{ "claims": [ { "claim": "When every selection probability lies in $[\\alpha,\\beta]$, the attainable values of the Hajek estimate form exactly $[\\hat\\mu^{\\mathrm L},\\hat\\mu^{\\mathrm H}]$. The endpoints are attained by threshold weight vectors whose cut points $k^{\\mathrm L}$ and $k^{\\mathrm H}$ are given by explicit rules.", "claim_id": "P025-H1", "related_corrections": [ "P025-E001", "P025-E004", "P025-N001" ], "tied_to": "Summary; Section 3.1; Proposition 1; equations (2)-(5); Appendix" }, { "claim": "The bounds depend on $\\alpha$ and $\\beta$ only through $\\gamma=\\beta/\\alpha$. At $\\gamma=1$ they equal the equal-probability estimate, so varying $\\gamma$ gives a nonparametric sensitivity analysis for departures from equal-probability sampling.", "claim_id": "P025-H2", "related_corrections": [], "tied_to": "Section 1; Section 3.1, equation (5) and the paragraph after it; Table 1" }, { "claim": "The threshold weight vectors assign equal weights to tied outcomes. For a binary outcome with sample proportion $p$, the interval is $[p/(\\gamma(1-p)+p),\\,\\gamma p/((1-p)+\\gamma p)]$.", "claim_id": "P025-H3", "related_corrections": [], "tied_to": "Section 3.1, Lemma 1 and the unnumbered display after it; supplement p. 1; underlies Table 1" }, { "claim": "Under Brewer’s replicated-population scaling, the population mean is identified up to $[\\mu_-,\\mu_+]$, obtained by the same optimization using class totals $\\Pi_j$. These bounds are sharp in the limit and have the same threshold form.", "claim_id": "P025-H4", "related_corrections": [ "P025-E001", "P025-E002", "P025-N002" ], "tied_to": "Section 1 ('yields sharp bounds on the population mean'); Section 3.2; Assumption 1; Proposition 2; equations (6)-(7); supplement pp. 1-2" }, { "claim": "Under Assumptions 1 and 2, the estimated bounds converge almost surely to $\\mu_+$ and $\\mu_-$, giving an asymptotically sharp interval estimator for the population mean.", "claim_id": "P025-H5", "related_corrections": [ "P025-E004", "P025-E005", "P025-N007" ], "tied_to": "Section 3.2; Proposition 3; Assumptions 1 and 2; supplement p. 2, equations (B3)-(B4)" }, { "claim": "Table 1 gives bounds on HIV prevalence in the shooting gallery, street and needle exchange samples for $\\gamma$ between $1$ and $4$.", "claim_id": "P025-H6", "related_corrections": [ "P025-N003" ], "tied_to": "Section 4; Table 1; dichotomous closed form" }, { "claim": "The needle data indicate that the gallery and exchange prevalence intervals overlap only when $\\gamma>3$, and the street and exchange intervals when $\\gamma>1.25$. The paper infers that equal prevalence in gallery and exchange needles would require infected gallery needles to be more than three times as likely to be tested as uninfected ones, with the reverse selection pattern for exchange needles.", "claim_id": "P025-H7", "related_corrections": [ "P025-E003", "P025-A001", "P025-N006" ], "tied_to": "Summary; Section 4, pp. 238-239; Table 1" } ], "date_published": "2026-10-25", "findings": [ { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the second half of Proposition 1, from An optimal solution to (3) to the displayed formula for the lower bound. A red outline surrounds the greater-than-or-equal sign that stands before y sub k plus 1 in the third line.", "caption": "Proposition 1, p. 236 (PDF p. 2). The outline marks the inequality sign in the rule that defines $k^{\\mathrm L}$.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-E001.png" }, { "alt": "Crop of the line and k minus is the smallest nonnegative integer k satisfying, followed by a displayed ratio of sums. A red outline surrounds the greater-than-or-equal sign between the ratio and y sub (k plus 1).", "caption": "Proposition 2, p. 238 (PDF p. 4), directly below Table 1. The outline marks the inequality sign in the rule that defines $k_-$.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-E001-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P025-T1", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T1" }, { "id": "P025-T2", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T2" }, { "id": "P025-T3", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T3" }, { "id": "P025-T4", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T4" }, { "id": "P025-T7", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T7" }, { "id": "P025-T9", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T9" }, { "id": "P025-T19", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T19" } ], "headline": "In the rules that define the lower cut points $k^{\\mathrm L}$ and $k_-$, the inequality $\\geqslant$ should read $\\leqslant$.", "id": "P025-E001", "location": "Proposition 1, the rule that defines $k^{\\mathrm L}$, p. 236 (PDF p. 2); Proposition 2, the rule that defines $k_-$, p. 238 (PDF p. 4)", "pdf_page": 2, "printed_page": "236", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P025-T1", "P025-T19", "P025-T2", "P025-T3", "P025-T4", "P025-T7", "P025-T9" ], "text": { "as_printed": "\\emph{Proposition 1, p.~236.} An optimal solution to (3) is $w^{\\mathrm L} = (1/\\alpha, \\ldots, 1/\\alpha, 1/\\beta, \\ldots, 1/\\beta)$, where the first $k^{\\mathrm L}$ entries equal $1/\\alpha$ and the remaining $n - k^{\\mathrm L}$ entries equal $1/\\beta$, with $k^{\\mathrm L}$ being the smallest nonnegative integer $k$ that satisfies $\\{\\gamma \\sum_{i=1}^{k} y_i + \\sum_{i=k+1}^{n} y_i\\}/\\{\\gamma k + (n - k)\\} \\hl{\\geqslant} y_{k+1}$. The lower bound on $\\hat\\mu$ is therefore\n\\[\n\\hat\\mu^{\\mathrm L} = \\frac{\\gamma \\sum_{i=1}^{k^{\\mathrm L}} y_i + \\sum_{i=k^{\\mathrm L}+1}^{n} y_i}{\\gamma k^{\\mathrm L} + (n - k^{\\mathrm L})}.\n\\]\n\n\\emph{Proposition 2, p.~238.} and $k_-$ is the smallest nonnegative integer $k$ satisfying\n\\[\n\\frac{\\gamma \\sum_{j=1}^{k} \\Pi_j y_{(j)} + \\sum_{j=k+1}^{|Y|} \\Pi_j y_{(j)}}{\\gamma \\sum_{j=1}^{k} \\Pi_j + \\sum_{j=k+1}^{|Y|} \\Pi_j} \\hl{\\geqslant} y_{(k+1)}.\n\\]", "corrected": "\\emph{Proposition 1, p.~236.} Let $\\gamma=\\beta/\\alpha\\geqslant 1$ and $y_1\\leqslant\\cdots\\leqslant y_n$. An optimal solution to (3) is $w^{\\mathrm L} = (1/\\alpha, \\ldots, 1/\\alpha, 1/\\beta, \\ldots, 1/\\beta)$, where the first $k^{\\mathrm L}$ entries equal $1/\\alpha$ and the remaining $n - k^{\\mathrm L}$ entries equal $1/\\beta$, with $k^{\\mathrm L}$ being the smallest \\hl{integer $k$ with $0\\leqslant k0$ be as in Section 3.2. Then $k_-$ is the smallest \\hl{integer $k$ with $0\\leqslant k<|Y|$} satisfying\n\\[\n\\frac{\\gamma \\sum_{j=1}^{k} \\Pi_j y_{(j)} + \\sum_{j=k+1}^{|Y|} \\Pi_j y_{(j)}}{\\gamma \\sum_{j=1}^{k} \\Pi_j + \\sum_{j=k+1}^{|Y|} \\Pi_j} \\hl{\\leqslant} y_{(k+1)}.\n\\]\nIn each rule the last index, $k=n-1$ or $k=|Y|-1$, satisfies the inequality, and the cut point exists.", "evidence": "The evidence is an exact counterexample for each proposition and a derivation of the corrected rule.\n\n\\emph{Counterexample.} Let $y=(0,1)$, $\\alpha=1/2$ and $\\beta=1$, so that $\\gamma=2$. I write $(u,v)=(\\beta w_1,\\beta w_2)\\in[1,2]^2$ for the scaled weights. The ratio in (3) is $v/(u+v)$. The printed rule accepts $k=0$, since $1/2\\geqslant 0$, and returns $w^{\\mathrm L}=(1,1)$ and $\\hat\\mu^{\\mathrm L}=1/2$. For all $(u,v)\\in[1,2]^2$, I have\n\\[\n\\frac{v}{u+v}-\\frac13=\\frac{(2-u)+2(v-1)}{3(u+v)}\\geqslant 0,\n\\]\nwith equality at $(u,v)=(2,1)$. The minimum of (3) is $1/3$, and the printed lower bound exceeds it by $1/6$. For Proposition 2, I consider a population of two units with outcomes $(0,1)$ and selection probabilities $(1/2,1)$. Then $(\\Pi_1,\\Pi_2)=(1/2,1)$, the ratio in (7) is $v/(u/2+v)$, and\n\\[\n\\frac{v}{u/2+v}-\\frac12=\\frac{(2-u)+2(v-1)}{2(u+2v)}\\geqslant 0 .\n\\]\nThe minimum of (7) is $1/2$, which is the population mean, and the ratio at the printed $W_-=(1,1)$ is $2/3$. The table lists the values at the four vertices.\n\n\\begin{audittable}{lcc}{Value of the ratio at the vertices of the box of scaled weights $[1,2]^2$, for outcomes $(0,1)$, $\\alpha=1/2$ and $\\beta=1$.}\nScaled weights $(u,v)$ & Proposition 1, ratio in (3) & Proposition 2 with $\\Pi=(1/2,1)$, ratio in (7) \\\\\n$(1,1)$, printed rule, $k=0$ & $1/2$ & $2/3$ \\\\\n$(1,2)$ & $2/3$ & $4/5$ \\\\\n$(2,1)$, corrected rule, $k=1$ & $1/3$ & $1/2$ \\\\\n$(2,2)$ & $1/2$ & $2/3$ \\\\\n\\end{audittable}\n\n\\emph{Derivation of the corrected rule.} Both propositions are instances of one problem. Let $x_1\\leqslant\\cdots\\leqslant x_m$, let $a_j>0$, and let $\\gamma\\geqslant 1$. Proposition 1 has $m=n$, $x_j=y_j$ and $a_j=1$, and Proposition 2 has $m=|Y|$, $x_j=y_{(j)}$ and $a_j=\\Pi_j$. After the weights are multiplied by $\\beta$, the lower problem is to minimize\n\\[\nR(q)=\\frac{\\sum_{j=1}^{m}a_jq_jx_j}{\\sum_{j=1}^{m}a_jq_j}\\qquad\\text{over } 1\\leqslant q_j\\leqslant\\gamma .\n\\]\nFor $0\\leqslant k\\leqslant m$, I define\n\\[\nD_k=\\gamma\\sum_{j=1}^{k}a_j+\\sum_{j=k+1}^{m}a_j,\\qquad\nB_k=\\gamma\\sum_{j=1}^{k}a_jx_j+\\sum_{j=k+1}^{m}a_jx_j,\\qquad F_k=\\frac{B_k}{D_k},\n\\]\nso that $F_k$ is the left side of the inequality in each rule. By these definitions, I have\n\\[\nF_{k+1}-F_k=\\frac{(\\gamma-1)\\,a_{k+1}\\,(x_{k+1}-F_k)}{D_{k+1}}\\qquad\\text{for } 0\\leqslant k1$, the value $F_k$ decreases at the step $k$ exactly when $F_k>x_{k+1}$. Let $k_*$ be the smallest $k\\in\\{0,\\ldots,m-1\\}$ such that $F_k\\leqslant x_{k+1}$. Such a $k$ exists, since $F_{m-1}$ is a weighted average of outcomes that are at most $x_m$. I set $\\lambda=F_{k_*}$, and $q^*_j=\\gamma$ for $j\\leqslant k_*$ and $q^*_j=1$ for $j>k_*$. If $k_*>0$, then $F_{k_*-1}>x_{k_*}$ because $k_*$ is the smallest such index, and\n\\[\nF_{k_*}-x_{k_*}=\\frac{D_{k_*-1}\\,(F_{k_*-1}-x_{k_*})}{D_{k_*}}>0 .\n\\]\nIt follows that $x_j<\\lambda$ for all $j\\leqslant k_*$, and $x_j\\geqslant x_{k_*+1}\\geqslant\\lambda$ for all $j>k_*$. Let $q$ be any feasible vector. For $j\\leqslant k_*$, I have $q_j-q^*_j\\leqslant 0$ and $x_j-\\lambda<0$, and for $j>k_*$, I have $q_j-q^*_j\\geqslant 0$ and $x_j-\\lambda\\geqslant 0$. Since $\\sum_ja_jq^*_j(x_j-\\lambda)=0$, I obtain\n\\[\nR(q)-\\lambda=\\frac{\\sum_{j}a_j(q_j-q^*_j)(x_j-\\lambda)}{\\sum_{j}a_jq_j}\\geqslant 0 .\n\\]\nThe vector $q^*$ minimizes $R$ over the whole box, including when outcomes are tied and when $\\gamma=1$.", "problem": "The outcomes are in increasing order. At $k=0$ the left side of the printed inequality is the sample mean in Proposition 1, and the $\\Pi$-weighted mean of the distinct outcomes in Proposition 2. Since neither is smaller than the smallest outcome, $k=0$ satisfies the printed inequality in every case, and both printed rules return the cut point $0$. The vector $w^{\\mathrm L}$, or $W_-$, then has every entry equal to $1/\\beta$. Evaluating these vectors in (3) and (7) gives the unweighted sample mean and the $\\Pi$-weighted mean, respectively. When $\\gamma>1$ and the outcomes are not all equal, this vector does not minimize (3) or (7), since a larger weight on the smallest outcome lowers the ratio. The left endpoint of (5), computed with the printed $k^{\\mathrm L}$, then leaves out values that $\\hat\\mu$ can take. The paper does not display the correct rule elsewhere. The Appendix says that the solution to (3) ``can be derived in a similar manner'', and the supplementary proof of Proposition 2 treats $\\mu_+$ only. When $\\gamma=1$ or all outcomes are equal, the ratio has the same value at every feasible weight vector, and the printed rules return a minimizer.", "what_survives": "The correction needs no new assumption. The definitions (2), (3), (6) and (7), the rules for $k^{\\mathrm H}$ and $k_+$, the upper bounds, and the closed form for dichotomous outcomes are unaffected by this sign correction. All 42 entries of Table 1 are reproduced from the reported percentages by the closed form, and this sign correction does not change the table. The estimated bounds still converge to the limits defined by (6) and (7). The rules for $k^{\\mathrm L}$ and $k_-$ change, and with them the left endpoint of (5) and computations of the lower bound that use either printed rule. The summation limits in the displayed formula for $\\mu_-$ are corrected separately in \\fid{P025-E002}." }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-E001", "work": { "arxiv_id": null, "doi": "10.1093/biomet/ass064", "title": "Interval Estimation of Population Means under Unknown but Bounded Probabilities of Sample Selection", "url": "https://pmaronow.github.io/llm-audit/works/P025/" }, "work_id": "P025" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the first line of Proposition 2 and the two displayed formulas for mu plus and mu minus. Red outlines surround the two lower summation limits j equals k minus, minus 1, in the numerator and in the denominator of the formula for mu minus, on the right.", "caption": "Proposition 2, p. 237 (PDF p. 3). The outlines mark the lower limits $j=k_--1$ of the second sums in the numerator and the denominator of the formula for $\\mu_-$.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-E002.png" } ], "formal_release": null, "formal_statements": [ { "id": "P025-T2", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T2" }, { "id": "P025-T4", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T4" }, { "id": "P025-T9", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T9" }, { "id": "P025-T10", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T10" } ], "headline": "In the displayed formula for $\\mu_-$, both tail sums should start at $j=k_-+1$, not at $j=k_--1$.", "id": "P025-E002", "location": "Proposition 2, displayed formula for $\\mu_-$, p. 237 (PDF p. 3)", "pdf_page": 3, "printed_page": "237", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P025-T10", "P025-T2", "P025-T4", "P025-T9" ], "text": { "as_printed": "\\[\n\\mu_- = \\frac{\\gamma \\sum_{j=1}^{k_-} \\Pi_j y_{(j)} + \\sum_{j=\\hl{\\scriptstyle k_- -1}}^{|Y|} \\Pi_j y_{(j)}}{\\gamma \\sum_{j=1}^{k_-} \\Pi_j + \\sum_{j=\\hl{\\scriptstyle k_- -1}}^{|Y|} \\Pi_j},\n\\]", "corrected": "With $k_-$ and $W_-$ as in Proposition 2, the value of (7) at $W_-$ is\n\\[\n\\mu_- = \\frac{\\gamma \\sum_{j=1}^{k_-} \\Pi_j y_{(j)} + \\sum_{j=\\hl{\\scriptstyle k_- +1}}^{|Y|} \\Pi_j y_{(j)}}{\\gamma \\sum_{j=1}^{k_-} \\Pi_j + \\sum_{j=\\hl{\\scriptstyle k_- +1}}^{|Y|} \\Pi_j},\n\\]\nwhere a sum over an empty range is zero.", "evidence": "The evidence is a substitution and an exact example. For $0\\leqslant k\\leqslant|Y|$, let $W^{(k)}$ have its first $k$ entries equal to $1/\\alpha$ and the others equal to $1/\\beta$. Substituting $W^{(k)}$ into the ratio in (7) and multiplying the numerator and the denominator by $\\beta$, I obtain\n\\[\n\\frac{\\gamma\\sum_{j=1}^{k}\\Pi_jy_{(j)}+\\sum_{j=k+1}^{|Y|}\\Pi_jy_{(j)}}{\\gamma\\sum_{j=1}^{k}\\Pi_j+\\sum_{j=k+1}^{|Y|}\\Pi_j},\n\\]\nwhich is the corrected display at $k=k_-$. For the example, let $\\alpha=1/2$, $\\beta=1$, and let the population have outcomes $(0,0,1,1,2,3)$ and selection probabilities $(1/2,1/2,1/2,1/2,1,1)$. The distinct outcomes are $(0,1,2,3)$ and $\\Pi=(1,1,1,1)$. With the weights scaled by $\\beta$, the ratio in (7) is $F(W)=(W_2+2W_3+3W_4)/(W_1+W_2+W_3+W_4)$ on $[1,2]^4$, and\n\\[\nF(W)-\\frac76=\\frac{7(2-W_1)+(2-W_2)+5(W_3-1)+11(W_4-1)}{6(W_1+W_2+W_3+W_4)}\\geqslant 0 ,\n\\]\nwith equality only at $W=(2,2,1,1)$. The minimum of (7) is $7/6$, and it is attained at the cut point $k_-=2$. At $k_-=2$ the printed display equals $\\{2(0+1)+(0+1+2+3)\\}/\\{2(1+1)+(1+1+1+1)\\}=1$, and the corrected display equals $\\{2(0+1)+(2+3)\\}/\\{2(1+1)+(1+1)\\}=7/6$.", "problem": "Proposition 2 states that the first $k_-$ entries of $W_-$ equal $1/\\alpha$ and the remaining $|Y|-k_-$ entries equal $1/\\beta$. The value of (7) at $W_-$ therefore has the classes $j\\leqslant k_-$ with the factor $\\gamma$ and the classes $j\\geqslant k_-+1$ without it. As printed, the second sums start at $k_--1$. For $k_-\\geqslant 2$, the classes $k_--1$ and $k_-$ are counted in both sums, and for $k_-=0$ or $k_-=1$, the lower limit is an index with no class. The displays for $\\mu_+$, for the cut points and in Assumption 2 start their second sums at $k+1$.", "what_survives": "The definitions (6) and (7), the description of $W_-$, the bounds of Section 3.1 and Table 1 are unaffected. The supplementary material works with the full weighted sums and does not use the printed limits. The rule that selects $k_-$ is corrected in \\fid{P025-E001}." }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-E002", "work": { "arxiv_id": null, "doi": "10.1093/biomet/ass064", "title": "Interval Estimation of Population Means under Unknown but Bounded Probabilities of Sample Selection", "url": "https://pmaronow.github.io/llm-audit/works/P025/" }, "work_id": "P025" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the top of page 240: the word Thus, a displayed formula in which k superscript H equals a minimum over k of one condition, which equals a minimum over k of a second condition, and the sentence The solution to (3) can be derived in a similar manner. A red outline surrounds the equality sign between the two minima.", "caption": "Appendix, end of the proof of Proposition 1, p. 240 (PDF p. 6). The outline marks the equality between the two minima that define $k^{\\mathrm H}$.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-E004.png" }, { "alt": "Crop of three lines of the Appendix proof, ending Then the difference of two ratios is less than or equal to 0 implies that, followed by a displayed inequality with y sub k plus 1 on the right. Red outlines surround the less-than-or-equal sign at the end of the second line and the words 0 implies that at the start of the third.", "caption": "Appendix, proof of Proposition 1, p. 239 (PDF p. 5). The outlines mark the words “$\\leqslant 0$ implies that”, which precede the displayed inequality.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-E004-b.png" }, { "alt": "Crop of four lines of the supplementary material: Then the condition mu(1/beta, 1/beta) greater than mu(1/alpha, 1/beta) is equivalent to, the displayed inequality (B1), and the condition mu(1/alpha, 1/alpha) greater than mu(1/alpha, 1/beta) is equivalent to, and the displayed inequality (B2). Red outlines surround the words is equivalent to in the first and third lines.", "caption": "Supplementary material, proof of Proposition 2, p. 2. The outlines mark the two occurrences of “is equivalent to” that introduce (B1) and (B2).", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-E004-c.png" } ], "formal_release": null, "formal_statements": [ { "id": "P025-T11", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T11" }, { "id": "P025-T12", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T12" } ], "headline": "The Appendix argument for $k^{\\mathrm H}$, and the steps (B1) and (B2) of the supplementary material, hold for $\\gamma>1$; the case $\\gamma=1$ needs a separate sentence.", "id": "P025-E004", "location": "Appendix, proof of Proposition 1, pp. 239–240 (PDF pp. 5–6); supplementary material, proof of Proposition 2, (B1) and (B2), p. 2", "pdf_page": 5, "printed_page": "239–240", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P025-T11", "P025-T12" ], "text": { "as_printed": "\\emph{Appendix, p.~239.} Let $w^k = (1/\\beta, \\ldots, 1/\\beta, 1/\\alpha, \\ldots, 1/\\alpha)$ where the first $k$ entries are $1/\\beta$ and the last $n - k$ are $1/\\alpha$. Then $(w^{k+1\\prime} y)/(w^{k+1\\prime} 1_n) - (w^{k\\prime} y)/(w^{k\\prime} 1_n) \\leqslant 0$ \\hl{implies that}\n\\[\n\\frac{\\sum_{i=1}^{k} y_i + \\gamma \\sum_{i=k+1}^{n} y_i}{k + \\gamma (n - k)} \\leqslant y_{k+1}.\n\\]\n\n\\emph{Appendix, p.~240.} Thus\n\\[\nk^{\\mathrm H} = \\min\\left\\{ k \\geqslant 0 : \\frac{w^{k\\prime} y}{w^{k\\prime} 1_n} \\geqslant \\frac{w^{k+1\\prime} y}{w^{k+1\\prime} 1_n} \\right\\} \\hl{=} \\min\\left\\{ k \\geqslant 0 : \\frac{\\sum_{i=1}^{k} y_i + \\gamma \\sum_{i=k+1}^{n} y_i}{k + \\gamma (n - k)} \\leqslant y_{k+1} \\right\\}.\n\\]\n\n\\emph{Supplementary material, p.~2.} Then the condition $\\mu(1/\\beta, 1/\\beta) > \\mu(1/\\alpha, 1/\\beta)$ \\hl{is equivalent to}\n\\[\nc_2 y_{(k)} - c_1 + \\beta^{-1} \\Pi_{k+1} (y_{(k)} - y_{(k+1)}) < 0 \\qquad \\text{(B1)}\n\\]\nand the condition $\\mu(1/\\alpha, 1/\\alpha) > \\mu(1/\\alpha, 1/\\beta)$ \\hl{is equivalent to}\n\\[\nc_1 - c_2 y_{(k+1)} + \\alpha^{-1} \\Pi_k (y_{(k)} - y_{(k+1)}) < 0. \\qquad \\text{(B2)}\n\\]", "corrected": "\\emph{Appendix, proof of Proposition 1.} \\hl{If $\\gamma=1$, then $\\alpha=\\beta$ and the only feasible weight vector is $(1/\\alpha,\\ldots,1/\\alpha)$. The bounds in (2) and (3) then both equal the sample mean, the cut point $k^{\\mathrm H}$ of Proposition 1 selects this vector, and (4) holds. For the rest of the proof, assume that $\\gamma>1$.} Let $w^k = (1/\\beta, \\ldots, 1/\\beta, 1/\\alpha, \\ldots, 1/\\alpha)$ where the first $k$ entries are $1/\\beta$ and the last $n - k$ are $1/\\alpha$. Then\\hl{, for $0\\leqslant k \\mu(1/\\alpha, 1/\\beta)$ is equivalent to (B1), and the condition $\\mu(1/\\alpha, 1/\\alpha) > \\mu(1/\\alpha, 1/\\beta)$ is equivalent to (B2).", "evidence": "The evidence is an exact counterexample to the four printed steps and the identities that show where $\\gamma>1$ is used. Let $n=N=2$, $y=(0,1)$ and $\\alpha=\\beta=1$, so that $\\gamma=1$ and $w^0=w^1=w^2=(1,1)$. Then $F_0=F_1=F_2=1/2$. At $k=0$, I have $F_1-F_0=0\\leqslant 0$, and the conclusion $F_0=1/2\\leqslant y_1=0$ is false. For $k\\in\\{0,1\\}$, the first minimum on p.~240 is $0$ and the second is $1$. In the supplementary material, the two classes have $y_{(1)}=0$, $y_{(2)}=1$ and $\\Pi_1=\\Pi_2=1$. With $k=1$, I have $c_1=c_2=0$, the left sides of (B1) and (B2) both equal $-1$, and $\\mu(1/\\beta,1/\\beta)=\\mu(1/\\alpha,1/\\beta)=\\mu(1/\\alpha,1/\\alpha)=1/2$. For the general case, let $a=1/\\beta$, $b=1/\\alpha$, $p=\\Pi_k$, $q=\\Pi_{k+1}$, and let $E_1$ and $E_2$ be the left sides of (B1) and (B2). A direct computation gives\n\\[\n\\mu(a,a)-\\mu(b,a)=\\frac{(a-b)\\,p\\,E_1}{(c_2+ap+aq)(c_2+bp+aq)},\\qquad\n\\mu(b,b)-\\mu(b,a)=\\frac{(a-b)\\,q\\,E_2}{(c_2+bp+bq)(c_2+bp+aq)} .\n\\]\nBoth denominators are positive. Since $a-b<0$ for $\\alpha<\\beta$, the two equivalences hold in this case. For $\\alpha=\\beta$, both differences vanish.", "problem": "Proposition 1 allows $\\gamma=1$, and Table 1 includes it. Let $F_k=(w^{k\\prime}y)/(w^{k\\prime}1_n)$ and $D_k=k+\\gamma(n-k)$. Then\n\\[\nF_{k+1}-F_k=\\frac{(1-\\gamma)\\,(y_{k+1}-F_k)}{D_{k+1}}\\qquad\\text{for } 0\\leqslant k1$, this identity implies that $F_{k+1}-F_k\\leqslant 0$ if and only if $F_k\\leqslant y_{k+1}$, and the two minima on p.~240 agree. For $\\gamma=1$, the vectors $w^k$ coincide and the difference is zero for every $k$. The printed implication and the equality of the two minima can then fail. The steps (B1) and (B2) have the same gap. Each difference of objective values equals $1/\\beta-1/\\alpha$ times a positive factor times the left side of (B1) or (B2), and for $\\alpha=\\beta$ the difference is zero whatever the sign of that left side. The results are not affected. For $\\gamma=1$ the only feasible weight vector has every entry equal to $1/\\alpha$, both bounds in (2) and (3) equal the sample mean, and both (6) and (7) equal $\\mu$.", "what_survives": "The upper bound (4) and its optimal weights hold for all $\\gamma\\geqslant 1$. Lemma 1, Propositions 2 and 3, the closed form for dichotomous outcomes and Table 1 are unaffected by this boundary case; the rows $\\gamma=1$ are the sample percentages. The identity for the sum of the left sides of (B1) and (B2) in the supplementary material is correct. One case is added to each proof, and no assumption or result changes." }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-E004", "work": { "arxiv_id": null, "doi": "10.1093/biomet/ass064", "title": "Interval Estimation of Population Means under Unknown but Bounded Probabilities of Sample Selection", "url": "https://pmaronow.github.io/llm-audit/works/P025/" }, "work_id": "P025" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop that begins By Proposition 2, shows Assumption 2 with its two displayed inequalities, and ends with Proposition 3. Red outlines surround four items: the words necessary to guarantee convergence in the second line, the words For any k greater than or equal to 0 at the start of Assumption 2, and in Proposition 3 the symbols mu sub h superscript H and mu sub h superscript L, neither of which has a hat.", "caption": "Section 3.2, from the paragraph before Assumption 2 to Proposition 3, p. 238 (PDF p. 4). The outlines mark the words “necessary to guarantee convergence”, the words “For any $k\\geqslant 0$”, and the symbols $\\mu_h^{\\mathrm H}$ and $\\mu_h^{\\mathrm L}$, which are printed without hats.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-E005.png" } ], "formal_release": null, "formal_statements": [ { "id": "P025-T13", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T13" }, { "id": "P025-T14", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T14" }, { "id": "P025-T21", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T21" } ], "headline": "Assumption 2 is sufficient for the proof of Proposition 3 and is not necessary for the estimated bounds to converge; Proposition 3 should carry hats on the estimators, and Assumption 2 is meant for $0\\leqslant k<|Y|$.", "id": "P025-E005", "location": "Section 3.2, the sentence before Assumption 2, Assumption 2, and Proposition 3, p. 238 (PDF p. 4)", "pdf_page": 4, "printed_page": "238", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P025-T13", "P025-T14", "P025-T21" ], "text": { "as_printed": "A mild technical condition on $k_+$ and $k_-$ is \\hl{necessary to guarantee convergence}.\n\nAssumption 2. \\hl{For any $k \\geqslant 0$},\n\\[\ny_{(k+1)} \\neq \\frac{\\sum_{j=1}^{k} \\Pi_j y_{(j)} + \\gamma \\sum_{j=k+1}^{|Y|} \\Pi_j y_{(j)}}{\\sum_{j=1}^{k} \\Pi_j + \\gamma \\sum_{j=k+1}^{|Y|} \\Pi_j}, \\qquad\ny_{(k+1)} \\neq \\frac{\\gamma \\sum_{j=1}^{k} \\Pi_j y_{(j)} + \\sum_{j=k+1}^{|Y|} \\Pi_j y_{(j)}}{\\gamma \\sum_{j=1}^{k} \\Pi_j + \\sum_{j=k+1}^{|Y|} \\Pi_j}.\n\\]\nPROPOSITION 3. If Assumptions 1 and 2 hold, then as $h \\to \\infty$, $\\hl{\\mu_h^{\\mathrm H}} \\to \\mu_+$ and $\\hl{\\mu_h^{\\mathrm L}} \\to \\mu_-$ almost everywhere.", "corrected": "A mild technical condition on $k_+$ and $k_-$ is \\hl{sufficient for the proof of convergence}.\n\nAssumption 2. \\hl{For every integer $k$ with $0 \\leqslant k < |Y|$},\n\\[\ny_{(k+1)} \\neq \\frac{\\sum_{j=1}^{k} \\Pi_j y_{(j)} + \\gamma \\sum_{j=k+1}^{|Y|} \\Pi_j y_{(j)}}{\\sum_{j=1}^{k} \\Pi_j + \\gamma \\sum_{j=k+1}^{|Y|} \\Pi_j}, \\qquad\ny_{(k+1)} \\neq \\frac{\\gamma \\sum_{j=1}^{k} \\Pi_j y_{(j)} + \\sum_{j=k+1}^{|Y|} \\Pi_j y_{(j)}}{\\gamma \\sum_{j=1}^{k} \\Pi_j + \\sum_{j=k+1}^{|Y|} \\Pi_j}.\n\\]\nPROPOSITION 3. If Assumptions 1 and 2 hold, then as $h \\to \\infty$, $\\hl{\\hat\\mu_h^{\\mathrm H}} \\to \\mu_+$ and $\\hl{\\hat\\mu_h^{\\mathrm L}} \\to \\mu_-$ almost everywhere.\n\n\\hl{The conclusion also holds without Assumption 2: if Assumption 1 holds, then as $h\\to\\infty$, $\\hat\\mu_h^{\\mathrm H}\\to\\mu_+$ and $\\hat\\mu_h^{\\mathrm L}\\to\\mu_-$ almost everywhere.}", "evidence": "The evidence is an exact example in which Assumption 2 fails and the bounds converge, a derivation that Assumption 1 suffices, and a check of the page for the hats and the range of $k$.\n\n\\emph{Example.} Let the population have outcomes $(0,0,2,2,3,3)$. Each replication selects one unit from each pair of units with equal outcomes, each with probability $1/2$, and the three choices are independent. Then $\\pi_i=1/2$ for all $i$, and I take $\\alpha=1/4$ and $\\beta=1/2$, so that $\\gamma=2$. The distinct outcomes are $(0,2,3)$ and $\\Pi=(1,1,1)$. After $h$ replications the sample contains exactly $h$ units at each outcome. With class weights $(w_0,w_2,w_3)\\in[2,4]^3$, the ratio is $f=(2w_2+3w_3)/(w_0+w_2+w_3)$ for the sample and in the limit, and\n\\[\n\\begin{aligned}\n4(2w_2+3w_3)-5(w_0+w_2+w_3)&=3(w_2-2)+7(w_3-2)+5(4-w_0)\\geqslant 0,\\\\\n2(w_0+w_2+w_3)-(2w_2+3w_3)&=2(w_0-2)+(4-w_3)\\geqslant 0 .\n\\end{aligned}\n\\]\nBy these two inequalities, I have $5/4\\leqslant f\\leqslant 2$, with equality at $(4,2,2)$ and at $(2,4,4)$. I conclude that $[\\hat\\mu_h^{\\mathrm L},\\hat\\mu_h^{\\mathrm H}]=[\\mu_-,\\mu_+]=[5/4,2]$ for every $h$ and every sample path. Assumption 2 fails at $k=1$, since the first ratio there equals $\\{1\\cdot 0+2(2+3)\\}/\\{1+2(1+1)\\}=2=y_{(2)}$.\n\n\\emph{Derivation.} Let $P_{h,j}$ be the number of sampled units of outcome $y_{(j)}$ after $h$ replications, divided by $h$. By Assumption 1, the counts of class $j$ in the successive replications are independent and identically distributed, they are bounded, and their mean is $\\Pi_j$. The strong law of large numbers implies that $P_{h,j}\\to\\Pi_j$ almost surely for every $j$. For a vector $p$ with nonnegative entries and positive sum, I define\n\\[\nf(W,p)=\\frac{\\sum_jW_jp_jy_{(j)}}{\\sum_jW_jp_j},\\qquad W\\in B=[1/\\beta,1/\\alpha]^{|Y|}.\n\\]\nIn (2) and (3) the weights of the units in one class can be replaced by their average, which lies in $[1/\\beta,1/\\alpha]$. Whenever the sample is nonempty, $\\hat\\mu_h^{\\mathrm H}=\\max_{W\\in B}f(W,P_h)$ and $\\hat\\mu_h^{\\mathrm L}=\\min_{W\\in B}f(W,P_h)$, and by (6) and (7), $\\mu_+=\\max_{W\\in B}f(W,\\Pi)$ and $\\mu_-=\\min_{W\\in B}f(W,\\Pi)$. Let $R=y_{(|Y|)}-y_{(1)}$ and $S=\\sum_j\\Pi_j$. With $m=f(W,p)$, I have\n\\[\nm-f(W,\\Pi)=\\frac{\\sum_jW_j(p_j-\\Pi_j)(y_{(j)}-m)}{\\sum_jW_j\\Pi_j},\n\\]\nand therefore\n\\[\n\\bigl|f(W,p)-f(W,\\Pi)\\bigr|\\leqslant\\frac{\\gamma R}{S}\\sum_j|p_j-\\Pi_j|\\qquad\\text{for all } W\\in B .\n\\]\nIn the last inequality, I used that $|y_{(j)}-m|\\leqslant R$ and $1/\\beta\\leqslant W_j\\leqslant 1/\\alpha$. The maximum and the minimum over $B$ change by at most the same amount. With $p=P_h$, I obtain $\\hat\\mu_h^{\\mathrm H}\\to\\mu_+$ and $\\hat\\mu_h^{\\mathrm L}\\to\\mu_-$ almost surely, including when the ratios of Assumption 2 tie.", "problem": "There are three points. First, the estimated bounds $\\hat\\mu_h^{\\mathrm H}$ and $\\hat\\mu_h^{\\mathrm L}$ converge almost surely to $\\mu_+$ and $\\mu_-$ under Assumption 1 alone. The supplementary material uses Assumption 2 to make the estimated cut point equal to $k_+$ for all large $h$, which the convergence of the bounds does not require. Proposition 3 is correct as a statement of sufficient conditions, with a hypothesis that it does not need. Second, Proposition 3 prints $\\mu_h^{\\mathrm H}$ and $\\mu_h^{\\mathrm L}$ without hats. Assumption 1 defines the estimators $\\hat\\mu_h^{\\mathrm H}$ and $\\hat\\mu_h^{\\mathrm L}$, and no quantity without a hat is defined. Third, $y_{(k+1)}$ is defined only for $0\\leqslant k<|Y|$, and ``for any $k\\geqslant 0$'' includes indices at which Assumption 2 has no meaning.", "what_survives": "The conclusion of Proposition 3 holds, under Assumption 1 alone. The optimization problems (2), (3), (6) and (7), Table 1 and the application are unaffected. The sentence before Assumption 2 and the notation of Proposition 3 and Assumption 2 change." }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-E005", "work": { "arxiv_id": null, "doi": "10.1093/biomet/ass064", "title": "Interval Estimation of Population Means under Unknown but Bounded Probabilities of Sample Selection", "url": "https://pmaronow.github.io/llm-audit/works/P025/" }, "work_id": "P025" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the last three lines of page 238, which contain the sentence on when the intervals overlap. A red outline surrounds the words This means that for the shooting gallery needles to possibly show the same, which begin the last sentence of Section 4 and run to the end of the page.", "caption": "Section 4, start of the last sentence, foot of p. 238 (PDF p. 4). The outline marks the words “This means that for the shooting gallery needles to possibly show the same”, which follow the two statements about overlap.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-E003.png" }, { "alt": "Crop of the first three lines of page 239, which complete the last sentence of Section 4. Red outlines surround the words must have been more than at the end of the first line and three times as likely at the start of the second, and the words with the converse holding for at the end of the second line and exchange needles at the start of the third.", "caption": "Section 4, end of the last sentence, p. 239 (PDF p. 5). The outlines mark the words “must have been more than three times as likely” and “with the converse holding for exchange needles”.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-E003-b.png" }, { "alt": "A plot with the shooting gallery ratio, infected against uninfected, on the horizontal axis and the needle exchange ratio, uninfected against infected, on the vertical axis, both from 1 to 6. A blue curve of constant product 1056/97, about 10.9, falls from the upper left to the lower right. A shaded square with a dashed orange border covers the region where both ratios exceed 3 and is labelled as printed: both ratios must exceed 3. The curve enters the shaded region through its left edge, passes through the open point near (3.30, 3.30), labelled the smallest common cap, and exits through the bottom edge. Two filled points on the curve lie outside the region: design B at gallery ratio 11/4 = 2.75 with both prevalences 4/5, and design A at exchange ratio 288/97, about 2.97, with both prevalences 3/4. The title reads Equal reweighted prevalence fixes the product of the two selection ratios; neither ratio need exceed three.", "caption": "The pairs of selection ratios for which the shooting gallery and needle exchange samples have equal reweighted prevalence, computed from the count-derived proportions $44/48$ and $291/579$. The curve is the set of pairs whose product is $1056/97$. The shaded region is what the printed sentence asserts. The filled points are designs A and B of the table, and the open point is the pair at which both ratios equal $\\sqrt{1056/97}\\approx 3.30$.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-E003-selection-ratios.png" } ], "formal_release": null, "formal_statements": [ { "id": "P025-T5", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T5" }, { "id": "P025-T6", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T6" } ], "headline": "For the shooting gallery and needle exchange intervals to overlap, the common bound $\\gamma$ must be at least about $3.30$; neither individual selection ratio needs to exceed three.", "id": "P025-E003", "location": "Section 4, last sentence, pp. 238–239 (PDF pp. 4–5)", "pdf_page": 4, "printed_page": "238–239", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P025-T5", "P025-T6" ], "text": { "as_printed": "This means that for the shooting gallery needles to possibly show the same HIV prevalence rate as the exchange needles, infected shooting gallery needles \\hl{must have been more than three times as likely} to be tested than uninfected shooting gallery needles, \\hl{with the converse holding for exchange needles}.", "corrected": "\\hl{For the shooting gallery and needle exchange interval estimates to overlap, the common bound on the ratios of selection probabilities must be at least about $3{\\cdot}30$. This does not require both populations' actual directional selection ratios to exceed three. With the printed proportions $0{\\cdot}917$ and $0{\\cdot}503$, equal reweighted estimates require that the product of the two effective ratios, for infected against uninfected shooting gallery needles and for uninfected against infected exchange needles, equal $455749/41749$, which is about $10{\\cdot}9$.}", "evidence": "The evidence is a derivation and two exact designs.\n\n\\emph{Derivation.} Let a sample have an infected fraction $p$, and write $\\pi_1$ and $\\pi_0$ for the effective probabilities obtained by averaging inverse selection probabilities within each sampled infection group and taking reciprocals. By (1), the Hajek estimate $q$ of the prevalence satisfies $q/(1-q)=\\{p/(1-p)\\}(\\pi_0/\\pi_1)$. I write $r_G=\\pi_{G,1}/\\pi_{G,0}$ for the shooting gallery and $r_E=\\pi_{E,0}/\\pi_{E,1}$ for the needle exchange. The two estimates are equal exactly when\n\\[\nr_G\\,r_E=\\frac{p_G(1-p_E)}{(1-p_G)\\,p_E}.