{ "claims": [ { "claim": "The paper defines homophily $h$ as the assortativity coefficient (correlation of a binary trait across the ends of edges) on the augmented recruitment-induced subgraph $G_{SU}$, and preferential recruitment $p$ as the realized average deviation of same-trait recruitment from recruitment uniformly at random among susceptible neighbours.", "claim_id": "P012-H1", "related_corrections": [ "P012-E006", "P012-N006" ], "tied_to": "Definition 1, equation (1); Definition 2, equation (2); Figure 1" }, { "claim": "The recruitment-induced subgraph $G_S$ and the augmented subgraph $G_{SU}$ are not identified by RDS data whenever two recruited vertices not joined by a recruitment edge both have reported degree above their recruitment degree.", "claim_id": "P012-H2", "related_corrections": [ "P012-E003", "P012-N005" ], "tied_to": "Result 1; Appendix, Proof of Result 1" }, { "claim": "Homophily $h$ is not point identified when two recruited vertices with different trait values both have reported degree above their recruitment degree.", "claim_id": "P012-H3", "related_corrections": [ "P012-E003", "P012-N006" ], "tied_to": "Result 2; Appendix, Proof of Result 2 and display (A1)" }, { "claim": "Preferential recruitment $p$ is not point identified when some vertex that recruited at least one other vertex has reported degree above its recruitment degree.", "claim_id": "P012-H4", "related_corrections": [ "P012-E002" ], "tied_to": "Result 3; Appendix, Proof of Result 3 and display (A2)" }, { "claim": "The paper states that $h$ and $p$ are generally not point identified in RDS studies unless the recruitment tree coincides with the underlying subgraph.", "claim_id": "P012-H5", "related_corrections": [], "tied_to": "Results 2 and 3; abstract; introduction (printed p. 154); text after Result 3 (printed pp. 155-156); discussion (printed pp. 157-158)" }, { "claim": "The paper defines identification regions for $h$ and $p$ as extrema over augmented subgraphs and traits compatible with the observed data $(G_R,\\mathbf d_R,\\mathbf t_R,\\mathbf Z_R)$. Its five-vertex example gives $h\\in[-0.625,0.752]$ and $p\\in[-0.138,0.408]$, and it states that a simulated-annealing search with a logarithmic cooling schedule converges in probability to an extremal compatible subgraph.", "claim_id": "P012-H6", "related_corrections": [ "P012-E001", "P012-E007", "P012-E006", "P012-N007", "P012-N008" ], "tied_to": "Definitions 3-4; Figure 1; Web Appendix 1, Proposition 1; Web Appendix 2" }, { "claim": "In the Hartford RDS-net study (527 recruited, 6 seeds, nomination network of 2,626 people), the paper reports that the full-network values of $h$ and $p$ for sex, crack use and homelessness lie inside the intervals computed from RDS data alone, that $h$ is positive for sex and crack use and negative for homelessness, that recruitment deviations are small, and that every interval covers zero.", "claim_id": "P012-H7", "related_corrections": [ "P012-N002", "P012-E001" ], "tied_to": "Table 1; Figure 2" }, { "claim": "The paper argues that claims of homophily or recruitment bias measured from RDS data alone may not be credible, while bounds permit credible statements under three assumptions: the network exists, recruitment follows its edges, and nobody is recruited twice. It reports that the RDS-net regions are considerably smaller than $[-1,1]$.", "claim_id": "P012-H8", "related_corrections": [ "P012-N002" ], "tied_to": "Results 1-3; Definition 4; Table 1; Discussion" }, { "claim": "Collecting additional network or recruitment information, with suitable assumptions, can tighten the identification intervals or permit point identification.", "claim_id": "P012-H9", "related_corrections": [], "tied_to": "Definitions 3-4; Figure 2; Table 1; introduction (printed p. 154); discussion (printed p. 158)" } ], "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": "Screenshot of the proof of Result 1. Red boxes mark the words: be any compatible subgraph, which run over a line break.", "caption": "Appendix, proof of Result 1, p. 160 (PDF p. 8), right column. The marked words allow any compatible starting graph that contains $\\{i,u\\}$ and $\\{j,u\\}$.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-E003.png" }, { "alt": "Screenshot of the proof of Result 2 up to the words The difference is. Red boxes mark the words: be any compatible subgraph, and the sentence: Clearly (G^2_SU, Z^2_SU) is in C(G_R, d_R, Z_R).", "caption": "Appendix, proof of Result 2, p. 160 (PDF p. 8), right column. The marked words allow any compatible starting pair and assert that the pair obtained by the swap is compatible.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-E003-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P012-T8", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T8" }, { "id": "P012-T9", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T9" }, { "id": "P012-T10", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T10" }, { "id": "P012-T11", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T11" } ], "headline": "The proofs of Results 1 and 2 start from “any” compatible graph with the named edges, for which the described edge moves can fail; both proofs are repaired by starting from a graph built with new unsampled vertices, and the two results stand as printed.", "id": "P012-E003", "location": "Appendix, proofs of Results 1 and 2, p. 160 (PDF p. 8), right column; the results are stated on p. 155 (PDF p. 3)", "pdf_page": 8, "printed_page": "160", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P012-T10", "P012-T11", "P012-T8", "P012-T9" ], "text": { "as_printed": "\\emph{Proof of Result 1:} Let $G^1_{SU}=(V^1_{SU},E^1_{SU})$ be \\hl{any compatible subgraph} with $\\{i,u\\}\\in E^1_{SU}$, $\\{j,u\\}\\in E^1_{SU}$, where $u\\notin V_R$ is an unsampled vertex. Let $G^2_{SU}=(V^2_{SU},E^2_{SU})$ be identical to $G^1_{SU}$ except that $\\{i,j\\}\\in E^2_{SU}$, so neither $i$ nor $j$ is connected to $u$.\n\n\\emph{Proof of Result 2:} Let $(G^1_{SU},\\mathbf Z^1_{SU})$ be \\hl{any compatible subgraph and trait set} with the property that $\\{i,u_1\\}\\in E^1_{SU}$ and $\\{j,u_2\\}\\in E^1_{SU}$, where $u_1$ and $u_2$ are unsampled vertices with $Z^1_{u_1}=0$ and $Z^1_{u_2}=1$. Let $(G^2_{SU},\\mathbf Z^2_{SU})$ be identical to $(G^1_{SU},\\mathbf Z^1_{SU})$ except that the edges connecting $i$ and $j$ to $u_1$ and $u_2$ are swapped: $\\{i,u_1\\}\\notin E^2_{SU}$, $\\{j,u_2\\}\\notin E^2_{SU}$, and $\\{i,u_2\\}\\in E^2_{SU}$ and $\\{j,u_1\\}\\in E^2_{SU}$. \\hl{Clearly} $(G^2_{SU},\\mathbf Z^2_{SU})\\in\\mathrm{C}(G_R,\\mathbf d_R,\\mathbf Z_R)$.", "corrected": "Results 1 and 2 are unchanged. In both proofs write $s_v=d_v-d_v^r$ for $v\\in V_R$.\n\n\\emph{Proof of Result 1:} \\hl{Construct $G^1_{SU}$ as follows.} Take the recruitment edges as undirected edges. Add one new unsampled vertex $u$ with the edges $\\{i,u\\}$ and $\\{j,u\\}$. Attach $s_i-1$ further new unsampled vertices to $i$, $s_j-1$ to $j$, and $s_v$ to every other $v\\in V_R$, each adjacent to that one sampled vertex only. Then $G^1_{SU}$ is compatible and \\hl{$\\{i,j\\}\\notin E^1_{SU}$}. Let $G^2_{SU}$ be obtained from $G^1_{SU}$ by removing $\\{i,u\\}$ and $\\{j,u\\}$, adding $\\{i,j\\}$, and deleting $u$. The proof continues as printed.\n\n\\emph{Proof of Result 2:} \\hl{Construct $(G^1_{SU},\\mathbf Z^1_{SU})$ as follows.} Take the recruitment edges as undirected edges and attach $s_v$ new unsampled vertices to each $v\\in V_R$, each adjacent to $v$ only. Let $u_1$ be one of those attached to $i$ and $u_2$ one of those attached to $j$, with $Z^1_{u_1}=0$ and $Z^1_{u_2}=1$; the traits of the other unsampled vertices are arbitrary. Then \\hl{$\\{i,u_2\\}\\notin E^1_{SU}$ and $\\{j,u_1\\}\\notin E^1_{SU}$}. Let $(G^2_{SU},\\mathbf Z^2_{SU})$ be identical to $(G^1_{SU},\\mathbf Z^1_{SU})$ except that $\\{i,u_1\\}$ and $\\{j,u_2\\}$ are replaced by $\\{i,u_2\\}$ and $\\{j,u_1\\}$. The swap preserves every degree, so $(G^2_{SU},\\mathbf Z^2_{SU})\\in\\mathcal C(G_R,\\mathbf d_R,\\mathbf Z_R)$, and the proof continues as printed.", "evidence": "The evidence is two exact counterexamples to the printed steps and a derivation of the corrected ones. Let $V_R=\\{1,2,3\\}$ with recruitments $1\\to2\\to3$, reported degrees $\\mathbf d_R=(3,2,3)$ and traits $\\mathbf Z_R=(0,0,1)$, and let $i=1$, $j=3$. I write an edge $\\{k,l\\}$ as $kl$.\n\\begin{itemize}\n\\item Result 1: with $u=4$, the graph with edges $12$, $23$, $13$, $14$, $34$ is compatible and contains $14$ and $34$. The printed step removes $14$ and $34$, adds $13$, which is already present, and deletes vertex 4. The result has edges $12$, $23$, $13$ and sampled degrees $(2,2,2)$ in place of $(3,2,3)$.\n\\item Result 2: with $u_1=4$, $u_2=5$, $Z_4=0$ and $Z_5=1$, the graph with edges $12$, $23$, $14$, $35$, $15$, $34$ is compatible and contains $14$ and $35$. The swap removes $14$ and $35$ and adds $15$ and $34$, which are already present. The result has edges $12$, $23$, $15$, $34$ and sampled degrees $(2,2,2)$.\n\\end{itemize}\nFor the corrected steps let $s_v=d_v-d_v^r\\ge0$. In the starting graph of the corrected proof of Result 1, the vertex $i$ has degree $d_i^r+1+(s_i-1)=d_i$, likewise for $j$, and every other sampled $v$ has degree $d_v^r+s_v=d_v$. Each unsampled vertex has a sampled neighbor, and no two unsampled vertices are adjacent. The graph is then compatible. Its edges among sampled vertices are the recruitment edges, and since $(i,j)\\notin E_R$ with $i0$ because $Q\\ge d_j>0$ and $2m-Q\\ge d_i>0$. The numerators differ by 2, and I obtain $h_1-h_2=2/D>0$, as (A1) states. On the instance $V_R=\\{1,2\\}$ with vertex 1 recruiting vertex 2, $\\mathbf d_R=(2,2)$ and $\\mathbf Z_R=(0,1)$, the two graphs have $h_1=1/3$ and $h_2=-1$, and $2/D=4/3$.", "problem": "Each proof takes as its starting graph any compatible graph that contains the named edges, changes some edges, and asserts that the result is compatible. This can fail. In the proof of Result 1 the edge $\\{i,j\\}$ may already belong to $E^1_{SU}$, since its absence from $E_R$ does not exclude it. Removing $\\{i,u\\}$ and $\\{j,u\\}$ then lowers the degrees of $i$ and $j$ by one, and the later claim that $\\{i,j\\}\\notin G^1_S$ is false. In the proof of Result 2 the edges $\\{i,u_2\\}$ and $\\{j,u_1\\}$ may already belong to $E^1_{SU}$. The swap then lowers the degrees of $i$ and $j$ by one, the pair $(G^2_{SU},\\mathbf Z^2_{SU})$ is not compatible, and display (A1), which uses one denominator for $h_1$ and $h_2$, does not apply.", "what_survives": "Results 1 and 2 hold exactly as printed, with the same hypotheses, and display (A1) is correct for the corrected swap. Result 2 needs no condition that excludes a recruitment edge between $i$ and $j$: in the last instance above $i$ recruits $j$. The correction restricts the choice of a witness in two existence proofs and adds no assumption on the data or the network. Definitions 1--4, Result 3, Figure 1 and the application do not depend on these steps." }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-E003", "work": { "arxiv_id": "1511.05397", "doi": "10.1093/aje/kwx208", "title": "Identification of homophily and preferential recruitment in respondent-driven sampling", "url": "https://pmaronow.github.io/llm-audit/works/P012/" }, "work_id": "P012" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Screenshot of the sentence that defines p_1 and p_2 and of display (A2). A red box marks the first equals sign, between p_1 minus p_2 and the bracket that contains the terms of the single recruit j.", "caption": "Appendix, proof of Result 3, display (A2), p. 160 (PDF p. 8), right column. The marked equality holds only when $i$ has a single recruit.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-E002.png" } ], "formal_release": null, "formal_statements": [ { "id": "P012-T12", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T12" }, { "id": "P012-T13", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T13" } ], "headline": "Display (A2) keeps one recruitment by $i$; when $i$ recruited two or more subjects, $p_1-p_2$ is a sum over all of them, with the same sign, and Result 3 is unchanged.", "id": "P012-E002", "location": "Appendix, proof of Result 3, display (A2), p. 160 (PDF p. 8), right column; Result 3 is stated on p. 155 (PDF p. 3)", "pdf_page": 8, "printed_page": "160", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P012-T12", "P012-T13" ], "text": { "as_printed": "\\[\n\\begin{aligned}\np_1-p_2&\\;\\hl{=}\\;\\frac{1}{n-|M|}\\left[\\left(1\\{Z_i=Z_j\\}-\\frac{K^1}{|S_i(t_j)|}\\right)-\\left(1\\{Z_i=Z_j\\}-\\frac{K^1-1}{|S_i(t_j)|}\\right)\\right]\\\\\n&=\\frac{-1}{(n-|M|)|S_i(t_j)|}\\ne0.\\qquad\\text{(A2)}\n\\end{aligned}\n\\]", "corrected": "Suppose, as in the printed proof, that $Z_i=1$, that $u$ is an unsampled vertex adjacent to $i$ and to no other vertex of $V_R$, and that $Z_u=1$ in $\\mathbf Z^1_{SU}$ and $Z_u=0$ in $\\mathbf Z^2_{SU}$. Then\n\\[\np_1-p_2=\\hl{-\\frac{1}{n-|M|}\\sum_{\\ell\\in V_R\\setminus M:\\,r_\\ell=i}\\frac{1}{|S_i(t_\\ell)|}}<0.\\qquad\\text{(A2)}\n\\]\nThe sum is not empty, because $i$ recruited $j$, and each $|S_i(t_\\ell)|$ is a positive integer. Therefore preferential recruitment is not point identified.", "evidence": "The evidence is a derivation and an exact counterexample. Under the printed assumption $Z_i=1$, the graph and the recruitment times are the same in the two pairs. Each set $S_{r_\\ell}(t_\\ell)$ and each indicator $1\\{Z_\\ell=Z_{r_\\ell}\\}$ is then the same in both. If $\\ell$ was not recruited by $i$, then $u\\notin S_{r_\\ell}(t_\\ell)$, because $u$ has no sampled neighbor other than $i$, and the summand of $\\ell$ does not change. If $\\ell$ was recruited by $i$, the number of vertices of $S_i(t_\\ell)$ with the trait of $i$ is smaller by one in the second pair. Subtracting the two averages term by term, I obtain the corrected display below. There exists a compatible pair of the kind the proof needs: it has the recruitment edges and, for each $v\\in V_R$, $d_v-d_v^r$ new unsampled vertices adjacent to $v$ only.\n\nFor the counterexample let $V_R=\\{1,2,3\\}$, let the seed 1 recruit 2 and then 3, let $\\mathbf d_R=(3,1,1)$ and $Z_1=Z_2=Z_3=1$, and let $u=4$ be the one unsampled neighbor of vertex 1. Then $S_1(t_2)=\\{2,3,4\\}$ and $S_1(t_3)=\\{3,4\\}$. With $Z_4=1$, $p_1=0$. With $Z_4=0$,\n\\[\np_2=\\frac12\\Bigl[\\Bigl(1-\\frac23\\Bigr)+\\Bigl(1-\\frac12\\Bigr)\\Bigr]=\\frac5{12},\n\\]\nand I find $p_1-p_2=-5/12$. Display (A2) gives $-1/6$ with $j=2$ and $-1/4$ with $j=3$.", "problem": "The proof defines $p_1$ and $p_2$ as the whole functional of equation (2) and changes the trait of an unsampled neighbor $u$ of $i$ from 1 to 0. Since the vertex $u$ is never recruited, I have $u\\in S_i(t_\\ell)$ for every $\\ell$ that $i$ recruited, and the change of $Z_u$ alters every one of those summands of (2). Display (A2) keeps the summand of the single recruit $j$. Its first equality is therefore false when $i$ has two or more recruits, which Result 3 allows (``at least 1 other vertex''). The omitted terms have the sign of the term kept, and $p_1-p_2$ is still nonzero. The printed minus sign is correct.", "what_survives": "Result 3 holds with its printed hypotheses, and display (A2) is exact when $i$ has exactly one recruit. Equation (2), Figure 1 and the application use the full functional $p$, and no later formula uses the magnitude in (A2). Nothing else changes." }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-E002", "work": { "arxiv_id": "1511.05397", "doi": "10.1093/aje/kwx208", "title": "Identification of homophily and preferential recruitment in respondent-driven sampling", "url": "https://pmaronow.github.io/llm-audit/works/P012/" }, "work_id": "P012" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Screenshot of Definition 1 with equation (1) for h as a ratio of two sums. A red box marks the denominator, the sum of (d_i delta_ij minus d_i d_j over 2|E_SU|) Z_i Z_j.", "caption": "Definition 1 and equation (1), p. 155 (PDF p. 3), left column. The marked denominator is zero when all vertices of positive degree have the same trait.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-E006.png" }, { "alt": "Screenshot of equation (2) for p. A red box marks the leading factor 1 over (n minus |M|).", "caption": "Equation (2), p. 155 (PDF p. 3), top of the right column. The marked factor $1/(n-|M|)$ is undefined for a sample that consists of seeds only.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-E006-b.png" }, { "alt": "Screenshot of Definition 4. A red box marks the words: every (G_SU, Z_SU) in C(G_R, d_R, Z_R).", "caption": "Definition 4, p. 155 (PDF p. 3), right column. The marked words take $h$ and $p$ at every compatible pair.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-E006-c.png" } ], "formal_release": null, "formal_statements": [ { "id": "P012-T14", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T14" }, { "id": "P012-T15", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T15" }, { "id": "P012-T24", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T24" } ], "headline": "Equations (1) and (2) are undefined at some compatible pairs, so Definition 4 needs the conditions under which $h$ and $p$ are defined on the whole compatible set.", "id": "P012-E006", "location": "Definitions 1, 2 and 4, equations (1) and (2), p. 155 (PDF p. 3); the same functionals are used in Web Appendix 1, p. 1", "pdf_page": 3, "printed_page": "155", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P012-T14", "P012-T15", "P012-T24" ], "text": { "as_printed": "\\emph{Definition 1:} Homophily is the correlation between trait values of vertices connected by an edge,\n\\[\nh(G_{SU},\\mathbf Z_{SU})=\\frac{\\sum_{i,j\\in V_{SU}}(A_{ij}-d_id_j/2|E_{SU}|)Z_iZ_j}{\\hl{\\sum_{i,j\\in V_{SU}}(d_i\\delta_{ij}-d_id_j/2|E_{SU}|)Z_iZ_j}},\\qquad(1)\n\\]\n\\emph{Definition 2:} Preferential recruitment is the average deviation from proportional recruitment, given knowledge of $G_{SU}$, $G_R$, $\\mathbf t_R$, and $\\mathbf Z_{SU}$,\n\\[\np(G_{SU},G_R,\\mathbf t_R,\\mathbf Z_{SU})=\\hl{\\frac{1}{n-|M|}}\\sum_{j\\notin M}\\left(1\\{Z_j=Z_{r_j}\\}-\\frac{|\\{k\\in S_{r_j}(t_j):Z_{r_j}=Z_k\\}|}{|S_{r_j}(t_j)|}\\right),\\qquad(2)\n\\]\nwhere $M$ is the set of seeds.\n\n\\emph{Definition 4:} Identification region: The identification regions for $h$ and $p$ are given by the smallest intervals that contain $h(G_{SU},\\mathbf Z_{SU})$ and $p(G_{SU},G_R,\\mathbf t_R,\\mathbf Z_{SU})$ for \\hl{every} $(G_{SU},\\mathbf Z_{SU})\\in\\mathrm{C}(G_R,\\mathbf d_R,\\mathbf Z_R)$.", "corrected": "\\emph{Definition 1} applies to pairs $(G_{SU},\\mathbf Z_{SU})$ with \\hl{$|E_{SU}|>0$ and $D_0D_1>0$}, where $D_z=\\sum_{i\\in V_{SU}:Z_i=z}d_i$; at any other pair $h$ is undefined. \\emph{Definition 2} applies when \\hl{$n>|M|$}; otherwise $p$ is undefined.\n\n\\emph{Definition 4:} Identification region. \\hl{Suppose that $\\sum_{i\\in V_R:Z_i=z}d_i>0$ for $z=0$ and for $z=1$.} Then $h$ is defined at every pair in $\\mathcal C(G_R,\\mathbf d_R,\\mathbf Z_R)$, and the identification region for $h$ is the smallest interval that contains $h(G_{SU},\\mathbf Z_{SU})$ for every $(G_{SU},\\mathbf Z_{SU})\\in\\mathcal C(G_R,\\mathbf d_R,\\mathbf Z_R)$. \\hl{Suppose that $n>|M|$.} Then $p$ is defined at every pair in $\\mathcal C(G_R,\\mathbf d_R,\\mathbf Z_R)$, and the identification region for $p$ is the smallest interval that contains $p(G_{SU},G_R,\\mathbf t_R,\\mathbf Z_{SU})$ for every such pair. If the condition for $h$ or for $p$ fails, the identification region of that functional is undefined.", "evidence": "The evidence is a derivation and three exact examples. The identities $Z_i^2=Z_i$ and $\\sum_id_i=2m$ imply that\n\\[\n\\sum_{i,j\\in V_{SU}}\\Bigl(d_i\\delta_{ij}-\\frac{d_id_j}{2m}\\Bigr)Z_iZ_j=q-\\frac{q^2}{2m}=\\frac{q(2m-q)}{2m}.\n\\]\nI write $D_1=q$ and $D_0=2m-q$ for the total degrees of the vertices with trait $1$ and trait $0$; the denominator is then $D_0D_1/(2m)$.\n\\begin{enumerate}\n\\item Let vertex 1 be a seed that recruits vertex 2, with $\\mathbf d_R=(1,1)$ and $\\mathbf Z_R=(0,0)$. The only compatible pair is the single edge $\\{1,2\\}$. Here $m=1$ and $q=0$, and (1) is $0/0$, while $p=0$.\n\\item Let vertices 1 and 2 both be seeds, with $\\mathbf d_R=(1,1)$ and $\\mathbf Z_R=(0,1)$. With the edge $\\{1,2\\}$ the pair is compatible and $h=-1$, and (2) is an empty sum divided by $n-|M|=0$ at every compatible pair.\n\\item Let vertex 1 recruit vertex 2, with $\\mathbf d_R=(1,4)$ and $\\mathbf Z_R=(1,1)$. Vertex 2 has three unsampled neighbors, and the eight assignments of their traits are the eight compatible pairs. Seven of them have $h\\in\\{-3/5,-1/3,-1/7\\}$; at the eighth, with all traits equal to 1, $h$ is undefined.\n\\end{enumerate}\nThe condition in the corrected Definition 4 is the exact one for $h$. If, for each trait, the reported degrees of the sampled vertices with that trait have a positive sum, then $D_0>0$ and $D_1>0$ at every compatible pair, because compatible graphs preserve the reported degrees. If all sampled vertices of positive degree share one trait, giving that trait to every unsampled vertex yields a compatible pair at which (1) is $0/0$, because every edge of $G_{SU}$ has a sampled endpoint.", "problem": "Definition 4 takes $h$ and $p$ at every compatible pair, but compatibility does not make the two formulas defined. Let $m=|E_{SU}|$ and $q=\\sum_{i\\in V_{SU}}d_iZ_i$. For $m>0$ the denominator of (1) equals $q(2m-q)/(2m)$. When all vertices of positive degree have the same trait ($q=0$ or $q=2m$), this denominator and the numerator both vanish, and (1) reads $0/0$. For $m=0$ the terms $d_id_j/2|E_{SU}|$ are themselves undefined. The factor $1/(n-|M|)$ in (2) is undefined when the sample consists of seeds only. The paper states no condition that excludes these cases.", "what_survives": "No result or number changes. The hypotheses of Result 2 imply that there exist sampled vertices of both traits with positive degree, and the hypothesis of Result 3 implies that at least one recruitment took place. Then $h$ and $p$ are defined on the whole compatible set where those results apply. Figure 1 meets both conditions, and the application has $527-6=521$ recruitments. What changes is the statement of Definition 4, the optimization of Web Appendix 1 over the compatible set, and the sentence on p.~158 that three assumptions suffice for the bounds: each needs the two conditions below." }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-E006", "work": { "arxiv_id": "1511.05397", "doi": "10.1093/aje/kwx208", "title": "Identification of homophily and preferential recruitment in respondent-driven sampling", "url": "https://pmaronow.github.io/llm-audit/works/P012/" }, "work_id": "P012" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Screenshot of lines 12 to 23 of a numbered listing. Red boxes mark line 13, if j in V_R then; line 21, end if; and line 22, Add {j, u} to E*_SU, which is indented at the level of line 13 and comes after the end if.", "caption": "Web Appendix 2, first proposal mechanism, lines 12–23, Web Appendix p. 3 (supplement PDF p. 3). Marked: the conditional opened at line 13, its end at line 21, and line 22, which follows the end.