf z\\in\\mathcal Z$. Equation (1) on p.~4 applies the inverse only on $\\mathcal Z$, and every definition is unchanged under this reading." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N003", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the display that defines f(z) for the school experiment and the line below it. A red box marks the condition of the first case, that the sum of z_k over k from 1 to N equals the ceiling of 0.2N.", "caption": "Section 3.1, p. 4 (PDF p. 4): the design of the school experiment. The marked condition defines the positive case.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N004.png" } ], "formal_release": null, "formal_statements": [], "headline": "The probability mass function of the school design is meant for binary assignment vectors, and its positive case needs the condition $\\mathbf z\\in\\{0,1\\}^N$.", "id": "P061-N004", "location": "Section 3.1, p. 4 (PDF p. 4), display at the top of the page; its lead-in is at the foot of p. 3", "pdf_page": 4, "printed_page": "4", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "``Letting $\\lceil 0.2N\\rceil$ be the number of observations in treatment, the design is''\n\\[\nf(\\mathbf z)=\\begin{cases}\\dfrac{(\\lceil 0.2N\\rceil)!(N-\\lceil 0.2N\\rceil)!}{N!} & :\\ \\hl{\\sum_{k=1}^{N}z_k=\\lceil 0.2N\\rceil}\\\\ 0 & :\\ \\text{otherwise,}\\end{cases}\n\\]\n``where the corresponding design space is $\\mathcal Z\\supseteq\\{0,1\\}^N$.''", "corrected": "``Letting $\\lceil 0.2N\\rceil$ be the number of observations in treatment, the design is''\n\\[\nf(\\mathbf z)=\\begin{cases}\\dfrac{(\\lceil 0.2N\\rceil)!(N-\\lceil 0.2N\\rceil)!}{N!} & :\\ \\hl{\\mathbf z\\in\\{0,1\\}^N\\text{ and }}\\sum_{k=1}^{N}z_k=\\lceil 0.2N\\rceil\\\\ 0 & :\\ \\text{otherwise,}\\end{cases}\n\\]\n``where the corresponding design space is $\\mathcal Z\\supseteq\\{0,1\\}^N$.''", "intended_meaning": "The text fixes the design: $\\lceil 0.2N\\rceil$ of the $N$ students are assigned to the program uniformly at random, and the exposure mappings on p.~5 take $\\mathbf z\\in\\{0,1\\}^N$. For binary vectors the display is correct. With $k=\\lceil 0.2N\\rceil$, there are $\\binom{N}{k}$ binary vectors with $k$ ones, and each has mass $k!(N-k)!/N!=1/\\binom{N}{k}$. As printed, the positive case also covers vectors that are not binary and have the same sum, since $f$ is defined on $\\mathbb R^N$ and the design space is only required to contain $\\{0,1\\}^N$. For $N=2$, the vectors $(0,1)$, $(1,0)$ and $(1/2,1/2)$ each receive mass $1/2$, and these three masses sum to $3/2$. For $N=10$ and $\\mathcal Z=\\{0,1,2\\}^{10}$, the 45 binary vectors with two ones and the 10 vectors with a single entry equal to 2 give total mass $55/45=11/9$. Nothing after the display depends on the omission." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N004", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "historical_lean_attribution", "exhibits": [ { "alt": "Crop of the heading '4. Inferential targets' and the first five lines of the section. A red box marks the words 'new inferential targets' in the first sentence.", "caption": "Section 4, p. 5 (PDF p. 5), first sentence. The marked words call the targets new.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N006.png" }, { "alt": "Crop of four lines of Section 3.1. A red box marks the words 'novel estimands' in the sentence 'However, we will use this setup to define novel estimands in Section 4.'", "caption": "Section 3.1, p. 3 (PDF p. 3). The marked words call the estimands novel.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N006-p3.png" }, { "alt": "Crop of five lines of the last paragraph of the Introduction. A red box marks the words 'new inferential targets' in the sentence about Section 4.", "caption": "Introduction, p. 2 (PDF p. 2): the outline of the article. The marked words repeat the description.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N006-p2.png" } ], "formal_release": null, "formal_statements": [ { "id": "P061-T6", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T6" } ], "headline": "The novelty claim for the targets of Section 4 needs qualification: the conditional mean of Equation (5) and the contrast of Equation (8) coincide on their shared domain with the expected exposure outcome and the expected exposure effect of Sävje (2024), and under an individualistic binary exposure the AEED is the average distributional shift effect of Sävje, Aronow, and Hudgens (2021).", "historical_evidence_type": "lean_proof", "id": "P061-N006", "location": "Section 4, p. 5 (PDF p. 5), first sentence; Section 3.1, p. 3 (PDF p. 3); Introduction, p. 2 (PDF p. 2), last paragraph", "pdf_page": 5, "printed_page": "5", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P061-T6" ], "text": { "as_printed": "Section 4, p.~5: ``We define \\hl{new} inferential targets for the design-based framework.''\n\nSection 3.1, p.~3: ``However, we will use this setup to define \\hl{novel} estimands in Section 4.''\n\nIntroduction, p.~2: ``In Section 4, we define a set of \\hl{new} inferential targets for the design-based setting.''", "corrected": "Section 4, p.~5: ``We \\hl{formulate design-conditional} inferential targets\\hl{, building on the expected exposure outcomes and effects of Sävje (2024). Under individualistic binary exposure, the AEED is the average distributional shift effect of Sävje, Aronow, and Hudgens (2021).}''\n\nSection 3.1, p.~3: ``However, we will use this setup to define \\del{novel} estimands in Section 4.''\n\nIntroduction, p.~2: ``In Section 4, we define a set of \\del{new} inferential targets for the design-based setting.''", "evidence": "The evidence is a comparison with the earlier definitions and a derivation of the identities. I read Sävje (2024) in its author manuscript of 13 March 2023 (arXiv:2103.06471v2); a manuscript of the same paper dated 26 November 2019 has the same two definitions. I did not obtain the typeset text of the journal article. I read Definition 4 of Sävje, Aronow, and Hudgens (2021) in the author manuscript of the published article and in arXiv:1711.06399v4.\n\n\\begin{audittable}{lll}{The targets of Section 4 beside the earlier definitions. The locators for Sävje (2024) are those of the author manuscript of 2023.}\nIn the article & Formula & Earlier definition \\\\\nEPO, Equation (5) & $\\overline{y}_i(d)=\\operatorname{E}[y_i(\\mathbf Z)\\mid D_i=d]$ & Sävje (2024), Section 4.3 \\\\\nAEED, Equation (8) & $N^{-1}\\sum_{i}[\\overline{y}_i(d)-\\overline{y}_i(d')]$ & Sävje (2024), Definition 1 \\\\\nAEED with $g_i(\\mathbf z)=z_i$ & $N^{-1}\\sum_{i}\\{\\operatorname{E}[Y_i\\mid Z_i=1]-\\operatorname{E}[Y_i\\mid Z_i=0]\\}$ & Sävje et al. (2021), Definition 4 \\\\\n\\end{audittable}\n\nThe identities hold for every design with finitely many assignments. By the definition of conditional probability, the ratio in Equation (5) equals $\\operatorname{E}[y_i(\\mathbf Z)\\mid D_i=d]$ whenever $\\pi_i(d)>0$. Substituting this identity into Equations (6)--(8), I obtain the contrast of Definition 1. If $g_i(\\mathbf z)=z_i$ and the treatment is binary, then $D_i=Z_i$ and $Y_i=y_i(\\mathbf Z)$, and the contrast $\\tau(1,0)$ is the ADSE. The three identities are also proved in Lean, as identities between weighted sums over an arbitrary finite set of assignments.", "problem": "The article calls the targets of Section 4 new or novel in three places. Earlier work defines the same conditional mean and the same contrast on their shared domain, and the article does not say so. Sävje (2024) defines the conditional mean of Equation (5), with the expectation taken over the design, and the average contrast of Equation (8), for an arbitrary exposure mapping. When the exposure of a unit is its own binary treatment, $g_i(\\mathbf z)=z_i$, the AEED is the average distributional shift effect (ADSE) of Sävje, Aronow, and Hudgens (2021). The article cites both works for other purposes, on p.~5 and in footnote 8, and footnote 7 credits Hudgens and Halloran (2008) with a special case of the EPO. The defect lies in the claim of novelty and in the missing attribution. Equations (5)--(8) are correct as definitions.", "what_survives": "Equations (5)--(8), the values of the targets, the analysis of NURVA and SUTVA, the examples, Tables 1--5, Figure 1 and the estimation results of the supplement do not depend on whether the targets are new, and none of them changes. I did not assess the novelty of the framework as a whole, of the survey-sampling interpretation or of NURVA. Footnotes 7 and 8 are not affected." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N006", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" }, { "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 from the top of page 7. A red box marks the word implies in the sentence: Our previously introduced exposure mapping for the survey setting implies that the number of members reported for this group should be unaffected by the act of surveying other groups.", "caption": "Section 4.1 (continued), p. 7 (PDF p. 7), first paragraph. The marked verb credits the exposure mapping with an invariance that only a stability assumption supplies.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N007.png" } ], "formal_release": null, "formal_statements": [], "headline": "The exposure mapping only labels assignments, and the invariance described in the sentence comes from a stability assumption on that mapping, not from the mapping itself.", "id": "P061-N007", "location": "Section 4.1 (continued), first paragraph of p. 7 (PDF p. 7), sentence beginning “Our previously introduced exposure mapping”", "pdf_page": 7, "printed_page": "7", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "While we do not typically think of surveys as requiring SUTVA-like assumptions, survey outcomes are not generally well-defined in their absence. Consider a quantity such as ``the potential outcome of rebel group three, when it is surveyed.'' Our previously introduced exposure mapping for the survey setting \\hl{implies} that the number of members reported for this group should be unaffected by the act of surveying other groups (no interference) and whether the group is surveyed first, second or third (no hidden treatment variations).", "corrected": "\\hl{For our previously introduced exposure mapping for the survey setting to yield a stable outcome for the surveyed exposure on the design support, one would have to assume} that the number of members reported for this group should be unaffected by the act of surveying other groups (no interference) and whether the group is surveyed first, second or third (no hidden treatment variations).", "intended_meaning": "The paragraph describes what a SUTVA-like assumption must assert for the outcome of a group ``when it is surveyed'' to be a single number, and that description is correct. The mapping of Section 3.3, $g_i(\\mathbf z)=1$ if $z_i>0$ and $g_i(\\mathbf z)=0$ otherwise, only labels each group as surveyed or not surveyed. For example, the survey orders $(1,2,3,0,\\ldots,0)$ and $(3,2,1,0,\\ldots,0)$ each have probability $1/720$ under the design of p.~3, and group three has the exposure $1$ under both. With the outcome rule $y_3(\\mathbf z)=z_3$, its outcomes are $3$ and $1$. The invariance holds once stability is assumed for this mapping, as in Condition 5.1 (NURVA). Nothing in the paper depends on the verb." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N007", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the middle of page 13, from the sentence that defines the corresponding AEED to part of the sentence on additional stability assumptions. Red boxes mark four terms: the italic subscript Z prime of tau, the italic z inside g i of z in the display, and two occurrences of y i of d that have no bar over the y.", "caption": "Section 6.3, p. 13 (PDF p. 13). Four terms are marked: the italic subscript in $\\tau_{Z'}(2,1)$, the italic argument of $g_i$, and the two EPOs printed as $y_i(d)$ without an overbar.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N008.png" } ], "formal_release": null, "formal_statements": [ { "id": "P061-T9", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T9" } ], "headline": "In Section 6.3 the two EPOs should carry their overbar, $\\overline y_i(d)$, and the argument of $g_i$ and the first subscript $\\mathbf Z'$ should be bold.", "id": "P061-N008", "location": "Section 6.3, p. 13 (PDF p. 13): the sentence defining the corresponding AEED, the display defining $g_i$, and the paragraph below it", "pdf_page": 13, "printed_page": "13", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P061-T9" ], "text": { "as_printed": "Using the same exposure mapping, define the corresponding AEED $\\tau_{\\hl{Z'}}(2,1)$.$^{10}$\n\nWe define the exposure mapping as\n\\[\ng_i(\\hl{z})=\\begin{cases}2&:z_i=2\\\\1&:z_i=1\\\\0&:z_i=0\\end{cases},\\ \\forall i\\in\\{1,\\ldots,N\\},\\mathbf z\\in\\mathcal Z.\n\\]\nEven if NURVA holds for both $\\mathbf Z$ and $\\mathbf Z'$, there is no guarantee that $\\tau_{\\mathbf Z}(2,1)$ will equal $\\tau_{\\mathbf Z'}(2,1)$: changing either $n$ to $n'$ or $k$ to $k'$ changes the distribution of other units' sampling or treatment statuses conditional on $D_i=d$, and hence may change the EPOs $\\hl{y_i(d)}$ that enter the AEPOs. Learning what would happen under alternative values of $(n,k)$ or in alternative target populations requires additional stability assumptions on how $\\hl{y_i(d)}$ varies with the sampling and treatment scheme, as in the broader generalizability and transportability literature.", "corrected": "Using the same exposure mapping, define the corresponding AEED $\\tau_{\\hl{\\mathbf Z'}}(2,1)$.$^{10}$\n\nWe define the exposure mapping as\n\\[\ng_i(\\hl{\\mathbf z})=\\begin{cases}2&:z_i=2\\\\1&:z_i=1\\\\0&:z_i=0\\end{cases},\\ \\forall i\\in\\{1,\\ldots,N\\},\\mathbf z\\in\\mathcal Z.\n\\]\n\\ldots\\ and hence may change the EPOs $\\hl{\\overline y_i(d)}$ that enter the AEPOs. Learning what would happen under alternative values of $(n,k)$ or in alternative target populations requires additional stability assumptions on how $\\hl{\\overline y_i(d)}$ varies with the sampling and treatment scheme, as in the broader generalizability and transportability literature.", "evidence": "The evidence is a check of p.~13 against Equation (5) on p.~6 and the vector notation on p.~4, including the fonts of the marked symbols.", "problem": "Equation (5) writes the EPO as $\\overline y_i(d)$, and the paper reserves $y_i(\\cdot)$ for raw potential outcomes of an assignment vector. The two quantities that this passage calls EPOs are printed without the overbar. In the same passage the argument of $g_i$ is an italic $z$ where the quantifier has the bold vector $\\mathbf z$, and the first subscript $Z'$ is italic where the next paragraph has the bold $\\mathbf Z'$.", "what_survives": "The argument of Section 6.3 is unaffected. NURVA for each of two designs does not make their EPOs or AEEDs equal, and additional stability assumptions are needed." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N008", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of five lines of text from the conclusion on page 14. Red boxes mark the phrase asymptotically equivalent and, on the next line, the phrase sufficiently local.", "caption": "Section 8 (Conclusion), p. 14 (PDF p. 14), second paragraph. The marked phrases assert an equivalence of inference under a condition, “sufficiently local”, that the paper does not specify.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N009.png" } ], "formal_release": null, "formal_statements": [ { "id": "P061-T4", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T4" }, { "id": "P061-T5", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T5" } ], "headline": "For the same estimator and uncertainty scale, the standardized statistics centred on the AEED and the average direct effect are asymptotically equal only when the target difference is $o(s_N)$; local dependence and local interference do not imply this, and the cited central limit theorem in arXiv version 2 is centred on the AEED.", "id": "P061-N009", "location": "Section 8 (Conclusion), second paragraph, sentence beginning “There is additional hope”, p. 14 (PDF p. 14)", "pdf_page": 14, "printed_page": "14", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P061-T4", "P061-T5" ], "text": { "as_printed": "There is additional hope for further refining inference on targets such as AEEDs from Sävje (2024); inference on AEEDs is \\hl{asymptotically equivalent} to inference on the ``average direct effect'' $N^{-1}\\sum_{i=1}^{N}\\bigl(\\mathrm E[y_i(d;\\mathbf D_{-i})-y_i(d';\\mathbf D_{-i})]\\bigr)$ if both statistical dependence in $\\mathbf D$ and unmodeled interference are \\hl{sufficiently local}.", "corrected": "There is additional hope for further refining inference on targets such as AEEDs from Sävje (2024). For the individualistic exposure mapping $g_i(\\mathbf z)=z_i$, let $\\tau_N=\\tau(d,d')$ be the AEED, let $A_N=N^{-1}\\sum_{i=1}^{N}\\mathrm E[y_i(d;\\mathbf Z_{-i})-y_i(d';\\mathbf Z_{-i})]$ be the ``average direct effect'', and let $\\widehat\\tau_N$ be an estimator of $\\tau_N$ with standard error $s_N>0$ and estimated standard error $\\widehat s_N>0$. \\hl{Assume that the assignments used to define $A_N$ are feasible and both target means are finite.} Suppose that $(\\widehat\\tau_N-\\tau_N)/s_N$ converges in distribution to a random variable $W$ \\hl{with a continuous distribution}, that $\\widehat s_N/s_N\\to1$ in probability, and that\n\\[\n\\hl{|\\tau_N-A_N|=o(s_N).}\n\\]\nThen $(\\widehat\\tau_N-A_N)/\\widehat s_N$ converges in distribution to $W$, so confidence intervals based on this limit have the same limiting coverage for either target. If $(\\tau_N-A_N)/s_N\\to c\\ne0$, the limit is $W+c$. \\hl{Local dependence in $\\mathbf Z$ and local unmodeled interference do not imply the displayed condition.} \\hl{Under the hypotheses of Lemma A20 of Sävje, Aronow, and Hudgens (2021), with its bound $k$ fixed,} the condition $\\alpha_{\\mathrm{int}}=o(Ns_N)$ is sufficient for it; when $s_N$ is of order $N^{-1/2}$, this reads $\\alpha_{\\mathrm{int}}=o(\\sqrt N)$.", "evidence": "The evidence is a derivation, an exact example, and a check of the cited works. For the derivation, let $\\tau_N$ be the AEED, $A_N$ the average direct effect, $\\delta_N=\\tau_N-A_N$, and $\\widehat\\tau_N$ an estimator with standard error $s_N>0$. Both targets are deterministic, and\n\\[\n\\frac{\\widehat\\tau_N-A_N}{s_N}-\\frac{\\widehat\\tau_N-\\tau_N}{s_N}=\\frac{\\delta_N}{s_N}.\n\\]\nIf $(\\widehat\\tau_N-\\tau_N)/s_N$ converges in distribution to $W$ and $\\widehat s_N/s_N\\to1$ in probability, then Slutsky's theorem implies that $(\\widehat\\tau_N-A_N)/\\widehat s_N$ converges in distribution to $W$ when $\\delta_N/s_N\\to0$, and to $W+c$ when $\\delta_N/s_N\\to c\\ne0$.\n\nFor the example, fix an integer $m\\ge2$ and take $m^8$ disjoint pairs of units, so that $N=2m^8$. The pairs are assigned independently. Within a pair the assignments $(0,0)$ and $(1,1)$ each have probability $(1+1/m)/4$, and $(0,1)$ and $(1,0)$ each have probability $(1-1/m)/4$. Let $g_i(\\mathbf z)=z_i$, and let $y_i(\\mathbf z)=1+z_j/m$, where $j$ is the other unit in the pair of $i$. Each unit has $\\Pr[D_i=1]=1/2$, the outcomes lie in $[1,3/2]$, and each assignment and each outcome depends on at most one other unit. The correlation of the two assignments in a pair is $1/m$, and the assignment of the other unit in the pair changes the outcome by $1/m$. The outcome does not depend on the unit's own assignment, and hence $A_N=0$. Since $\\Pr[D_j=1\\mid D_i=1]=(1+1/m)/2$ and $\\Pr[D_j=1\\mid D_i=0]=(1-1/m)/2$, I have $\\tau_N=1/m^2$ and $\\delta_N=1/m^2\\to0$. The Horvitz--Thompson estimator of the AEED is unbiased, with variance\n\\[\ns_N^2=\\frac{V_m}{4m^8},\\qquad V_m=8+\\frac{16}{m}+\\frac{14}{m^2}+\\frac{2}{m^3}-\\frac{4}{m^4},\\qquad 8\\le V_m<20 .\n\\]\nA direct computation gives $\\delta_N^2/s_N^2=4m^4/V_m\\ge m^4/5$. The difference between the two targets is of order $N^{-1/4}$ and the standard error is of order $N^{-1/2}$. By Chebyshev's inequality, the interval $\\widehat\\tau_N\\pm5s_N$ covers the AEED with probability at least $24/25$, and for $m\\ge5$ it covers the average direct effect with probability at most $V_m/m^4\\le20/m^4$.\n\\begin{audittable}{rrcccc}{Exact values in the example with $N=2m^8$ units. The last column is the upper bound $V_m/m^4$ on the probability that $\\widehat\\tau_N\\pm5s_N$ covers the average direct effect, which applies for $m\\ge5$; decimal values are rounded.}\n$m$ & $N$ & $\\delta_N$ & $\\delta_N^2/s_N^2$ & $\\delta_N/s_N$ & Bound on coverage \\\\\n$2$ & $512$ & $1/4$ & $128/39$ & $1.81$ & --- \\\\\n$3$ & $13122$ & $1/9$ & $6561/302$ & $4.66$ & --- \\\\\n$5$ & $781250$ & $1/25$ & $390625/1839$ & $14.6$ & $7356/390625\\approx0.019$ \\\\\n$10$ & $2\\times10^{8}$ & $1/100$ & $5\\times10^{7}/12177$ & $64.1$ & $12177/12500000\\approx0.00097$ \\\\\n$100$ & $2\\times10^{16}$ & $1/10^{4}$ & $10^{16}/204035049$ & $7000.8$ & $204035049/(2.5\\times10^{15})\\approx8.2\\times10^{-8}$ \\\\\n\\end{audittable}\nThe example violates NURVA. It satisfies the conditions of the central limit theorem in the version of Sävje (2024) that was read, which is Proposition S8 of the supplement to arXiv version 2. The outcomes are bounded, the exposure probabilities equal $1/2$, and the variance is of order $N^{-1}$. Each unit depends on its own pair only, and hence Condition S6, $d_{\\max}=o(n^{1/4})$, holds. Propositions S7 and S8 of that supplement keep the limit centred on the expected exposure effect. Lemma A20 of Sävje, Aronow, and Hudgens (2021), read in Supplement A of arXiv version 4, bounds the difference between the corresponding two targets of that paper by $6k^2n^{-1}\\alpha_{\\mathrm{int}}$ in its notation, and this bound tends to zero only when $\\alpha_{\\mathrm{int}}=o(n)$. Both works were read as preprints, and the published journal versions were not compared.", "problem": "The sentence cites Sävje (2024) for an equivalence between inference on AEEDs and inference on the average direct effect when the dependence in the design and the unmodeled interference are ``sufficiently local'', and neither the paper nor its supplement says what that requires. I take the individualistic exposure mapping $g_i(\\mathbf z)=z_i$, with feasible assignments and finite means defining the average direct effect (see \\fid{P061-N019}), and an estimator of the AEED with standard error $s_N$. The statistic standardized at the AEED and the statistic standardized at the average direct effect differ by $\\delta_N/s_N$, where $\\delta_N$ is the difference between the two targets. These statistics are asymptotically equal if and only if $\\delta_N=o(s_N)$. It is not enough that $\\delta_N$ tends to zero. In arXiv version 2 of Sävje (2024), including its supplement, the central limit theorem under local dependence is centred on the AEED, and I found no result there transferring inference to the average direct effect. Locality does not imply the rate. In the example below, each unit depends on one other unit only and the conditions of that theorem hold, while $\\delta_N/s_N$ diverges. Conditions stronger than locality can make the printed sentence true, and the sentence is not shown to be false under every interpretation of ``sufficiently local''.", "what_survives": "The definitions of the AEPO and the AEED, their interpretation, the examples, the finite-sample unbiasedness of the Horvitz--Thompson estimators, and the variance and consistency results stated under NURVA do not use this comparison. The closing remark changes. Using the same estimator and uncertainty scale for the average direct effect requires the rate condition below to preserve the first-order limit. Whether the journal version of Sävje (2024) contains a result of this kind was not checked." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N009", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of three lines of text and Condition 1.1 on page 5 of the supplement. A red box marks the words 'the variance bound is o(N^2).' at the end of the sentence that begins 'To provide intuition for Condition 1.3'.", "caption": "Supplementary Appendix, p. 5: the sentence that gives the intuition for Condition 1.3, and Condition 1.1. The marked words state the rate.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N010.png" } ], "formal_release": null, "formal_statements": [ { "id": "P061-T21", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T21" } ], "headline": "The requirement behind Condition 1.3 is that the bound on the variance of the Horvitz–Thompson estimator is $o(1)$; the quantity that is $o(N^2)$ is the bound $b_{(N)}c_{(N)}^2$ before division by $N^2$.", "id": "P061-N010", "location": "Supplementary Appendix, p. 5 (PDF p. 5 of the supplement), sentence before Condition 1.1", "pdf_page": 5, "printed_page": "5", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [ "P061-T21" ], "text": { "as_printed": "``To provide intuition for Condition 1.3, what we require jointly from our regularity conditions is that \\hl{the variance bound is $o(N^2)$}.''\n\n``\\textbf{Condition 1.3.} (Interaction condition) $b_{(N)}c_{(N)}^2=o(N^2)$''", "corrected": "``To provide intuition for Condition 1.3, what we require jointly from our regularity conditions is that \\hl{the bound on the variance of the Horvitz--Thompson mean estimator is $o(1)$, equivalently $b_{(N)}c_{(N)}^2=o(N^2)$}.''", "intended_meaning": "Under NURVA, Condition 1.1 implies that $|\\overline{y}_i(d)|/\\pi_i(d)\\le c_{(N)}$ for every $i$. Inserting this bound into supplement Equation (5) and using Condition 1.2, I obtain\n\\[\n\\operatorname{Var}[\\widehat{\\overline{y}}(d)]\\le\\frac{c_{(N)}^2}{N^2}\\sum_{i=1}^{N}\\sum_{j=1}^{N}|\\pi_{ij}(d)-\\pi_i(d)\\pi_j(d)|=\\frac{b_{(N)}c_{(N)}^2}{N^2}.\n\\]\nBy Condition 1.3, the right side is $o(1)$, and Chebyshev's inequality gives Proposition 1. The sentence therefore refers to $b_{(N)}c_{(N)}^2$, which bounds the variance of the total $N\\widehat{\\overline{y}}(d)$. A bound of order $o(N^2)$ on the variance of $\\widehat{\\overline{y}}(d)$ itself would not give consistency. Condition 1.3 and Proposition 1 are correct as stated." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N010", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" }, { "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 from page 6 of the supplement. A red box marks the rate little o of n to the power one quarter in the sentence: the maximum degree d max grows at a rate of o of n to the one quarter.", "caption": "Supplementary Material, p. 6 (PDF p. 6), “Central limit theorems” paragraph. The marked rate is written in $n$, while the population size in the supplement is $N$.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N011.png" }, { "alt": "Crop of the last four lines of text on page 6 of the supplement. A red box marks the rate capital O of n to the power one half, at the end of the sentence: the maximum degree d max is O of n to the one half.", "caption": "Supplementary Material, p. 6 (PDF p. 6), last paragraph. The second rate is also written in $n$.", "url": "https://pmaronow.github.io/llm-audit/works/P061/exhibits/P061-N011-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P061-T10", "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T10" } ], "headline": "The two degree rates on p. 6 of the supplement should be written in the population size $N$, as $o(N^{1/4})$ and $O(N^{1/2})$.", "id": "P061-N011", "location": "Supplementary Material, p. 6 (PDF p. 6 of the supplement), “Central limit theorems” paragraph and last paragraph of the page", "pdf_page": 6, "printed_page": "6", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P061-T10" ], "text": { "as_printed": "Recent advancements in this area have been made under weakened assumptions about the growth of the maximum degree in the dependency graph, specifically that the maximum degree $d_{\\max}$ grows at a rate of $o(\\hl{n}^{1/4})$ (Ogburn et al. 2024; Han, Basse, and Bojinov 2024). \\ldots\n\nHowever, there are relevant limitations to note. Sävje, Aronow, and Hudgens (2021) prove that Chebyshev's inequality remains asymptotically sharp under strong restrictions on the growth of the degree sequence, even when the average degree $d_{\\mathrm{avg}}$ is $O(1)$ and the maximum degree $d_{\\max}$ is $O(\\hl{n}^{1/2})$.", "corrected": "\\ldots\\ specifically that the maximum degree $d_{\\max}$ grows at a rate of $o(\\hl{N}^{1/4})$ (Ogburn et al. 2024; Han, Basse, and Bojinov 2024). \\ldots\\ even when the average degree $d_{\\mathrm{avg}}$ is $O(1)$ and the maximum degree $d_{\\max}$ is $O(\\hl{N}^{1/2})$.", "evidence": "The evidence is a check of pp.~1, 5 and 6 of the supplement.", "problem": "Both rates are written in $n$, which the passage does not define. The supplement indexes its populations by their size $N$ and lets $N$ tend to infinity (p.~5), and on p.~1 it uses $n$ for the number of units sampled into exposure $d$. The rates refer to the number of units in the dependency graph, which is $N$.", "what_survives": "Nothing depends on the symbol. The correction concerns the notation only. It does not assess whether the exponents are those of the cited works." }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-N011", "work": { "arxiv_id": "2505.10519", "doi": "10.1017/pan.2026.10045", "title": "On the Foundations of the Design-Based Approach", "url": "https://pmaronow.github.io/llm-audit/works/P061/" }, "work_id": "P061" } ], "lean": { "applicable": true, "guide": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/docs/works/P061.md", "repository": "https://github.com/pmaronow/llm-audit-lean", "targets": [ { "declarations": [ { "claim_revision_id": "catalog:b38dcb36d01aa956a6dbfea9e1d24fb737350f5d205a549f7efac64f5463968d", "conditional_on": [], "credit_reason": "This is the explicit witness family for T1(a)–(e), with the remaining named components supplied by the other T1 rows. 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main p. 14, with the mapping of Section 6.2, pp. 11–12. Independent uniform treatment versions give AEED 3/2, both raw-coordinate direct averages 1, a persistent gap, an exposure-schedule obstruction, and the exact HT moments and tail bound. This refutes the raw-assignment-coordinate reading of the comparison under general exposure mappings.