\n\\]\nWith the printed proportions $0.917$ and $0.503$, the right side is $455749/41749\\approx 10.92$, and with the count-derived proportions $44/48$ and $291/579$, it is $1056/97\\approx 10.89$. By the closed form on p.~237, the two intervals overlap exactly when $\\gamma^2$ is at least this number. The condition is $\\gamma\\geqslant\\sqrt{455749/41749}\\approx 3.304$ with the printed proportions and $\\gamma\\geqslant\\sqrt{1056/97}\\approx 3.299$ with the count-derived proportions. At the smallest such $\\gamma$ both ratios equal $\\gamma$, and at a larger bound, such as the row $\\gamma=4$ of Table 1, they can differ.\n\n\\emph{Two designs.} In each design below, a fixed number of needles is drawn uniformly from the infected and from the uninfected needles of each population. The samples have the sizes $48$ and $579$ reported in Section 4 and positive counts $44$ and $291$ implied by rounding its percentages to the nearest tenth of a percent. The two populations have the same prevalence, the two Hajek estimates equal this prevalence on every draw, and all selection probabilities lie in $[1/4,1]$, so that $\\gamma=4$. In design A the ratio for the needle exchange is below three, and in design B the ratio for the shooting gallery is below three. The conclusion is the same for two further choices of the selection probabilities, with the proportions $0.917$ and $0.503$ taken as exact. There the ratios $(r_G,r_E)$ are $(917/249,\\ 1491/503)$ at prevalence $3/4$, with $1491/503<3$, and $(917/332,\\ 1988/503)$ at prevalence $4/5$, with $917/332<3$.\n\n\\begin{audittable}{llllll}{Two exact designs with equal prevalence in which one selection ratio is below three. Pairs are (infected, uninfected). The ratio is $r_G$ for the shooting gallery and $r_E$ for the needle exchange.}\nDesign & Population & Tested & Selection probabilities & Ratio & Prevalence \\\\\nA, gallery & $(48,16)$ & $(44,4)$ & $(11/12,\\ 1/4)$ & $11/3\\approx 3.67$ & $3/4$ \\\\\nA, exchange & $(864,288)$ & $(291,288)$ & $(97/288,\\ 1)$ & $288/97\\approx 2.97$ & $3/4$ \\\\\nB, gallery & $(48,12)$ & $(44,4)$ & $(11/12,\\ 1/3)$ & $11/4=2.75$ & $4/5$ \\\\\nB, exchange & $(1156,289)$ & $(291,288)$ & $(291/1156,\\ 288/289)$ & $384/97\\approx 3.96$ & $4/5$ \\\\\n\\end{audittable}", "problem": "The sentence makes two claims about the selection of the samples. It says that infected shooting gallery needles were more than three times as likely to be tested as uninfected ones, and that uninfected exchange needles were more than three times as likely to be tested as infected ones. Table 1 does not show this. The parameter $\\gamma$ is a common bound on how far the selection probabilities may differ within a population. On p.~236 the paper says that $\\gamma=5$ ``indicates that some units may be up to five times more likely to be sampled than others''. The table shows that the two intervals overlap only when this bound is larger than three. It says nothing about the ratio that each sample attains. Equal reweighted estimates fix the product of the two effective ratios, which must equal the odds ratio of the two sample proportions, about $10.9$. The product can be split unevenly, and one ratio can be below three when the other is larger. The paper does not assume that the two ratios are equal.", "what_survives": "Table 1 is unchanged, and so is the statement that the shooting gallery and needle exchange intervals overlap only when $\\gamma>3$, with the first overlap in the table at $\\gamma=4$. The comparison of the street and needle exchange samples and the method of sensitivity analysis are unaffected. I replace the inference in the last sentence about the two selection processes by a weaker statement about the common bound, without adding an assumption." }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-E003", "work": { "arxiv_id": null, "doi": "10.1093/biomet/ass064", "title": "Interval Estimation of Population Means under Unknown but Bounded Probabilities of Sample Selection", "url": "https://pmaronow.github.io/llm-audit/works/P025/" }, "work_id": "P025" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of the reference list of the article, from Brewer (1979) to Rosenbaum (2002). Red outlines surround six items: the names C.-E. Wretman in the entry for Cassel, Sarndal and Wretman; the words The Foundations at the end of one line and of Survey Sampling at the start of the next in the entry for Hajek; the pages 2 to 8 in the entry for Kaplan; and the words Statistical Methods in Epidemiology, Ed. E. Halloran, and pp. 1 to 92 on the next line, in the entry for Robins, Rotnitzky and Scharfstein.", "caption": "References, p. 240 (PDF p. 6). The outlines mark “C.-E. Wretman”, where the separator is missing, the volume title “The Foundations of Survey Sampling”, the pages “2–8”, and the clause “Statistical Methods in Epidemiology, Ed. E. Halloran, pp. 1–92”.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-E007.png" } ], "formal_release": null, "formal_statements": [], "headline": "Four entries of the reference list need a correction: the separator before Wretman, the title of the volume that contains the comment of Hajek, the pages of Kaplan (1991), and the title, editors and pages of the volume that contains Robins et al.", "id": "P025-E007", "location": "References, entries for Cassel et al. (1976), Hajek (1971), Kaplan (1991) and Robins et al. (1999), p. 240 (PDF p. 6)", "pdf_page": 6, "printed_page": "240", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "CASSEL, C. M., SÄRNDAL, \\hl{C.-E. WRETMAN}, J. H. (1976). Some results on generalized difference estimation and generalized regression estimation for finite populations. \\emph{Biometrika} \\textbf{63}, 615--20.\n\nHAJEK, J. (1971). Comment on ``An essay on the logical foundations of survey sampling, part one.''. In \\hl{\\emph{The Foundations of Survey Sampling}}, Ed. V. P. Godambe \\& D. A. Sprott, p. 246. Toronto: Holt, Rinehart, and Winston.\n\nKAPLAN, E. H. (1991). Evaluating needle-exchange programs via syringe tracking and testing (STT). \\emph{AIDS Public Pol. J.} \\textbf{6}, \\hl{2--8}.\n\nROBINS, J. M., ROTNITZKY, A. \\& SCHARFSTEIN, D. O. (1999). Sensitivity analysis for selection bias and unmeasured confounding in missing data and causal inference models. In \\hl{\\emph{Statistical Methods in Epidemiology}, Ed. E. Halloran, pp. 1--92}. New York: Springer.", "corrected": "CASSEL, C. M., SÄRNDAL, C.-E. \\hl{\\&} WRETMAN, J. H. (1976). Some results on generalized difference estimation and generalized regression estimation for finite populations. \\emph{Biometrika} \\textbf{63}, 615--20.\n\nHAJEK, J. (1971). Comment on ``An essay on the logical foundations of survey sampling, part one.''. In \\hl{\\emph{Foundations of Statistical Inference}}, Ed. V. P. Godambe \\& D. A. Sprott, p. 246. Toronto: Holt, Rinehart, and Winston.\n\nKAPLAN, E. H. (1991). Evaluating needle-exchange programs via syringe tracking and testing (STT). \\emph{AIDS Public Pol. J.} \\textbf{6}, \\hl{109--15}.\n\nROBINS, J. M., ROTNITZKY, A. \\& SCHARFSTEIN, D. O. (1999). Sensitivity analysis for selection bias and unmeasured confounding in missing data and causal inference models. In \\hl{\\emph{Statistical Models in Epidemiology, the Environment, and Clinical Trials}, Eds. M. E. Halloran \\& D. Berry, pp. 1--94}. New York: Springer.", "evidence": "I compared the printed entries with public bibliographic records. The publisher's record for the volume of Halloran and Berry gives its title, its editors and the pages $1$--$94$ of the chapter (\\url{https://link.springer.com/book/10.1007/978-1-4612-1284-3}). E.~H. Kaplan's list of his papers gives \\emph{AIDS and Public Policy Journal} 6:109--115, 1991 (\\url{https://faculty.som.yale.edu/edkaplan/papers/}). Two library catalogues give the title \\emph{Foundations of statistical inference: a symposium} for the volume of 1971 edited by Godambe and Sprott (\\url{https://ci.nii.ac.jp/ncid/BA8882938X}). No record that locates the comment of Hajek within that volume was found, and the page $246$ is left as printed. The publisher released the volume of Halloran and Berry in 1999 and dates its copyright 2000, and the year 1999 is not counted as an error.", "problem": "The entry for Cassel et al.\\ lacks the separator before the third author. The volume edited by Godambe and Sprott is \\emph{Foundations of Statistical Inference}. The article of Kaplan is on pp.~109--115 of its volume. The chapter of Robins et al.\\ is on pp.~1--94 of \\emph{Statistical Models in Epidemiology, the Environment, and Clinical Trials}, edited by M.~E. Halloran and D.~Berry. Each cited work can still be identified, and no result or number depends on these entries.", "what_survives": "The mathematics, the calculations and the citations in the text are unaffected." }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-E007", "work": { "arxiv_id": null, "doi": "10.1093/biomet/ass064", "title": "Interval Estimation of Population Means under Unknown but Bounded Probabilities of Sample Selection", "url": "https://pmaronow.github.io/llm-audit/works/P025/" }, "work_id": "P025" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of five lines of the second paragraph of Section 4, from the interval estimates for each population to the foot of the page. Red outlines surround the words overlap only when gamma is greater than 3 in the fourth line and overlap when gamma is greater than 1.25 in the fifth line.", "caption": "Section 4, second paragraph, p. 238 (PDF p. 4). The outlines mark the clauses “overlap only when $\\gamma>3$” and “overlap when $\\gamma>1{\\cdot}25$”.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-A001.png" } ], "formal_release": null, "formal_statements": [], "headline": "The values $3$ and $1.25$ are rows of Table 1; the two pairs of intervals begin to overlap at about $\\gamma=3.30$ and $\\gamma=1.43$.", "id": "P025-A001", "location": "Section 4, second paragraph, fourth sentence, p. 238 (PDF p. 4)", "pdf_page": 4, "printed_page": "238", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "The estimates for the shooting gallery and needle exchange populations \\hl{overlap only when $\\gamma > 3$}; the estimates for the street needle and needle exchange populations \\hl{overlap when $\\gamma > 1{\\cdot}25$}.", "corrected": "\\hl{Among the values of $\\gamma$ shown in Table 1,} the estimates for the shooting gallery and needle exchange populations overlap only when $\\gamma > 3$; the estimates for the street needle and needle exchange populations overlap when $\\gamma > 1{\\cdot}25$. \\hl{For continuous $\\gamma$, the intervals overlap when $\\gamma$ is at least about $3{\\cdot}30$ and $1{\\cdot}43$, respectively.}", "intended_meaning": "I read the sentence as a summary of Table 1, whose rows are $\\gamma=1$, $1.25$, $1.5$, $2$, $2.5$, $3$ and $4$. On these rows both clauses are correct. The shooting gallery and needle exchange intervals are disjoint at $\\gamma=3$ and overlap at $\\gamma=4$, and the street and needle exchange intervals are disjoint at $\\gamma=1.25$ and overlap at $\\gamma=1.5$. For a continuous $\\gamma$, the first clause is a correct necessary condition, and the second clause would fail between $1.25$ and about $1.43$. At $\\gamma=1.3$, the lower bound for the street sample is $270/439$ and the upper bound for the needle exchange sample is $6539/11509$, which is smaller. For sample proportions $p_A>p_B$, the two intervals overlap exactly when $\\gamma^2\\geqslant p_A(1-p_B)/\\{p_B(1-p_A)\\}$. With the printed proportions $0.917$, $0.675$ and $0.503$, the thresholds are $\\sqrt{455749/41749}\\approx 3.304$ and $\\sqrt{13419/6539}\\approx 1.433$. The entries of the table are unchanged." }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-A001", "work": { "arxiv_id": null, "doi": "10.1093/biomet/ass064", "title": "Interval Estimation of Population Means under Unknown but Bounded Probabilities of Sample Selection", "url": "https://pmaronow.github.io/llm-audit/works/P025/" }, "work_id": "P025" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of four lines of the second paragraph of the Introduction, beginning Our approach serves two primary purposes for applied practitioners. Red outlines surround the words sharp bounds on the population mean, which run from the end of the second line to the start of the third.", "caption": "Section 1, second paragraph, p. 235 (PDF p. 1). The outlines mark the words “sharp bounds on the population mean”.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-N001.png" } ], "formal_release": null, "formal_statements": [], "headline": "At a finite sample size the bounds are sharp bounds on the Hajek estimator, and they bound the population mean only in the limit of Section 3.2.", "id": "P025-N001", "location": "Section 1, second paragraph, second sentence, p. 235 (PDF p. 1)", "pdf_page": 1, "printed_page": "235", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "First, when there is a theoretical basis for bounding the probabilities of sample selection, the proposed method yields \\hl{sharp bounds on the population mean}.", "corrected": "First, when there is a theoretical basis for bounding the probabilities of sample selection, the proposed method yields sharp bounds on \\hl{the Hajek estimator of} the population mean.", "intended_meaning": "I read the sentence as referring to the interval estimate of Section 3.1, which is the set of values that the Hajek estimator $\\hat\\mu$ can take when every $\\pi_i$ lies in $[\\alpha,\\beta]$. The Summary (p.~235) and Section 3.1 (p.~236) state the claim in this form, and it is correct. At a finite sample size the interval need not contain the population mean $\\mu$. For the population $(0,10)$ with one unit selected uniformly and $\\alpha=\\beta=1/2$, the interval is $[0,0]$ or $[10,10]$, and $\\mu=5$. The limiting statement about $\\mu$ is the subject of \\fid{P025-N002}." }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-N001", "work": { "arxiv_id": null, "doi": "10.1093/biomet/ass064", "title": "Interval Estimation of Population Means under Unknown but Bounded Probabilities of Sample Selection", "url": "https://pmaronow.github.io/llm-audit/works/P025/" }, "work_id": "P025" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop that begins However, W sub j is actually only identified up to the interval, continues through the displayed equations (6) and (7) for mu plus and mu minus, and ends with the first line of Proposition 2. Red outlines surround the words so in the limit, mu can be identified up to the interval bounded by, on two lines, and the words on mu are sharp in Proposition 2.", "caption": "Section 3.2, from the sentence before (6) to the first line of Proposition 2, p. 237 (PDF p. 3). The outlines mark the words “so in the limit, $\\mu$ can be identified up to the interval bounded by” and “on $\\mu$ are sharp”.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-N002.png" }, { "alt": "Crop of the heading 3.2 Asymptotic sharpness of estimated bounds and the sentence below it. A red outline surrounds the words asymptotically sharp bounds on mu itself.", "caption": "Section 3.2, heading and first sentence, p. 237 (PDF p. 3). The outline marks the words “asymptotically sharp bounds on $\\mu$ itself”.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-N002-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P025-T15", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T15" }, { "id": "P025-T16", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T16" } ], "headline": "The bounds $\\mu_-$ and $\\mu_+$ are sharp for the population mean relative to the limiting distribution of outcomes among sampled units, with the population size unrestricted, and the statement should name this information.", "id": "P025-N002", "location": "Section 3.2, first sentence, the sentence before (6), and the first sentence of Proposition 2, p. 237 (PDF p. 3)", "pdf_page": 3, "printed_page": "237", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P025-T15", "P025-T16" ], "text": { "as_printed": "The proposed interval estimator yields \\hl{asymptotically sharp bounds on $\\mu$ itself} under the scaling proposed by Brewer (1979) and articulated in Assumption 1.\n\nHowever, $W_j$ is actually only identified up to the interval $\\beta^{-1} \\leqslant W_j \\leqslant \\alpha^{-1}$; \\hl{so in the limit, $\\mu$ can be identified up to the interval bounded by}\n\\[\n\\mu_+ = \\max_W \\left\\{ \\frac{\\sum_{j=1}^{|Y|} W_j \\Pi_j y_{(j)}}{\\sum_{j=1}^{|Y|} W_j \\Pi_j} : 1/\\beta \\leqslant W_1, \\ldots, W_{|Y|} \\leqslant 1/\\alpha \\right\\} \\qquad (6)\n\\]\nand\n\\[\n\\mu_- = \\min_W \\left\\{ \\frac{\\sum_{j=1}^{|Y|} W_j \\Pi_j y_{(j)}}{\\sum_{j=1}^{|Y|} W_j \\Pi_j} : 1/\\beta \\leqslant W_1, \\ldots, W_{|Y|} \\leqslant 1/\\alpha \\right\\}. \\qquad (7)\n\\]\nPROPOSITION 2. In the limit $h \\to \\infty$, the bounds $\\mu_+$ and $\\mu_-$ \\hl{on $\\mu$ are sharp}. Specifically,", "corrected": "The proposed interval estimator yields asymptotically sharp bounds on $\\mu$ itself under the scaling proposed by Brewer (1979) and articulated in Assumption 1\\hl{, relative to the limiting distribution of the outcomes among sampled units}.\n\nHowever, $W_j$ is actually only identified up to the interval $\\beta^{-1} \\leqslant W_j \\leqslant \\alpha^{-1}$; so in the limit, \\hl{given only the shares $q_j=\\Pi_j/\\sum_{k=1}^{|Y|}\\Pi_k$ and with the population size $N$ unrestricted,} $\\mu$ can be identified up to the interval bounded by $\\mu_+$ in (6) and $\\mu_-$ in (7).\n\nPROPOSITION 2. \\hl{Let $0<\\alpha\\leqslant\\beta\\leqslant 1$, and let $q_j=\\Pi_j/\\sum_{k=1}^{|Y|}\\Pi_k$ for $j=1,\\ldots,|Y|$ be the limiting shares of the outcomes among sampled units. Suppose that only $q$ is retained from the limit $h\\to\\infty$ and that the size $N$ of the original population is unrestricted. Then} the bounds $\\mu_+$ and $\\mu_-$ on $\\mu$ are sharp\\hl{, in the following sense. Every finite population and sampling design with $\\alpha\\leqslant\\pi_i\\leqslant\\beta$ for all $i$ and with shares $q$ has $\\mu_-\\leqslant\\mu\\leqslant\\mu_+$, and $\\mu_-$ and $\\mu_+$ are the infimum and the supremum of the means of such populations.} Specifically, $\\mu_+$ and $\\mu_-$ are given by the two displayed formulas, with the corrections of \\fid{P025-E001} and \\fid{P025-E002}. \\hl{If $N$ is known, or if the totals $\\Pi_j$ are retained together with fixed numerical values of $\\alpha$ and $\\beta$, then $[\\mu_-,\\mu_+]$ still contains $\\mu$, and it can be strictly wider than the set of compatible means.}", "evidence": "The evidence is two exact counterexamples to the unqualified claim and a derivation of the qualified one.\n\n\\emph{Totals retained, $N$ unknown.} Let the population have outcomes $(0,1,1)$, and let each replication select one of the three samples of two units with probability $1/3$. Every $\\pi_i$ equals $2/3$, and I take $\\alpha=1/2$ and $\\beta=3/4$. The totals for the outcomes $0$ and $1$ are $2/3$ and $4/3$. The sizes $m_0$ and $m_1$ of the two classes satisfy $8/9\\leqslant m_0\\leqslant 4/3$ and $16/9\\leqslant m_1\\leqslant 8/3$, which implies that $(m_0,m_1)=(1,2)$ and $\\mu=2/3$. Equations (6) and (7) give $[\\mu_-,\\mu_+]=[4/7,3/4]$. The gaps $2/3-4/7=2/21$ and $3/4-2/3=1/12$ do not depend on $h$.\n\n\\emph{$N$ known.} Let $\\Pi=(50,100)$ for the outcomes $0$ and $1$, $\\alpha=1/10$, $\\beta=1/5$ and $N=1000$. The class sizes satisfy $250\\leqslant m_0\\leqslant 500$, $500\\leqslant m_1\\leqslant 1000$ and $m_0+m_1=1000$. The compatible means are $m_1/1000$ with $500\\leqslant m_1\\leqslant 750$, and each is attained by a design that selects $50$ zeros and $100$ ones uniformly. The sharp interval is $[1/2,3/4]$, and the interval defined by (6) and (7) is $[1/2,4/5]$.\n\n\\emph{Shares only, $N$ unrestricted.} Let $r_j=|\\Upsilon_j|/N$ be the population proportions, so that $\\mu=\\sum_jr_jy_{(j)}$ and $r_j=W_jq_j/\\sum_kW_kq_k$. As $W$ ranges over the box, $r$ ranges over the set\n\\[\n\\mathcal C_\\gamma(q)=\\Bigl\\{r:\\ r_j\\geqslant 0,\\ \\sum_jr_j=1,\\ r_jq_k\\leqslant\\gamma\\,r_kq_j \\text{ for all } j,k\\Bigr\\},\n\\]\nand $\\mu_-$ and $\\mu_+$ are the minimum and the maximum of $\\sum_jr_jy_{(j)}$ over $\\mathcal C_\\gamma(q)$. Since every compatible population has $r\\in\\mathcal C_\\gamma(q)$, it satisfies $\\mu_-\\leqslant\\mu\\leqslant\\mu_+$. Conversely, let $r\\in\\mathcal C_\\gamma(q)$ have rational coordinates, and let $d$ be an integer such that every $dr_j$ is an integer. I consider a population with $dr_j$ units of outcome $y_{(j)}$, each selected with probability $tq_j/r_j$, where $t$ satisfies\n\\[\n\\alpha\\max_j\\frac{r_j}{q_j}\\leqslant t\\leqslant\\beta\\min_j\\frac{r_j}{q_j}.\n\\]\nSuch a $t$ exists because $r\\in\\mathcal C_\\gamma(q)$. These probabilities lie in $[\\alpha,\\beta]$, the shares of the outcomes among sampled units are exactly $q$, and the population mean is $\\sum_jr_jy_{(j)}$. For $\\gamma>1$, every inequality that defines $\\mathcal C_\\gamma(q)$ is strict at $r=q$, which implies that the rational points are dense in $\\mathcal C_\\gamma(q)$. I conclude that the closure of the set of compatible population means is $[\\mu_-,\\mu_+]$. For $\\gamma=1$, the set $\\mathcal C_\\gamma(q)$ is the point $q$. In the first example, the shares are $q=(1/3,2/3)$, and both bounds are attained when only $q$ is retained. A population of three zeros, each selected with probability $1/2$, and four ones, each selected with probability $3/4$, has the mean $4/7$. A population of one zero, selected with probability $3/4$, and three ones, each selected with probability $1/2$, has the mean $3/4$.", "problem": "Proposition 2 and the first sentence of Section 3.2 call the interval $[\\mu_-,\\mu_+]$ sharp for $\\mu$ and do not say relative to which information. The supplementary material shows that $\\mu_-$ and $\\mu_+$ are the minimum and the maximum of the ratio over the box $1/\\beta\\leqslant W_j\\leqslant 1/\\alpha$. It does not show that populations compatible with what is observed in the limit have means that reach or approach these values, and whether they do depends on what is observed. The identification display on p.~237 names only the limiting shares $q_j=\\Pi_j/\\sum_k\\Pi_k$ of the outcomes among sampled units. Relative to these shares, with the population size $N$ unrestricted, the claim holds when sharp means that $\\mu_-$ and $\\mu_+$ are the infimum and the supremum of the compatible population means. The claim fails for richer information. The class sizes $|\\Upsilon_j|=W_j\\Pi_j$ are positive integers. If the totals $\\Pi_j$ themselves are retained, with fixed numerical $\\alpha$ and $\\beta$, these integers lie in $[\\Pi_j/\\beta,\\Pi_j/\\alpha]$, and if $N$ is known they sum to $N$. The interval can then be strictly narrowed. The same qualification applies to the conclusion drawn from Proposition 2 on p.~238 and to the sentence of Section 1 discussed in \\fid{P025-N001}. In every case $[\\mu_-,\\mu_+]$ contains $\\mu$.", "what_survives": "The formulas (6) and (7), the form of the optimal weights, the bounds on the Hajek estimator, the convergence of the estimated bounds to $\\mu_-$ and $\\mu_+$, Table 1 and the sensitivity analysis do not change. The interval $[\\mu_-,\\mu_+]$ is a valid bound on $\\mu$ whatever else is known. The stronger claim does not hold. When $N$ or the totals per replication are also known, the interval can be narrowed. Sharpness also does not mean that every point of the interval is the mean of a finite population." }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-N002", "work": { "arxiv_id": null, "doi": "10.1093/biomet/ass064", "title": "Interval Estimation of Population Means under Unknown but Bounded Probabilities of Sample Selection", "url": "https://pmaronow.github.io/llm-audit/works/P025/" }, "work_id": "P025" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of Table 1 with its caption, Estimated bounds for prevalence rates, in percent, of HIV proviral DNA in syringes collected as part of the New Haven syringe tracking and testing programme. The table has three panels, Shooting gallery n = 48, Street n = 160 and Needle exchange n = 579, each with columns gamma, lower bound and upper bound and seven rows for gamma from 1 to 4. Red outlines surround six entries: in the shooting gallery panel the lower bounds 84.7 at gamma 2 and 73.4 at gamma 4, and in the needle exchange panel the upper bound 55.9 at gamma 1.25, the lower bound 40.3 and the upper bound 60.3 at gamma 1.5, and the upper bound 71.7 at gamma 2.5.", "caption": "Table 1, p. 238 (PDF p. 4). The outlines mark the six entries that are $0.1$ too high: $84{\\cdot}7$ and $73{\\cdot}4$ for the shooting gallery, and $55{\\cdot}9$, $40{\\cdot}3$, $60{\\cdot}3$ and $71{\\cdot}7$ for the needle exchange.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-N003.png" } ], "formal_release": null, "formal_statements": [ { "id": "P025-T17", "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T17" } ], "headline": "Six entries of Table 1 are $0.1$ percentage point above the bounds computed from the counts that the sample sizes and percentages imply.", "id": "P025-N003", "location": "Table 1, p. 238 (PDF p. 4): shooting gallery, $\\hat\\mu^{\\mathrm L}$ at $\\gamma=2$ and $\\gamma=4$; needle exchange, $\\hat\\mu^{\\mathrm H}$ at $\\gamma=1{\\cdot}25$, both bounds at $\\gamma=1{\\cdot}5$, and $\\hat\\mu^{\\mathrm H}$ at $\\gamma=2{\\cdot}5$", "pdf_page": 4, "printed_page": "238", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P025-T17" ], "text": { "as_printed": "Table 1, estimated bounds (\\%). Shooting gallery, $n = 48$: at $\\gamma = 2$, $\\hat\\mu^{\\mathrm L} = \\hl{84{\\cdot}7}$; at $\\gamma = 4$, $\\hat\\mu^{\\mathrm L} = \\hl{73{\\cdot}4}$. Needle exchange, $n = 579$: at $\\gamma = 1{\\cdot}25$, $\\hat\\mu^{\\mathrm H} = \\hl{55{\\cdot}9}$; at $\\gamma = 1{\\cdot}5$, $\\hat\\mu^{\\mathrm L} = \\hl{40{\\cdot}3}$ and $\\hat\\mu^{\\mathrm H} = \\hl{60{\\cdot}3}$; at $\\gamma = 2{\\cdot}5$, $\\hat\\mu^{\\mathrm H} = \\hl{71{\\cdot}7}$.", "corrected": "Table 1, estimated bounds (\\%), computed from $44$ of $48$, $108$ of $160$ and $291$ of $579$ positive tests and rounded at the last step. Shooting gallery, $n = 48$: at $\\gamma = 2$, $\\hat\\mu^{\\mathrm L} = \\hl{84{\\cdot}6}$; at $\\gamma = 4$, $\\hat\\mu^{\\mathrm L} = \\hl{73{\\cdot}3}$. Needle exchange, $n = 579$: at $\\gamma = 1{\\cdot}25$, $\\hat\\mu^{\\mathrm H} = \\hl{55{\\cdot}8}$; at $\\gamma = 1{\\cdot}5$, $\\hat\\mu^{\\mathrm L} = \\hl{40{\\cdot}2}$ and $\\hat\\mu^{\\mathrm H} = \\hl{60{\\cdot}2}$; at $\\gamma = 2{\\cdot}5$, $\\hat\\mu^{\\mathrm H} = \\hl{71{\\cdot}6}$. The other 36 entries are unchanged.", "evidence": "I recalculated all 42 entries in exact rational arithmetic from the closed form $[p/\\{\\gamma(1-p)+p\\},\\ \\gamma p/\\{(1-p)+\\gamma p\\}]$. The counts are inferred from the sample sizes and percentages printed in Section 4 and not from the data of Kaplan (1991). The other 36 entries agree with the counts.\n\n\\begin{audittable}{llrrr}{The six entries of Table 1 that change, in percent.}\nSample & Entry & Exact value from the counts & Rounded & Printed \\\\\nShooting gallery & $\\hat\\mu^{\\mathrm L}$ at $\\gamma=2$ & $1100/13=84.615\\ldots$ & $84.6$ & $84.7$ \\\\\nShooting gallery & $\\hat\\mu^{\\mathrm L}$ at $\\gamma=4$ & $220/3=73.333\\ldots$ & $73.3$ & $73.4$ \\\\\nNeedle exchange & $\\hat\\mu^{\\mathrm H}$ at $\\gamma=1.25$ & $48500/869=55.811\\ldots$ & $55.8$ & $55.9$ \\\\\nNeedle exchange & $\\hat\\mu^{\\mathrm L}$ at $\\gamma=1.5$ & $9700/241=40.248\\ldots$ & $40.2$ & $40.3$ \\\\\nNeedle exchange & $\\hat\\mu^{\\mathrm H}$ at $\\gamma=1.5$ & $9700/161=60.248\\ldots$ & $60.2$ & $60.3$ \\\\\nNeedle exchange & $\\hat\\mu^{\\mathrm H}$ at $\\gamma=2.5$ & $48500/677=71.639\\ldots$ & $71.6$ & $71.7$ \\\\\n\\end{audittable}", "problem": "Section 4 reports $48$, $160$ and $579$ tested needles, of which $91.7\\%$, $67.5\\%$ and $50.3\\%$ tested positive. Each percentage is compatible with exactly one count of positive tests, namely $44$ of $48$, $108$ of $160$ and $291$ of $579$. The closed form on p.~237 defines $p$ as the sample proportion. With $p=44/48$ and $p=291/579$, six bounds round to a value $0.1$ below the printed one. All 42 printed entries are reproduced when the rounded values $0.917$, $0.675$ and $0.503$ are used as $p$. The six entries are therefore bounds for rounded inputs and not, at the precision shown, bounds for the samples described.", "what_survives": "No qualitative conclusion of Section 4 changes. For each pair of samples and each value of $\\gamma$ in the table, the two intervals overlap after the correction exactly when they overlap in the printed table. The closed form and the propositions do not depend on the table." }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-N003", "work": { "arxiv_id": null, "doi": "10.1093/biomet/ass064", "title": "Interval Estimation of Population Means under Unknown but Bounded Probabilities of Sample Selection", "url": "https://pmaronow.github.io/llm-audit/works/P025/" }, "work_id": "P025" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the last three lines of page 238. A red outline surrounds the words possibly show the same at the end of the last line, where the sentence breaks across the page.", "caption": "Section 4, start of the last sentence, foot of p. 238 (PDF p. 4). The outline marks the words “possibly show the same”.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-N006.png" }, { "alt": "Crop of the first three lines of page 239, which complete the last sentence of Section 4. Red outlines surround the words HIV prevalence rate at the start of the first line and the words must have been later in the same line.", "caption": "Section 4, end of the last sentence, p. 239 (PDF p. 5). The outlines mark the words “HIV prevalence rate” and “must have been”.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-N006-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "The comparison in Section 4 is between the estimated bounds for the samples in hand, and it does not allow for sampling error.", "id": "P025-N006", "location": "Section 4, last sentence, pp. 238–239 (PDF pp. 4–5)", "pdf_page": 4, "printed_page": "238–239", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "This means that for the shooting gallery needles to \\hl{possibly show the same HIV prevalence rate} as the exchange needles, infected shooting gallery needles \\hl{must have been} more than three times as likely to be tested than uninfected shooting gallery needles, with the converse holding for exchange needles.", "corrected": "For the shooting gallery and needle exchange \\hl{interval estimates} to overlap, the common bound on the ratios of selection probabilities must be at least about $3{\\cdot}30$. \\hl{This comparison is conditional on the observed samples and does not account for sampling uncertainty.}", "intended_meaning": "I read the paragraph as comparing the estimated bounds of Table 1 with the observed samples held fixed, and ``the same HIV prevalence rate'' refers to the estimates. On this reading the comparison of the intervals is correct. The paper attaches no confidence level to Table 1, and the shooting gallery sample has $48$ needles. Read as a condition for equal true prevalences, the sentence would not follow. Let $n$ be the size of a sample and $k$ its number of positive tests. Suppose that the population consists of $32$ blocks of $n$ needles, that $16$ blocks contain $k$ positives and $16$ contain $n-k$, and that one block is selected uniformly and tested in full. Every needle is tested with probability $1/32$, and the true prevalence is $1/2$. With independent selections in the three populations, the reported counts $44$ of $48$, $108$ of $160$ and $291$ of $579$ occur together with probability $1/8$. The claim about each selection ratio is corrected in \\fid{P025-E003}." }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-N006", "work": { "arxiv_id": null, "doi": "10.1093/biomet/ass064", "title": "Interval Estimation of Population Means under Unknown but Bounded Probabilities of Sample Selection", "url": "https://pmaronow.github.io/llm-audit/works/P025/" }, "work_id": "P025" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of the last three lines of the proof of Proposition 3 in the supplementary material. A red outline surrounds the words along each sample path omega in the second line, after the statement that the estimated upper bound tends to mu plus.", "caption": "Supplementary material, end of the proof of Proposition 3, p. 2. The outline marks the words “along each sample path $\\omega$”.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-N007.png" }, { "alt": "Crop of the heading Proof of Proposition 3 and the two lines below it. A red outline surrounds the opening words Fix a sample realization omega in Omega = {0, 1} to the power of the natural numbers.", "caption": "Supplementary material, start of the proof of Proposition 3, p. 2. The outline marks the words “Fix a sample realization $\\omega\\in\\Omega=\\{0,1\\}^{\\mathbb N}$”.", "url": "https://pmaronow.github.io/llm-audit/works/P025/exhibits/P025-N007-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "In the proof of Proposition 3, the estimated bound converges along almost every sample path, as the proposition states, and not along each one.", "id": "P025-N007", "location": "Supplementary material, proof of Proposition 3, first and last sentences, p. 2", "pdf_page": 2, "printed_page": "2", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "Fix a sample realization $\\hl{\\omega \\in \\Omega} = \\{0, 1\\}^{\\mathbb N}$ so that $I_1(\\omega), I_2(\\omega), \\ldots$ is a deterministic binary sequence. [\\ldots] By the strong law of large numbers $P_{h,j} \\to \\Pi_j$ as $h \\to \\infty$. [\\ldots] Then (B4) implies that $\\hat\\mu_h^{H} \\to \\mu_+$ \\hl{along each sample path $\\omega$}.", "corrected": "Fix a sample realization $\\omega \\in \\hl{\\Omega_0}$, \\hl{where $\\Omega_0 = \\{\\omega \\in \\Omega : P_{h,j}(\\omega) \\to \\Pi_j \\text{ for every } j\\}$ has probability one by the strong law of large numbers}, so that $I_1(\\omega), I_2(\\omega), \\ldots$ is a deterministic binary sequence. [\\ldots] Then (B4) implies that $\\hat\\mu_h^{H} \\to \\mu_+$ along \\hl{almost every} sample path $\\omega$.", "intended_meaning": "I apply the argument only to the event of probability one on which the strong law of large numbers gives $P_{h,j}\\to\\Pi_j$ for every $j$. Its conclusion is the convergence almost everywhere that Proposition 3 states. On a path outside the event the convergence can fail. For the population $(0,1)$ with one unit selected uniformly in each replication, $\\alpha=1/4$ and $\\beta=3/4$, Assumption 2 holds and $\\mu_+=3/4$, and on the path that selects the unit with outcome $0$ in every replication the upper bound equals $0$ for all $h$. This path has probability zero. The sample space $\\{0,1\\}^{\\mathbb N}$ consists of infinite sequences and needs no change." }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-N007", "work": { "arxiv_id": null, "doi": "10.1093/biomet/ass064", "title": "Interval Estimation of Population Means under Unknown but Bounded Probabilities of Sample Selection", "url": "https://pmaronow.github.io/llm-audit/works/P025/" }, "work_id": "P025" } ], "lean": { "applicable": true, "guide": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/docs/works/P025.md", "repository": "https://github.com/pmaronow/llm-audit-lean", "targets": [ { "declarations": [ { "claim_revision_id": "catalog:8d5b542dd90f9651cd4f9f17ffcdfd51fc949bcaf221219b64f415926ceb92aa", "conditional_on": [], "credit_reason": "The numerical witness is exact for the printed >= stopping rule on the valid cut domain. 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p. 236 (PDF p. 2). The realizable two-unit sample refutes the printed lower rule: its true range is [1/3,2/3], with the displayed residual certificates and independent inclusion indicators. For every sorted nonempty sample the printed rule stops at zero; the sample-mean residual and exact one-coordinate change explain its strict nonoptimality whenever the bound ratio exceeds one and the outcomes vary.