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-E001.png" }, { "alt": "Screenshot of the end of the first listing and the two-line second listing. Red boxes mark line 27, Remove any isolated vertices from V*_SU, and line 1 of the second listing, Choose u in the set of unsampled vertices.", "caption": "Web Appendix 2, line 27 and the second proposal mechanism, Web Appendix p. 3 (supplement PDF p. 3). Marked: the removal of “any isolated vertices” and the choice of an unsampled vertex.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-E001-b.png" }, { "alt": "Three small graph drawings. Left: sampled vertices 1 and 2 joined by an arrow, and unsampled vertex 3 joined to vertex 1. Center: vertices 1 and 2 as before, a new unsampled vertex 4 joined to vertex 1, and a dashed orange edge between the unsampled vertices 3 and 4; vertex 3 has no edge to a sampled vertex. Right: vertices 1, 2 and 4 only, with vertex 4 joined to vertex 1.", "caption": "The counterexample. Left: a compatible state. Center: the output of the listing as printed for $i=1$, $j=3$, with the edge $\\{3,4\\}$ between two unsampled vertices. Right: the output of the same choices with line 22 moved inside the conditional of line 13.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-E001-counterexample.png" } ], "formal_release": null, "formal_statements": [ { "id": "P012-T1", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T1" }, { "id": "P012-T2", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T2" }, { "id": "P012-T19", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T19" }, { "id": "P012-T24", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T24" } ], "headline": "Line 22 of the edge proposal lies outside the conditional opened at line 13, so the listing as printed can return a graph outside the compatible set; moving line 22 inside that conditional, with two boundary rules, repairs it.", "id": "P012-E001", "location": "Web Appendix 2, first proposal mechanism, lines 13–22 and 27, and second proposal mechanism, Web Appendix p. 3 (supplement PDF p. 3); the listing begins on Web Appendix p. 2", "pdf_page": 3, "printed_page": "3", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P012-T1", "P012-T19", "P012-T2", "P012-T24" ], "text": { "as_printed": "\\begin{audittable}{rl}{Web Appendix 2, first proposal mechanism, line 2 on Web Appendix p.~2, lines 3--4, 12--23 and 27 on p.~3, as printed (indentation as in the listing).}\nLine & Statement \\\\\n2: & Randomly choose $i\\in V_R$ and $j\\in V_{SU}$ with $i\\neq j$. \\\\\n3: & \\textbf{if} $\\{i,j\\}\\in E_{SU}$ and $\\{i,j\\}\\notin E_R$ \\textbf{then} \\\\\n4: & ~~~Remove $\\{i,j\\}$ from $E^*_{SU}$ \\\\\n12: & ~~~Add $\\{i,u\\}$ to $E^*_{SU}$ \\\\\n13: & ~~~\\textbf{if} $j\\in V_R$ \\textbf{then} \\\\\n14: & ~~~~~~$B\\sim\\operatorname{Bernoulli}(1/2)$ \\\\\n15: & ~~~~~~\\textbf{if} $B<0.5$ and $U_{-j}\\neq\\emptyset$ \\textbf{then} \\\\\n16: & ~~~~~~~~~Randomly choose $u\\in U_{-j}$ \\\\\n17: & ~~~~~~\\textbf{else} \\\\\n18: & ~~~~~~~~~Add a new vertex $u$ to $V^*_{SU}$ \\\\\n19: & ~~~~~~~~~Randomly choose a trait $Z^*_u\\in\\{0,1\\}$ \\\\\n20: & ~~~~~~\\textbf{end if} \\\\\n21: & ~~~\\textbf{end if} \\\\\n22: & ~~~\\hl{Add $\\{j,u\\}$ to $E^*_{SU}$} \\\\\n23: & \\textbf{else if} $\\{i,j\\}\\notin E_{SU}$ and $\\exists u_1,u_2\\in U:\\{i,u_1\\}\\in E_{SU}$ and $\\{j,u_2\\}\\in E_{SU}$ \\textbf{then} \\\\\n27: & Remove \\hl{any isolated vertices} from $V^*_{SU}$ \\\\\n\\end{audittable}\nSecond proposal mechanism: 1: \\hl{Choose $u\\in\\{u\\in V_{SU}:u\\notin V_R\\}$}. 2: Set $Z^*_u=1-Z_u$.", "corrected": "\\begin{audittable}{rl}{Web Appendix 2, first proposal mechanism, lines 13--22 and 27, corrected.}\nLine & Statement \\\\\n13: & ~~~\\textbf{if} $j\\in V_R$ \\textbf{then} \\\\\n14: & ~~~~~~$B\\sim\\operatorname{Bernoulli}(1/2)$ \\\\\n15: & ~~~~~~\\textbf{if} $B<0.5$ and $U_{-j}\\neq\\emptyset$ \\textbf{then} \\\\\n16: & ~~~~~~~~~Randomly choose $u\\in U_{-j}$ \\\\\n17: & ~~~~~~\\textbf{else} \\\\\n18: & ~~~~~~~~~Add a new vertex $u$ to $V^*_{SU}$ \\\\\n19: & ~~~~~~~~~Randomly choose a trait $Z^*_u\\in\\{0,1\\}$ \\\\\n20: & ~~~~~~\\textbf{end if} \\\\\n21: & ~~~~~~\\hl{Add $\\{j,u\\}$ to $E^*_{SU}$} \\\\\n22: & ~~~\\hl{\\textbf{end if}} \\\\\n27: & Remove any isolated \\hl{unsampled} vertices from $V^*_{SU}$ \\\\\n\\end{audittable}\nSecond proposal mechanism: 1: \\hl{If there is no unsampled vertex, return the current state. Otherwise} choose $u\\in\\{u\\in V_{SU}:u\\notin V_R\\}$. 2: Set $Z^*_u=1-Z_u$.", "evidence": "I checked an exact counterexample and a derivation for the corrected listing. Let $V_R=\\{1,2\\}$ with vertex 1 recruiting vertex 2, $\\mathbf d_R=(2,1)$ and $\\mathbf Z_R=(0,1)$. The state with $V_{SU}=\\{1,2,3\\}$, $E_{SU}=\\{\\{1,2\\},\\{1,3\\}\\}$ and $Z_3=0$ is compatible. Let $i=1$ and $j=3$. Line 4 removes $\\{1,3\\}$. Since the set $U_{-1}$ is empty, lines 9 and 10 create a new vertex $u=4$, say with $Z_4=1$, and line 12 adds $\\{1,4\\}$. The condition of line 13 is false. Line 22 adds $\\{3,4\\}$, and line 27 removes nothing. The output has the edges $\\{1,2\\}$, $\\{1,4\\}$ and $\\{3,4\\}$: the edge $\\{3,4\\}$ joins two unsampled vertices, and vertex 3 has no sampled neighbor. With line 22 moved inside the conditional, the same choices produce the vertices $\\{1,2,4\\}$ and the edges $\\{1,2\\}$ and $\\{1,4\\}$, a compatible state. A second example, with three sampled vertices, fails in the same way for every outcome of the coin and of the new trait.\n\nFor the corrected listing every execution returns a compatible pair. If an edge $\\{i,j\\}$ with $j$ unsampled is removed, $i$ loses one edge and gains $\\{i,u\\}$, and $j$ is deleted if it becomes isolated. If $j$ is sampled, each of $i$ and $j$ loses one edge and gains one edge to an unsampled vertex that was not its neighbor. In lines 23 to 25 each of $i$ and $j$ loses one edge to an unsampled vertex and gains $\\{i,j\\}$. In every case each remaining edge has a sampled endpoint. An unsampled vertex then has no sampled neighbor exactly when it is isolated, and line 27 removes it. The trait flip changes the trait of one unsampled vertex only. The sets $U_{-i}$ and $U_{-j}$ of the current state need no update between lines 12 and 16, because the replacement at $i$ does not change which existing unsampled vertices are adjacent to $j$.", "problem": "In the listing the conditional opened at line 13, ``if $j\\in V_R$'', is closed at line 21, and line 22, which adds the edge $\\{j,u\\}$, comes after it. Line 22 runs also when $j$ is an unsampled vertex. In that case $u$ is still the unsampled vertex attached to $i$ at line 12, and line 22 joins two unsampled vertices. The graph $G_{SU}$ has no such edges by definition. The step can also leave $j$ with no sampled neighbor, against condition 3 of Definition 3, and line 27 does not remove $j$, because $j$ is not isolated. The listing is the paper's only specification of the kernel $K$ of Web Appendix 1, which is defined on pairs of compatible states. Proposition 1 therefore does not cover the listing as printed, and bounds computed by an implementation that follows it literally need not be the bounds of Definition 4.\n\nTwo boundary cases also need a rule. Line 27 removes ``any isolated vertices'', which would delete a sampled seed of reported degree zero. The second proposal mechanism has no vertex to choose when the current state has no unsampled vertex. The materials available to me do not include the code used for the Hartford bounds, and I do not know whether it followed the listing.", "what_survives": "The definitions of $h$, $p$, the compatible set and the identification intervals do not depend on the listing, and neither do Results 1--3 or the full-network values of Table 1. I have not established a change to any reported number. Proposition 1 requires a compatible proposal and its other hypotheses (see \\fid{P012-N008}). The corrected listing keeps every proposal inside the compatible set; this alone does not establish convergence, and it does not show what the code behind the intervals of Table 1 did." }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-E001", "work": { "arxiv_id": "1511.05397", "doi": "10.1093/aje/kwx208", "title": "Identification of homophily and preferential recruitment in respondent-driven sampling", "url": "https://pmaronow.github.io/llm-audit/works/P012/" }, "work_id": "P012" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Screenshot of three lines of Web Appendix 1. A red box marks the word non-decreasing in the sentence: Let T_t be a positive non-decreasing sequence indexed by t, with limit zero.", "caption": "Web Appendix 1, Web Appendix p. 1 (supplement PDF p. 1). The marked word should read “non-increasing”.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-E007.png" }, { "alt": "Screenshot of display (2), a chain of four infinite sums ending in a comparison with the sum of 1 over t. Red boxes mark the lower limit t=1 under each of the four summation signs.", "caption": "Proof of Proposition 1, display (2), Web Appendix p. 2 (supplement PDF p. 2). The four marked sums start at $t=1$, where $T_t=(\\epsilon\\log t)^{-1}$ is undefined.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-E007-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P012-T16", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T16" }, { "id": "P012-T24", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T24" } ], "headline": "The cooling schedule is called “non-decreasing” where “non-increasing” is meant, and the schedule and the sums of display (2) start at $t=1$, where $\\log t=0$.", "id": "P012-E007", "location": "Web Appendix 1, Web Appendix p. 1 (supplement PDF p. 1), sentence after the definition of the kernel $K$; Proposition 1 and display (2), Web Appendix p. 2", "pdf_page": 1, "printed_page": "1", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P012-T16", "P012-T24" ], "text": { "as_printed": "Let $T_t$ be a positive \\hl{non-decreasing} sequence indexed by $t$, with $\\lim_{t\\to\\infty}T_t=0$. (Web Appendix p.~1.)\n\nProposition 1. Let the cooling schedule be given by $T_t=\\bigl(\\epsilon\\log(t)\\bigr)^{-1}$ where $\\epsilon>0$ is a constant. (Web Appendix p.~2.)\n\\[\n\\sum_{\\hl{\\scriptstyle t=1}}^{\\infty}\\exp[-D^*/T_t]=\\sum_{\\hl{\\scriptstyle t=1}}^{\\infty}\\exp[-D^*\\epsilon\\log(t)]=\\sum_{\\hl{\\scriptstyle t=1}}^{\\infty}\\frac{1}{t^{D^*\\epsilon}}\\ge\\sum_{\\hl{\\scriptstyle t=1}}^{\\infty}\\frac1t=\\infty\\qquad(2)\n\\]", "corrected": "Let $T_t$ be a positive \\hl{non-increasing} sequence indexed by $t$, with $\\lim_{t\\to\\infty}T_t=0$.\n\nProposition 1. Let the cooling schedule be given by \\hl{a finite $T_1\\ge T_2$ and} $T_t=\\bigl(\\epsilon\\log(t)\\bigr)^{-1}$ \\hl{for $t\\ge2$}, where $\\epsilon>0$ is a constant.\n\\[\n\\sum_{\\hl{\\scriptstyle t=2}}^{\\infty}\\exp[-D^*/T_t]=\\sum_{\\hl{\\scriptstyle t=2}}^{\\infty}\\exp[-D^*\\epsilon\\log(t)]=\\sum_{\\hl{\\scriptstyle t=2}}^{\\infty}\\frac{1}{t^{D^*\\epsilon}}\\ge\\sum_{\\hl{\\scriptstyle t=2}}^{\\infty}\\frac1t=\\infty\\qquad(2)\n\\]", "evidence": "The evidence is a derivation. For $t>1$, differentiation gives\n\\[\n\\frac{d}{dt}\\,\\frac{1}{\\epsilon\\log t}=-\\frac{1}{\\epsilon\\,t\\,(\\log t)^2}<0.\n\\]\nFor $t\\ge2$, the bound $D^*\\epsilon\\le1$ implies that $t^{-D^*\\epsilon}\\ge t^{-1}$, and $\\sum_{t\\ge2}1/t=\\infty$. The comparison in display (2) is unchanged when the first term is dropped.", "problem": "A positive non-decreasing sequence satisfies $T_t\\ge T_1>0$ and cannot tend to zero; the schedule of Proposition 1 is decreasing for $t>1$, and ``non-increasing'' is meant. The schedule is also undefined at the first index, since $\\log1=0$, although the sums of display (2) start at $t=1$.", "what_survives": "The bound $D^*\\epsilon\\le1$ and the comparison with the harmonic series stand. This correction concerns the wording and the first index only; the other hypotheses of the convergence theorem are the subject of \\fid{P012-N008}, and the proposal kernel that of \\fid{P012-E001}." }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-E007", "work": { "arxiv_id": "1511.05397", "doi": "10.1093/aje/kwx208", "title": "Identification of homophily and preferential recruitment in respondent-driven sampling", "url": "https://pmaronow.github.io/llm-audit/works/P012/" }, "work_id": "P012" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Screenshot of four lines of the introduction. A red box marks the words: the estimators used by these programs are not documented.", "caption": "Introduction, p. 154 (PDF p. 2), right column. The marked clause says that the estimators of RDSAT and RDS Analyst are not documented.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-N001.png" } ], "formal_release": null, "formal_statements": [], "headline": "The homophily estimators of RDS Analyst were publicly documented before the paper was accepted; the statement that the estimators of both programs “are not documented” is wrong for that program.", "id": "P012-N001", "location": "Introduction, p. 154 (PDF p. 2), right column, sentence beginning “Two software tools”; reference 39 is on p. 160 (PDF p. 8)", "pdf_page": 2, "printed_page": "154", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "Two software tools for analysis of RDS data produce estimates of homophily or preferential recruitment---RDSAT (38) and RDS Analyst (39)---but \\hl{the estimators used by these programs are not documented}.", "corrected": "Two software tools for analysis of RDS data produce estimates of homophily or preferential recruitment---RDSAT (38) and RDS Analyst (39)\\hl{; the RDS Analyst estimators are described in the help page of the function \\texttt{homophily.estimates} of the RDS package (39)}.", "evidence": "A check of public sources: the help file of \\texttt{homophily.estimates} in RDS 0.5 (\\url{https://github.com/cran/RDS/blob/0.5/man/homophily.estimates.Rd}), the function's source in the same release, the help pages of RDS Analyst for recruitment homophily and population homophily (last modified in August and September 2013), and the source of the RDS Analyst interface, in which the two dialogs for homophily call \\texttt{homophily.estimates}.", "problem": "The sentence says of both programs that their estimators are not documented. For RDS Analyst this is not the case. Reference 39 is the RDS package, version 0.5, released on 28 November 2013. Its help page for the function \\texttt{homophily.estimates} says that the function ``computes an estimate of the population homophily and the recruitment homophily based on a categorical variable'' and defines both ratios, and RDS Analyst obtains its homophily output by calling this function. The paper was accepted on 9 March 2017. I could not obtain the RDSAT 7.1 manual, and the half of the sentence that concerns RDSAT is not settled either way. The citation of reference 39 for RDS Analyst is apt and is not an error.", "what_survives": "No definition, result or number of the paper depends on this sentence. That the estimators are documented does not mean that they identify $h$ or $p$ as the paper defines them; the paper's argument on that point is untouched." }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-N001", "work": { "arxiv_id": "1511.05397", "doi": "10.1093/aje/kwx208", "title": "Identification of homophily and preferential recruitment in respondent-driven sampling", "url": "https://pmaronow.github.io/llm-audit/works/P012/" }, "work_id": "P012" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "historical_lean_attribution", "exhibits": [ { "alt": "Screenshot of Table 1 with columns Trait, h, I_h, p and I_p and rows Sex, Crack cocaine use and Homelessness. Red boxes mark the three lower limits of I_h: -0.7027, -0.6918 and -0.7179.", "caption": "Table 1, p. 158 (PDF p. 6). The three lower limits of $I_h$ are marked.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-N002.png" }, { "alt": "Two-panel chart. Upper panel: three horizontal blue bars for sex, crack cocaine use and homelessness span the printed intervals from about -0.70 to about 0.83 on an axis from -1 to 1, with a dot at each reported point value of h; an orange dashed vertical line at -0.7373 lies to the left of all three bars. Lower panel: the same lower ends enlarged, with the printed limits -0.7027, -0.6918 and -0.7179 to the right of the dashed line and orange arrows labeled at least 0.0345, 0.0454 and 0.0193 spanning the gaps.", "caption": "The printed intervals $I_h$ of Table 1 on the whole range of $h$ (upper panel) and their lower ends enlarged (lower panel). The dots show the point values after excluding vertices with missing traits and their incident edges. The dashed line is the bound $-1462/1983$; for nomination-network degrees, the true lower limit of each interval lies at or to the left of it.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-N002-lower-limits.png" } ], "formal_release": null, "formal_statements": [ { "id": "P012-T3", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T3" }, { "id": "P012-T4", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T4" }, { "id": "P012-T5", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T5" }, { "id": "P012-T20", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T20" } ], "headline": "The three lower limits of $I_h$ in Table 1 cannot be the lower limits of the identification intervals of Definition 4 for the network described on p. 156: a compatible graph has $h\\le-1462/1983\\approx-0.7373$, which is below all three.", "historical_evidence_type": "lean_proof", "id": "P012-N002", "location": "Table 1, p. 158 (PDF p. 6), column $I_h$, the three lower limits; the network counts are on p. 156 (PDF p. 4)", "pdf_page": 6, "printed_page": "158", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P012-T20", "P012-T3", "P012-T4", "P012-T5" ], "text": { "as_printed": "\\begin{audittable}{lrl}{Table 1, homophily columns, as printed.}\nTrait & $h$ & $I_h$ \\\\\nSex & $0.2638$ & $\\hl{-0.7027}$, $0.8292$ \\\\\nCrack cocaine use & $0.2711$ & $\\hl{-0.6918}$, $0.8231$ \\\\\nHomelessness & $-0.2154$ & $\\hl{-0.7179}$, $0.8242$ \\\\\n\\end{audittable}\nThe RDS-net study included $|V_{SU}|=2{,}626$ people, of whom $n=|V_R|=527$ were either seeds or recruited subjects, and 2,099 were nominated but never-recruited subjects. There are $|E_{SU}|=3{,}307$ edges in $G_{SU}$, of which $|E_S|=1{,}180$ link recruited subjects to recruited subjects, and 2,127 link recruited subjects to unrecruited subjects.", "corrected": "\\begin{audittable}{lrl}{Table 1, homophily columns, corrected as far as the published information allows for nomination-network degrees.}\nTrait & $h$ & $I_h$ \\\\\nSex & $0.2638$ & $\\hl{\\text{lower limit at most } -1462/1983}$, $0.8292$ \\\\\nCrack cocaine use & $0.2711$ & $\\hl{\\text{lower limit at most } -1462/1983}$, $0.8231$ \\\\\nHomelessness & $-0.2154$ & $\\hl{\\text{lower limit at most } -1462/1983}$, $0.8242$ \\\\\n\\end{audittable}\nTo retain the stated nomination-network interpretation, the three lower limits are to be recomputed from \\hl{that network's sampled degrees} and the traits of the recruited subjects. For those degrees, the value $-1462/1983\\approx-0.7373$ is an upper bound on each true lower limit and is not itself the limit. The degree vector $\\mathbf d_R$ used for the bounds, and its relation to the nomination network, should be stated. If a degree vector other than that of the nomination network was used, the intervals \\hl{must be checked for that input}, and the nomination network is not a member of the corresponding compatible set. The exact minimum has a closed form, given in \\fid{P012-N007}.", "evidence": "The evidence is a derivation, with its algebra checked by exact arithmetic and by a Lean proof. For a graph with $a$ edges joining two vertices of trait 1, $b$ edges joining two vertices of trait 0 and $c$ edges joining unlike vertices, equation (1) reduces to\n\\[\nh=\\phi(a,b,c)=\\frac{4ab-c^2}{(2a+c)(2b+c)}.\n\\]\nIn the graph constructed above let $a$, $b$ and $c_R$ count the recruitment edges of the three kinds; then $a+b+c_R=r=521$. All $S$ added edges join unlike vertices, and I have $h=\\phi(a,b,c)$ with $c=c_R+S$. I set $T=r+S=3{,}966$. A direct computation gives $(2a+c)(2b+c)=T^2-(a-b)^2>0$ and\n\\[\n\\frac{r-S}{T}\\,(2a+c)(2b+c)-(4ab-c^2)=2T\\,c_R+\\frac{2S}{T}\\,(a-b)^2\\ge0,\n\\]\nand dividing by this positive product I obtain $h\\le(r-S)/T=-1462/1983$. The bound uses only the degree total $4{,}487$ and the fact that each $d_i$ is at least the recruitment degree of $i$; it does not depend on the unknown traits, individual degrees or recruitment graph. The constructed graph has $3{,}445$ unsampled vertices, more than the $2{,}099$ of the nomination network. Definition 3 allows this, since it does not fix the number of unsampled vertices, and Web Appendix 2 itself creates new ones.\n\nEvaluating $\\phi(a,b,c_R+S)$ exactly at all $136{,}503$ triples of nonnegative integers with $a+b+c_R=521$, I find the largest value $-178409/241987\\approx-0.73727$, at $(a,b,c_R)=(260,261,0)$ and $(261,260,0)$. A Lean proof covers the inequality above for nonnegative rational $a$, $b$, $c_R$ with sum $521$ and its comparison with the three printed limits; it also covers the graph construction, its compatibility and the reduction of equation (1).\n\\begin{audittable}{lrrr}{The printed lower limits of $I_h$ against the bound $-1462/1983$ (exact differences).}\nTrait & Printed lower limit & Printed limit minus bound & Decimal (rounded) \\\\\nSex & $-0.7027$ & $685459/19830000$ & $0.034567$ \\\\\nCrack cocaine use & $-0.6918$ & $450803/9915000$ & $0.045467$ \\\\\nHomelessness & $-0.7179$ & $384043/19830000$ & $0.019367$ \\\\\n\\end{audittable}", "problem": "Definition 4 makes $I_h$ the smallest interval that contains $h$ at every compatible pair. The paper identifies the nomination network with $G_{SU}$ and defines $d_i$ as the number of edges at $i$. The counts on p.~156 then fix the total of the reported degrees: $2(1{,}180)+2{,}127=4{,}487$. The $527-6=521$ recruitment edges use $1{,}042$ of these edge-ends and leave $S=3{,}445$. I attach each of the $S$ spare edge-ends to a new unsampled vertex of its own, and give that vertex the trait opposite to the trait of its sampled neighbor. The resulting graph satisfies the four conditions of Definition 3, and whatever the traits of the recruited subjects its homophily is at most $(521-3{,}445)/(521+3{,}445)=-1462/1983\\approx-0.7373$. Each printed lower limit exceeds this value, by at least $0.0345$ (sex), $0.0454$ (crack cocaine use) and $0.0193$ (homelessness), which rounding to four decimals cannot explain. Either the search that produced the limits stopped short of the minimum, or the degrees given to it were not those of the nomination network. In the second case the nomination network violates condition 4 of Definition 3 and is not in the compatible set, and the comparison of the point values with the intervals is not the one the paper describes. The published materials do not supply the Hartford data or analysis code, so I cannot say which case holds or give the correct limits.", "what_survives": "The point values of $h$ and $p$, the upper limits of $I_h$ and the intervals $I_p$ are not shown wrong by this finding. What the paper reads from the table holds with any wider intervals: each point value lies inside its interval and every interval covers zero. Definitions 1--4 and Results 1--3 do not depend on the table. For nomination-network degrees, the three lower limits of $I_h$ and the widths of those intervals change, and the remark on p.