{ "claims": [ { "claim": "An OLS regression on all elections that controls for vote share through indicator variables for bins of width $2k$, one of them centred on the threshold, returns exactly the RD estimate defined as the difference in means within bandwidth $k$ (degree-0 local regression with a rectangular kernel), whether or not the outcome model is correct.", "claim_id": "P023-H1", "related_corrections": [ "P023-E006", "P023-N001", "P023-N004" ], "tied_to": "Proposition 1; Appendix A; Table 1 rows (d) and (e); abstract ('exactly reproduces'); Section 3.1, pp. 5–6" }, { "claim": "An OLS regression on all elections with a polynomial in vote share approximates the RD estimate in large samples, provided the polynomial degree is large enough to capture the conditional expectation function; sufficient richness is described as no meaningful remaining improvement in model fit.", "claim_id": "P023-H2", "related_corrections": [ "P023-E002" ], "tied_to": "Proposition 2 and its Remark (Section 3.2, pp. 6–7); Appendix B; Table 1 rows (b), (c) and (e)–(h); abstract ('or approximates')" }, { "claim": "All flexibly specified OLS studies of the party incumbency advantage are mathematically equivalent to studies of close elections; with a covariate control for vote share the resulting estimates rely solely on data from close elections.", "claim_id": "P023-H3", "related_corrections": [ "P023-E001" ], "tied_to": "Propositions 1 and 2 taken together; Section 1, p. 3; end of Section 3.2, p. 7; first sentence of Section 5, p. 8" }, { "claim": "The paper argues that these equivalences make concerns about close elections, including sorting at the threshold and limited external validity, equally applicable to regression analyses of all elections that control for vote share. It describes a focus on close elections as, in a sense, unavoidable for causal inference about the party incumbency advantage.", "claim_id": "P023-H4", "related_corrections": [ "P023-E001", "P023-E002" ], "tied_to": "Propositions 1 and 2; Table 1; abstract; Section 1, pp. 1–3; Section 5, p. 8" }, { "claim": "In the Caughey–Sekhon data for US House elections, 1942–2008, every estimator except the naive one ($34.40$ percentage points) gives a substantively similar party incumbency effect on next-election Democratic vote share ($9.97$ to $13.15$ percentage points), and coarsened-control OLS equals degree-0 rectangular-kernel RD ($10.54$ percentage points).", "claim_id": "P023-H5", "related_corrections": [ "P023-E003", "P023-E004", "P023-N002" ], "tied_to": "Table 1; Section 4, pp. 7–8" }, { "claim": "Close House elections are rare: $0.6\\%$ of elections between 1944 and 2006 were decided by less than $0.5$ percentage points.", "claim_id": "P023-H6", "related_corrections": [ "P023-E005", "P023-N002" ], "tied_to": "Section 1, p. 1; no table or figure (a count from the Caughey–Sekhon data)" }, { "claim": "The findings may carry over, as a first approximation, to related estimands such as the Gelman–King estimator of the personal incumbency advantage.", "claim_id": "P023-H7", "related_corrections": [], "tied_to": "Section 5, pp. 8–9; no result in the paper; rests on Caughey and Sekhon (2011, §5.2)" }, { "claim": "Without stronger modelling assumptions, vote-share adjustment offers little basis for estimating the party incumbency effect away from close elections, because winning and losing parties do not overlap on vote margin there.", "claim_id": "P023-H8", "related_corrections": [ "P023-E001" ], "tied_to": "Section 2, pp. 3–4; Section 5, pp. 8–9; the deterministic treatment setup underlying Conditions 1–3 and Propositions 1–2" } ], "date_published": "2026-10-25", "findings": [ { "before_review": { "class": "error", "severity": "Major" }, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Screenshot of seven lines of the paragraph continued from page 2 onto page 3. Red boxes mark the phrases: are mathematically equivalent to studies of close elections; rely solely on data from close elections.", "caption": "Section 1, p. 3 (PDF p. 3), paragraph continued from p. 2. The two marked phrases say that all flexibly specified OLS studies are mathematically equivalent to studies of close elections, and that the estimates rely solely on data from close elections. Only the exclusive-outcome-use assertion is rejected here; the equivalence results are not