", "headline": "Proposition 1, second half (the rule for kLk^{\\mathrm L}kL, the lower optimiser wLw^{\\mathrm L}wL and the lower bound μ^L\\hat\\mu^{\\mathrm L}μ^​L), as printed: a two-unit sample in which the printed rule gives 1/21/21/2 while the minimum of (3) is 1/31/31/3", "location": "p. 236 (PDF p. 2)" }, "id": "P025-T1", "text": { "body": "p. 236 (PDF p. 2). The realizable two-unit sample refutes the printed lower rule: its true range is [1/3,2/3], with the displayed residual certificates and independent inclusion indicators. For every sorted nonempty sample the printed rule stops at zero; the sample-mean residual and exact one-coordinate change explain its strict nonoptimality whenever the bound ratio exceeds one and the outcomes vary.", "headline": "Proposition 1, second half (the rule for $k^{\\mathrm L}$, the lower optimiser $w^{\\mathrm L}$ and the lower bound $\\hat\\mu^{\\mathrm L}$), as printed: a two-unit sample in which the printed rule gives $1/2$ while the minimum of (3) is $1/3$", "location": "p. 236 (PDF p. 2)" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T1" }, { "declarations": [ { "claim_revision_id": "catalog:252bdeef02bb2fbda135167e68950a4467c05691c4b964c80e1b41b79c060a9c", "conditional_on": [], "credit_reason": "The witness concerns the printed optimizer in (7), not the separately mistyped lower-tail display. Design realization is read in R160-05; two singleton classes make the supplied marginal vector exactly the class-total vector.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.class_sign_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L162", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:252bdeef02bb2fbda135167e68950a4467c05691c4b964c80e1b41b79c060a9c", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P025-T2", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P025-T2" }, "html": { "body": "