~158 that the regions ``were considerably smaller than the outcome space $[-1,1]$'' has to be reassessed once the limits are recomputed; this finding does not show it false." }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-N002", "work": { "arxiv_id": "1511.05397", "doi": "10.1093/aje/kwx208", "title": "Identification of homophily and preferential recruitment in respondent-driven sampling", "url": "https://pmaronow.github.io/llm-audit/works/P012/" }, "work_id": "P012" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Screenshot of the end of the proof of Result 1. Red boxes mark the words: under the conditions of definition 3, which run over a line break.", "caption": "Appendix, end of the proof of Result 1, p. 160 (PDF p. 8), right column. The marked words cite Definition 3.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-N005.png" } ], "formal_release": null, "formal_statements": [ { "id": "P012-T17", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T17" } ], "headline": "The last step of the proof of Result 1 cites Definition 3 where it needs the compatibility of recruitment-induced subgraphs defined at the start of the same proof.", "id": "P012-N005", "location": "Appendix, proof of Result 1, p. 160 (PDF p. 8), right column, sentence beginning “Clearly both”", "pdf_page": 8, "printed_page": "160", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P012-T17" ], "text": { "as_printed": "Clearly both $G^1_S$ and $G^2_S$ are compatible with the observed data \\hl{under the conditions of definition 3}, but $\\{i,j\\}\\notin G^1_S$ and $\\{i,j\\}\\in G^2_S$.", "corrected": "Clearly both $G^1_S$ and $G^2_S$ are compatible with the observed data \\hl{in the sense defined at the start of this proof}, but $\\{i,j\\}\\notin G^1_S$ and $\\{i,j\\}\\in G^2_S$.", "evidence": "A check of the page and a small example. Let vertices 1 and 2 be seeds with $d_1=d_2=1$ and no recruitment, and let the augmented graph have the edges $\\{1,u\\}$ and $\\{2,u\\}$. Removing $u$ leaves degrees $(0,0)$, which fail condition 4 of Definition 3 and satisfy the opening definition.", "problem": "Definition 3 concerns augmented graphs, and its condition 4 requires each sampled vertex to have exactly its reported degree. Removing the unsampled vertices lowers those degrees, and $G^1_S$ and $G^2_S$ need not satisfy Definition 3. They satisfy the definition given at the opening of the proof, which asks that the degree of $i$ in $G_S$ be at most $d_i$, and this is what the step needs.", "what_survives": "The step is valid with the right reference, and Result 1 is unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-N005", "work": { "arxiv_id": "1511.05397", "doi": "10.1093/aje/kwx208", "title": "Identification of homophily and preferential recruitment in respondent-driven sampling", "url": "https://pmaronow.github.io/llm-audit/works/P012/" }, "work_id": "P012" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Screenshot of the proof of Result 2 up to the words The difference is. Four red boxes mark the argument G_R inside h(G^1_SU, G_R, Z^1_SU) and h(G^2_SU, G_R, Z^2_SU), each of which appears twice.", "caption": "Appendix, proof of Result 2, p. 160 (PDF p. 8), right column. The four marked occurrences of $G_R$ are redundant arguments of $h$.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-N006.png" }, { "alt": "Screenshot of the sentence beginning To ease notation. A red box marks the bold Z in h(G_SU, Z), which has no subscript.", "caption": "p. 155 (PDF p. 3), right column, below equation (2). The marked $\\mathbf Z$ lacks the subscript $SU$.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-N006-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P012-T18", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T18" } ], "headline": "In the proof of Result 2 the function $h$ is written four times with a redundant argument $G_R$, and on p. 155 once with an unsubscripted $\\mathbf Z$; the function $h(G_{SU},\\mathbf Z_{SU})$ of equation (1) is meant.", "id": "P012-N006", "location": "Appendix, proof of Result 2, p. 160 (PDF p. 8), right column, four occurrences; and p. 155 (PDF p. 3), sentence beginning “To ease notation”", "pdf_page": 8, "printed_page": "160", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P012-T18" ], "text": { "as_printed": "[\\ldots] such that $h(G^1_{SU},\\hl{G_R,}\\,\\mathbf Z^1_{SU})\\ne h(G^2_{SU},\\hl{G_R,}\\,\\mathbf Z^2_{SU})$. [\\ldots] Let $h_1=h(G^1_{SU},\\hl{G_R,}\\,\\mathbf Z^1_{SU})$ and $h_2=h(G^2_{SU},\\hl{G_R,}\\,\\mathbf Z^2_{SU})$ be the calculated values of homophily. (p.~160.)\n\nTo ease notation, we will often use $h$ and $p$ to refer to $h(G_{SU},\\hl{\\mathbf Z})$ and $p(G_{SU},G_R,\\mathbf t_R,\\mathbf Z_{SU})$ respectively. (p.~155.)", "corrected": "[\\ldots] such that $\\hl{h(G^1_{SU},\\mathbf Z^1_{SU})}\\ne\\hl{h(G^2_{SU},\\mathbf Z^2_{SU})}$. [\\ldots] Let $h_1=\\hl{h(G^1_{SU},\\mathbf Z^1_{SU})}$ and $h_2=\\hl{h(G^2_{SU},\\mathbf Z^2_{SU})}$ be the calculated values of homophily.\n\nTo ease notation, we will often use $h$ and $p$ to refer to $h(G_{SU},\\hl{\\mathbf Z_{SU}})$ and $p(G_{SU},G_R,\\mathbf t_R,\\mathbf Z_{SU})$ respectively.", "evidence": "A check of the two pages against Definition 1, equation (1) and Definition 4. Display (A1) uses adjacency, degrees and traits only, in agreement with equation (1).", "problem": "Definition 1 gives $h$ two arguments, the augmented graph and its traits, and $h$ does not depend on the recruitment graph. The symbol $\\mathbf Z$ without a subscript is not defined in the paper. Both are slips of notation.", "what_survives": "The calculation in (A1) and Result 2 are unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-N006", "work": { "arxiv_id": "1511.05397", "doi": "10.1093/aje/kwx208", "title": "Identification of homophily and preferential recruitment in respondent-driven sampling", "url": "https://pmaronow.github.io/llm-audit/works/P012/" }, "work_id": "P012" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Screenshot of the heading Web Appendix 1 and its first three lines. A red box marks the words: there are no general closed-form expressions for the extrema of h and p.", "caption": "Web Appendix 1, first sentence, Web Appendix p. 1 (supplement PDF p. 1). The marked words deny that the extrema of $h$ and $p$ have closed forms.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-N007.png" } ], "formal_release": null, "formal_statements": [ { "id": "P012-T6", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T6" }, { "id": "P012-T7", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T7" }, { "id": "P012-T21", "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T21" } ], "headline": "The extrema of $h$ and $p$ over the compatible set have closed forms that are computed in one pass over the data; the opening sentence of Web Appendix 1 says that there are none.", "id": "P012-N007", "location": "Web Appendix 1, first sentence, Web Appendix p. 1 (supplement PDF p. 1)", "pdf_page": 1, "printed_page": "1", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P012-T21", "P012-T6", "P012-T7" ], "text": { "as_printed": "Unfortunately \\hl{there are no general closed-form expressions} for the extrema of $h$ and $p$ on $\\mathcal C(G_R,\\mathbf d_R,\\mathbf Z_R)$.", "corrected": "\\hl{The extrema of $h$ and $p$ on $\\mathcal C(G_R,\\mathbf d_R,\\mathbf Z_R)$ have closed-form expressions.} For \\hl{observed data with a nonempty compatible set}, let $s_i=d_i-d_i^r$ for $i\\in V_R$, let $S_z=\\sum_{i\\in V_R:Z_i=z}s_i$ for $z=0,1$ and $S=S_0+S_1$, and let $a$, $b$ and $c_R$ be the numbers of recruitment edges that join two vertices of trait 1, two vertices of trait 0, and unlike vertices. For a nonseed $j$ let $b_j$ be the number of vertices recruited by $r_j$ at or after time $t_j$, and $a_j$ the number of those with trait $Z_{r_j}$. Suppose that $\\sum_{i\\in V_R:Z_i=z}d_i>0$ for $z=0$ and for $z=1$, and that $n>|M|$. Let $\\phi(x,y,c)=(4xy-c^2)/((2x+c)(2y+c))$. Then\n\\[\n\\begin{aligned}\n\\min_{\\mathcal C}h&=\\phi(a,b,c_R+S),\\\\\n\\max_{\\mathcal C}h&=\\phi(a+S_1,b+S_0,c_R),\\\\\n\\min_{\\mathcal C}p&=\\frac{1}{n-|M|}\\sum_{j\\notin M}\\Bigl[\\mathbf 1\\{Z_j=Z_{r_j}\\}-\\frac{a_j+s_{r_j}}{b_j+s_{r_j}}\\Bigr],\\\\\n\\max_{\\mathcal C}p&=\\frac{1}{n-|M|}\\sum_{j\\notin M}\\Bigl[\\mathbf 1\\{Z_j=Z_{r_j}\\}-\\frac{a_j}{b_j+s_{r_j}}\\Bigr].\n\\end{aligned}\n\\]\nEach extremum is attained at a compatible pair in which every spare edge-end of a sampled vertex has its own unsampled neighbor: of the opposite trait for the minimum of $h$ and the maximum of $p$, of the same trait for the maximum of $h$ and the minimum of $p$. These four endpoints are computed directly from the data; simulated annealing remains available for other objectives or for additional constraints on the graph.", "evidence": "The evidence is a complete derivation, with exact enumeration as a check. The notation is that of the corrected statement below.\n\n\\emph{Homophily.} Let $x$, $y$ and $c$ be the numbers of edges of a compatible graph that join two vertices of trait 1, two vertices of trait 0, and unlike vertices. Equation (1) reduces to\n\\[\nh=\\phi(x,y,c)=\\frac{4xy-c^2}{(2x+c)(2y+c)}=1-\\frac{c}{2x+c}-\\frac{c}{2y+c}.\n\\]\nEvery edge outside the recruitment graph uses at least one spare edge-end of a sampled vertex, of trait 1 if the edge joins two vertices of trait 1 and of trait 0 if it joins two vertices of trait 0. Then $a\\le x\\le a+S_1$, $b\\le y\\le b+S_0$ and $c_R\\le c\\le c_R+S$. The function $\\phi$ is nondecreasing in $x$ and in $y$ and nonincreasing in $c$: for $X\\ge x$ and $C\\ge c$,\n\\[\n\\begin{aligned}\n\\phi(X,y,c)-\\phi(x,y,c)&=\\frac{2c(X-x)}{(2X+c)(2x+c)}\\ge0,\\\\\n\\phi(x,y,c)-\\phi(x,y,C)&=\\frac{2x(C-c)}{(2x+c)(2x+C)}+\\frac{2y(C-c)}{(2y+c)(2y+C)}\\ge0,\n\\end{aligned}\n\\]\nand the first identity holds with $x$ and $y$ exchanged. Under the two conditions on the reported degrees every denominator that occurs is positive. I conclude that $\\phi(a,b,c_R+S)\\le h\\le\\phi(a+S_1,b+S_0,c_R)$. Both bounds are attained. To see this, I keep the recruitment edges and attach to each sampled vertex $i$ exactly $s_i$ new unsampled neighbors of degree one; the graph is compatible. If each new vertex has the trait opposite to that of its neighbor, the counts are $(a,b,c_R+S)$; if each has the trait of its neighbor, they are $(a+S_1,b+S_0,c_R)$.\n\n\\emph{Preferential recruitment.} Fix a nonseed $j$ with recruiter $r=r_j$. The $b_j$ vertices recruited by $r$ at or after $t_j$ are neighbors of $r$ that are susceptible just before $t_j$, and they are the only susceptible neighbors of $r$ fixed by the recruitment graph; $a_j$ of them have the trait of $r$. Any other susceptible neighbor of $r$ uses a spare edge-end of $r$. Let $y_j\\le s_r$ be the number of these and $q_j\\le y_j$ the number with the trait of $r$. The fraction subtracted in the summand of $j$ in equation (2) is $(a_j+q_j)/(b_j+y_j)$, and by cross-multiplication I find\n\\[\n\\frac{a_j}{b_j+s_r}\\le\\frac{a_j+q_j}{b_j+y_j}\\le\\frac{a_j+s_r}{b_j+s_r}.\n\\]\nIn the graph with one new unsampled neighbor for every spare edge-end, these neighbors are never recruited, and $y_j=s_{r_j}$ for every $j$ at once. With equal traits, $q_j=s_{r_j}$ for every $j$, which is the minimum of $p$; with opposite traits, $q_j=0$ for every $j$, which is the maximum.\n\n\\emph{Checks.} For the example of Figure 1, I have $a=3$, $b=0$, $c_R=1$, $S_1=7$, $S_0=2$ and $S=9$, and from the formulas I obtain the four values in the table. They equal the extrema over all $26{,}182$ compatible pairs of that example, counted up to the labeling of the unsampled vertices and found by exhaustive enumeration with exact arithmetic. The formulas also agree with exhaustive enumeration on $5{,}008$ further small data sets, with $86{,}988$ compatible pairs in all.\n\\begin{audittable}{llrr}{Figure 1 of the paper: the closed forms against the printed extrema.}\nEndpoint & Closed form & Exact value & Printed in Figure 1 \\\\\nMinimum of $h$ & $\\phi(3,0,10)$ & $-5/8$ & $-0.625$ \\\\\nMaximum of $h$ & $\\phi(10,2,1)$ & $79/105$ & $0.752$ \\\\\nMinimum of $p$ & $\\tfrac14\\bigl(\\tfrac15+0-\\tfrac34+0\\bigr)$ & $-11/80$ & $-0.138$ \\\\\nMaximum of $p$ & $\\tfrac14\\bigl(\\tfrac45+\\tfrac13+0+\\tfrac12\\bigr)$ & $49/120$ & $0.408$ \\\\\n\\end{audittable}", "problem": "The sentence is the premise of the computational procedure of Web Appendixes 1 and 2, that is, of the simulated-annealing search and of Proposition 1. It is false under the conditions stated below. All four endpoints of the identification intervals are explicit functions of quantities read from the observed data: the recruitment edges counted by the traits of their ends, the spare degrees $d_i-d_i^r$, and the order of recruitment. Each spare edge-end of a sampled vertex can be given its own unsampled neighbor, and the extreme values of $h$ and of $p$ are attained when the traits of these neighbors are all opposite to, or all equal to, the trait of the sampled vertex. A search is unnecessary for these four endpoints. For them, the appendix search, its convergence argument and the search for extremizing compatible graphs can all be replaced by the formulas and the two constructions with separate unsampled neighbors. A finite run of a search can return an interval that is too narrow. The formulas give the endpoints exactly. This finding does not by itself show any reported number to be wrong.", "what_survives": "This correction leaves Definitions 1--4, the identification regions themselves and Results 1--3 unchanged; the four exact extrema round to the values printed in Figure 1. The formulas also give short proofs of the nonidentification conclusions in Results 2 and 3. The central motivation survives: incomplete network observation can leave $h$ and $p$ undetermined even when their sharp bounds are easy to compute. The full-network values in Table 1 do not use the search. Simulated annealing may still serve for another objective $J(h,p)$ or for a compatible set with further constraints, and whether Proposition 1 is true is not at issue here. The premise of Web Appendixes 1 and 2 changes: the four endpoints should be computed from the formulas. The intervals of Table 1 can be recomputed in this way once the data are available; if the printed endpoints came from compatible graphs, the exact intervals are no narrower and still cover zero." }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-N007", "work": { "arxiv_id": "1511.05397", "doi": "10.1093/aje/kwx208", "title": "Identification of homophily and preferential recruitment in respondent-driven sampling", "url": "https://pmaronow.github.io/llm-audit/works/P012/" }, "work_id": "P012" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Screenshot of Proposition 1 and the first paragraph of its proof. Red boxes mark epsilon greater than 0 in the statement, 0 less than epsilon less than 1 in the proof, and the sentence that begins: Theorem 1 of Hajek (3) states that.", "caption": "Proposition 1 and the start of its proof, Web Appendix p. 2 (supplement PDF p. 2). Marked: the range of $\\epsilon$ in the statement, the narrower range in the proof, and the restatement of Hajek’s theorem.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-N008.png" } ], "formal_release": null, "formal_statements": [], "headline": "Proposition 1 holds for the intended proposal, but its proof quotes Hajek’s theorem without the theorem’s standing hypotheses, restricts $\\epsilon$ to $(0,1)$ without need, and cites for connectivity an argument made for a smaller state space.", "id": "P012-N008", "location": "Web Appendix 1, Proposition 1 and its proof, Web Appendix p. 2 (supplement PDF p. 2); the sentence on connectivity is in Web Appendix 2, Web Appendix p. 3", "pdf_page": 2, "printed_page": "2", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "Proposition 1. Let the cooling schedule be given by $T_t=\\bigl(\\epsilon\\log(t)\\bigr)^{-1}$ where \\hl{$\\epsilon>0$} is a constant. Then $\\lim_{t\\to\\infty}\\Pr\\bigl((G_{SU},\\mathbf Z_{SU})_t\\in\\mathcal M\\bigr)=1$.\n\nProof. Let $J(h,p)=1/(1+\\epsilon+h)$ for \\hl{$0<\\epsilon<1$} and let $\\mathcal M$ be the set of $(G_{SU},\\mathbf Z_{SU})$ that achieve the global maximum of $J$ on $\\mathcal C(G_R,\\mathbf d_R,\\mathbf Z_R)$. [\\ldots] \\hl{Theorem 1 of Hajek (3) states that} if $T_t\\to0$ and $\\sum_{t=1}^{\\infty}\\exp[-D^*/T_t]$ diverges, then the sequence $(G_{SU},\\mathbf Z_{SU})_t$ converges in probability to an element of $\\mathcal M$.\n\nThe space $\\mathcal C(G_R,\\mathbf d_R,\\mathbf Z_R)$ is connected via proposals of this type (see 6, for explanation). (Web Appendix p.~3.)", "corrected": "Proposition 1. \\hl{Assume that $h$ and $p$ are defined throughout $\\mathcal C(G_R,\\mathbf d_R,\\mathbf Z_R)$} (see \\fid{P012-E006}). \\hl{Let $K$ be the repaired proposal kernel of Web Appendix 2 on this finite set} (see \\fid{P012-E001}). \\hl{It is fixed, irreducible and weakly reversible in the sense of Hajek (3).} Let the cooling schedule be given by \\hl{a finite $T_1\\ge T_2$ and} $T_t=\\bigl(\\epsilon\\log(t)\\bigr)^{-1}$ \\hl{for $t\\ge2$}, where $\\epsilon>0$ is a constant. Then $\\lim_{t\\to\\infty}\\Pr\\bigl((G_{SU},\\mathbf Z_{SU})_t\\in\\mathcal M\\bigr)=1$.\n\nProof. Let $J(h,p)=1/(1+\\epsilon+h)$ for \\hl{$\\epsilon>0$} and let $\\mathcal M$ be the set of $(G_{SU},\\mathbf Z_{SU})$ that achieve the global maximum of $J$ on $\\mathcal C(G_R,\\mathbf d_R,\\mathbf Z_R)$. [\\ldots] Theorem 1 of Hajek (3) states that\\hl{, for a fixed proposal that is irreducible and weakly reversible and for positive nonincreasing temperatures} $T_t\\to0$, if $\\sum_{t=2}^{\\infty}\\exp[-D^*/T_t]$ diverges, then \\hl{the probability that $(G_{SU},\\mathbf Z_{SU})_t$ belongs to $\\mathcal M$ tends to one}.", "intended_meaning": "Proposition 1 applies Theorem 1 of Hajek (1988) to the annealing chain on the finite set $\\mathcal C$ that the proposals of Web Appendix 2 define, with the proposal repairs in \\fid{P012-E001} and with $h$ and $p$ defined throughout $\\mathcal C$ (see \\fid{P012-E006}). A check of the published article shows that the theorem assumes a fixed proposal that is irreducible and weakly reversible, and positive temperatures that are nonincreasing with limit zero. Under these hypotheses the probability that the chain is in the set of global optima tends to one exactly when $\\sum_k\\exp(-d^*/T_k)=+\\infty$. The summary in the proof omits the hypotheses.\n\nThe intended chain satisfies them. From any compatible pair a sequence of proposals leads to the pair in which the recruitment edges are the only edges among sampled vertices and every spare edge-end has its own unsampled neighbor of trait 0. Each proposal can be undone by a path of proposals. These two facts imply that the chain is irreducible. The path can be chosen so that $J$ stays at or above its smaller value at the two ends, which is weak reversibility. The schedule $(\\epsilon\\log t)^{-1}$ is decreasing for $t\\ge2$. The bound $D^*\\le1/\\epsilon$ and the comparison with $\\sum_t1/t$ hold for every $\\epsilon>0$, and the restriction to $0<\\epsilon<1$ in the proof is unnecessary. The argument cited from reference 6, which I could check in the preprint only, concerns graphs on the sampled vertices with upper bounds on the degrees; this is a smaller space than the pairs $(G_{SU},\\mathbf Z_{SU})$ with unsampled vertices and traits, and the argument above covers the larger one. The convergence is to the set $\\mathcal M$ and not to one maximizing pair. The undefined $T_1$ is the subject of \\fid{P012-E007}." }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-N008", "work": { "arxiv_id": "1511.05397", "doi": "10.1093/aje/kwx208", "title": "Identification of homophily and preferential recruitment in respondent-driven sampling", "url": "https://pmaronow.github.io/llm-audit/works/P012/" }, "work_id": "P012" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Screenshot of the heading References and entries 1 and 2. Red boxes mark the author list Kirkpatrick C. D., Vecchi M. P. and the word simmulated in entry 1.", "caption": "Web Appendix, References, entry [1], Web Appendix p. 3 (supplement PDF p. 3). The author list and the misspelled word are marked.", "url": "https://pmaronow.github.io/llm-audit/works/P012/exhibits/P012-N009.png" } ], "formal_release": null, "formal_statements": [], "headline": "Reference [1] of the Web Appendix omits the second author, gives the first author the wrong initials and misspells “simulated”.", "id": "P012-N009", "location": "Web Appendix, References, entry [1], Web Appendix p. 3 (supplement PDF p. 3)", "pdf_page": 3, "printed_page": "3", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "[1] \\hl{Kirkpatrick C. D., Vecchi M. P.}. Optimization by \\hl{simmulated} annealing \\emph{Science}. 1983;220:671--680.", "corrected": "[1] \\hl{Kirkpatrick S, Gelatt CD Jr, Vecchi MP}. Optimization by \\hl{simulated} annealing. \\emph{Science}. 1983;220(4598):671--680.", "evidence": "A check of the published article: its first page carries the title ``Optimization by Simulated Annealing'' and the byline ``S. Kirkpatrick, C. D. Gelatt, Jr., M. P. Vecchi''.", "problem": "The article is by S.~Kirkpatrick, C.~D.~Gelatt Jr. and M.~P.~Vecchi. 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The printed claim is false: P012-T1; Web Appendix pp. 2–3 (supplement PDF pp. 2–3); the kernel K: Web Appendix p. 1; augmented graphs and Definition 3: p. 155 (PDF p. 3). All four specified counterexamples hold, including their functional values. In (a) and (b), for the fixed chosen i,j and either coin outcome, every execution uses a fresh vertex of either trait and produces exactly the stated old graph with the removed sampled edge replaced by a sampled–fresh edge and an old-unsampled–fresh edge. Cleanup deletes nothing. The old unsampled vertex has no sampled neighbour; the output is incompatible, while conditions 1,2,4 remain true in (a). Every fresh name gives the named output up to renaming unsampled identities. In (c) both branches fail and cleanup deletes precisely sampled seed 3, violating vertex preservation and inclusion of the observed active trait assignment. In (d) every compatible pair is the displayed dyad on its active domain, so the trait proposal has no vertex to choose.