refuted.", "url": "https://pmaronow.github.io/llm-audit/works/P023/exhibits/P023-E001.png" }, { "alt": "Screenshot of four lines at the end of Section 3.2. A red box marks the phrase: will also only use information from close elections.", "caption": "Section 3.2, p. 7 (PDF p. 7), last paragraph. The marked phrase draws the same conclusion for any well-specified OLS estimator.", "url": "https://pmaronow.github.io/llm-audit/works/P023/exhibits/P023-E001-b.png" }, { "alt": "Screenshot of the first six lines of Section 5. A red box marks the words: therefore equally applicable.", "caption": "Section 5, p. 8 (PDF p. 8), first two sentences. The first sentence says only that OLS may rely solely on close elections, which is correct; the second broadly transfers concerns about bias and external validity to full-sample regressions. Shared-cutoff concerns can survive clarification of the claim about exclusive outcome use.", "url": "https://pmaronow.github.io/llm-audit/works/P023/exhibits/P023-E001-c.png" }, { "alt": "Two panels of vertical stems, one pair per election, ordered by vote margin. Blue circles (polynomial regression) sit away from zero at every election, including the most distant ones; in the left panel they range from about minus 0.72 to 0.72 and in the right panel from about minus 1.8 to 1.8. Orange squares (RD estimate and coarsened regression) are at minus 1 and plus 1 for the two elections in the shaded close-election window and exactly at zero for every other election.", "caption": "Weights of the outcomes in the two exact examples. Left: linear control, six elections, window $(-1/200,1/200]$. Right: degree-7 control, sixteen elections, window $(-1/5,1/5]$. Circles are the weights in the coefficient of the polynomial regression, equation (2), which are nonzero at every election. Squares are the weights in the RD estimate, which equals the coefficient of the coarsened regression of Proposition 1; they are $\\pm1$ at the two elections inside the window (shaded) and zero elsewhere.", "url": "https://pmaronow.github.io/llm-audit/works/P023/exhibits/P023-E001-weights.png" }, { "alt": "Line chart with a logarithmic horizontal axis, the absolute vote margin m from 0.02 to 100 percentage points, and a vertical axis, the share of the absolute weight on elections with margin at most m, from 0 to 100 per cent. A solid blue line (linear control) and a dashed orange line (degree-7 control) stay below 5 per cent up to m equal to 0.5, where circles mark 1.7 and 4.0 per cent. They rise to about 30 and 53 per cent at 10 points and to about 83 and 87 per cent at 50 points, and step up to 100 per cent at 100 points. A green dash-dotted line at 100 per cent, starting from a square at m equal to 0.5, is the coarsened regression. A dotted vertical line at 0.5 is labelled: window of rows (d) and (e).", "caption": "The 8,610 complete House-election observations in this calculation, using the 2014 release of Caughey and Sekhon’s (2011) replication data: the share of the absolute weight $\\sum_i|w_i|$ that falls on elections with absolute margin at most $m$ percentage points. Solid line: linear specification of row (b) of Table 1. Dashed line: degree-7 specification of row (c). At $m=0.5$, the window of rows (d) and (e), the two shares are 1.7% and 4.0% (circles). Dash-dotted line and square: the coarsened specification of row (d), whose weight lies entirely on the 85 elections within 0.5 points. The step at $m=100$ is the 922 elections with a margin of exactly 100 points.", "url": "https://pmaronow.github.io/llm-audit/works/P023/exhibits/P023-E001-house-weights.png" } ], "formal_release": null, "formal_statements": [ { "id": "P023-T1", "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T1" }, { "id": "P023-T2", "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T2" }, { "id": "P023-T3", "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T3" }, { "id": "P023-T10", "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T10" } ], "headline": "Full-sample OLS with the coarsened control of Proposition 1 relies solely on outcomes from close elections; a polynomial regression can estimate the same jump at the threshold while also using distant outcomes.", "id": "P023-E001", "location": "Section 1, p. 3 (PDF p. 3), paragraph continued from p. 2; repeated in the last paragraph of Section 3.2, p. 7, and in the second sentence of Section 5, p. 8", "pdf_page": 3, "printed_page": "3", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P023-T1", "P023-T10", "P023-T2", "P023-T3" ], "text": { "as_printed": "[p.