p. 238 (PDF p. 4), below Table 1; (7), p. 237 (PDF p. 3). The two-class design, with independent Brewer replication, has singleton classes, totals (1/2,1), true weights (2,1), and mean 1/2. The printed lower cut gives 2/3, while the true extrema are 1/2 and 4/5, certified by the two displayed residuals; the upper first cut is one and Assumption 2 holds. For all strictly ordered positive class totals the printed rule stops at zero and is strictly nonoptimal when the bound ratio exceeds one and there are at least two classes.

", "headline": "Proposition 2, the rule for k−k_-k−​ and 'The optimal solution to (7) is W−W_-W−​', as printed: a two-class population in which the printed W−W_-W−​ gives 2/32/32/3 while the minimum of (7), and the population mean, is 1/21/21/2", "location": "p. 238 (PDF p. 4), directly below Table 1; (7) on p. 237 (PDF p. 3)" }, "id": "P025-T2", "text": { "body": "p. 238 (PDF p. 4), below Table 1; (7), p. 237 (PDF p. 3). The two-class design, with independent Brewer replication, has singleton classes, totals (1/2,1), true weights (2,1), and mean 1/2. The printed lower cut gives 2/3, while the true extrema are 1/2 and 4/5, certified by the two displayed residuals; the upper first cut is one and Assumption 2 holds. For all strictly ordered positive class totals the printed rule stops at zero and is strictly nonoptimal when the bound ratio exceeds one and there are at least two classes.", "headline": "Proposition 2, the rule for $k_-$ and 'The optimal solution to (7) is $W_-$', as printed: a two-class population in which the printed $W_-$ gives $2/3$ while the minimum of (7), and the population mean, is $1/2$", "location": "p. 238 (PDF p. 4), directly below Table 1; (7) on p. 237 (PDF p. 3)" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T2" }, { "declarations": [ { "claim_revision_id": "catalog:c8d62c2a2b90f02342719c651377517b3162b2d831fe222c5b50a19094f6a6fd", "conditional_on": [], "credit_reason": "The selected corrected_weighted_bounds theorem exports the sample result together with the prescribed vectors and endpoint identities.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.corrected_proposition_one", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L846", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:c8d62c2a2b90f02342719c651377517b3162b2d831fe222c5b50a19094f6a6fd", "conditional_on": [], "credit_reason": "This is the described auxiliary denominator lemma, not a standalone proof of Proposition 1; the target supplies nonemptiness and strict positivity.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.denominator_pos", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L196", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:c8d62c2a2b90f02342719c651377517b3162b2d831fe222c5b50a19094f6a6fd", "conditional_on": [], "credit_reason": "The auxiliary average bound applies to the last outcome in sorted data, giving the last valid cut test for either threshold orientation.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.ratio_le", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L203", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:c8d62c2a2b90f02342719c651377517b3162b2d831fe222c5b50a19094f6a6fd", "conditional_on": [], "credit_reason": "The numerical and general counterexample rows already contain the needed lower-mean conclusions.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.le_ratio", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L214", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:c8d62c2a2b90f02342719c651377517b3162b2d831fe222c5b50a19094f6a6fd", "conditional_on": [], "credit_reason": "Judged as the described weight-scaling auxiliary fact. 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p. 236 (PDF p. 2); Appendix, pp. 239-240 (PDF pp. 5-6). Both corrected first cuts exist. Their prescribed feasible threshold vectors attain the global minimum and maximum and the full interval of ratios. The objective is continuous on their feasible connecting segment, which attains every intermediate value. Both increment identities and the lower separator and termwise residual certificate hold, including original-scale weights, tied outcomes, singleton samples, and equal bounds.