", "headline": "Web Appendix 2, the two proposal mechanisms, as printed (said to propose another compatible pair)", "location": "Web Appendix pp. 2–3 (supplement PDF pp. 2–3); the kernel KKK on C\\mathcal{C}C: Web Appendix p. 1; GSUG_{SU}GSU​ and Definition 3: p. 155 (PDF p. 3)" }, "id": "P012-T1", "text": { "body": "The printed claim is false: P012-T1; Web Appendix pp. 2–3 (supplement PDF pp. 2–3); the kernel K: Web Appendix p. 1; augmented graphs and Definition 3: p. 155 (PDF p. 3). All four specified counterexamples hold, including their functional values. In (a) and (b), for the fixed chosen i,j and either coin outcome, every execution uses a fresh vertex of either trait and produces exactly the stated old graph with the removed sampled edge replaced by a sampled–fresh edge and an old-unsampled–fresh edge. Cleanup deletes nothing. The old unsampled vertex has no sampled neighbour; the output is incompatible, while conditions 1,2,4 remain true in (a). 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Web Appendix pp. 2–3 (supplement PDF pp. 2–3). Every execution of either corrected mechanism preserves compatibility. In the unsampled deletion branch exactly one sampled degree is restored, and cleanup deletes the old unsampled endpoint exactly when its only neighbour was the selected sampled vertex. In the sampled deletion branch both degrees are restored and the original second replacement pool remains nonadjacent. In the addition branch the second endpoint is sampled, two distinct non-recruitment edges are removed even for a shared unsampled endpoint, both sampled degrees are restored, and cleanup deletes exactly the selected vertices left isolated. If neither branch applies the state is unchanged. Recruitment edges, sampled vertices and traits, and the augmented graph rules are preserved throughout the returned states. The trait mechanism flips one unsampled trait or returns the same state when the unsampled set is empty.