~3] ``In fact, we demonstrate that all flexibly specified empirical studies of the party incumbency advantage using ordinary least squares (OLS) regression are mathematically equivalent to studies of close elections. The mechanics of linear regression with a covariate control for vote share are such that the resulting estimates \\hl{rely solely on data from close elections}. It follows that any concerns about the study of close elections are equally applicable to regression analyses including the entire dataset of all elections.''\n\n[p.~7] ``Since the RD estimator only uses information from close elections to estimate causal effects, it is also generally the case that the OLS estimator, when well-specified, \\hl{will also only use information from close elections} to estimate causal effects.''\n\n[p.~8] ``We have demonstrated that OLS estimation of the party incumbency advantage may produce estimates that rely solely on data from close elections. The limitations of studying close elections to estimate causal effects -- both in terms of bias and of external validity -- are therefore equally applicable to regression analyses that include the entire dataset.''", "corrected": "The sentences on p.~3 should read: ``In fact, \\hl{under Conditions 1--3 of Proposition 1,} the OLS estimate from all elections equals the \\hl{degree-0, rectangular-kernel} RD estimate and relies solely on \\hl{outcomes} from close elections. When equation (2) exactly specifies the conditional mean, \\hl{under conditions giving shared consistency, the polynomial OLS and shrinking-bandwidth RD estimates converge to the same cutoff jump and their difference converges to zero in probability}. Outcomes of elections outside the close-election window \\hl{can also enter the polynomial OLS estimate}.''\n\nThe sentence on p.~7 should read: ``The OLS estimator \\hl{with the coarsened control of Proposition 1} uses \\hl{outcomes} from close elections only. An exactly specified polynomial regression \\hl{can also use distant outcomes}. \\hl{Under shared consistency, its difference from a shrinking-bandwidth RD estimate converges to zero in probability}; this does not imply equal observation weights, sampling variance or finite-sample bias.''\n\nThe second sentence of Section 5, p.~8, should read: ``The limitations of studying close elections apply in full \\hl{to the coarsened regression}. For a polynomial regression, \\hl{concerns about causal interpretation and scope can be shared when they arise from the common cutoff target and the same identifying assumptions}; equal targets alone do not imply equal bias.''", "evidence": "The evidence is a derivation, two exact counterexamples and a recalculation of the weights on the available House replication data. Let $r$ be the residual of the vector $\\mathit{DemWin}$ after least-squares projection on the constant and the powers $\\mathit{DemVot},\\ldots,\\mathit{DemVot}^J$. For an identified coefficient in equation (2), I obtain from the normal equations\n\\[\n\\hat\\beta_{OLS2}=\\sum_i w_i\\,\\mathit{DV}_{i(t+1)},\\qquad w_i=\\frac{r_i}{\\sum_j r_j^2},\n\\]\nwhich implies that raising one outcome by $\\delta$ changes the estimate by $w_i\\delta$.\n\nFor the linear control ($J=1$), take six elections with margins $-3/4$, $-1/4$, $-1/400$, $1/400$, $1/4$, $3/4$ and outcomes $\\mathit{DV}=1/2+\\mathit{DemWin}/10+\\mathit{DemVot}/20$. Equation (2) then holds exactly with $\\beta=1/10$. In the order of the margins, the weights are\n\\[\nw=\\frac{1}{139202}\\,(20299,\\,-59901,\\,-99600,\\,99600,\\,59901,\\,-20299).\n\\]\nWith $k=1/200$, only the elections at $\\pm1/400$ lie in the window. For the degree-7 control, take sixteen elections with margins $\\pm1/8,\\pm2/8,\\ldots,\\pm1$ and outcomes $1/2+\\mathit{DemWin}/10+\\mathit{DemVot}/20+\\mathit{DemVot}^7/100$. Equation (2) with $J=7$ holds exactly, again with $\\beta=1/10$, and the sixteen weights, in increasing order of margin, are\n\\[\n\\begin{aligned}\nw=\\frac{1}{1607}\\,(&{-}175,\\,725,\\,-685,\\,-643,\\,554,\\,1289,\\,239,\\,-2911,\\\\\n&2911,\\,-239,\\,-1289,\\,-554,\\,643,\\,685,\\,-725,\\,175),\n\\end{aligned}\n\\]\nnone of them zero. With $k=1/5$, only the elections at $\\pm1/8$ lie in the window. The table shows what happens when the outcome of the election with the largest margin ($3/4$ in the first example, $1$ in the second) is raised by $1/100$.