", "headline": "Proposition 1 and equation (5), corrected: ⩽\\leqslant⩽ in the rule for kLk^{\\mathrm L}kL", "location": "p. 236 (PDF p. 2); proof in the Appendix, pp. 239-240 (PDF pp. 5-6), which omits the lower half ('The solution to (3) can be derived in a similar manner.')" }, "id": "P025-T3", "text": { "body": "p. 236 (PDF p. 2); Appendix, pp. 239-240 (PDF pp. 5-6). Both corrected first cuts exist. Their prescribed feasible threshold vectors attain the global minimum and maximum and the full interval of ratios. The objective is continuous on their feasible connecting segment, which attains every intermediate value. Both increment identities and the lower separator and termwise residual certificate hold, including original-scale weights, tied outcomes, singleton samples, and equal bounds.", "headline": "Proposition 1 and equation (5), corrected: $\\leqslant$ in the rule for $k^{\\mathrm L}$", "location": "p. 236 (PDF p. 2); proof in the Appendix, pp. 239-240 (PDF pp. 5-6), which omits the lower half ('The solution to (3) can be derived in a similar manner.')" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T3" }, { "declarations": [ { "claim_revision_id": "catalog:b9e9e446ea8962b40f3ae28ff8e5aed6e7d7c0bbd8afb85bf7214b8397f9ad0c", "conditional_on": [], "credit_reason": "Faithful to the credited endpoint/first-cut component; this does not provide full coverage of the target, which additionally requests increment identities and explicit last-cut inequalities. No assumption of separation, uniqueness, or the desired optimizer conclusion is hidden in the definitions.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.corrected_proposition_two", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L861", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:b9e9e446ea8962b40f3ae28ff8e5aed6e7d7c0bbd8afb85bf7214b8397f9ad0c", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P025-T4", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P025-T4" }, "html": { "body": "