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The printed claim is false: P012-T3; Table 1: p. 158 (PDF p. 6); network counts and seeds: p. 156 (PDF p. 4); bounds: pp. 156–157 (PDF pp. 4–5); Definitions 3–4: p. 155 (PDF p. 3). For every admissible data set with 527 sampled vertices, six seeds and sampled degree total 4487, the opposite-trait private-leaf completion is compatible, has 521 recruitment edges, 3445 private degree-one leaves and 3972 vertices, and has positive equation-(1) denominator. Its edge counts and quotient value obey the uniform real bound -1462/1983 and the sharp integer bound -178409/241987, attained by the two stated integer triples. The displayed polynomial certificate is nonnegative. All three printed lower bounds are violated, with the exact stated gaps exceeding four-decimal rounding error; even a rounded lower endpoint cannot enclose every compatible homophily value. The larger completion is allowed because compatibility fixes no population size.

", "headline": "Table 1, the three lower limits of IhI_hIh​, as printed, read with the network counts of p. 156", "location": "Table 1: p. 158 (PDF p. 6); network counts and number of seeds: p. 156 (PDF p. 4); the bounds: pp. 156–157 (PDF pp. 4–5); Definitions 3–4: p. 155 (PDF p. 3)" }, "id": "P012-T3", "text": { "body": "The printed claim is false: P012-T3; Table 1: p. 158 (PDF p. 6); network counts and seeds: p. 156 (PDF p. 4); bounds: pp. 156–157 (PDF pp. 4–5); Definitions 3–4: p. 155 (PDF p. 3). For every admissible data set with 527 sampled vertices, six seeds and sampled degree total 4487, the opposite-trait private-leaf completion is compatible, has 521 recruitment edges, 3445 private degree-one leaves and 3972 vertices, and has positive equation-(1) denominator. Its edge counts and quotient value obey the uniform real bound -1462/1983 and the sharp integer bound -178409/241987, attained by the two stated integer triples. The displayed polynomial certificate is nonnegative. All three printed lower bounds are violated, with the exact stated gaps exceeding four-decimal rounding error; even a rounded lower endpoint cannot enclose every compatible homophily value. The larger completion is allowed because compatibility fixes no population size.", "headline": "Table 1, the three lower limits of $I_h$, as printed, read with the network counts of p. 156", "location": "Table 1: p. 158 (PDF p. 6); network counts and number of seeds: p. 156 (PDF p. 4); the bounds: pp. 156–157 (PDF pp. 4–5); Definitions 3–4: p. 155 (PDF p. 3)" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T3" }, { "declarations": [ { "claim_revision_id": "catalog:e23fe94b844dc00d783909d2cfc2593239b0ddfd103e5defc0575d3b601aa5f8", "conditional_on": [], "credit_reason": "Assuming IsGLB characterizes the endpoint already named by the claim; it does not exclude a permissible dataset here because T=3445 gives a defined attaining completion. All rational gaps were checked exactly. Claim-stage verdict unchanged after reading the record.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P012.corrected_table_bound", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/P012_Identification.lean#L2548", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e23fe94b844dc00d783909d2cfc2593239b0ddfd103e5defc0575d3b601aa5f8", "conditional_on": [], "credit_reason": "The inclusive rounding interval is slightly broader than any particular tie-breaking rule and includes every case the target needs. Claim-stage verdict unchanged after reading the record.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P012.table_degree_converse", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/P012_Identification.lean#L2569", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:e23fe94b844dc00d783909d2cfc2593239b0ddfd103e5defc0575d3b601aa5f8", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P012-T4", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P012-T4" }, "html": { "body": "