\n\n\\begin{audittable}{lrrr}{Estimates before and after one distant outcome is raised by $1/100$.}\nEstimate & Before & After & Change \\\\\n\\hline\nPolynomial regression, $J=1$ & $1/10$ & $1371721/13920200$ & $-20299/13920200$ \\\\\nRD, coarsened regression, $k=1/200$ & $401/4000$ & $401/4000$ & $0$ \\\\\nPolynomial regression, $J=7$ & $1/10$ & $3249/32140$ & $7/6428$ \\\\\nRD, coarsened regression, $k=1/5$ & $11796481/104857600$ & $11796481/104857600$ & $0$ \\\\\n\\end{audittable}\n\nIn each example the two outcome vectors are realizations of one correctly specified model. Take errors independent of one another and of the margins, with probabilities $1/4$, $1/2$ and $1/4$ at $-1/100$, $0$ and $1/100$. Conditional on the stated margins, the baseline and perturbed error vectors have probabilities $1/64$ and $1/128$ in the six-election example, and $1/65536$ and $1/131072$ in the sixteen-election example. These probabilities establish that the outcomes are possible under the same conditional-mean model; they do not measure the probability of an OLS--RD discrepancy exceeding a fixed positive amount. The finite designs themselves do not test a shrinking-bandwidth asymptotic claim.\n\nThe targets do agree. For margins uniform on $[-1,1]$ and the same two conditional means, the population coefficient of equation (2) is $1/10$, and the difference in means within bandwidth $k$ is $1/10+k/20$ for $J=1$ and $1/10+k/20+k^7/400$ for $J=7$, which tends to $1/10$ as $k\\to0$. With independent observations, the bounded mean-zero errors just specified and, for example, $k_N=N^{-1/3}$, both estimators are consistent for $1/10$. The finite grids are used for conditional sensitivity calculations; no positive probability of drawing those exact grids from a continuous distribution is asserted.\n\nMore generally, whenever both estimators converge in probability to the same $\\beta$, for each fixed $\\epsilon>0$ the triangle inequality gives\n\\[\n\\begin{aligned}\nP\\bigl(|\\hat\\beta_{OLS2}-\\hat\\tau_{RD}|>\\epsilon\\bigr)\n&\\le P\\bigl(|\\hat\\beta_{OLS2}-\\beta|>\\epsilon/2\\bigr)\\\\\n&\\quad+P\\bigl(|\\hat\\tau_{RD}-\\beta|>\\epsilon/2\\bigr)\\longrightarrow0.\n\\end{aligned}\n\\]\nThis requires no independence between the estimators. It does not assert equality at finite sample sizes, a probability for a particular sample size, or equivalence after division by a shrinking standard error.\n\nTheir variance contributions can still differ. With homoskedastic errors, elections whose margins have absolute value at least $1/2$ contribute $1/8$ of the leading error variance of the coefficient of equation (2) for $J=1$ and $572158921/10276044800$ (about 5.6\\%) for $J=7$. The leading contribution from the errors to the variance is $16\\sigma^2/N$ for $J=1$ and $(65536/1225)\\sigma^2/N$ for $J=7$, against $800\\sigma^2/N$ for the difference in means with $k=1/200$. The polynomial contributions vanish at order $1/N$, so their positive distant shares do not imply a persistent discrepancy between the estimates.\n\nOn the available House replication data, the two polynomial specifications of Table 1 also put most of their weight on distant elections. In the 2014 release of the replication data of Caughey and Sekhon (2011), 8,610 elections from 1942--2008 have the outcome (\\texttt{DPctNxt}), the margin (\\texttt{DifDPct}) and the winner (\\texttt{DemWin}) recorded, and a recalculation in exact arithmetic gives the weights $w_i$ for the specifications of rows (b) and (c) on these elections. The available codebook defines this margin as the leading Democrat's percentage minus the leading non-Democrat's, rather than a two-party renormalization. The 85 elections within 0.5 points of the threshold form the window of rows (d) and (e). They carry 1.7\\% of $\\sum_i|w_i|$ for the linear control and 4.0\\% for the degree-7 control. Each of the other 8,525 elections has a nonzero weight in both regressions, and the elections decided by more than 10 points carry 70\\% and 47\\% of $\\sum_i|w_i|$. In the coarsened specification of row (d), the 85 elections carry all of the weight. The three coefficients are nonetheless close: 11.15, 9.95 and 10.54 points. With the sample restricted to the 8,594 elections flagged \\texttt{Use=1} for Caughey