pp. 237-238 (PDF pp. 3-4); proof in the Supplement, pp. 1-2. Both corrected first class cuts exist. Their prescribed original-scale threshold vectors are feasible, have the displayed disjoint prefix/tail values, and attain the global minimum and maximum. Both last-cut inequalities and both class-weight increment identities hold, including equal probability bounds. Every displayed denominator is positive; optimality does not assert uniqueness.

", "headline": "Proposition 2, the closed forms of μ+\\mu_+μ+​ and μ−\\mu_-μ−​ and the optimisers W+W_+W+​, W−W_-W−​, corrected: ⩽\\leqslant⩽ in the rule for k−k_-k−​ (and tail sums of μ−\\mu_-μ−​ from j=k−+1j = k_- + 1j=k−​+1)", "location": "pp. 237-238 (PDF pp. 3-4); proof in the Supplement, pp. 1-2, which omits the lower half ('the result for μ−\\mu_-μ−​ can be established in the same fashion')" }, "id": "P025-T4", "text": { "body": "pp. 237-238 (PDF pp. 3-4); proof in the Supplement, pp. 1-2. Both corrected first class cuts exist. Their prescribed original-scale threshold vectors are feasible, have the displayed disjoint prefix/tail values, and attain the global minimum and maximum. Both last-cut inequalities and both class-weight increment identities hold, including equal probability bounds. Every displayed denominator is positive; optimality does not assert uniqueness.", "headline": "Proposition 2, the closed forms of $\\mu_+$ and $\\mu_-$ and the optimisers $W_+$, $W_-$, corrected: $\\leqslant$ in the rule for $k_-$ (and tail sums of $\\mu_-$ from $j = k_- + 1$)", "location": "pp. 237-238 (PDF pp. 3-4); proof in the Supplement, pp. 1-2, which omits the lower half ('the result for $\\mu_-$ can be established in the same fashion')" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T4" }, { "declarations": [ { "claim_revision_id": "catalog:e26caf67c9bb0a8fb2fb37603ae2cbd3106ea19b10a48246253973fbfe5b361f", "conditional_on": [], "credit_reason": "The arithmetic witness matches its credited numerical component. 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pp. 238-239 (PDF pp. 4-5); Table 1 on p. 238 (PDF p. 4). These two finite population pairs refute the reading of the application sentence that equal gallery and exchange population prevalences require both actual directional selection ratios to exceed three: the pairs have equal prevalences and the reported samples, while respectively the exchange and gallery selection ratio is below three. Their joint independent uniform fixed-quota sampling designs are normalized, have the stated actual unit marginals in [1/4,1], and on every draw yield the reported counts and equal Hajek estimates. The interval memberships, rounding bounds, ratio identities, and both exact rounded-input variants hold.

", "headline": "Section 4, last sentence ('infected shooting gallery needles must have been more than three times as likely to be tested ... with the converse holding for exchange needles'), as printed: exact designs with equal prevalence in which one of the two selection ratios is below three", "location": "pp. 238-239 (PDF pp. 4-5); Table 1 on p. 238 (PDF p. 4)" }, "id": "P025-T5", "text": { "body": "pp. 238-239 (PDF pp. 4-5); Table 1 on p. 238 (PDF p. 4). These two finite population pairs refute the reading of the application sentence that equal gallery and exchange population prevalences require both actual directional selection ratios to exceed three: the pairs have equal prevalences and the reported samples, while respectively the exchange and gallery selection ratio is below three. Their joint independent uniform fixed-quota sampling designs are normalized, have the stated actual unit marginals in [1/4,1], and on every draw yield the reported counts and equal Hajek estimates. The interval memberships, rounding bounds, ratio identities, and both exact rounded-input variants hold.", "headline": "Section 4, last sentence ('infected shooting gallery needles must have been more than three times as likely to be tested ... with the converse holding for exchange needles'), as printed: exact designs with equal prevalence in which one of the two selection ratios is below three", "location": "pp. 238-239 (PDF pp. 4-5); Table 1 on p. 238 (PDF p. 4)" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T5" }, { "declarations": [ { "claim_revision_id": "catalog:ee7f192fb42a84978d301f62110663580e88a54d426b0af85a7168bdc2744469", "conditional_on": [], "credit_reason": "Closed intervals include equality at the threshold. 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pp. 238-239 (PDF pp. 4-5). The binary intervals overlap at the exact odds-ratio threshold, with both numerical thresholds, the reweighted-odds and ratio-product identities, and the common-cap consequence.

", "headline": "Section 4, last sentence, corrected: overlap of the gallery and exchange intervals constrains a common bound on the selection ratios (about 3·30), equivalently the product of the two ratios, and not each ratio", "location": "pp. 238-239 (PDF pp. 4-5)" }, "id": "P025-T6", "text": { "body": "pp. 238-239 (PDF pp. 4-5). The binary intervals overlap at the exact odds-ratio threshold, with both numerical thresholds, the reweighted-odds and ratio-product identities, and the common-cap consequence.", "headline": "Section 4, last sentence, corrected: overlap of the gallery and exchange intervals constrains a common bound on the selection ratios (about 3·30), equivalently the product of the two ratios, and not each ratio", "location": "pp. 238-239 (PDF pp. 4-5)" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T6" }, { "declarations": [ { "claim_revision_id": "catalog:ad3474bea059a0219840a18e5c61039a262a6de0cf4fc8a3ff43532d4b2272e0", "conditional_on": [], "credit_reason": "Full refutation of the printed optimizer/range conjunction; no positive claim is credited as proved.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.printed_proposition_one_refuted", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_PrintedRefutations.lean#L7", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:ad3474bea059a0219840a18e5c61039a262a6de0cf4fc8a3ff43532d4b2272e0", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P025/P025_Interval.lean:2607 supplies the least upper cut satisfying the printed next-outcome test and asserts that its value is the greatest element of the full physical-box attainable set. Expanding upperValue and upperWeights at lines 75 and 65 gives exactly the upper formula (4), with the claim's full domain of nonempty sorted samples and 0 < alpha <= beta <= 1. Multiplying the normalized threshold vector by 1/beta yields the printed feasible optimizer, with entries 1/beta and 1/alpha and the same objective value. Thus the scaling convention loses none of the upper component credited to this row.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.printed_proposition_one_upper", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L2586", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:ad3474bea059a0219840a18e5c61039a262a6de0cf4fc8a3ff43532d4b2272e0", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P025-T7", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P025-T7" }, "html": { "body": "

The printed claim is false: P025-T7; p. 236 (PDF p. 2); proof of the upper half in the Appendix, pp. 239-240 (PDF pp. 5-6). The printed sample lower optimizer and range assertion fail, while the upper first-cut formula remains valid.

", "headline": "Proposition 1 and equation (5), as printed", "location": "p. 236 (PDF p. 2); proof of the upper half in the Appendix, pp. 239-240 (PDF pp. 5-6)" }, "id": "P025-T7", "text": { "body": "The printed claim is false: P025-T7; p. 236 (PDF p. 2); proof of the upper half in the Appendix, pp. 239-240 (PDF pp. 5-6). The printed sample lower optimizer and range assertion fail, while the upper first-cut formula remains valid.", "headline": "Proposition 1 and equation (5), as printed", "location": "p. 236 (PDF p. 2); proof of the upper half in the Appendix, pp. 239-240 (PDF pp. 5-6)" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T7" }, { "declarations": [ { "claim_revision_id": "catalog:f6d4bdc2577d38b3567230e0885a92e8631a2a2165ec0cc207e2ccee974455dc", "conditional_on": [], "credit_reason": "The old exported negation is vulnerable to an undefined Option display and is not used for the new feasible non-minimizer proof.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.printed_proposition_two_formula_refuted", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_PrintedRefutations.lean#L29", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:f6d4bdc2577d38b3567230e0885a92e8631a2a2165ec0cc207e2ccee974455dc", "conditional_on": [], "credit_reason": "Concerns separate sharpness with fixed absolute totals, outside this lower-optimizer refutation.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.printed_sharpness_absolute_information_refuted", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_PrintedRefutations.lean#L53", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:f6d4bdc2577d38b3567230e0885a92e8631a2a2165ec0cc207e2ccee974455dc", "conditional_on": [], "credit_reason": "The printed upper cut, maximum and endpoint equality are proved for arbitrary positive class masses, without needing finite-population realizability constraints.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.printed_proposition_two_upper", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L2595", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:f6d4bdc2577d38b3567230e0885a92e8631a2a2165ec0cc207e2ccee974455dc", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P025-T8", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P025-T8" }, "html": { "body": "

The printed claim is false: P025-T8; pp. 237-238 (PDF pp. 3-4); proof in the Supplement, pp. 1-2. The printed lower first-cut rule returns zero. At a realizable six-unit, four-outcome design with γ = 2, the resulting weight vector has value 3/2, while the attainable minimum is 7/6, so it is not the optimizer of (7). The separate lower display is undefined at that printed cut; this observation does not carry the refutation. The upper first-cut rule, optimizer, and endpoint formula remain valid.

", "headline": "Proposition 2, as printed", "location": "pp. 237-238 (PDF pp. 3-4); proof in the Supplement, pp. 1-2" }, "id": "P025-T8", "text": { "body": "The printed claim is false: P025-T8; pp. 237-238 (PDF pp. 3-4); proof in the Supplement, pp. 1-2. The printed lower first-cut rule returns zero. At a realizable six-unit, four-outcome design with γ = 2, the resulting weight vector has value 3/2, while the attainable minimum is 7/6, so it is not the optimizer of (7). The separate lower display is undefined at that printed cut; this observation does not carry the refutation. The upper first-cut rule, optimizer, and endpoint formula remain valid.", "headline": "Proposition 2, as printed", "location": "pp. 237-238 (PDF pp. 3-4); proof in the Supplement, pp. 1-2" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T8" }, { "declarations": [ { "claim_revision_id": "catalog:79f2534de905e195d3c7ad83599f79992d885d62c2f57ba79cfc645120c1ac28", "conditional_on": [], "credit_reason": "The original-probability-weight partition identity holds for all real outcome and mass vectors and all natural cuts, encompassing every admissible class cut.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.lower_partition_original_box", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L1227", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:79f2534de905e195d3c7ad83599f79992d885d62c2f57ba79cfc645120c1ac28", "conditional_on": [], "credit_reason": "The disjoint partition identity is proved for arbitrary real threshold weights and vectors, including all weights and cuts needed by the target.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.threshold_partition", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L876", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:79f2534de905e195d3c7ad83599f79992d885d62c2f57ba79cfc645120c1ac28", "conditional_on": [], "credit_reason": "The supplied k is a Fin J first corrected cut, so 0<=kp. 237 (PDF p. 3). The corrected lower display partitions every threshold vector correctly, equals the minimum at a corrected first cut, and gives the exact duplicated-class error and undefined-index cases.