Table 1: p. 158 (PDF p. 6); network counts: p. 156 (PDF p. 4). The true lower endpoint obeys both certified bounds and all three gap bounds, and printed endpoints force a degree discrepancy with the nomination network.

", "headline": "Table 1, the three lower limits of IhI_hIh​, corrected claim: the bound that each lower limit must satisfy", "location": "Table 1: p. 158 (PDF p. 6); network counts: p. 156 (PDF p. 4)" }, "id": "P012-T4", "text": { "body": "Table 1: p. 158 (PDF p. 6); network counts: p. 156 (PDF p. 4). The true lower endpoint obeys both certified bounds and all three gap bounds, and printed endpoints force a degree discrepancy with the nomination network.", "headline": "Table 1, the three lower limits of $I_h$, corrected claim: the bound that each lower limit must satisfy", "location": "Table 1: p. 158 (PDF p. 6); network counts: p. 156 (PDF p. 4)" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T4" }, { "declarations": [ { "claim_revision_id": "catalog:04d6d2b358d4c238eaff9a811435f1b95a0aaf1dc6af13634e9bd7ef54abfef6", "conditional_on": [], "credit_reason": "The endpoint-domain premise is exactly the target's, including one-trait sampled data with positive spare total. Undefined compatible states are excluded from hValues as required by T5. Claim-stage verdict unchanged after reading the record.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P012.exact_h_lower", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/P012_Identification.lean#L2711", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:04d6d2b358d4c238eaff9a811435f1b95a0aaf1dc6af13634e9bd7ef54abfef6", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P012-T5", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P012-T5" }, "html": { "body": "

Table 1: p. 158 (PDF p. 6); network counts: p. 156 (PDF p. 4); the computation described: pp. 156–157 (PDF pp. 4–5). The opposite-trait leaf completion attains the exact lower homophily endpoint on its stated domain.

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Web Appendix p. 1 (supplement PDF p. 1); Definitions 1, 3 and 4: p. 155 (PDF p. 3). For admissible observed data with positive sampled degree total in each trait group, every compatible graph has the positive displayed denominator and the displayed edge-count and homophily bounds. Opposite-trait and same-trait private degree-one leaf completions attain the bounds, so their endpoints give the smallest containing interval. An explicit algorithm scans each sampled-vertex record and each directed recruitment-edge record once, in any order within each stream, accumulating the two degree totals and three edge-type counts. It recovers the spare totals without truncated subtraction and returns the two exact rational endpoints. The instrumented scan has exactly n+|E_R| visits and accumulator additions, followed by 26 arithmetic operations, with indexed trait/degree/parent access and unit-cost exact arithmetic.

", "headline": "Web Appendix 1, first sentence, corrected: closed forms for the extrema of hhh on C(GR,dR,ZR)\\mathcal{C}(G_R, \\mathbf{d}_R, \\mathbf{Z}_R)C(GR​,dR​,ZR​)", "location": "Web Appendix p. 1 (supplement PDF p. 1); Definitions 1, 3 and 4: p. 155 (PDF p. 3)" }, "id": "P012-T6", "text": { "body": "Web Appendix p. 1 (supplement PDF p. 1); Definitions 1, 3 and 4: p. 155 (PDF p. 3). For admissible observed data with positive sampled degree total in each trait group, every compatible graph has the positive displayed denominator and the displayed edge-count and homophily bounds. Opposite-trait and same-trait private degree-one leaf completions attain the bounds, so their endpoints give the smallest containing interval. An explicit algorithm scans each sampled-vertex record and each directed recruitment-edge record once, in any order within each stream, accumulating the two degree totals and three edge-type counts. It recovers the spare totals without truncated subtraction and returns the two exact rational endpoints. The instrumented scan has exactly n+|E_R| visits and accumulator additions, followed by 26 arithmetic operations, with indexed trait/degree/parent access and unit-cost exact arithmetic.", "headline": "Web Appendix 1, first sentence, corrected: closed forms for the extrema of $h$ on $\\mathcal{C}(G_R, \\mathbf{d}_R, \\mathbf{Z}_R)$", "location": "Web Appendix p. 1 (supplement PDF p. 1); Definitions 1, 3 and 4: p. 155 (PDF p. 3)" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T6" }, { "declarations": [ { "claim_revision_id": "catalog:aa178acdd52072eb05a04be267e5853dcc50bb0cf054fc090823e4751757228e", "conditional_on": [], "credit_reason": "The row covers the corrected mathematical region and endpoint assertions, not a runtime guarantee. 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Web Appendix p. 1 (supplement PDF p. 1); Definitions 2–4: p. 155 (PDF p. 3). The observed recruitment history and spare degrees give the sharp attained preferential-recruitment interval. Obligatory child sets are nonempty; every compatible susceptible fraction has the stated decomposition and sharp bounds. Same-trait private degree-one neighbours attain p_min, and opposite-trait private degree-one neighbours attain p_max. The smallest containing interval is [p_min,p_max]. An exact rational forward algorithm computes both endpoints in one pass over the supplied chronological records, with positive event denominators, one read per record, at most n stored cells and at most 32n+4 unit arithmetic and indexed-cell operations, including the final divisions.

", "headline": "Web Appendix 1, first sentence, corrected: closed forms for the extrema of ppp on C(GR,dR,ZR)\\mathcal{C}(G_R, \\mathbf{d}_R, \\mathbf{Z}_R)C(GR​,dR​,ZR​)", "location": "Web Appendix p. 1 (supplement PDF p. 1); Definitions 2, 3 and 4: p. 155 (PDF p. 3)" }, "id": "P012-T7", "text": { "body": "Web Appendix p. 1 (supplement PDF p. 1); Definitions 2–4: p. 155 (PDF p. 3). The observed recruitment history and spare degrees give the sharp attained preferential-recruitment interval. Obligatory child sets are nonempty; every compatible susceptible fraction has the stated decomposition and sharp bounds. Same-trait private degree-one neighbours attain p_min, and opposite-trait private degree-one neighbours attain p_max. The smallest containing interval is [p_min,p_max]. An exact rational forward algorithm computes both endpoints in one pass over the supplied chronological records, with positive event denominators, one read per record, at most n stored cells and at most 32n+4 unit arithmetic and indexed-cell operations, including the final divisions.", "headline": "Web Appendix 1, first sentence, corrected: closed forms for the extrema of $p$ on $\\mathcal{C}(G_R, \\mathbf{d}_R, \\mathbf{Z}_R)$", "location": "Web Appendix p. 1 (supplement PDF p. 1); Definitions 2, 3 and 4: p. 155 (PDF p. 3)" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T7" }, { "declarations": [ { "claim_revision_id": "catalog:2e74942add27ad4ade0da90c8e249d30ebec3cef702aa35d3a615bbfdba909f6", "conditional_on": [], "credit_reason": "This refutes the universally permitted proof construction. It does not refute the existential assertion of Result 1. Claim-stage verdict unchanged after reading the record.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P012.result1_proof_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/P012_Identification.lean#L809", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:2e74942add27ad4ade0da90c8e249d30ebec3cef702aa35d3a615bbfdba909f6", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P012-T8", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P012-T8" }, "html": { "body": "

The printed claim is false: P012-T8; Appendix, p. 160 (PDF p. 8); Result 1: p. 155 (PDF p. 3). An allowed starting graph in the printed Result 1 proof yields an incompatible output with the same induced sampled graph.

", "headline": "Proof of Result 1, the choice of GSU1G^1_{SU}GSU1​ as "any compatible subgraph" and the step to GSU2G^2_{SU}GSU2​, as printed", "location": "Appendix, p. 160 (PDF p. 8); Result 1: p. 155 (PDF p. 3)" }, "id": "P012-T8", "text": { "body": "The printed claim is false: P012-T8; Appendix, p. 160 (PDF p. 8); Result 1: p. 155 (PDF p. 3). An allowed starting graph in the printed Result 1 proof yields an incompatible output with the same induced sampled graph.", "headline": "Proof of Result 1, the choice of $G^1_{SU}$ as \"any compatible subgraph\" and the step to $G^2_{SU}$, as printed", "location": "Appendix, p. 160 (PDF p. 8); Result 1: p. 155 (PDF p. 3)" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T8" }, { "declarations": [ { "claim_revision_id": "catalog:41a4d8c28f4ed4b7f2d42bc71fbdee59da55cf928f259d7e68324029dff2475d", "conditional_on": [], "credit_reason": "This refutes the printed swap construction on an allowed input. It does not refute Result 2 itself. Claim-stage verdict unchanged after reading the record.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P012.result2_proof_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/P012_Identification.lean#L838", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:41a4d8c28f4ed4b7f2d42bc71fbdee59da55cf928f259d7e68324029dff2475d", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P012-T9", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P012-T9" }, "html": { "body": "

The printed claim is false: P012-T9; Appendix, p. 160 (PDF p. 8); Result 2: p. 155 (PDF p. 3). The specified compatible starting pair has both cross edges already present. The printed swap leaves degrees (2,2,2,1,1) instead of (3,2,3,2,2), and four edges instead of six, so its output is incompatible. Its equation-(1) denominator is 15/8, whereas the starting denominator is 35/12; both are positive and unequal. This refutes the printed proof step.

", "headline": "Proof of Result 2, the choice of the starting pair as "any compatible subgraph and trait set" and the sentence "Clearly (GSU2,ZSU2)∈C(GR,dR,ZR)(G^2_{SU}, \\mathbf{Z}^2_{SU}) \\in \\mathrm{C}(G_R, \\mathbf{d}_R, \\mathbf{Z}_R)(GSU2​,ZSU2​)∈C(GR​,dR​,ZR​)", as printed", "location": "Appendix, p. 160 (PDF p. 8); Result 2: p. 155 (PDF p. 3)" }, "id": "P012-T9", "text": { "body": "The printed claim is false: P012-T9; Appendix, p. 160 (PDF p. 8); Result 2: p. 155 (PDF p. 3). The specified compatible starting pair has both cross edges already present. The printed swap leaves degrees (2,2,2,1,1) instead of (3,2,3,2,2), and four edges instead of six, so its output is incompatible. Its equation-(1) denominator is 15/8, whereas the starting denominator is 35/12; both are positive and unequal. This refutes the printed proof step.", "headline": "Proof of Result 2, the choice of the starting pair as \"any compatible subgraph and trait set\" and the sentence \"Clearly $(G^2_{SU}, \\mathbf{Z}^2_{SU}) \\in \\mathrm{C}(G_R, \\mathbf{d}_R, \\mathbf{Z}_R)$\", as printed", "location": "Appendix, p. 160 (PDF p. 8); Result 2: p. 155 (PDF p. 3)" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T9" }, { "declarations": [ { "claim_revision_id": "catalog:964d22cf7e1b0b7d43597c228b5d58fe93b180ba6bf24470fbc7bc11d23d01a7", "conditional_on": [], "credit_reason": "The covered component is Result 1 and its repaired shared-neighbour existence argument. The signature does not assert the full recipe's privacy of every other unsampled vertex or that the common neighbour has degree two; those details are unnecessary for the stated non-identification witnesses and are not claimed by row_says. Corrected_statement status is consistent with an unchanged Result 1 and a repaired proof.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P012.result1", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/P012_Identification.lean#L3829", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:964d22cf7e1b0b7d43597c228b5d58fe93b180ba6bf24470fbc7bc11d23d01a7", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P012-T10", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P012-T10" }, "html": { "body": "

Result 1: p. 155 (PDF p. 3); corrected proof: Appendix, p. 160 (PDF p. 8). Two time-ordered sampled vertices not joined by a recruitment edge, each with positive spare degree, give two distinct compatible augmented graphs and two distinct compatible sampled induced graphs. The first graph has exactly one shared unsampled neighbour of the two chosen vertices, s_i-1 and s_j-1 further private leaves there, and s_v private leaves at every other sampled vertex. The shared vertex has exactly the two chosen neighbours; every private leaf has degree one. For every binary assignment extending the observed traits both graphs are compatible. The second deletes precisely the shared vertex and its two edges, adds {i,j}, and keeps all remaining traits. The induced edge sets are B and B union {{i,j}}. The graphs remain distinct under every renaming of unsampled identities, so neither graph is identified.