and Sekhon's (2011) RD analysis, the two shares are 1.8\\% and 4.0\\%. These are shares of absolute outcome weights, not probabilities of an OLS--RD discrepancy or causal population weights. These calculations do not establish the authors' exact sample or code, the population fit of either polynomial, or a difference in causal bias or external validity.", "problem": "The paper distinguishes two results. Proposition 1 concerns the regression whose controls are indicators for bins of width $2k$ of the margin $\\mathit{DemVot}_{it}$, the ``coarsened control'' of Table 1; I call it the coarsened regression. Its OLS coefficient equals the difference in means within the window $(-k,k]$, and every outcome outside the window has weight zero in it. Proposition 2 concerns regressions with a polynomial in the margin, such as the linear and degree-7 controls of Table 1. Appendix B compares the polynomial and shrinking-window RD estimates through the cutoff jump $\\beta=f(0)-\\lim_{x\\uparrow0}g(x)$, rather than through their observation weights.\n\nRead as a whole, the paper separates exact coarsened equivalence from approximate polynomial equivalence in large samples. The phrase ``mathematically equivalent'' on p.~3 should be read in that context. Nevertheless, the explanation in terms of the mechanics of regression, followed by ``rely solely on data'' on p.~3 and ``only use information'' on p.~7, naturally asserts exclusive dependence on close-election outcomes. That assertion is false for the permitted finite polynomial controls. Its failure does not demonstrate failure of either equivalence result. The identified coefficient of a polynomial regression of degree $J$ is a weighted sum $\\sum_i w_i\\,\\mathit{DV}_i$ over all elections. The weight $w_i$ is proportional to the residual of $\\mathit{DemWin}$ on the polynomial terms, and this residual is nonzero at all but at most $2J$ distinct margins. Raising the outcome of a distant election with nonzero weight then changes the estimate, even when the polynomial model is exactly correct, while the RD estimate and the coefficient of the coarsened regression do not move.\n\nTwo estimators with the same limit need not have the same weights, the same sampling variance or the same bias at a fixed bandwidth. Under shared consistency their difference nevertheless converges to zero in probability; nonzero distant weights do not refute that approximation. For the coarsened regression, equality of the estimators transfers the properties of the specified RD estimate directly. For polynomial controls, Appendix B already supplies the common-target argument, and Section 2 describes assumptions needed for its causal interpretation. Different outcome weights do not remove concerns that arise from that common target and those assumptions. They establish neither different causal bias nor broader external validity in the application.", "what_survives": "Proposition 1 and Section 5's statement that OLS ``may'' rely solely on close-election outcomes are unaffected. When the conditional mean has exactly the form of equation (2), its coefficient and a consistent shrinking-bandwidth RD estimator estimate the same jump; under shared consistency their difference converges to zero in probability. The paper's distinction between exact and approximate equivalence and Appendix B's common-target argument are compatible with the weight calculations. Those calculations do not adjudicate the general conditions for polynomial approximation; see also \\fid{P023-E002}.\n\nThis clarification does not change any entry of Table 1 (for the entries, see \\fid{P023-E003} and \\fid{P023-E004}), and the advice to examine regression assumptions stands. The exclusive-data explanation requires qualification for polynomial controls. Concerns about causal identification and external validity can remain shared when they arise from the common cutoff target and the same identifying assumptions; the weight calculation establishes neither different causal bias nor broader causal scope in the application." }, "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-E001", "work": { "arxiv_id": null, "doi": "10.1515/spp-2014-0003", "title": "A Note on Close Elections and Regression Analysis of the Party Incumbency Advantage", "url": "https://pmaronow.github.io/llm-audit/works/P023/" }, "work_id": "P023" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Screenshot