", "headline": "Section 8 (Conclusion), second paragraph, the ‘average direct effect’ sentence: counterexample with coarsened treatment versions", "location": "p. 14 (PDF p. 14); the ingredients of the example are those of Section 6.2, pp. 11–12 (PDF pp. 11–12)" }, "id": "P061-T1", "text": { "body": "main p. 14, with the mapping of Section 6.2, pp. 11–12. Independent uniform treatment versions give AEED 3/2, both raw-coordinate direct averages 1, a persistent gap, an exposure-schedule obstruction, and the exact HT moments and tail bound. This refutes the raw-assignment-coordinate reading of the comparison under general exposure mappings.", "headline": "Section 8 (Conclusion), second paragraph, the ‘average direct effect’ sentence: counterexample with coarsened treatment versions", "location": "p. 14 (PDF p. 14); the ingredients of the example are those of Section 6.2, pp. 11–12 (PDF pp. 11–12)" }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T1" }, { "declarations": [ { "claim_revision_id": "catalog:17853e5436dfb5568cf38b8d6d2686ef7f1b0c18ce75556cabcfb1e6a1025bdc", "conditional_on": [], "credit_reason": "This is a witness for the domain defect, not a claimed numerical counterexample to an equality. 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main p. 14 and Equation (3), p. 5. The adjacent-pair exposure map has positive target exposures and stable outcomes, but its hybrid arguments fail the exposure or raw assignment domain tests, also for independent copies of the pair. This refutes well-definedness of the displayed hybrid readings, rather than an asserted numerical equality.