", "headline": "Result 1, with its proof corrected", "location": "Result 1: p. 155 (PDF p. 3); proof: Appendix, p. 160 (PDF p. 8)" }, "id": "P012-T10", "text": { "body": "Result 1: p. 155 (PDF p. 3); corrected proof: Appendix, p. 160 (PDF p. 8). Two time-ordered sampled vertices not joined by a recruitment edge, each with positive spare degree, give two distinct compatible augmented graphs and two distinct compatible sampled induced graphs. The first graph has exactly one shared unsampled neighbour of the two chosen vertices, s_i-1 and s_j-1 further private leaves there, and s_v private leaves at every other sampled vertex. The shared vertex has exactly the two chosen neighbours; every private leaf has degree one. For every binary assignment extending the observed traits both graphs are compatible. The second deletes precisely the shared vertex and its two edges, adds {i,j}, and keeps all remaining traits. The induced edge sets are B and B union {{i,j}}. The graphs remain distinct under every renaming of unsampled identities, so neither graph is identified.", "headline": "Result 1, with its proof corrected", "location": "Result 1: p. 155 (PDF p. 3); proof: Appendix, p. 160 (PDF p. 8)" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T10" }, { "declarations": [ { "claim_revision_id": "catalog:07ac86b80054bd1c77db4d4de942644ebecb4b79bc1f30dc8bc036481b2c677c", "conditional_on": [], "credit_reason": "The starting graph is existentially constructed; restricting that chosen witness to a permissible trait assignment does not restrict the universal observed-data claim. 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Result 2: p. 155 (PDF p. 3); corrected proof and (A1): Appendix, p. 160 (PDF p. 8). Opposite-trait sampled vertices with spare degree make homophily not point identified. Attach exactly s_v private degree-one neighbours to each sampled vertex. Any selected private neighbours of the normalized trait-0 and trait-1 vertices can be assigned traits 0 and 1, with all other unsampled traits arbitrary. The cross edges are absent and the swap is compatible and preserves vertices, traits, all degrees and edge count. With Q the trait-weighted degree sum, both denominators equal Q-Q^2/(2m)=Q(2m-Q)/(2m)>0, because Q≥d_j>0 and 2m-Q≥d_i>0. Both forms of (A1) give h_1-h_2=2/D≠0. The stated exact two-subject instance has m=3, Q=3, D=3/2, h_1=1/3, h_2=-1 and difference 4/3.

", "headline": "Result 2 and display (A1), with the proof corrected", "location": "Result 2: p. 155 (PDF p. 3); proof and (A1): Appendix, p. 160 (PDF p. 8)" }, "id": "P012-T11", "text": { "body": "Result 2: p. 155 (PDF p. 3); corrected proof and (A1): Appendix, p. 160 (PDF p. 8). Opposite-trait sampled vertices with spare degree make homophily not point identified. Attach exactly s_v private degree-one neighbours to each sampled vertex. Any selected private neighbours of the normalized trait-0 and trait-1 vertices can be assigned traits 0 and 1, with all other unsampled traits arbitrary. The cross edges are absent and the swap is compatible and preserves vertices, traits, all degrees and edge count. With Q the trait-weighted degree sum, both denominators equal Q-Q^2/(2m)=Q(2m-Q)/(2m)>0, because Q≥d_j>0 and 2m-Q≥d_i>0. Both forms of (A1) give h_1-h_2=2/D≠0. The stated exact two-subject instance has m=3, Q=3, D=3/2, h_1=1/3, h_2=-1 and difference 4/3.", "headline": "Result 2 and display (A1), with the proof corrected", "location": "Result 2: p. 155 (PDF p. 3); proof and (A1): Appendix, p. 160 (PDF p. 8)" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T11" }, { "declarations": [ { "claim_revision_id": "catalog:9851474ccc0b5d9b159a8e87438323a653f0937ef61913547e4f3013962f7afa", "conditional_on": [], "credit_reason": "A witness refuting display A2 for the entire functional, not a refutation of the positive non-identification conclusion of Result 3. Although the total trait functions differ at unused names, on the displayed vertex set only the private unsampled neighbour changes trait.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P012.a2_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/P012_Identification.lean#L865", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:9851474ccc0b5d9b159a8e87438323a653f0937ef61913547e4f3013962f7afa", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P012-T12", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P012-T12" }, "html": { "body": "

The printed claim is false: P012-T12; Appendix, p. 160 (PDF p. 8); Result 3: p. 155 (PDF p. 3). The two-recruit witness has preferential-recruitment difference minus five twelfths, disagreeing with both printed (A2) right-hand sides.

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Result 3, p. 155; corrected (A2), Appendix p. 160. A recruiter with spare degree admits two compatible completions with distinct preferential recruitment and the corrected signed full sum, sharp bound and sole-child equality criterion, which also hold at any compatible private-leaf start.

", "headline": "Result 3, with display (A2) corrected", "location": "Result 3: p. 155 (PDF p. 3); proof and (A2): Appendix, p. 160 (PDF p. 8)" }, "id": "P012-T13", "text": { "body": "Result 3, p. 155; corrected (A2), Appendix p. 160. A recruiter with spare degree admits two compatible completions with distinct preferential recruitment and the corrected signed full sum, sharp bound and sole-child equality criterion, which also hold at any compatible private-leaf start.", "headline": "Result 3, with display (A2) corrected", "location": "Result 3: p. 155 (PDF p. 3); proof and (A2): Appendix, p. 160 (PDF p. 8)" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T13" }, { "declarations": [ { "claim_revision_id": "catalog:9fc862da16fbc8b4e501d79ef52179f5ba41ce80bf365effd821e423d3180610", "conditional_on": [], "credit_reason": "These are witnesses to failure of the printed unqualified domain, not a claim that the displayed partial functions are themselves inconsistent. Uniqueness of A/C, graph-shape uniqueness of D and all permutations of D's selected numerical values are not exported, but the row claims the counterexample witnesses and does not need those stronger narrative facts.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P012.domain_counterexamples", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/P012_Identification.lean#L912", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:9fc862da16fbc8b4e501d79ef52179f5ba41ce80bf365effd821e423d3180610", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P012-T14", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P012-T14" }, "html": { "body": "

The printed claim is false: P012-T14; Definitions 1, 2 and 4, p. 155 (PDF p. 3); Web Appendix 1, p. 1. The four admissible examples A--D exhibit undefined printed quotients. A and C have unique displayed compatible pairs; B has raw numerator -1/2 and denominator 1/2 and an empty recruitment average at every pair. Every D graph is a dyad with three private leaves at vertex 2; all eight trait assignments are compatible, and their trait sum gives h=-3/5,-1/3,-1/7 or an undefined 0/0, while p=0. A and D have no whole-compatible-set h interval.

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Definitions 1, 2 and 4, p. 155 (PDF p. 3); Web Appendix 1, p. 1. The denominator identities give the exact individual domains of h and p. On admissible data the whole-compatible-set domains are exactly positive sampled degree in each trait class for h and at least one nonseed for p. Under each condition separately, unique attained endpoints give the smallest containing interval. If the h condition fails no whole-set h region exists; the h-defined-only set can be empty (example A), while p still has region {0}. The hypotheses of Results 2 and 3 imply the respective domain conditions.

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Web Appendix pp. 1-2, cooling schedule and display (2). The corrected logarithmic schedule is positive, nonincreasing, strictly decreasing on its tail and tends to zero, while its depth series diverges for 0 ≤ D*ε ≤ 1; the two printed schedule defects are also certified.

", "headline": "Web Appendix 1, the cooling schedule and display (2) of the proof of Proposition 1, corrected", "location": "Web Appendix p. 1 (the sentence after the display of KKK) and p. 2 (Proposition 1 and display (2)); supplement PDF pp. 1–2" }, "id": "P012-T16", "text": { "body": "Web Appendix pp. 1-2, cooling schedule and display (2). The corrected logarithmic schedule is positive, nonincreasing, strictly decreasing on its tail and tends to zero, while its depth series diverges for 0 ≤ D*ε ≤ 1; the two printed schedule defects are also certified.", "headline": "Web Appendix 1, the cooling schedule and display (2) of the proof of Proposition 1, corrected", "location": "Web Appendix p. 1 (the sentence after the display of $K$) and p. 2 (Proposition 1 and display (2)); supplement PDF pp. 1–2" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T16" }, { "declarations": [ { "claim_revision_id": "catalog:f56dc7d58b05c7f72dc9372356eef27d10d6e2765fbe80b6fd9c26a86895df54", "conditional_on": [], "credit_reason": "GraphCompatible omits trait preservation exactly as the opening of the printed proof does; InducedCompatible requires degree upper bounds, not full Definition 3 compatibility.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P012.induced_compatibility", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/P012_Identification.lean#L1063", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:f56dc7d58b05c7f72dc9372356eef27d10d6e2765fbe80b6fd9c26a86895df54", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P012-T17", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P012-T17" }, "html": { "body": "

Appendix p. 160 (PDF p. 8), proof of Result 1; Definition 3, p. 155. Deleting unsampled vertices preserves compatibility with upper degree bounds. Exact degrees survive precisely when there is no sampled-to-unsampled edge, equivalently no unsampled vertex, equivalently the graph equals its induced graph. The two seeds with one common unsampled neighbour have degrees (1,1) but induced degrees (0,0); the edge-between-seeds comparison has both (1,1).

", "headline": "Proof of Result 1, the compatibility of GS1G^1_SGS1​ and GS2G^2_SGS2​, corrected", "location": "Appendix, p. 160 (PDF p. 8), last sentences of the proof of Result 1; Definition 3: p. 155 (PDF p. 3)" }, "id": "P012-T17", "text": { "body": "Appendix p. 160 (PDF p. 8), proof of Result 1; Definition 3, p. 155. Deleting unsampled vertices preserves compatibility with upper degree bounds. Exact degrees survive precisely when there is no sampled-to-unsampled edge, equivalently no unsampled vertex, equivalently the graph equals its induced graph. The two seeds with one common unsampled neighbour have degrees (1,1) but induced degrees (0,0); the edge-between-seeds comparison has both (1,1).", "headline": "Proof of Result 1, the compatibility of $G^1_S$ and $G^2_S$, corrected", "location": "Appendix, p. 160 (PDF p. 8), last sentences of the proof of Result 1; Definition 3: p. 155 (PDF p. 3)" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T17" }, { "declarations": [ { "claim_revision_id": "catalog:e6675fc3201db395f729ae8bfe60f21cd612f4100fe22856c14adc7ff27f2f67", "conditional_on": [], "credit_reason": "The row covers h-invariance. The target also contrasts this with p’s dependence on recruitment data, which this equality does not establish; see targets.json.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P012.homophily_arguments", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/P012_Identification.lean#L1121", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:e6675fc3201db395f729ae8bfe60f21cd612f4100fe22856c14adc7ff27f2f67", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P012-T18", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P012-T18" }, "html": { "body": "

Appendix p. 160 (PDF p. 8), proof of Result 2; p. 155 (PDF p. 3), after equation (2). Homophily depends only on the displayed augmented graph and its traits, including agreement of undefined partial values. Preferential recruitment can change with the recruiter map alone or with entry order alone: on the fixed four-vertex example the star, chain and reordered star histories give the defined values -1/12, 1/6 and -1/3 respectively.

", "headline": "The arguments of hhh in the proof of Result 2 and in the sentence after equation (2), corrected", "location": "Appendix, p. 160 (PDF p. 8), proof of Result 2 (four occurrences); p. 155 (PDF p. 3), sentence after equation (2)" }, "id": "P012-T18", "text": { "body": "Appendix p. 160 (PDF p. 8), proof of Result 2; p. 155 (PDF p. 3), after equation (2). Homophily depends only on the displayed augmented graph and its traits, including agreement of undefined partial values. Preferential recruitment can change with the recruiter map alone or with entry order alone: on the fixed four-vertex example the star, chain and reordered star histories give the defined values -1/12, 1/6 and -1/3 respectively.", "headline": "The arguments of $h$ in the proof of Result 2 and in the sentence after equation (2), corrected", "location": "Appendix, p. 160 (PDF p. 8), proof of Result 2 (four occurrences); p. 155 (PDF p. 3), sentence after equation (2)" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T18" }, { "declarations": [ { "claim_revision_id": "catalog:aa9ff214c364c5a5f89e11d2d3c1da3cfba7ce167697afafb98ba549bdbe3e98", "conditional_on": [], "credit_reason": "The positive printed target and counterexample_proved/exact_closed_contract_refutation metadata consistently identify a refutation. The execution relation contains no compatibility filter and matches the specified indentation and original-state replacement pools; connectivity and mixing are outside the target’s explicit formal preservation assertion.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P012.printed_proposal_preserves_refuted", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/P012_PrintedRefutations.lean#L7", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:aa9ff214c364c5a5f89e11d2d3c1da3cfba7ce167697afafb98ba549bdbe3e98", "conditional_on": [], "credit_reason": "Supporting declaration is already used inside the selected aggregate rows and is not separately invoked.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P012.pair_edge_eq", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/P012_Identification.lean#L36", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:aa9ff214c364c5a5f89e11d2d3c1da3cfba7ce167697afafb98ba549bdbe3e98", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P012-T19", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P012-T19" }, "html": { "body": "

The printed claim is false: P012-T19; Web Appendix 2, pp. 2-3; kernel on Web Appendix p. 1. The literal printed proposal mechanisms do not preserve compatibility for every execution from admissible compatible data.

", "headline": "Web Appendix 2, the two proposal mechanisms, as printed", "location": "Web Appendix pp. 2–3 (supplement PDF pp. 2–3); the kernel KKK: Web Appendix p. 1 (supplement PDF p. 1)" }, "id": "P012-T19", "text": { "body": "The printed claim is false: P012-T19; Web Appendix 2, pp. 2-3; kernel on Web Appendix p. 1. The literal printed proposal mechanisms do not preserve compatibility for every execution from admissible compatible data.", "headline": "Web Appendix 2, the two proposal mechanisms, as printed", "location": "Web Appendix pp. 2–3 (supplement PDF pp. 2–3); the kernel $K$: Web Appendix p. 1 (supplement PDF p. 1)" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T19" }, { "declarations": [ { "claim_revision_id": "catalog:1ff6d17a1be0520510bfd87deafb902e9cf9a7fcf1a52d164218637b19869787", "conditional_on": [], "credit_reason": "Exact full refutation of the joint homophily interval assertion, not a claim to determine the actual upper bounds, preferential intervals, point estimates, or corrected empirical endpoints. In particular -1462/1983 < -0.7027-1/20000, so any asserted least sex endpoint within the printed rounding band contradicts a compatible defined value. No target-kind/status inconsistency is present. Claim-stage verdict unchanged after reading the record.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P012.Wave1.printed_table_intervals_refuted", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P008/Wave1/Gap/WretiredunstatedB1.lean#L82", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:1ff6d17a1be0520510bfd87deafb902e9cf9a7fcf1a52d164218637b19869787", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P012-T20", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P012-T20" }, "html": { "body": "

The printed claim is false: P012-T20; Table 1, p. 158; study totals, pp. 156-157. No admissible shared study inputs with the reported totals can have all three printed homophily intervals, even allowing their four-decimal rounding tolerance.