", "headline": "Proposition 2, displayed formula for μ−\\mu_-μ−​, corrected: both second sums start at j=k−+1j = k_- + 1j=k−​+1", "location": "p. 237 (PDF p. 3)" }, "id": "P025-T9", "text": { "body": "p. 237 (PDF p. 3). The corrected lower display partitions every threshold vector correctly, equals the minimum at a corrected first cut, and gives the exact duplicated-class error and undefined-index cases.", "headline": "Proposition 2, displayed formula for $\\mu_-$, corrected: both second sums start at $j = k_- + 1$", "location": "p. 237 (PDF p. 3)" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T9" }, { "declarations": [ { "claim_revision_id": "catalog:2cb6a3596995cd397415dda46747ede1a203a3094e4eeae3d9bfa9e0dda96c40", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P025/P025_Interval.lean:951 states the corrected first lower cut 2, its attained value 7/6, and a lower bound of 7/6 for every weight vector in [1,2]^4, establishing the whole-box minimum required for this numerical counterexample. It evaluates the printed overlapping-tail expression at that same cut as 1, and separately states that the printed stopping rule selects 0, where the literal display is undefined. The six-unit design and its inclusion probabilities, class totals, class sizes and population mean are explicitly supplied by P025/P025_Interval.lean:1326. Using the independently correct cut to isolate the tail-index defect matches parts (c)-(e) of T10 exactly.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.tail_index_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L950", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:2cb6a3596995cd397415dda46747ede1a203a3094e4eeae3d9bfa9e0dda96c40", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.tail_design_realizes_example", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L1321", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:2cb6a3596995cd397415dda46747ede1a203a3094e4eeae3d9bfa9e0dda96c40", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P025-T10", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P025-T10" }, "html": { "body": "

p. 237 (PDF p. 3). The six-unit design realizes the four-class counterexample. Over the whole real weight box its minimum is 7/6, uniquely at (2,2,1,1), as certified by the displayed residual equality and its nonnegative slack terms. The corrected first cut is two, where the printed display gives 1 instead of 7/6; the printed stopping rule gives zero, where that display is undefined.

", "headline": "Proposition 2, displayed formula for μ−\\mu_-μ−​ with tail sums from j=k−−1j = k_- - 1j=k−​−1, as printed: four classes in which the display evaluates to 111 while the minimum of (7) is 7/67/67/6", "location": "p. 237 (PDF p. 3)" }, "id": "P025-T10", "text": { "body": "p. 237 (PDF p. 3). The six-unit design realizes the four-class counterexample. Over the whole real weight box its minimum is 7/6, uniquely at (2,2,1,1), as certified by the displayed residual equality and its nonnegative slack terms. The corrected first cut is two, where the printed display gives 1 instead of 7/6; the printed stopping rule gives zero, where that display is undefined.", "headline": "Proposition 2, displayed formula for $\\mu_-$ with tail sums from $j = k_- - 1$, as printed: four classes in which the display evaluates to $1$ while the minimum of (7) is $7/6$", "location": "p. 237 (PDF p. 3)" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T10" }, { "declarations": [ { "claim_revision_id": "catalog:5c7fa6e94867be5883b14601a1436d97d5cdf72da4685cf42dbf0682150b20c6", "conditional_on": [], "credit_reason": "The two residual equivalences, existence of an improving move, and coincident-bound zero differences hold for general positive real weight bounds.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.pair_boundary_repair", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L1185", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:5c7fa6e94867be5883b14601a1436d97d5cdf72da4685cf42dbf0682150b20c6", "conditional_on": [], "credit_reason": "The gamma-one threshold and singleton normalized-box components hold for arbitrary outcomes and even empty finite index types.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.equal_probability_thresholds", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L1089", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:5c7fa6e94867be5883b14601a1436d97d5cdf72da4685cf42dbf0682150b20c6", "conditional_on": [], "credit_reason": "The record's repair-history and approval text supplies no additional mathematical premise; the current statement includes gamma=1 in the identity exactly as required.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.Wave1.gamma_stationarity_certificate", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/Wave1/Gap/FidP025.lean#L42", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:5c7fa6e94867be5883b14601a1436d97d5cdf72da4685cf42dbf0682150b20c6", "conditional_on": [], "credit_reason": "The monotonicity premise matches the paper setting, although this single-step inequality itself does not need it. Target-level first-index and global-maximum conclusions are not stated here.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.upper_cut_persistence", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L1135", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:5c7fa6e94867be5883b14601a1436d97d5cdf72da4685cf42dbf0682150b20c6", "conditional_on": [], "credit_reason": "The cross-multiplied identities and strictly negative residual sum hold for all positive ordered real weight bounds, including the probability-weight specialization.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.pair_difference_identities", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L1159", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:5c7fa6e94867be5883b14601a1436d97d5cdf72da4685cf42dbf0682150b20c6", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P025-T11", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P025-T11" }, "html": { "body": "

pp. 239-240 (PDF pp. 5-6); Supplement p. 2. The repaired proof includes the original-scale threshold formulas and increments. At equal bounds the sample box is a singleton and both endpoints are the sample mean; its outcome-test first cut exists, while the first stationary index is zero. At strict bounds the stationary and outcome-test first cuts agree and attain the global threshold maximum. The exact two-coordinate identities give strict improvement only for strict bounds. At equal inclusion probabilities the actual class totals are probability times class size, and both class optimization problems attain the finite-population mean.

", "headline": "Appendix, proof of Proposition 1 (the implication on p. 239 and the display 'Thus kH=min⁡{…}=min⁡{…}k^{\\mathrm H} = \\min\\{\\ldots\\} = \\min\\{\\ldots\\}kH=min{…}=min{…}' on p. 240), and (B1)-(B2) in the Supplement's proof of Proposition 2, corrected: the case γ=1\\gamma = 1γ=1 treated separately", "location": "pp. 239-240 (PDF pp. 5-6); Supplement p. 2" }, "id": "P025-T11", "text": { "body": "pp. 239-240 (PDF pp. 5-6); Supplement p. 2. The repaired proof includes the original-scale threshold formulas and increments. At equal bounds the sample box is a singleton and both endpoints are the sample mean; its outcome-test first cut exists, while the first stationary index is zero. At strict bounds the stationary and outcome-test first cuts agree and attain the global threshold maximum. The exact two-coordinate identities give strict improvement only for strict bounds. At equal inclusion probabilities the actual class totals are probability times class size, and both class optimization problems attain the finite-population mean.", "headline": "Appendix, proof of Proposition 1 (the implication on p. 239 and the display 'Thus $k^{\\mathrm H} = \\min\\{\\ldots\\} = \\min\\{\\ldots\\}$' on p. 240), and (B1)-(B2) in the Supplement's proof of Proposition 2, corrected: the case $\\gamma = 1$ treated separately", "location": "pp. 239-240 (PDF pp. 5-6); Supplement p. 2" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T11" }, { "declarations": [ { "claim_revision_id": "catalog:f35d67320c69e530b01afd15614198c5facf7d2509e0d6c14348f9ccea517033", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P025/P025_Interval.lean:988 states that the three threshold values are 1/2, the nonpositive-increment implication fails, and the stationary and next-outcome rules first stop at 0 and 1 respectively, as T12 specifies. It also evaluates both Supplement residuals as -1 and denies strict improvement of the unit-weight ratio over itself. At alpha=beta=1 both replacement pairs and the original pair are (1,1), so this single strict self-comparison covers both failed Supplement equivalences. The same singleton box forces both true optimization endpoints to equal 1/2, so the endpoint-preservation context introduces no missing nontrivial case. The census realization is stated at P025/P025_Interval.lean:1356.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.equal_probability_proof_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L987", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f35d67320c69e530b01afd15614198c5facf7d2509e0d6c14348f9ccea517033", "conditional_on": [], "credit_reason": "The record's generic stronger-difference template is inapplicable: this theorem concerns one fixed two-unit census, with no generalized population or design quantifiers.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.census_realizes_boundary", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L1351", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:f35d67320c69e530b01afd15614198c5facf7d2509e0d6c14348f9ccea517033", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P025-T12", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P025-T12" }, "html": { "body": "

pp. 239-240 (PDF pp. 5-6); Supplement p. 2. The printed claim is false: the two-unit census with outcomes (0,1) and equal probability bounds has threshold values 1/2 and different least stopping indices. It refutes the Appendix implication and cut equivalence and both Supplement equivalences. Its feasible box is a single point; all sample and class endpoints and both means are 1/2. The first upper cut is one and its physical optimizer and displayed value still give 1/2.

", "headline": "Appendix, proof of Proposition 1: 'Then (wk+1′y)/(wk+1′1n)−(wk′y)/(wk′1n)⩽0(w^{k+1\\prime}y)/(w^{k+1\\prime}1_n) - (w^{k\\prime}y)/(w^{k\\prime}1_n) \\leqslant 0(wk+1′y)/(wk+1′1n​)−(wk′y)/(wk′1n​)⩽0 implies that ...' (p. 239) and 'Thus kH=min⁡{…}=min⁡{…}k^{\\mathrm H} = \\min\\{\\ldots\\} = \\min\\{\\ldots\\}kH=min{…}=min{…}' (p. 240); Supplement, the equivalences with (B1) and (B2): all fail at γ=1\\gamma = 1γ=1 with y=(0,1)y = (0, 1)y=(0,1)", "location": "pp. 239-240 (PDF pp. 5-6); Supplement p. 2" }, "id": "P025-T12", "text": { "body": "pp. 239-240 (PDF pp. 5-6); Supplement p. 2. The printed claim is false: the two-unit census with outcomes (0,1) and equal probability bounds has threshold values 1/2 and different least stopping indices. It refutes the Appendix implication and cut equivalence and both Supplement equivalences. Its feasible box is a single point; all sample and class endpoints and both means are 1/2. The first upper cut is one and its physical optimizer and displayed value still give 1/2.", "headline": "Appendix, proof of Proposition 1: 'Then $(w^{k+1\\prime}y)/(w^{k+1\\prime}1_n) - (w^{k\\prime}y)/(w^{k\\prime}1_n) \\leqslant 0$ implies that ...' (p. 239) and 'Thus $k^{\\mathrm H} = \\min\\{\\ldots\\} = \\min\\{\\ldots\\}$' (p. 240); Supplement, the equivalences with (B1) and (B2): all fail at $\\gamma = 1$ with $y = (0, 1)$", "location": "pp. 239-240 (PDF pp. 5-6); Supplement p. 2" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T12" }, { "declarations": [ { "claim_revision_id": "catalog:f8b7fbef25be2883968c76fe9e842668668786c1b10c7ca5c848f69e78f726f5", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'P025/P025_Interval.lean:2173 asserts almost-sure convergence of both actual pooled-sample extrema to the class-total extrema, together with eventual nonemptiness, exactly as corrected T13(iv) requires. FiniteDesign permits every distribution over subsets of a finite population, and its canonical iid product therefore loses no sampling-design law or within-sample dependence. Surjective class labels and strictly increasing class outcomes encode precisely the distinct nonempty outcome classes. 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p. 238 (PDF p. 4); proof in the Supplement, p. 2. Under independent Brewer replication, both actual pooled extrema converge almost surely to the class optimization endpoints without separation, and hence also with the finite-index Assumption 2.

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p. 238 (PDF p. 4). The printed claim is false: the six-unit random pair design satisfies independent Brewer replication and violates Assumption 2, yet every nonempty pooled sample has endpoints 5/4 and 2 on every path. Both polynomial certificates and all three optimizer evaluations hold. Pooled class sums lie in [2h,4h], give the displayed ratio and its normalization by h, and map onto the entire class weight box.

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p. 237 (PDF p. 3), echoed on pp. 235 and 238; Supplement pp. 1-2. Conditional only on positive limiting shares and with population size unrestricted, the class box endpoints are the sharp infimum and supremum of compatible finite population means. The box maps onto the proportion region; its rational points are precisely finite population proportions, with the stated endpoint attainment criteria. Compatibility is automatic at ratio greater than one and is equivalent to rational shares at ratio one. Positive scaling preserves the objectives and endpoints. Bounded class marginals are realized by finite sampling designs. Absolute totals and known size give the stated integer calibration and outer bounds; sharpness can fail with totals retained or with shares and size retained.

", "headline": "Proposition 2, first sentence ('In the limit h→∞h \\to \\inftyh→∞, the bounds μ+\\mu_+μ+​ and μ−\\mu_-μ−​ on μ\\muμ are sharp'), with the sentence before (6) and the opening sentence of Section 3.2, corrected: sharp as infimum and supremum of the population means compatible with the limiting shares of outcomes among sampled units, the population size being unrestricted", "location": "p. 237 (PDF p. 3); echoed in Section 1, p. 235 (PDF p. 1) and on p. 238 (PDF p. 4); proof in the Supplement, pp. 1-2" }, "id": "P025-T15", "text": { "body": "p. 237 (PDF p. 3), echoed on pp. 235 and 238; Supplement pp. 1-2. Conditional only on positive limiting shares and with population size unrestricted, the class box endpoints are the sharp infimum and supremum of compatible finite population means. The box maps onto the proportion region; its rational points are precisely finite population proportions, with the stated endpoint attainment criteria. Compatibility is automatic at ratio greater than one and is equivalent to rational shares at ratio one. Positive scaling preserves the objectives and endpoints. Bounded class marginals are realized by finite sampling designs. Absolute totals and known size give the stated integer calibration and outer bounds; sharpness can fail with totals retained or with shares and size retained.", "headline": "Proposition 2, first sentence ('In the limit $h \\to \\infty$, the bounds $\\mu_+$ and $\\mu_-$ on $\\mu$ are sharp'), with the sentence before (6) and the opening sentence of Section 3.2, corrected: sharp as infimum and supremum of the population means compatible with the limiting shares of outcomes among sampled units, the population size being unrestricted", "location": "p. 237 (PDF p. 3); echoed in Section 1, p. 235 (PDF p. 1) and on p. 238 (PDF p. 4); proof in the Supplement, pp. 1-2" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T15" }, { "declarations": [ { "claim_revision_id": "catalog:a77859a8b747b6a113060e5d7794efb71e864e7e0597288d8657d7181cf5412f", "conditional_on": [], "credit_reason": "Faithful to T16’s expressly fixed absolute-total information set. This certificate does not refute sharpness conditional only on normalized shares, and is not credited as such.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.integrality_mean_set_and_gaps", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L2565", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:a77859a8b747b6a113060e5d7794efb71e864e7e0597288d8657d7181cf5412f", "conditional_on": [], "credit_reason": "Only T16(A)(i) is assigned to this supporting finite-design lemma.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.integrality_design_realizes_example", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L1369", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:a77859a8b747b6a113060e5d7794efb71e864e7e0597288d8657d7181cf5412f", "conditional_on": [], "credit_reason": "The row is supporting witness arithmetic despite status counterexample_proved; design realization and global box extrema are certified by separate rows. 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p. 237 (PDF p. 3); Supplement p. 2. These examples refute the reading of the printed sharpness assertion that retains absolute class totals, with population size also retained in the second example: retained absolute totals can force the singleton mean set {2/3} strictly within [4/7,3/4], even when Assumption 2 holds. With totals (50,100) and size 1000 the exact means are k/1000 for 500 ≤ k ≤ 750, whose maximum 3/4 is below the continuous upper endpoint 4/5. All four specified weight vectors give the stated endpoints. Every such integer k is realized by independent uniform quota sampling with the specified probabilities and totals; the 500-zero, 500-one example has mean 1/2 and probabilities 1/10 and 1/5. Independent replication holds, outer bounds remain valid, and shares alone with unrestricted size have the corrected sharp endpoints.

", "headline": "Proposition 2, first sentence ('In the limit h→∞h \\to \\inftyh→∞, the bounds μ+\\mu_+μ+​ and μ−\\mu_-μ−​ on μ\\muμ are sharp'), as printed and read with the limiting information of the replication experiment of Assumption 1: populations in which the compatible means form a strict subset of [μ−,μ+][\\mu_-, \\mu_+][μ−​,μ+​]", "location": "p. 237 (PDF p. 3); Supplement p. 2 (Ph,j→ΠjP_{h,j} \\to \\Pi_jPh,j​→Πj​)" }, "id": "P025-T16", "text": { "body": "p. 237 (PDF p. 3); Supplement p. 2. These examples refute the reading of the printed sharpness assertion that retains absolute class totals, with population size also retained in the second example: retained absolute totals can force the singleton mean set {2/3} strictly within [4/7,3/4], even when Assumption 2 holds. With totals (50,100) and size 1000 the exact means are k/1000 for 500 ≤ k ≤ 750, whose maximum 3/4 is below the continuous upper endpoint 4/5. All four specified weight vectors give the stated endpoints. Every such integer k is realized by independent uniform quota sampling with the specified probabilities and totals; the 500-zero, 500-one example has mean 1/2 and probabilities 1/10 and 1/5. Independent replication holds, outer bounds remain valid, and shares alone with unrestricted size have the corrected sharp endpoints.", "headline": "Proposition 2, first sentence ('In the limit $h \\to \\infty$, the bounds $\\mu_+$ and $\\mu_-$ on $\\mu$ are sharp'), as printed and read with the limiting information of the replication experiment of Assumption 1: populations in which the compatible means form a strict subset of $[\\mu_-, \\mu_+]$", "location": "p. 237 (PDF p. 3); Supplement p. 2 ($P_{h,j} \\to \\Pi_j$)" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T16" }, { "declarations": [ { "claim_revision_id": "catalog:e29a323421f62457b646541778afb7eab2a58b2c285ea072872e62de5d3711a3", "conditional_on": [], "credit_reason": "The full literal table transcription was compared with all 42 entries supplied in definitions_needed; the theorem certifies its numerical rounding rather than assuming corrected entries are correct.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.table_all_cells", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L714", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e29a323421f62457b646541778afb7eab2a58b2c285ea072872e62de5d3711a3", "conditional_on": [], "credit_reason": "The closed error interval agrees with T17(a). Its endpoint convention differs from roundsTenth but no integer count lies on a tie, so the count inference is unchanged.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.reported_counts_unique", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L636", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e29a323421f62457b646541778afb7eab2a58b2c285ea072872e62de5d3711a3", "conditional_on": [], "credit_reason": "R163-15 separately certifies the half-open rounding intervals and six one-tenth changes; this row is only the six exact rational evaluations.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.table_six_exact", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L668", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e29a323421f62457b646541778afb7eab2a58b2c285ea072872e62de5d3711a3", "conditional_on": [], "credit_reason": "The row’s stated scope is the two grid comparisons at 3 and 4. For the target’s other grid values, R163-15 already supplies strictly separated rounded endpoint ranges.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.table_grid_comparison", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L727", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:e29a323421f62457b646541778afb7eab2a58b2c285ea072872e62de5d3711a3", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P025-T17", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P025-T17" }, "html": { "body": "

p. 238 (PDF p. 4). The inferred counts determine all 42 corrected Table 1 cells, reproduce the printed cells from rounded inputs, and preserve all gallery/exchange grid comparisons.