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main p. 14 and footnote 8, p. 7. Own-coordinate independence identifies the AEED with the feasible raw direct contrast, and exposure independence does the same for a consistent response schedule on exposure vectors.

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main p. 14, supplement's HT contrast. Independent correlated pairs on the full binary cube have exactly one assignment-dependence neighbor and one interference neighbor per unit. All clauses (a)–(f) hold: positivity, bounded responses, the exact AEED, zero direct effect, unbiased HT moments, both variance expressions and coverage bounds. The target gap vanishes but its standardized value diverges; its exact population-size normalization and the standard-error limit give the claimed orders. The usual measure-theoretic conditions are assumed; here every space is finite. Limits use m=index+2, along the claimed populations N=2m^8.

", "headline": "Section 8 (Conclusion), second paragraph, the claim that inference on AEEDs is asymptotically equivalent to inference on the average direct effect under local dependence and interference: counterexample to the transfer of inference", "location": "p. 14 (PDF p. 14)" }, "id": "P061-T4", "text": { "body": "main p. 14, supplement's HT contrast. Independent correlated pairs on the full binary cube have exactly one assignment-dependence neighbor and one interference neighbor per unit. All clauses (a)–(f) hold: positivity, bounded responses, the exact AEED, zero direct effect, unbiased HT moments, both variance expressions and coverage bounds. The target gap vanishes but its standardized value diverges; its exact population-size normalization and the standard-error limit give the claimed orders. The usual measure-theoretic conditions are assumed; here every space is finite. 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main p. 14, Equation (8) and footnote 8, p. 7, and supplement p. 5. For individualistic assignments, a negligible AEED--ADE gap in positive standard-error units transfers the limiting law under consistent scale estimation. Both Wald coverages tend to L([-q,q]) for each fixed q>0 with zero limiting mass at both endpoints; a continuous limiting law automatically satisfies this condition. A nonzero limiting standardized drift shifts the law. The T4 design has a vanishing raw gap but a diverging absolute standardized gap and failing direct-effect coverage. For arbitrary discrete feasible designs with finite neighborhood support, the range-times-TV envelope bounds the target gap, and an envelope o(s) suffices. The usual measure-theoretic conditions are assumed. Only the hybrid differences need finite means for transfer, and estimated scales may be nonpositive on asymptotically negligible events.

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main pp. 2–3, 5 and Equations (5)–(8), pp. 6–7. On the common finite domain, EPOs are conditional expected exposure outcomes and the AEED is their average contrast, specializing to the binary average distributional shift functional.

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main Equations (5)–(8), pp. 6–7, supplement Equation (3), p. 2. The one-unit real-assignment geometric design is a probability measure on the power set, with identity assignment vector, nonnegative-integer support, finite raw responses and both exposure probabilities 1/2. Its weighted positive-integer terms alternate +1,-1, giving all stated finite sign-part sums and no limit of the ordered signed series. The exposure-one numerator and conditional mean have both sign parts +infinity, whereas the exposure-zero ordinary EPO is zero. The exposure-one EPO, AEPO, EED and AEED have no ordinary value. The pointwise finite HT estimator has positive and negative expectations +infinity and no ordinary finite expectation. The usual measure-theoretic conditions are assumed for existing quantities; nonexistence of the claimed means is the conclusion being demonstrated.

", "headline": "Condition 4.1 and Equation (5), with Equations (6)–(8) and supplement Equation (3): positivity alone does not make the targets well-defined", "location": "p. 6 (PDF p. 6) for Condition 4.1 and Equation (5); p. 7 (PDF p. 7) for Equations (6)–(8); Supplementary Appendix p. 2 (publisher-supplement.pdf p. 2) for supplement Equation (3)" }, "id": "P061-T7", "text": { "body": "main Equations (5)–(8), pp. 6–7, supplement Equation (3), p. 2. The one-unit real-assignment geometric design is a probability measure on the power set, with identity assignment vector, nonnegative-integer support, finite raw responses and both exposure probabilities 1/2. Its weighted positive-integer terms alternate +1,-1, giving all stated finite sign-part sums and no limit of the ordered signed series. The exposure-one numerator and conditional mean have both sign parts +infinity, whereas the exposure-zero ordinary EPO is zero. The exposure-one EPO, AEPO, EED and AEED have no ordinary value. The pointwise finite HT estimator has positive and negative expectations +infinity and no ordinary finite expectation. The usual measure-theoretic conditions are assumed for existing quantities; nonexistence of the claimed means is the conclusion being demonstrated.", "headline": "Condition 4.1 and Equation (5), with Equations (6)–(8) and supplement Equation (3): positivity alone does not make the targets well-defined", "location": "p. 6 (PDF p. 6) for Condition 4.1 and Equation (5); p. 7 (PDF p. 7) for Equations (6)–(8); Supplementary Appendix p. 2 (publisher-supplement.pdf p. 2) for supplement Equation (3)" }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T7" }, { "declarations": [ { "claim_revision_id": "catalog:5a7e6b353249232c832eb6fdf7966c7244a0b1384d1340b50207ea95401e37d3", "conditional_on": [], "credit_reason": "This row proves T8(i)–(iii); the automatic-moment branches of (iv) are separate. 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main pp. 6–7 and supplement Equation (3), p. 2. Positive target cells with finite targeted absolute moments give ordinary conditional means and unbiased integrable HT estimates of AEPO and AEED, and the moments are automatic on finite support or under stability within the targeted exposure cells, with their exact constant-cell values.