of a displayed approximation of the conditional expectation and the four lines after it. A red box marks the sentence: E[u given DemVot] is approximately 0 and therefore E of the OLS estimator is approximately beta.", "caption": "Appendix B, p. 10 (PDF p. 10). The marked sentence passes from a small approximation error of the conditional mean to a small error of the coefficient.", "url": "https://pmaronow.github.io/llm-audit/works/P023/exhibits/P023-E002.png" }, { "alt": "Screenshot of the Remark to Proposition 2 and two lines of the next paragraph. Red boxes mark the words: model fit cannot be meaningfully improved by including additional terms.", "caption": "Remark to Proposition 2, p. 7 (PDF p. 7). The marked words state the test of fit.", "url": "https://pmaronow.github.io/llm-audit/works/P023/exhibits/P023-E002-b.png" }, { "alt": "Line chart with a logarithmic horizontal axis, the upper bound on the gain in fit from further terms, from about 0.00001 to 0.8 squared percentage points, and a vertical axis, the OLS coefficient on DemWin, from 0 to 10 percentage points. A solid blue line (linear control) and a dashed orange line (degree-7 control) lie at about 10 for small values of the bound and fall to about 8 and 4.5 at the right edge. A grey dashed line at 0 is labelled as the jump at the threshold and the limit of the RD estimate. An open circle at a bound of 1/1200 marks coefficients of 9.998 and 9.993.", "caption": "The population coefficient on $\\mathit{DemWin}$ in the linear and the degree-7 regression (2), for margins uniform on $[-1,1]$, no jump at the threshold and smooth part $h_\\eta(x)=(1/20)\\operatorname{clip}(x/\\eta,-1,1)$, as $\\eta$ runs from $10^{-6}$ to $10^{-1}$. The horizontal axis is the bound $10^4a^2\\eta/3$, in squared percentage points, on the reduction in mean squared error that any further terms could give. The coefficient tends to 10 percentage points as the bound tends to zero, while the jump, to which the RD estimate converges, is 0 (dashed line). The circle marks the example of the text, $\\eta=1/10000$.", "url": "https://pmaronow.github.io/llm-audit/works/P023/exhibits/P023-E002-fit-and-coefficient.png" } ], "formal_release": null, "formal_statements": [ { "id": "P023-T4", "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T4" }, { "id": "P023-T5", "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T5" }, { "id": "P023-T6", "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T6" }, { "id": "P023-T7", "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T7" }, { "id": "P023-T8", "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T8" }, { "id": "P023-T12", "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T12" } ], "headline": "A small error in the polynomial approximation of the conditional mean does not by itself make the coefficient on $\\mathit{DemWin}$ close to the jump $\\beta$: Proposition 2 holds for a sufficiently large degree under a density condition and a Lipschitz condition, and the test of fit in its Remark does not certify a given degree.", "id": "P023-E002", "location": "Appendix B, p. 10 (PDF p. 10), the sentence beginning “$E[u\\mid \\mathit{DemVot}]\\approx0$ and therefore”; Proposition 2 and its Remark, p. 7 (PDF p. 7)", "pdf_page": 10, "printed_page": "10", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Moderate", "subject": "paper", "target_ids": [ "P023-T12", "P023-T4", "P023-T5", "P023-T6", "P023-T7", "P023-T8" ], "text": { "as_printed": "[p.~7] ``Proposition 2. Under weak regularity conditions, if $N\\to\\infty$, $k\\to0$, $Nk\\to\\infty$, and $J$ is ``sufficiently large,'' the OLS estimator of $\\beta$ in (2), $\\hat\\beta_{OLS2}\\approx\\hat\\tau_{RD}$. Remark. \\ldots\\ Put simply, in order for the result to hold, we require that \\hl{the model fit cannot be meaningfully improved by including additional terms} in the polynomial expansion of $\\mathit{DemVot}_{it}$.''\n\n[p.