", "headline": "Table 1, the column IhI_hIh​, as printed, with the text that presents it", "location": "Table 1: p. 158 (PDF p. 6); text: pp. 156–157 (PDF pp. 4–5)" }, "id": "P012-T20", "text": { "body": "The printed claim is false: P012-T20; Table 1, p. 158; study totals, pp. 156-157. No admissible shared study inputs with the reported totals can have all three printed homophily intervals, even allowing their four-decimal rounding tolerance.", "headline": "Table 1, the column $I_h$, as printed, with the text that presents it", "location": "Table 1: p. 158 (PDF p. 6); text: pp. 156–157 (PDF pp. 4–5)" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T20" }, { "declarations": [ { "claim_revision_id": "catalog:68764cfba3c9e8f33d000c09db0ad782c0716ae0804686c8b07c670e18024d47", "conditional_on": [], "credit_reason": "Historical excluded association.", "credit_status": null, "fidelity_status_baseline": null, "lean_name": null, "manifest_use": "not_applicable", "named_axioms": [], "says": null, "source": null, "status": "not_applicable", "support_scope": "historical declaration", "used_in_solution": null } ], "formalization": { "claim_revision_id": "catalog:68764cfba3c9e8f33d000c09db0ad782c0716ae0804686c8b07c670e18024d47", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P012-T21", "decision": "omitted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P012-T21" }, "html": { "body": "

No current theorem is supplied for this target.

", "headline": "Web Appendix 1, first sentence, as printed", "location": "Web Appendix p. 1 (supplement PDF p. 1)" }, "id": "P012-T21", "text": { "body": "No current theorem is supplied for this target.", "headline": "Web Appendix 1, first sentence, as printed", "location": "Web Appendix p. 1 (supplement PDF p. 1)" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T21" }, { "declarations": [ { "claim_revision_id": "catalog:297e3952443e26fc48157d98a6b98898a0e24164eacc8ac4fb4518f14699decf", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P012_Identification.lean:3426 states the joint point-identification equivalence, equivalence with every compatible graph being the recruitment skeleton, non-identification of h when any sampled degree is spare, and uniqueness of graph and active traits when none is spare. 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abstract p. 153, introduction p. 154, and text after Result 3, pp. 155--156 (PDF pp. 1--4). On the stated whole-set domain, joint point identification is equivalent to zero spare degree and to every compatible graph being the undirected recruitment graph with its observed traits. A same-trait private-leaf flip witnesses h variation, with the exact positive edge-count difference. Preferential recruitment alone is identified exactly when actual recruiters have no spare degree. In the (1,2)-degree, (0,1)-trait example every graph has one private leaf at vertex 2: p=0 and h is -1 or -1/3 according to its trait. Thus the exact value sets are {0} and {-1,-1/3}, and the smallest containing intervals are {0} and [-1,-1/3], with attained endpoints.

", "headline": "The statement that hhh and ppp are not point identified unless the recruitment graph is identical to the underlying subgraph (abstract; introduction; text after Result 3)", "location": "Abstract: p. 153 (PDF p. 1); introduction: p. 154 (PDF p. 2); text after Result 3: pp. 155–156 (PDF pp. 3–4)" }, "id": "P012-T22", "text": { "body": "abstract p. 153, introduction p. 154, and text after Result 3, pp. 155--156 (PDF pp. 1--4). On the stated whole-set domain, joint point identification is equivalent to zero spare degree and to every compatible graph being the undirected recruitment graph with its observed traits. A same-trait private-leaf flip witnesses h variation, with the exact positive edge-count difference. Preferential recruitment alone is identified exactly when actual recruiters have no spare degree. In the (1,2)-degree, (0,1)-trait example every graph has one private leaf at vertex 2: p=0 and h is -1 or -1/3 according to its trait. Thus the exact value sets are {0} and {-1,-1/3}, and the smallest containing intervals are {0} and [-1,-1/3], with attained endpoints.", "headline": "The statement that $h$ and $p$ are not point identified unless the recruitment graph is identical to the underlying subgraph (abstract; introduction; text after Result 3)", "location": "Abstract: p. 153 (PDF p. 1); introduction: p. 154 (PDF p. 2); text after Result 3: pp. 155–156 (PDF pp. 3–4)" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T22" }, { "declarations": [ { "claim_revision_id": "catalog:fc10133b9b062995c18f30e0116e19d4e004680eaa2e1d6f35e857b3c16c35cb", "conditional_on": [], "credit_reason": "Expanded FullHomophilyRegion and FullPreferentialRegion assert each marginal extremum is attained and bound every compatible pair, without assuming the result or asserting joint interior attainability. Claim-stage verdict unchanged after reading the record.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P012.figure1_extrema", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/P012_Identification.lean#L3169", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:fc10133b9b062995c18f30e0116e19d4e004680eaa2e1d6f35e857b3c16c35cb", "conditional_on": [], "credit_reason": "An accurately labelled arithmetic auxiliary fact; no full target credit is assigned to it alone. Claim-stage verdict unchanged after reading the record.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P012.figure1_endpoint_arithmetic", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/P012_Identification.lean#L1317", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:fc10133b9b062995c18f30e0116e19d4e004680eaa2e1d6f35e857b3c16c35cb", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P012-T23", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P012-T23" }, "html": { "body": "

Figure 1 and caption, p. 156 (PDF p. 4). For the five sampled vertices with degrees (5,4,3,3,2), recruitments 1 to 2, 2 to 3, 1 to 4, 2 to 5, and traits (1,1,1,0,1), the attained marginal extrema are -5/8, 79/105, -11/80 and 49/120. Every compatible pair lies in their rectangle inside [-1,1]^2. Some extremal augmented subgraphs are not unique, even after renaming unsampled vertices. The stated recruitment degrees, spare degrees, edge and stub counts, phi evaluations and individual preferential summands hold. The listed trait-specific edge and stub counts use the displayed coding (1,1,1,0,1). All arithmetic and statistical conclusions hold for every strictly increasing recruitment time order. The extrema, bounds and nonuniqueness conclusions hold for either coding of the binary traits; exchanging both labels preserves compatibility and both defined statistics.

", "headline": "Figure 1: the extrema of hhh and ppp for the five-vertex example", "location": "Figure 1 and its caption: p. 156 (PDF p. 4)" }, "id": "P012-T23", "text": { "body": "Figure 1 and caption, p. 156 (PDF p. 4). For the five sampled vertices with degrees (5,4,3,3,2), recruitments 1 to 2, 2 to 3, 1 to 4, 2 to 5, and traits (1,1,1,0,1), the attained marginal extrema are -5/8, 79/105, -11/80 and 49/120. Every compatible pair lies in their rectangle inside [-1,1]^2. Some extremal augmented subgraphs are not unique, even after renaming unsampled vertices. The stated recruitment degrees, spare degrees, edge and stub counts, phi evaluations and individual preferential summands hold. The listed trait-specific edge and stub counts use the displayed coding (1,1,1,0,1). All arithmetic and statistical conclusions hold for every strictly increasing recruitment time order. The extrema, bounds and nonuniqueness conclusions hold for either coding of the binary traits; exchanging both labels preserves compatibility and both defined statistics.", "headline": "Figure 1: the extrema of $h$ and $p$ for the five-vertex example", "location": "Figure 1 and its caption: p. 156 (PDF p. 4)" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T23" }, { "declarations": [ { "claim_revision_id": "catalog:54a8b97e1d5b7c7bcda90a70a31f9dc693a63f7e38ba706bb61463d1a81b5d7a", "conditional_on": [], "credit_reason": "Excluded: proves convergence for a materially repaired proposal process, not the canonical printed process.", "credit_status": "wrong_credit", "fidelity_status_baseline": "wrong_credit", "lean_name": "AronowAudit.P012.Wave1.annealing_convergence_proof", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/Wave1/Gap/GLP013P012P012B1c.lean#L600", "status": "proved", "support_scope": "withdrawn", "used_in_solution": false }, { "claim_revision_id": "catalog:54a8b97e1d5b7c7bcda90a70a31f9dc693a63f7e38ba706bb61463d1a81b5d7a", "conditional_on": [], "credit_reason": "Purpose: the scalar bound and attained maximum in display (1), not the asymptotic conclusion. 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The all-vertex positive-degree conditions are equivalent to the correlation’s ordinary domain in a finite simple graph and exclude no defined correlation case.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P012.homophily_global_range", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/P012_Identification.lean#L4475", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:54a8b97e1d5b7c7bcda90a70a31f9dc693a63f7e38ba706bb61463d1a81b5d7a", "conditional_on": [], "credit_reason": "Not needed: the two ok rows give the graph estimate using all-vertex trait-degree positivity, without its stronger sampled-trait positivity premises.", "credit_status": "narrower", "fidelity_status_baseline": "narrower", "lean_name": "AronowAudit.P012.compatible_objective_bounds", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P012/P012_Identification.lean#L4498", "status": "proved", "support_scope": "limited", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:54a8b97e1d5b7c7bcda90a70a31f9dc693a63f7e38ba706bb61463d1a81b5d7a", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P012-T24", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P012-T24" }, "html": { "body": "

Web Appendix 1, p. 2, display (1); set-up on p. 1. The objective is positive and has attained scalar maximum 1/epsilon on the square, and its value at any ordinary binary graph correlation is positive and bounded by 1/epsilon.

", "headline": "Web Appendix 1, Proposition 1", "location": "Web Appendix p. 2 (supplement PDF p. 2); set-up on Web Appendix p. 1" }, "id": "P012-T24", "text": { "body": "Web Appendix 1, p. 2, display (1); set-up on p. 1. The objective is positive and has attained scalar maximum 1/epsilon on the square, and its value at any ordinary binary graph correlation is positive and bounded by 1/epsilon.", "headline": "Web Appendix 1, Proposition 1", "location": "Web Appendix p. 2 (supplement PDF p. 2); set-up on Web Appendix p. 1" }, "url": "https://pmaronow.github.io/llm-audit/works/P012/#P012-T24" } ] }, "missing_sources": [], "reproduction": { "agreement": "agrees_in_part", "status": "selected_recalculation", "targets": [ { "agreement": "to_rounding", "location": "p. 156 (PDF p. 4), panels A-F and caption", "method": "own recalculation from printed diagrams: exact equations and exhaustive compatible-pair enumeration; four extrema agree to rounding", "status": "recalculated", "target": "Figure 1" }, { "agreement": "not_compared", "location": "p. 158 (PDF p. 6); interpretation on p. 157 (PDF p. 5) and discussion p. 158", "method": "inspection of printed point values and interval arithmetic only; Hartford values and endpoints not recomputed because data and code are unavailable; P012-N002 gives a conditional lower-limit inconsistency", "status": "blocked", "target": "Table 1" }, { "agreement": "not_compared", "location": "Figure 2 p. 157 (PDF p. 5); in-text counts p. 156 (PDF p. 4)", "method": "printed count arithmetic only; network layout and underlying counts not reproduced because Hartford data and layout code are unavailable", "status": "blocked", "target": "Figure 2 and Hartford network counts" } ] }, "summary": "The paper defines homophily $h$ and preferential recruitment $p$ in respondent-driven sampling, shows why both generally fail to be point identified from the data such a study observes, and bounds both over all compatible graphs, with an application to injection drug users in Hartford. 11 entries are classified as errors (3 Moderate, 8 Minor), and 1 is a clarity note; 3 other concerns were not confirmed. Results 1--3 hold once their proofs are repaired. On the functionals' defined domain, the closed forms in \\fid{P012-N007} replace the search and convergence argument for the four marginal endpoints, supply extremizing graphs and shorten the proofs of Results 2--3; the identification problem remains. The three lower homophily limits in Table 1 are too high by at least $0.019$ if the bounds use the nomination-network degrees; otherwise the comparison concerns different networks.", "what_was_checked": "I checked the following, with the qualifications below.\n\\begin{itemize}\n\\item Equation (1) is the correlation of the trait across the two ends of an edge; for a binary trait it reduces to $(4e_{11}e_{00}-c^2)/((2e_{11}+c)(2e_{00}+c))$ in the numbers of edges of the three kinds, and it lies in $[-1,1]$ where it is defined. Each summand of equation (2) lies in $(-1,1)$ and has conditional mean zero when recruitment is uniform among susceptible neighbors.\n\\item Results 1, 2 and 3, using the corrected proofs and the defined domains in \\fid{P012-E006}, by explicit construction and by exhaustive checks of small exact cases (104, 92 and 184 cases that meet the respective hypotheses); display (A1) for a swap that preserves the degrees.\n\\item Figure 1: the degrees $(5,4,3,3,2)$ implied by panel A, and the four extrema $-5/8$, $79/105$, $-11/80$ and $49/120$, which round to the printed values and are the extrema over all $26{,}182$ compatible pairs, counted up to the labeling of the unsampled vertices.\n\\item The arithmetic of the reported application counts: $2{,}626-527=2{,}099$, $3{,}307-1{,}180=2{,}127$ and $527-6=521$.\n\\item Table 1 against the text: the signs of the point values, each point value inside its printed interval, every printed interval covering zero.\n\\item The objective bound and harmonic series comparison in the proof of Proposition 1, after correcting the cooling direction and initial index, and the cited theorem of Hajek (1988) against the published article. Applying that theorem requires a fixed proposal on a finite state space, irreducibility and weak reversibility.\n\\item The proposals of Web Appendix 2 with the repairs in \\fid{P012-E001}: every proposal stays in the compatible set.\n\\item The remark that further correct information can only narrow the identification intervals, since it shrinks the compatible set.\n\\item The page numbers given with the quotations from references 7 and 22--26, which fall inside the page ranges of the reference list.\n\\end{itemize}\nI could not recompute the Hartford point values or bounds, or reconstruct Figure 2, because no public copy of the Hartford RDS-net data or code for the bounds was found; the full-network values, the upper limits of $I_h$ and the intervals $I_p$ are neither confirmed nor shown wrong. The lower-limit inconsistency in \\fid{P012-N002} follows from the printed network totals if the bounds use the nomination-network degrees; otherwise the comparison concerns different networks. It does not give the actual corrected endpoints or establish a failure of the unavailable implementation. The quotations from the empirical papers, and the cited sources other than those named in the corrections, were not compared with the originals. The RDSAT manual could not be obtained, and reference 6 of the Web Appendix was read in its preprint only. No search was made for corrections or supplements beyond the article and its official Web Appendix. That a passage has no correction here does not mean that it was found correct.\n\nWith a nonempty compatible set, positive total reported degree in each sampled trait group and at least one recruitment, I checked the closed forms in \\fid{P012-N007} for all four sharp marginal endpoints and the graphs with private unsampled neighbors that attain them. These replace the appendix search and convergence argument for these endpoints and shorten the proofs of Results 2--3; the identification problem from incomplete networks remains. I did not address arbitrary objectives or additional graph constraints, or independently reproduce the Hartford bounds.", "work": { "arxiv_id": "1511.05397", "audit_complete": true, "authors": [ "Forrest W. Crawford", "P. M. Aronow", "Li Zeng", "Jianghong Li" ], "category": "Articles - Theory and Methodology", "citation": "Forrest W. Crawford, P. M. Aronow, Li Zeng, and Jianghong Li (2018). Identification of homophily and preferential recruitment in respondent-driven sampling. American Journal of Epidemiology 187(1):153–160.", "doi": "10.1093/aje/kwx208", "edition": { "pages_in_audited_file": 8, "sha256": "dcff3579cb1914c3cc7bb6bbf7220fe05aa5c8afbe74eff8bfe0cafe3a7283dc", "source_url": "https://doi.org/10.1093/aje/kwx208", "venue": "American Journal of Epidemiology", "year": "2018" }, "id": "P012", "later_materials": null, "title": "Identification of homophily and preferential recruitment in respondent-driven sampling", "type": "Journal article", "url": "https://pmaronow.github.io/llm-audit/works/P012/", "venue": "American Journal of Epidemiology", "year": "2018" } }