", "headline": "Condition 4.1 and Equations (5)–(8), with supplement Equation (3): corrected well-definedness and first-moment unbiasedness under a targeted first-moment condition", "location": "p. 6 (PDF p. 6) and p. 7 (PDF p. 7); Supplementary Appendix p. 2 (publisher-supplement.pdf p. 2); the general claims qualified are on p. 2 (PDF p. 2) and p. 14 (PDF p. 14)" }, "id": "P061-T8", "text": { "body": "main pp. 6–7 and supplement Equation (3), p. 2. Positive target cells with finite targeted absolute moments give ordinary conditional means and unbiased integrable HT estimates of AEPO and AEED, and the moments are automatic on finite support or under stability within the targeted exposure cells, with their exact constant-cell values.", "headline": "Condition 4.1 and Equations (5)–(8), with supplement Equation (3): corrected well-definedness and first-moment unbiasedness under a targeted first-moment condition", "location": "p. 6 (PDF p. 6) and p. 7 (PDF p. 7); Supplementary Appendix p. 2 (publisher-supplement.pdf p. 2); the general claims qualified are on p. 2 (PDF p. 2) and p. 14 (PDF p. 14)" }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T8" }, { "declarations": [ { "claim_revision_id": "catalog:69a6a137e7a56cb115f92fe7eb3b18c330e61ae1a341d70a13ccf0342987de1d", "conditional_on": [], "credit_reason": "The target's formal content is existential, so choosing N=4 and the two specified sample sizes loses no case; this is a witness to design dependence rather than a universal formula for all sampling designs.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.sampling_treatment_designs_differ", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L1566", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:69a6a137e7a56cb115f92fe7eb3b18c330e61ae1a341d70a13ccf0342987de1d", "conditional_on": [], "credit_reason": "Auxiliary or specialized result; not needed separately for the retained clauses, which are supplied by the rows used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.sum_vector_succ", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L1505", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:69a6a137e7a56cb115f92fe7eb3b18c330e61ae1a341d70a13ccf0342987de1d", "conditional_on": [], "credit_reason": "The restriction n=2 or n=3 is exactly the pair of witnesses covered by this auxiliary row; (n-1 : real) uses real subtraction.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.sampling_treatment_properties", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L1527", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:69a6a137e7a56cb115f92fe7eb3b18c330e61ae1a341d70a13ccf0342987de1d", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P061-T9", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P061-T9" }, "html": { "body": "

main Section 6.3, p. 13 and footnote 10. The four-unit sampling/treatment designs with sample sizes two and three satisfy positivity and support stability, but have AEEDs two and three for the same raw response schedule, which fails global stability.

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Supplementary Appendix p. 6, the central-limit and limitations paragraphs. The three corrected population-indexed rate conditions have their standard little-o and big-O characterizations for nonnegative maximum- and average-degree sequences.

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main Equations (4)–(8), pp. 6–7, and the existence assertions on pp. 2, 6 and 14. Positive target exposures on finite positive-probability support give ordinary conditional exposure means and their unit and population identities, without SUTVA or stability.

", "headline": "Equations (4)–(8) with Condition 4.1: the design-conditional targets (EPO, AEPO, EED, AEED) and the claim that they are well-defined without SUTVA", "location": "p. 6 (PDF p. 6) for Equation (4), Condition 4.1 and Equation (5); p. 7 (PDF p. 7) for Equations (6)–(8); the claim is stated on p. 2 (PDF p. 2), p. 6 (PDF p. 6) and p. 14 (PDF p. 14)" }, "id": "P061-T11", "text": { "body": "main Equations (4)–(8), pp. 6–7, and the existence assertions on pp. 2, 6 and 14. Positive target exposures on finite positive-probability support give ordinary conditional exposure means and their unit and population identities, without SUTVA or stability.", "headline": "Equations (4)–(8) with Condition 4.1: the design-conditional targets (EPO, AEPO, EED, AEED) and the claim that they are well-defined without SUTVA", "location": "p. 6 (PDF p. 6) for Equation (4), Condition 4.1 and Equation (5); p. 7 (PDF p. 7) for Equations (6)–(8); the claim is stated on p. 2 (PDF p. 2), p. 6 (PDF p. 6) and p. 14 (PDF p. 14)" }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T11" }, { "declarations": [ { "claim_revision_id": "catalog:0b55a1cdc862bc0cfaaac6b96e853f8e5c8e7778f1523f7bdf21f95c1b0d80f4", "conditional_on": [], "credit_reason": "Global factorization is equivalent to the common-value encoding in the target; it assumes stability, not the EPO or AEED equalities being established.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.sutva_design_free", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L476", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:0b55a1cdc862bc0cfaaac6b96e853f8e5c8e7778f1523f7bdf21f95c1b0d80f4", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.household_sutva", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L734", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:0b55a1cdc862bc0cfaaac6b96e853f8e5c8e7778f1523f7bdf21f95c1b0d80f4", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.sutva_epo_global", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L2863", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:0b55a1cdc862bc0cfaaac6b96e853f8e5c8e7778f1523f7bdf21f95c1b0d80f4", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.sutva_design_invariance", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L2874", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:0b55a1cdc862bc0cfaaac6b96e853f8e5c8e7778f1523f7bdf21f95c1b0d80f4", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P061-T12", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P061-T12" }, "html": { "body": "

main text pp. 2, 6, 8–10, Condition 5.2 and Table 3. Global stability identifies the positive-cell potential outcomes and their contrasts independently of design, and housemate exposure gives AEED one for every positive household design.

", "headline": "Condition 5.2 (SUTVA) and the claim that the contrasts coincide with the usual average treatment effects when SUTVA holds; Table 3 (Exposure mapping that satisfies SUTVA)", "location": "p. 8 (PDF p. 8) for Condition 5.2 and the paragraph after it; p. 2 (PDF p. 2) and p. 6 (PDF p. 6) for the claim; p. 9 (PDF p. 9) and Table 3, p. 10 (PDF p. 10), for the example" }, "id": "P061-T12", "text": { "body": "main text pp. 2, 6, 8–10, Condition 5.2 and Table 3. Global stability identifies the positive-cell potential outcomes and their contrasts independently of design, and housemate exposure gives AEED one for every positive household design.", "headline": "Condition 5.2 (SUTVA) and the claim that the contrasts coincide with the usual average treatment effects when SUTVA holds; Table 3 (Exposure mapping that satisfies SUTVA)", "location": "p. 8 (PDF p. 8) for Condition 5.2 and the paragraph after it; p. 2 (PDF p. 2) and p. 6 (PDF p. 6) for the claim; p. 9 (PDF p. 9) and Table 3, p. 10 (PDF p. 10), for the example" }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T12" }, { "declarations": [ { "claim_revision_id": "catalog:f930acbaa70392c69cf95c23814c73a3ab454ef00f3cd326c7e4ea9d9df7fbc9", "conditional_on": [], "credit_reason": "This is the existence direction of observational equivalence; R290-04 supplies the converse inclusion. A concrete assignment vector can be identified with its abstract feasible-assignment label without losing cases.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.nurva_sutva_observational_equivalence", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L531", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f930acbaa70392c69cf95c23814c73a3ab454ef00f3cd326c7e4ea9d9df7fbc9", "conditional_on": [], "credit_reason": "The definition of SUTVA omits the printed EPO clause in accord with S10(d); on positive cells that clause follows from constancy and normalized event weights, as proved by R290-01.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.sutva_implies_nurva", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L329", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f930acbaa70392c69cf95c23814c73a3ab454ef00f3cd326c7e4ea9d9df7fbc9", "conditional_on": [], "credit_reason": "Covers the EPO component of T13(i); instantiate it at the two exposure cells and unfold eed to obtain the printed unit contrast. The positive-cell moment is implied by on-support constancy.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.nurva_epo", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L494", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f930acbaa70392c69cf95c23814c73a3ab454ef00f3cd326c7e4ea9d9df7fbc9", "conditional_on": [], "credit_reason": "An auxiliary law-invariance component, not a SUTVA-existence assertion. The arbitrary event quantifier describes the full joint observed-data law.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.observed_law_support", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L514", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:f930acbaa70392c69cf95c23814c73a3ab454ef00f3cd326c7e4ea9d9df7fbc9", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P061-T13", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P061-T13" }, "html": { "body": "

main text pp. 2 and 8, Condition 5.1. Support stability identifies supported-cell outcomes and contrasts, is implied by global stability, and gives exactly the same possible observed-data laws as global stability under a fixed design.