~10] ``Our assumptions about $g()$ and $f()$ imply that $h()$ is continuous on $[-1,1]$ and $h(0)=0$. By the Weierstrass Approximation Theorem, for any $\\epsilon>0$, there exist some $J^*$ and $\\gamma_1^*,\\ldots,\\gamma_{J^*}^*$ such that $|h(x)-\\sum_{j=1}^{J^*}\\gamma_j^*x^j|<\\epsilon$ for all $x\\in[-1,1]$. We assume that $J$ in (2) is sufficiently large to ensure that there exist $\\gamma_1,\\ldots,\\gamma_J$ such that $|h(x)-\\sum_{j=1}^{J}\\gamma_jx^j|\\approx0$ for all $x\\in[-1,1]$. Then\n\\[\nE[\\mathit{DV}\\mid \\mathit{DemVot}]\\approx\\alpha+\\beta I[\\mathit{DemVot}\\ge0]+\\sum_{j=1}^{J}\\gamma_j\\mathit{DemVot}^j,\n\\]\n$E[u\\mid \\mathit{DemVot}]\\approx0$ \\hl{and therefore} $E[\\hat\\beta_{OLS2}]\\approx\\beta$. If the sample is drawn i.i.d. from a population $U$ with $\\operatorname{Supp}(\\mathit{DemVot})\\supseteq[-k,k]$, where $k>0$, then for large enough $N$, we have $\\hat\\beta_{OLS2}\\approx\\beta$.''", "corrected": "Proposition 2 should read as follows. Let the observations $(X,Y)=(\\mathit{DemVot}_{it},\\mathit{DV}_{i(t+1)})$ be i.i.d. with $E[Y^2]<\\infty$, and let $D=I[X\\ge0]$. Suppose that the law of $X$ is supported on $[-1,1]$, \\hl{has no atom at $0$, and satisfies $P(X\\in A)\\ge c\\,|A|$ for every Borel set $A\\subseteq[-\\delta,\\delta]$}, for some $c>0$ and $\\delta>0$, where $|A|$ is the Lebesgue measure of $A$; that $E[Y\\mid X=x]=\\alpha+\\beta I[x\\ge0]+h(x)$ with $h(0)=0$ and \\hl{$|h(x)-h(y)|\\le L|x-y|$ for all $x,y\\in[-1,1]$}; and that $\\operatorname{Var}(Y\\mid X)$ is bounded on a neighbourhood of $0$. Let $b_J$ be the coefficient on $D$ in the population least-squares regression of $Y$ on $D,1,X,\\ldots,X^J$. Then for every $J\\ge1$,\n\\[\n\\hl{|b_J-\\beta|\\le\\frac{8KL}{\\sqrt{c\\,\\delta\\,(J+1)}},\\qquad K=\\frac{3(1+\\pi^4)}{4\\pi}.}\n\\]\nFor each fixed $J$, $\\hat\\beta_{OLS2}\\to b_J$ in probability as $N\\to\\infty$, and $\\hat\\tau_{RD}\\to\\beta$ in probability as $N\\to\\infty$, $k\\to0$ and $Nk\\to\\infty$. Hence for every $\\epsilon>0$ there exists $J_0$ such that, for each fixed $J\\ge J_0$, the difference $\\hat\\beta_{OLS2}-\\hat\\tau_{RD}$ converges in probability to $b_J-\\beta$, and \\hl{$|b_J-\\beta|<\\epsilon$}. If $J=J_N$ grows with $N$, the difference converges in probability to zero provided $J_N\\to\\infty$ and $(J_N+2)^2/(N\\lambda_{J_N}^3)\\to0$, where $\\lambda_J$ is the smallest eigenvalue of the second-moment matrix of $(1,D,X,\\ldots,X^J)$.\n\nThe Remark should read: if $h$ is a polynomial of degree at most $J$, then $b_J=\\beta$. If $\\sup_x|h(x)-p(x)|\\le\\epsilon$ for some polynomial $p$ of degree at most $J$, then \\hl{$|b_J-\\beta|\\le\\kappa_J\\epsilon$}, where $\\kappa_J=E|r_J|/E[r_J^2]$ and $r_J$ is the residual of $D$ on $1,X,\\ldots,X^J$; for margins uniform on $[-1,1]$, $\\kappa_1=10/3$ and $\\kappa_7\\approx4.764$. That the fit cannot be meaningfully improved by further terms \\hl{does not imply} that $b_J$ is close to $\\beta$.\n\nIn Appendix B, the sentence should read: ``$E[u\\mid \\mathit{DemVot}]\\approx0$, \\hl{and the coefficient $b_J$ satisfies $b_J-\\beta=E[r_J(h-p_J)]/E[r_J^2]$, which is small when $\\|h-p_J\\|_2$ is small relative to $\\|r_J\\|_2$}.''", "evidence": "The evidence is a derivation and exact counterexamples. Let $X=\\mathit{DemVot}$, $D=I[X\\ge0]$ and $Y=\\mathit{DV}$, with $E[Y\\mid X]=\\alpha+\\beta D+h(X)$ as in Appendix B. Let $r_J$ be the residual of $D$ after projection in $L^2$ on $1,X,\\ldots,X^J$, and set $v_J=E[r_J^2]$, which I assume positive. Then $E[r_J\\,p(X)]=0$ for every polynomial $p$ of degree at most $J$, and $E[r_JD]=v_J$. Let $b_J$ be the coefficient on $D$ in the population regression (2). Subtracting from the normal equation for $D$ the combination of the other normal equations that defines the projection, I obtain $E[r_J(Y-b_JD)]=0$. Then\n\\[\nb_J-\\beta=\\frac{E[r_J\\,h(X)]}{v_J}=\\frac{E[r_J\\,(h(X)-p(X))]}{v_J}.\n\\]\nBy the triangle inequality and the Cauchy--Schwarz inequality, I have\n\\[\n|b_J-\\beta|\\le\\kappa_J\\,\\|h-p\\|_\\infty,\\qquad \\kappa_J=\\frac{E|r_J|}{v_J},\\qquad\\text{and}\\qquad |b_J-\\beta|\\le\\frac{\\|h-p\\|_2}{\\sqrt{v_J}}.\n\\]\nSince $E|r_J|\\ge E[r_J(2D-1)]=2v_J$, the factor $\\kappa_J$ is at least 2. For fixed $J$ and an i.i.d. sample with finite second moments, $\\hat\\beta_{OLS2}$ converges in probability to $b_J$, and $b_J-\\beta$ is then the difference between the two estimators in large samples.