", "headline": "Table 1, six entries, corrected: the bounds computed from the counts that the reported sample sizes and percentages imply (44/48, 108/160, 291/579), rounded at the last step", "location": "p. 238 (PDF p. 4)" }, "id": "P025-T17", "text": { "body": "p. 238 (PDF p. 4). The inferred counts determine all 42 corrected Table 1 cells, reproduce the printed cells from rounded inputs, and preserve all gallery/exchange grid comparisons.", "headline": "Table 1, six entries, corrected: the bounds computed from the counts that the reported sample sizes and percentages imply (44/48, 108/160, 291/579), rounded at the last step", "location": "p. 238 (PDF p. 4)" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T17" }, { "declarations": [ { "claim_revision_id": "catalog:76047d7310e751572b2aa5bf87423fcc0945411b8ee36a4a431d276ee0e41029", "conditional_on": [], "credit_reason": "The primed interval imposes no loss on the singleton assertion: choose it equal to the first. Choosing alpha prime=alpha/beta,beta prime=1 gives the normalized set in T18(a).", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.ratio_only_and_equal_probabilities", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L890", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:76047d7310e751572b2aa5bf87423fcc0945411b8ee36a4a431d276ee0e41029", "conditional_on": [], "credit_reason": "The hypotheses imply alpha=beta>0 and a nonempty point box, so no total-division or empty-infimum artifact is used.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.equal_probability_box_and_endpoints", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L2765", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:76047d7310e751572b2aa5bf87423fcc0945411b8ee36a4a431d276ee0e41029", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P025-T18", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P025-T18" }, "html": { "body": "

p. 236 (PDF p. 2). The attainable set and its extrema depend only on the ratio of probability bounds and reduce to the sample mean at ratio one.

", "headline": "Section 3.1, paragraph after (5): 'The bounds do not depend on the actual values of α\\alphaα and β\\betaβ, only on the ratio γ\\gammaγ. When γ=1\\gamma = 1γ=1, we have equal-probability sample selection.'", "location": "p. 236 (PDF p. 2)" }, "id": "P025-T18", "text": { "body": "p. 236 (PDF p. 2). The attainable set and its extrema depend only on the ratio of probability bounds and reduce to the sample mean at ratio one.", "headline": "Section 3.1, paragraph after (5): 'The bounds do not depend on the actual values of $\\alpha$ and $\\beta$, only on the ratio $\\gamma$. When $\\gamma = 1$, we have equal-probability sample selection.'", "location": "p. 236 (PDF p. 2)" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T18" }, { "declarations": [ { "claim_revision_id": "catalog:5c9bf434d7cf7b6f297d0f77082a0942e793d562ac87daff2cc386e24a6d5613", "conditional_on": [], "credit_reason": "The corrected lower cut is explicitly part of the target’s formal content. First-cut predicates assume only the prescribed stopping rule, not equality of tied weights or optimality.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.tied_outcomes_equal_weights", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L912", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:5c9bf434d7cf7b6f297d0f77082a0942e793d562ac87daff2cc386e24a6d5613", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P025-T19", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P025-T19" }, "html": { "body": "

p. 237 (PDF p. 3); proof in the Supplement, p. 1. For a sorted sample of at least two outcomes and 0 < alpha <= beta <= 1, the upper and corrected lower least cuts exist uniquely and select feasible optimal physical weights. Every tied pair receives equal weights. When gamma = beta/alpha > 1, equality of adjacent weights is equivalent to the cut not separating that pair, so neither cut separates tied observations. All finite threshold formulas have positive denominators. For constant outcomes the upper cut is zero and every upper weight is 1/alpha; otherwise the zero-cut upper value is the sample mean, strictly above the first outcome, and the upper cut is positive. At each adjacent tie at or before either cut, the preceding test fails strictly, the corresponding Supplement numerator/residual recurrence propagates strict failure, and the next cut index is still at or before the selected cut.

", "headline": "Lemma 1", "location": "p. 237 (PDF p. 3); proof in the Supplement, p. 1" }, "id": "P025-T19", "text": { "body": "p. 237 (PDF p. 3); proof in the Supplement, p. 1. For a sorted sample of at least two outcomes and 0 < alpha <= beta <= 1, the upper and corrected lower least cuts exist uniquely and select feasible optimal physical weights. Every tied pair receives equal weights. When gamma = beta/alpha > 1, equality of adjacent weights is equivalent to the cut not separating that pair, so neither cut separates tied observations. All finite threshold formulas have positive denominators. For constant outcomes the upper cut is zero and every upper weight is 1/alpha; otherwise the zero-cut upper value is the sample mean, strictly above the first outcome, and the upper cut is positive. At each adjacent tie at or before either cut, the preceding test fails strictly, the corresponding Supplement numerator/residual recurrence propagates strict failure, and the next cut index is still at or before the selected cut.", "headline": "Lemma 1", "location": "p. 237 (PDF p. 3); proof in the Supplement, p. 1" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T19" }, { "declarations": [ { "claim_revision_id": "catalog:191bda124820e479eed3cff38a6381640b06428dd1c3544f52b654dcec1f1742", "conditional_on": [], "credit_reason": "The conclusion is equality of the entire attainable set, so it includes attainability of every interior point and both endpoints. Omitting an outcome ordering loses no case: the claim is invariant under jointly permuting outcomes and weights, and its explicit formal content quantifies over every binary sample. beta<=1 is already a standing probability bound. No change from claim-stage.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.dichotomous_closed_form", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L1512", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:191bda124820e479eed3cff38a6381640b06428dd1c3544f52b654dcec1f1742", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P025-T20", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P025-T20" }, "html": { "body": "

p. 237 (PDF p. 3). Every nonempty binary sample has exactly the displayed attainable interval and optimization endpoints, including zero, one, and equal probability cases. The classes have counts np and n(1-p); their normalized weight sums A and C give the ratio A/(A+C), satisfy the stated bounds, and make that ratio nondecreasing in A and nonincreasing in C.

", "headline": "Section 3.1, closed form of the bounds for a dichotomous outcome (unnumbered display after Lemma 1)", "location": "p. 237 (PDF p. 3)" }, "id": "P025-T20", "text": { "body": "p. 237 (PDF p. 3). Every nonempty binary sample has exactly the displayed attainable interval and optimization endpoints, including zero, one, and equal probability cases. The classes have counts np and n(1-p); their normalized weight sums A and C give the ratio A/(A+C), satisfy the stated bounds, and make that ratio nondecreasing in A and nonincreasing in C.", "headline": "Section 3.1, closed form of the bounds for a dichotomous outcome (unnumbered display after Lemma 1)", "location": "p. 237 (PDF p. 3)" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T20" }, { "declarations": [ { "claim_revision_id": "catalog:1b3944205ceb4f8affc6f78c5bf900bd1b554a1561e9401e871b748c7f90594e", "conditional_on": [], "credit_reason": "FiniteDesign assumes only a sampling map and valid probabilities, not convergence. Surjectivity and StrictMono encode all nonempty distinct-outcome classes; separation has precisely the valid cut range. Both limits concern the actual pooled extrema and the full h-to-infinity sequence under a nonzero probability law. The stronger theorem used by the proof does not alter this statement's hypotheses or its fidelity. No change from claim-stage.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.printed_proposition_three", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L2212", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:1b3944205ceb4f8affc6f78c5bf900bd1b554a1561e9401e871b748c7f90594e", "conditional_on": [], "credit_reason": "Auxiliary deterministic cut certificate; printed_proposition_three supplies the stated almost-sure conclusion directly.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P025.upper_cut_from_strict_neighbors", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P025/P025_Interval.lean#L2806", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:1b3944205ceb4f8affc6f78c5bf900bd1b554a1561e9401e871b748c7f90594e", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P025-T21", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P025-T21" }, "html": { "body": "

p. 238 (PDF p. 4); proof in the Supplement, p. 2. Under Assumptions 1 and 2 on the valid finite cut range, both actual pooled endpoints converge almost surely to the limiting class extrema.

", "headline": "Proposition 3, as printed (with Assumptions 1 and 2)", "location": "p. 238 (PDF p. 4); proof in the Supplement, p. 2" }, "id": "P025-T21", "text": { "body": "p. 238 (PDF p. 4); proof in the Supplement, p. 2. Under Assumptions 1 and 2 on the valid finite cut range, both actual pooled endpoints converge almost surely to the limiting class extrema.", "headline": "Proposition 3, as printed (with Assumptions 1 and 2)", "location": "p. 238 (PDF p. 4); proof in the Supplement, p. 2" }, "url": "https://pmaronow.github.io/llm-audit/works/P025/#P025-T21" } ] }, "missing_sources": [], "reproduction": { "agreement": "agrees_in_part", "status": "selected_recalculation", "targets": [ { "agreement": "agrees_in_part", "location": "p. 238 (PDF p. 4)", "method": "I recalculated all 42 bounds from the printed proportions 0.917, 0.675 and 0.503; they agree to one decimal place. Using the counts instead changes six cells by 0.1 percentage points (P025-N003).", "status": "recalculated", "target": "Table 1" }, { "agreement": "agrees", "location": "pp. 238–239 (PDF pp. 4–5), end of Section 4", "method": "I compared the intervals on the published table grid: they are disjoint at $\\gamma=3$ and overlap at $\\gamma=4$. The continuous touching threshold from the printed proportions is about $3.304$, consistent with “only when $\\gamma>3$”. The separate inference about selection ratios is corrected in P025-E003.", "status": "recalculated", "target": "Shooting gallery versus needle exchange overlap" }, { "agreement": "agrees", "location": "p. 238 (PDF p. 4), end of Section 4", "method": "I compared the intervals on the published table grid: they are disjoint at $\\gamma=1.25$ and overlap at $\\gamma=1.5$. The continuous touching threshold from the printed proportions is about $1.433$; the grid qualification is explained in P025-A001.", "status": "recalculated", "target": "Street versus needle exchange overlap" } ] }, "summary": "This paper asks what can be learned about a population mean when the probabilities of sample selection are unknown but lie in an interval $[\\alpha,\\beta]$. It derives sharp bounds on the Hajek estimator, shows that the estimated bounds converge to bounds on the population mean, and applies the method to HIV prevalence in needles from New Haven. 8 entries are classified as errors (2 Moderate, 6 Minor), and 4 are clarity notes. In \\fid{P025-E001}, the printed rule for the lower cut point uses $\\geqslant$ where $\\leqslant$ is needed.", "what_was_checked": "I checked the following claims; they held up with the corrections and qualifications noted below.\n\\begin{itemize}\n\\item Equation (1). For a nonempty sample the Hajek estimator equals $w'y/(w'1_n)$ with $w_i=1/\\pi_i$. The Horvitz--Thompson estimator of $\\mu$ is unbiased when $\\pi_i>0$ for all $i$, and $\\hat 1_{\\mathrm{HT}}$ has expectation $1$.\n\\item Proposition 1, upper bound. By the Appendix argument, there exists an optimal vertex of the stated form, and the rule for $k^{\\mathrm H}$ is correct for $\\gamma>1$ (see \\fid{P025-E004} for $\\gamma=1$). The printed rule agreed with an exact search over the vertices of the weight box in 9000 random cases with $n\\leqslant 8$. The lower rule as corrected in \\fid{P025-E001} agreed with an exact search in 1844 cases.\n\\item Equation (5). Every value between the two bounds is attained, because the ratio is continuous on the weight box.\n\\item The bounds depend on $\\alpha$ and $\\beta$ only through $\\gamma$, both bounds equal the sample mean at $\\gamma=1$, and the interval widens as $\\gamma$ increases.\n\\item Lemma 1 and its proof in the supplementary material, including the case of equal outcomes.\n\\item The closed form for dichotomous outcomes, including $p=0$, $p=1$ and $\\gamma=1$.\n\\item The setup of Section 3.2. The shares of the outcomes among sampled units converge almost surely to $\\Pi_j/\\sum_k\\Pi_k$, the identity for $\\mu$ in terms of $W_j$ holds, and $\\mu$ lies in $[\\mu_-,\\mu_+]$. The sense in which the interval is sharp is the subject of \\fid{P025-N002}.\n\\item Proposition 2, the formula for $\\mu_+$ and the rule for $k_+$. The conditions (B1) and (B2) and the identity for their sum are correct for $\\alpha<\\beta$ (see \\fid{P025-E004}). The rule for $k_+$ agreed with an exact search in 3000 cases, and the bounds of Proposition 2 after the corrections of \\fid{P025-E001} and \\fid{P025-E002} agreed with an exact search in 480 cases.\n\\item Proposition 3. The estimated bounds converge almost surely to $\\mu_+$ and $\\mu_-$, also when Assumption 2 fails (see \\fid{P025-E005}).\n\\item Table 1. All 42 entries are reproduced from the proportions $0.917$, $0.675$ and $0.503$, and 36 of them from the counts (see \\fid{P025-N003}). The rows $\\gamma=1$ equal the reported percentages.\n\\item Section 4, on the rows of Table 1. The shooting gallery and needle exchange intervals are disjoint at $\\gamma=3$ and overlap at $\\gamma=4$, and the street and needle exchange intervals are disjoint at $\\gamma=1.25$ and overlap at $\\gamma=1.5$. The direction of selection described in the last sentence is the right one.\n\\item Section 5 presents its three extensions as proposals, and no contradiction with the results was found.\n\\item The numbered equations, assumptions, lemma, propositions and table are cited consistently in the article and in the supplementary material, and the supplementary material contains the proofs that the article says it contains.\n\\end{itemize}\nMy check did not cover the following. The article of Kaplan (1991) could not be obtained. The numbers of needles tested and of positive tests in Section 4 agree with indexed passages of a later report by the same investigator, and the numbers of needles returned were not compared with any source. Brewer (1979), Hajek (1971) and the other cited works were not read, and four reference entries were compared with public bibliographic records only (see \\fid{P025-E007}). The page of the comment of Hajek was not located. Whether the selection of the needle samples satisfies a bound $\\gamma$ cannot be checked from the article. I could not check replication code or original data; neither was available to me. The article reports no simulations.", "work": { "arxiv_id": null, "audit_complete": true, "authors": [ "P. M. Aronow", "Donald K. K. Lee" ], "category": "Articles - Theory and Methodology", "citation": "P. M. Aronow and Donald K. K. Lee (2013). Interval Estimation of Population Means under Unknown but Bounded Probabilities of Sample Selection. Biometrika 100(1):235–240.", "doi": "10.1093/biomet/ass064", "edition": { "pages_in_audited_file": 6, "sha256": "72730733bdc2c144723e6c3c091165ec5e65c5c0b3bf67bf0fc474c8a8ab401a", "source_url": "https://doi.org/10.1093/biomet/ass064", "venue": "Biometrika", "year": "2013" }, "id": "P025", "later_materials": null, "title": "Interval Estimation of Population Means under Unknown but Bounded Probabilities of Sample Selection", "type": "Journal article", "url": "https://pmaronow.github.io/llm-audit/works/P025/", "venue": "Biometrika", "year": "2013" } }