", "headline": "Condition 5.1 (NURVA): the EPO under NURVA, NURVA as the support-restricted variant of SUTVA, and their observational equivalence for a given design", "location": "p. 8 (PDF p. 8); the claim is announced on p. 2 (PDF p. 2)" }, "id": "P061-T13", "text": { "body": "main text pp. 2 and 8, Condition 5.1. Support stability identifies supported-cell outcomes and contrasts, is implied by global stability, and gives exactly the same possible observed-data laws as global stability under a fixed design.", "headline": "Condition 5.1 (NURVA): the EPO under NURVA, NURVA as the support-restricted variant of SUTVA, and their observational equivalence for a given design", "location": "p. 8 (PDF p. 8); the claim is announced on p. 2 (PDF p. 2)" }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T13" }, { "declarations": [ { "claim_revision_id": "catalog:75db229e9703c9aeead2952ba7bba1f36823fa70600bf1627cbc7d9d682256d1", "conditional_on": [], "credit_reason": "Exact formal content of T14; the claim only requests NURVA under the original design, and no additional NURVA assertion about the uniform alternative is smuggled in.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.household_tables", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L713", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:75db229e9703c9aeead2952ba7bba1f36823fa70600bf1627cbc7d9d682256d1", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P061-T14", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P061-T14" }, "html": { "body": "

main text pp. 8–10, Section 5.2 and Tables 1–2. The household voting schedule has AEEDs minus one, one and zero under the three printed designs, with the printed AEPOs, support stability, and an explicit off-support failure of global stability.

", "headline": "Section 5.2 (Toy example: NURVA holds but SUTVA does not), Table 1 and Table 2 (Different designs yield different AEEDs)", "location": "pp. 8–9 (PDF pp. 8–9) for the text; Table 1, p. 9 (PDF p. 9); Table 2, p. 10 (PDF p. 10)" }, "id": "P061-T14", "text": { "body": "main text pp. 8–10, Section 5.2 and Tables 1–2. The household voting schedule has AEEDs minus one, one and zero under the three printed designs, with the printed AEPOs, support stability, and an explicit off-support failure of global stability.", "headline": "Section 5.2 (Toy example: NURVA holds but SUTVA does not), Table 1 and Table 2 (Different designs yield different AEEDs)", "location": "pp. 8–9 (PDF pp. 8–9) for the text; Table 1, p. 9 (PDF p. 9); Table 2, p. 10 (PDF p. 10)" }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T14" }, { "declarations": [ { "claim_revision_id": "catalog:5b4230846ec5774259e664264224e23b709b23bf4c70c649dd253bc49d4972fb", "conditional_on": [], "credit_reason": "The finite spaces, zero-mass assignments, both raw schedules and both designs are exactly the printed Table 4 objects.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.equilibrium_tables", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L1395", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:5b4230846ec5774259e664264224e23b709b23bf4c70c649dd253bc49d4972fb", "conditional_on": [], "credit_reason": "The full schedule y_i=z_i is fixed by the target’s no-interference interpretation of Table 5. The no-interference property is an exported conjunct, not just a fact of the witness definition.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.hidden_version_tables", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L1454", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:5b4230846ec5774259e664264224e23b709b23bf4c70c649dd253bc49d4972fb", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P061-T15", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P061-T15" }, "html": { "body": "

main text pp. 11–12, Sections 6.1–6.2 and Tables 4–5. The printed attenuation, amplification and hidden-version schedules give the stated contrasts under both designs and satisfy support stability while failing global stability.

", "headline": "Section 6.1, Table 4 (General equilibria) and Section 6.2, Table 5 (Hidden treatment variations): a contrast under one design need not equal the contrast under another, even when NURVA holds", "location": "p. 11 (PDF p. 11) for the text of Sections 6.1 and 6.2; Tables 4 and 5 and the paragraph after them, p. 12 (PDF p. 12)" }, "id": "P061-T15", "text": { "body": "main text pp. 11–12, Sections 6.1–6.2 and Tables 4–5. The printed attenuation, amplification and hidden-version schedules give the stated contrasts under both designs and satisfy support stability while failing global stability.", "headline": "Section 6.1, Table 4 (General equilibria) and Section 6.2, Table 5 (Hidden treatment variations): a contrast under one design need not equal the contrast under another, even when NURVA holds", "location": "p. 11 (PDF p. 11) for the text of Sections 6.1 and 6.2; Tables 4 and 5 and the paragraph after them, p. 12 (PDF p. 12)" }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T15" }, { "declarations": [ { "claim_revision_id": "catalog:6ef3d198ff9cc0ff320c20a84a831dd9b16097b5ae11faaa184c9887b32aac89", "conditional_on": [], "credit_reason": "Refutes unconditional mean existence from positivity, which does not contradict the printed identity under the required existence-of-expectations reading.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.Refutations.printed_general_ht_unbiased_refuted", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Refutations.lean#L34", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:6ef3d198ff9cc0ff320c20a84a831dd9b16097b5ae11faaa184c9887b32aac89", "conditional_on": [], "credit_reason": "{'source': 'dispute', 'reason': \"The row expressly credits a refutation plus an explicit correction, and P061_PaperDomains.lean:218–223 concludes both ¬PrintedClaims.printed_general_ht_unbiased and integrable AEPO unbiasedness under TargetedMoment. T16's claim text gives the affirmative result for every design under individual positivity, without NURVA or a first-moment assumption ([supplement p. 2, Equation (3)](https://static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS104719872610045X/resource/name/S104719872610045Xsup001.pdf#page=2)). The negated universal proposition at P061_PrintedClaims.lean:19–25 asserts ordinary integrability and the level/contrast expectation equalities without assuming moments, and P061_Refutations.lean:34–38 refutes it using a permitted countably supported real design with both exposure probabilities equal to 1/2 and a nonintegrable HT estimator (P061_Foundations.lean:820–857,904–910). Thus the refutation establishes a failure of existence in the unrestricted printed domain, rather than claiming a biased estimator with an existing finite mean. TargetedMoment at P061_Foundations.lean:76–78 requires each unit's exposure-restricted outcome to have a finite absolute design moment, and the correction actually proves integrability and unbiasedness on that domain, including infinite support (P061_PaperDomains.lean:27–31; P061_Foundations.lean:265–277,374–377). This disclosed hypothesis limits the additional positive result, while the closed universal proposition being refuted remains unchanged, so neither the statement, claim text, nor description requires repair for the credited component.\", 'defensible_statement': \"The unrestricted assertion that individual positivity guarantees integrable, unbiased HT level and contrast estimators is false. For N > 0 and positive probabilities of exposure d for every unit, if each unit's exposure-restricted outcome has a finite absolute design moment, then the HT estimator at d is integrable and its expectation equals the AEPO, on finite or countably infinite positive support and without NURVA or SUTVA.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.PaperDomains.ht_unbiased_T16_resolved", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_PaperDomains.lean#L189", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:6ef3d198ff9cc0ff320c20a84a831dd9b16097b5ae11faaa184c9887b32aac89", "conditional_on": [], "credit_reason": "Final status wrong_credit; not invoked. 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supplement pp. 1–2, Equations (1) and (3); main text pp. 13–14. The usual measure-theoretic existence conditions are assumed, expressed as exposure-restricted absolute first moments for every unit at each exposure used. With positive exposure probabilities, the HT mean and contrast have finite ordinary expectations and are unbiased for the AEPO and AEED without stability assumptions.

", "headline": "Supplementary Appendix, Equation (3): the Horvitz–Thompson estimator is finite-N unbiased for the AEPO, without NURVA", "location": "Supplementary Appendix p. 2 (publisher-supplement.pdf p. 2); the estimator is supplement Equation (1), p. 1; announced in the main text on pp. 13–14 (PDF pp. 13–14)" }, "id": "P061-T16", "text": { "body": "supplement pp. 1–2, Equations (1) and (3); main text pp. 13–14. The usual measure-theoretic existence conditions are assumed, expressed as exposure-restricted absolute first moments for every unit at each exposure used. With positive exposure probabilities, the HT mean and contrast have finite ordinary expectations and are unbiased for the AEPO and AEED without stability assumptions.", "headline": "Supplementary Appendix, Equation (3): the Horvitz–Thompson estimator is finite-N unbiased for the AEPO, without NURVA", "location": "Supplementary Appendix p. 2 (publisher-supplement.pdf p. 2); the estimator is supplement Equation (1), p. 1; announced in the main text on pp. 13–14 (PDF pp. 13–14)" }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T16" }, { "declarations": [ { "claim_revision_id": "catalog:02d4614b5d0b50aad3463f677fc88080e2bc7a750d69f97ed2683b86f21f207f", "conditional_on": [], "credit_reason": "Final status wrong_credit for T17: the survey conclusion loses within-mask variation; the usable general experiment row supplies the retained experiment.", "credit_status": "wrong_credit", "fidelity_status_baseline": "wrong_credit", "lean_name": "AronowAudit.P061.complete_randomization_unbiased", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L4784", "status": "proved", "support_scope": "withdrawn", "used_in_solution": false }, { "claim_revision_id": "catalog:02d4614b5d0b50aad3463f677fc88080e2bc7a750d69f97ed2683b86f21f207f", "conditional_on": [], "credit_reason": "The row and T17(a) both distinguish the fixed-divisor identity from being a realized sample mean. 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supplement pp. 1–2, Equation (2) and the take-away sentence. Equal positive exposure margins reduce HT to the fixed-denominator exposed mean, which is the sample mean when the exposed count is fixed. A simple random sample without replacement, including the census, has sample mean unbiased for the AEPO on arbitrary raw assignment spaces. A completely randomized binary experiment on any feasible set containing the binary cube has difference in means unbiased for the AEED. No stability assumption is needed. The usual measure-theoretic conditions are assumed; survey exposure first moments make the conditional means defined, and the experiment supplies its moments.

", "headline": "Supplementary Appendix, Equation (2) and the ‘key take-away’ after Equation (3): reduction to the sample mean and the difference in means, unbiased without NURVA", "location": "Supplementary Appendix p. 1 (publisher-supplement.pdf p. 1) for Equation (2); p. 2 (publisher-supplement.pdf p. 2) for the take-away sentence" }, "id": "P061-T17", "text": { "body": "supplement pp. 1–2, Equation (2) and the take-away sentence. Equal positive exposure margins reduce HT to the fixed-denominator exposed mean, which is the sample mean when the exposed count is fixed. A simple random sample without replacement, including the census, has sample mean unbiased for the AEPO on arbitrary raw assignment spaces. A completely randomized binary experiment on any feasible set containing the binary cube has difference in means unbiased for the AEED. No stability assumption is needed. 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supplement pp. 2–4, Equations (4), (5) and (9). Under support stability, the revealed-value reduction and both printed variance expressions, the covariance expression and the variance-of-contrast identity hold on general atomic designs.

", "headline": "Supplementary Appendix, Equations (4), (5) and (9): the estimator and its exact variance under NURVA, and the variance of the AEED estimator", "location": "Supplementary Appendix p. 2 (publisher-supplement.pdf p. 2) for Equation (4); p. 3 (publisher-supplement.pdf p. 3) for Equation (5); p. 4 (publisher-supplement.pdf p. 4) for Equation (9)" }, "id": "P061-T18", "text": { "body": "supplement pp. 2–4, Equations (4), (5) and (9). Under support stability, the revealed-value reduction and both printed variance expressions, the covariance expression and the variance-of-contrast identity hold on general atomic designs.", "headline": "Supplementary Appendix, Equations (4), (5) and (9): the estimator and its exact variance under NURVA, and the variance of the AEED estimator", "location": "Supplementary Appendix p. 2 (publisher-supplement.pdf p. 2) for Equation (4); p. 3 (publisher-supplement.pdf p. 3) for Equation (5); p. 4 (publisher-supplement.pdf p. 4) for Equation (9)" }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T18" }, { "declarations": [ { "claim_revision_id": "catalog:f086e47fa21f3738b788c2ea71d6679054284ce1008d3894e2aa87a9c15d8d8a", "conditional_on": [], "credit_reason": "NURVA on a finite population makes each targeted revealed outcome a fixed cell value times an indicator, giving the first and second moments even on infinite assignment support. All ordered pairs, including i=j, and all zero-joint terms match the printed formulas; conservativeness is only in expectation.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.nurva_variance_estimators", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L3773", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f086e47fa21f3738b788c2ea71d6679054284ce1008d3894e2aa87a9c15d8d8a", "conditional_on": [], "credit_reason": "Auxiliary inequality stated for all signed real inputs, then applicable to the two EPO values. It is not being credited as the full variance estimator theorem.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.young_missing_pair", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L2458", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f086e47fa21f3738b788c2ea71d6679054284ce1008d3894e2aa87a9c15d8d8a", "conditional_on": [], "credit_reason": "The selected variance-estimator theorem supplies conservativeness and the Young row supplies the printed bound; exact slack is not a separate target clause.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.missing_pair_slack", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L2465", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:f086e47fa21f3738b788c2ea71d6679054284ce1008d3894e2aa87a9c15d8d8a", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P061-T19", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P061-T19" }, "html": { "body": "

supplement pp. 3–4, Equations (6)–(8). Under individual positivity and NURVA, the joint-positive HT variance estimator is unbiased with the printed covariance-sum expectation, and the Young-corrected estimator is conservative in expectation. Each pair's Young bound admits an integrable unbiased estimator using only marginal exposure probabilities. If a joint exposure probability is zero, the two target cells are never jointly revealed on support, and no single observed-data statistic is unbiased for their EPO product for every NURVA outcome schedule. All sums include ordered diagonal pairs. The usual measure-theoretic conditions are assumed; all displayed moments follow from positivity and NURVA even with infinite raw support.

", "headline": "Supplementary Appendix, Equations (6), (7) and (8): the unbiased Horvitz–Thompson variance estimator when all joint exposure probabilities are positive, and the conservative variance estimator under NURVA", "location": "Supplementary Appendix p. 3 (publisher-supplement.pdf p. 3) for Equation (6); p. 4 (publisher-supplement.pdf p. 4) for Equations (7) and (8)" }, "id": "P061-T19", "text": { "body": "supplement pp. 3–4, Equations (6)–(8). Under individual positivity and NURVA, the joint-positive HT variance estimator is unbiased with the printed covariance-sum expectation, and the Young-corrected estimator is conservative in expectation. Each pair's Young bound admits an integrable unbiased estimator using only marginal exposure probabilities. If a joint exposure probability is zero, the two target cells are never jointly revealed on support, and no single observed-data statistic is unbiased for their EPO product for every NURVA outcome schedule. All sums include ordered diagonal pairs. The usual measure-theoretic conditions are assumed; all displayed moments follow from positivity and NURVA even with infinite raw support.", "headline": "Supplementary Appendix, Equations (6), (7) and (8): the unbiased Horvitz–Thompson variance estimator when all joint exposure probabilities are positive, and the conservative variance estimator under NURVA", "location": "Supplementary Appendix p. 3 (publisher-supplement.pdf p. 3) for Equation (6); p. 4 (publisher-supplement.pdf p. 4) for Equations (7) and (8)" }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T19" }, { "declarations": [ { "claim_revision_id": "catalog:da87224565ff7861e4fcdbc031e0dfbae2d24e5a89d360e7a335f80e1afb80d1", "conditional_on": [], "credit_reason": "The target definitions expressly assume existence of the displayed expectations. Its AEPO requires the targeted outcome means; prediction integrability supplies the targeted prediction moments, making hFd redundant. Hence these are not added restrictions relative to the supplied target, and no independence or stability is assumed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.PaperDomains.covariate_adjustment_T20", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_PaperDomains.lean#L14", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:da87224565ff7861e4fcdbc031e0dfbae2d24e5a89d360e7a335f80e1afb80d1", "conditional_on": [], "credit_reason": "The selected PaperDomains wrapper supplies the same complete moment-qualified adjustment result.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.covariate_adjustment", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L1120", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:da87224565ff7861e4fcdbc031e0dfbae2d24e5a89d360e7a335f80e1afb80d1", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P061-T20", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P061-T20" }, "html": { "body": "

supplement p. 7, Equation (10); main text p. 14. With the ordinary moments required by the claim's definitions, calibrated predictions give integrable unbiased adjustment without stability, including independent or mean-independent predictions.

", "headline": "Supplementary Appendix, ‘Unbiased covariate adjustment (without NURVA)’, Equation (10)", "location": "Supplementary Appendix p. 7 (publisher-supplement.pdf p. 7); announced in the main text on p. 14 (PDF p. 14)" }, "id": "P061-T20", "text": { "body": "supplement p. 7, Equation (10); main text p. 14. With the ordinary moments required by the claim's definitions, calibrated predictions give integrable unbiased adjustment without stability, including independent or mean-independent predictions.", "headline": "Supplementary Appendix, ‘Unbiased covariate adjustment (without NURVA)’, Equation (10)", "location": "Supplementary Appendix p. 7 (publisher-supplement.pdf p. 7); announced in the main text on p. 14 (PDF p. 14)" }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T20" }, { "declarations": [ { "claim_revision_id": "catalog:f1ebe40034b29de4f783acad68d0a83fa3c0d1401e0e4e466fb4e08185527ec7", "conditional_on": [], "credit_reason": "The conclusion uses every positive real ε and the atTop filter on the population size, with no subsequence or common-sample-space restriction. The two ratio limits are exactly the printed little-o conditions; N=0 is a harmless extendable initial index. The exact finite bounds are separately exported by R291-18 and R291-20.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.ht_consistency", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L3613", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f1ebe40034b29de4f783acad68d0a83fa3c0d1401e0e4e466fb4e08185527ec7", "conditional_on": [], "credit_reason": "Supporting identity or bound already incorporated in the selected aggregate theorem; not separately invoked.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.event_atom_le", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L2398", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f1ebe40034b29de4f783acad68d0a83fa3c0d1401e0e4e466fb4e08185527ec7", "conditional_on": [], "credit_reason": "Supporting identity or bound already incorporated in the selected aggregate theorem; not separately invoked.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.nurva_reweighted_bound", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L2418", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f1ebe40034b29de4f783acad68d0a83fa3c0d1401e0e4e466fb4e08185527ec7", "conditional_on": [], "credit_reason": "Supporting identity or bound already incorporated in the selected aggregate theorem; not separately invoked.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.design_chebyshev", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L2438", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f1ebe40034b29de4f783acad68d0a83fa3c0d1401e0e4e466fb4e08185527ec7", "conditional_on": [], "credit_reason": "Supporting identity or bound already incorporated in the selected aggregate theorem; not separately invoked.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.nurva_variance_bound", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L3557", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f1ebe40034b29de4f783acad68d0a83fa3c0d1401e0e4e466fb4e08185527ec7", "conditional_on": [], "credit_reason": "Supporting identity or bound already incorporated in the selected aggregate theorem; not separately invoked.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P061.nurva_chebyshev_bound", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P061/P061_Foundations.lean#L3590", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:f1ebe40034b29de4f783acad68d0a83fa3c0d1401e0e4e466fb4e08185527ec7", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P061-T21", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P061-T21" }, "html": { "body": "

supplement pp. 5–6, Conditions 1.1–1.3 and Proposition 1. The three printed boundedness and dependence conditions together with support stability imply convergence in probability of the HT estimation error to zero along growing positive populations.

", "headline": "Supplementary Appendix, Proposition 1 (consistency of the Horvitz–Thompson estimator under NURVA and Conditions 1.1–1.3)", "location": "Supplementary Appendix p. 5 (publisher-supplement.pdf p. 5); the proof sketch is on p. 6 (publisher-supplement.pdf p. 6)" }, "id": "P061-T21", "text": { "body": "supplement pp. 5–6, Conditions 1.1–1.3 and Proposition 1. The three printed boundedness and dependence conditions together with support stability imply convergence in probability of the HT estimation error to zero along growing positive populations.", "headline": "Supplementary Appendix, Proposition 1 (consistency of the Horvitz–Thompson estimator under NURVA and Conditions 1.1–1.3)", "location": "Supplementary Appendix p. 5 (publisher-supplement.pdf p. 5); the proof sketch is on p. 6 (publisher-supplement.pdf p. 6)" }, "url": "https://pmaronow.github.io/llm-audit/works/P061/#P061-T21" } ] }, "missing_sources": [], "reproduction": { "agreement": "agrees", "status": "selected_recalculation", "targets": [ { "agreement": "exact", "location": "p. 9 (PDF p. 9); associated AEPOs in text p. 8 (PDF p. 8)", "method": "own recalculation from reported numbers", "status": "recalculated", "target": "Table 1" }, { "agreement": "exact", "location": "p. 10 (PDF p. 10)", "method": "own recalculation from reported numbers", "status": "recalculated", "target": "Table 2" }, { "agreement": "exact", "location": "p. 10 (PDF p. 10)", "method": "own recalculation from reported numbers", "status": "recalculated", "target": "Table 3" }, { "agreement": "exact", "location": "p. 12 (PDF p. 12)", "method": "own recalculation from reported numbers", "status": "recalculated", "target": "Table 4" }, { "agreement": "exact", "location": "p. 12 (PDF p. 12)", "method": "own recalculation from reported numbers", "status": "recalculated", "target": "Table 5" } ] }, "summary": "The paper asks what design-based inference targets when SUTVA is not assumed. It defines design-conditional expected potential outcomes and their contrasts under unrestricted interference. Under NURVA, which requires stability only on the support of the design, these contrasts are causal for the implemented design, with no guarantee beyond it. 7 entries are classified as errors (1 Moderate, 6 Minor), and 10 are clarity notes. The finding that matters most is that the closing remark on inference for an average direct effect is not correct for general exposure mappings (\\fid{P061-N019}).", "what_was_checked": "I checked the following items, which held up subject to the corrections cited.\n\\begin{itemize}\n\\item The two stylized designs, pp.~3--4. The design of the rebel-group survey has 720 ordered samples, each with probability $7!/10!=1/720$, and each group is surveyed with probability $3/10$. The design of the school experiment sums to one on binary vectors for $N=2,\\ldots,10$; its display is the subject of \\fid{P061-N004}.\n\\item Equations (1)--(8). The definitions are consistent with one another, Equation (4) equals $\\Pr[D_i=d]$, and Equations (5)--(8) are correct under individual positivity for every design with finitely many assignments. Designs with infinitely many assignments are the subject of \\fid{P061-N014}.\n\\item Section 5.1. Under NURVA, each unit has a single outcome at each exposure on the support of the design. For a given design, NURVA and SUTVA are observationally equivalent, because the distribution of $(\\mathbf Z,\\mathbf Y)$ depends only on the outcomes at assignments in the support. The display of Condition 5.2 is the subject of \\fid{P061-N016}.\n\\item Tables 1--5. I recalculated every AEPO and AEED from the printed outcomes and designs with exact arithmetic, and all values agree: $\\tau(1,0)=-1$ in Table 1; $1$ and $0$ in Table 2; $1$ under both designs in Table 3; $1$ and $0$ in panel (a) and $0$ and $1$ in panel (b) of Table 4; $1$ and $2$ in Table 5. In Table 1, NURVA holds and SUTVA fails, as stated.\n\\item Section 5.2 and footnote 9. For binary treatment with exposure equal to own treatment and feasible all-treated and all-control assignments, if treatment is constant within clusters and the relevant exposures have positive probability, then partial interference makes the outcomes stable on the support and identifies the contrast of all units treated against none. The wording is the subject of \\fid{P061-N018} and \\fid{P061-N017}.\n\\item Section 6.3 and footnote 10. In an example with $N=4$, NURVA holds under two sampling-treatment designs and the AEEDs are $2$ and $3$. The supports in footnote 10 are as described. Misprints in the section are listed in \\fid{P061-N008}.\n\\item Figure 1. In a recalculation from the vector graphics of the published PDF, the exposure colours of the right panel agree with Equation (3) applied to the left panel at all 148 nodes, and the two panels have the same 203 edges.\n\\item Supplement, Equations (1)--(5). The Horvitz--Thompson estimator reduces to the sample mean under simple random sampling. With finite target means, it is unbiased for the AEPO without NURVA, and the difference in means is unbiased for the AEED under complete randomization, by derivation and by exact enumeration with interference (see \\fid{P061-N020} and \\fid{P061-N014}). The variance formula of Equation (5) under NURVA is exact in every enumerated case (see \\fid{P061-N021}).\n\\item Supplement, Equations (6)--(11) and Proposition 1. Equation (6) is unbiased for the variance when all joint exposure probabilities are positive. The estimator of Equations (7)--(8) is conservative in expectation under NURVA, including when some joint probabilities are zero, and it can fail to be conservative without NURVA, as the supplement says. Equation (9), the covariance of two Horvitz--Thompson estimators and the variance of the estimator of the AEED, is correct. Proposition 1 follows from Conditions 1.1 and 1.3, NURVA and Chebyshev's inequality (see \\fid{P061-N010}). Equation (10) is correct under the moment condition displayed before it when the target and prediction means are finite. The variance expression in Equation (11) is correct for a fixed limiting parameter, and it reduces to a constant multiple of the least-squares residual sum of squares when the exposure indicators are independent with a common probability.\n\\item Supplement, p.~6. The statement that Chebyshev's inequality remains asymptotically sharp under restrictions on the degree sequence matches Proposition 11 of Sävje, Aronow, and Hudgens (2021). The rate $o(n^{1/4})$ for the maximum degree is the condition of the central limit theorem of Han, Basse, and Bojinov (2024); Ogburn et al. (2024) use a weaker condition on a degree that is defined differently. These works were read in author manuscripts and preprints.\n\\item Cross-references. The references to equations, conditions, sections, tables and figures point to the right items, in the article and in the supplement, and every citation in the text has an entry in the reference list.\n\\end{itemize}\n\nThe following were not checked. I did not read most of the cited works. Sävje (2024) and Sävje, Aronow, and Hudgens (2021) were read in author manuscripts and preprints, not in the typeset versions (\\fid{P061-N006}, \\fid{P061-N009}). The texts of Rubin (1974) and of Rubin's comment of 1980 could not be read, and the source of the two quotations on pp.~1--2 is unchecked. The text of Bickel and Freedman (1981) could not be read, and whether it is the right reference for the central limit theorem for stratified designs cited on p.~6 of the supplement is unchecked. Which of the two orders of the authors of Harshaw et al. (2022) is correct was not determined (\\fid{P061-N002}). I did not assess the statement in Section 7 that the result on covariate adjustment without NURVA is new. The article states no precise conditions for the closing remark on the average direct effect. Conditions under which the remark holds, other than those in \\fid{P061-N019} and \\fid{P061-N009}, were not assessed.\n\nI had no code for Figure 1 or replication code or data. The tables were recalculated from the printed outcomes and designs. These checks cover the items listed, and they are not a verification of the paper as a whole.", "work": { "arxiv_id": "2505.10519", "audit_complete": true, "authors": [ "P. M. Aronow", "Austin Jang", "Molly Offer-Westort" ], "category": "Articles - Theory and Methodology", "citation": "P. M. Aronow, Austin Jang, and Molly Offer-Westort (2026). On the Foundations of the Design-Based Approach. Political Analysis.", "doi": "10.1017/pan.2026.10045", "edition": { "pages_in_audited_file": 16, "sha256": "a5e0ceabf204cb06ddee865caa4194cdbac1084df7895271fb57d9431f5102a4", "source_url": "https://doi.org/10.1017/pan.2026.10045", "venue": "Political Analysis", "year": "2026" }, "id": "P061", "later_materials": null, "title": "On the Foundations of the Design-Based Approach", "type": "Journal article", "url": "https://pmaronow.github.io/llm-audit/works/P061/", "venue": "Political Analysis", "year": "2026" } }