\n\nFor margins uniform on $[-1,1]$, I find by a direct computation that $v_J=c_n^2/4$ with $c_n=\\binom{2n}{n}4^{-n}$ and $n=\\lfloor(J+1)/2\\rfloor$, so that $v_J$ is of order $1/J$, and that $\\kappa_J=(4/\\pi)\\log J+O(1)$. The table lists these constants for the two degrees of Table 1, together with the following example. Let $h_\\eta(x)=a\\,\\operatorname{clip}(x/\\eta,-1,1)$ with $a=1/20$ and $\\eta=1/10000$, and let $Y=1/2+h_\\eta(X)+U$, where $U=\\pm1/10$ with equal probability, independently of $X$. Then $h_\\eta$ is Lipschitz, $h_\\eta(0)=0$, the jump is $\\beta=0$, and the RD estimate converges to 0. The predictor $1/2+a(2D-1)$ lies in the span of the regressors of (2), and its mean squared error exceeds the noise variance $1/100$ by $a^2\\eta/3=1/12{,}000{,}000$. No further terms can lower the mean squared error by more than this amount.\n\n\\begin{audittable}{lrrrrr}{Uniform margins on $[-1,1]$: the constants, and the coefficient $b_J$ in percentage points for the example with $\\beta=0$, $a=1/20$ and $\\eta=1/10000$, with the largest reduction in mean squared error that further terms could give.}\nDegree & $v_J$ & $\\kappa_J$ & $1/\\sqrt{v_J}$ & $b_J$ (points) & Gain in fit at most \\\\\n\\hline\n$J=1$ & $1/16$ & $10/3$ & $4$ & $9.998$ & $1/12{,}000{,}000$ \\\\\n$J=7$ & $1225/65536$ & $4.7642$ & $256/35$ & $9.993$ & $1/12{,}000{,}000$ \\\\\n\\end{audittable}\n\nIn exact terms, $b_1=99980001/1000000000$. As $\\eta\\to0$ with $J$ fixed, the bound $a^2\\eta/3$ tends to zero and $b_J$ tends to $2a=1/10$. A second example concentrates the margins: all but a fraction $10^{-12}$ of the mass lies in two windows of half-width $1/2000$ around $\\pm1/2$, the density is continuous and positive on $[-1,1]$, and $h$ is Lipschitz with $\\sup|h|=1/2000$ and vanishes near 0. Then $\\beta=0$, further terms can lower the mean squared error by at most $1/4{,}000{,}000$, and yet $b_1=-499999999999503/500000999999500$. With the fraction $10^{-96}$, the degree-7 coefficient lies between $-1001/1000$ and $-999/1000$.\n\nContinuity alone fails even for one fixed design. Since $\\kappa_J$ is unbounded for uniform margins, a uniformly convergent series of polynomials defines a continuous $h$ with $h(0)=0$ and $\\sup|h|\\le1/12$ such that $b_{J_m}\\ge m/4$ along a sequence of degrees $J_m\\to\\infty$, while $\\beta=0$.\n\nThe corrected statement below rests on two estimates. Comparing with uniform margins on $[-\\delta,\\delta]$, I have $v_J\\ge c\\,\\delta/(4(J+1))$ under its density condition. Under its Lipschitz condition, convolution with a Jackson kernel yields a polynomial $p_J$ of degree at most $J$ with $p_J(0)=0$ and $\\|h-p_J\\|_\\infty\\le4KL/(J+1)$, where $K=3(1+\\pi^4)/(4\\pi)$. Inserting these into the second bound above, I obtain the rate of the corrected statement. The limit of $\\hat\\tau_{RD}$ follows from $|E[h(X)\\mid0p. 3 (PDF p. 3) and p. 7 (PDF p. 7). The six-election linear design has the stated rank, exact model, weights and estimator values, and changing a distant outcome changes polynomial OLS while preserving the close-window and coarsened estimates.

", "headline": "Section 1, last paragraph (‘the resulting estimates rely solely on data from close elections’), and Section 3.2, last paragraph (‘will also only use information from close elections’), as printed; the witness is the validators’ construction (linear control) and is not in the paper", "location": "p. 3 (PDF p. 3) and p. 7 (PDF p. 7)" }, "id": "P023-T1", "text": { "body": "p. 3 (PDF p. 3) and p. 7 (PDF p. 7). The six-election linear design has the stated rank, exact model, weights and estimator values, and changing a distant outcome changes polynomial OLS while preserving the close-window and coarsened estimates.", "headline": "Section 1, last paragraph (‘the resulting estimates rely solely on data from close elections’), and Section 3.2, last paragraph (‘will also only use information from close elections’), as printed; the witness is the validators’ construction (linear control) and is not in the paper", "location": "p. 3 (PDF p. 3) and p. 7 (PDF p. 7)" }, "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T1" }, { "declarations": [ { "claim_revision_id": "catalog:71eea86e9ad5395ba239e26066862464cc04a0acb0d8c579abd8df0eddf151a8", "conditional_on": [], "credit_reason": "Witness credit only; the exact degree-seven baseline and unchanged close outcomes are both exported.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.degree7_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/P023_Audit.lean#L973", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:71eea86e9ad5395ba239e26066862464cc04a0acb0d8c579abd8df0eddf151a8", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P023-T2", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P023-T2" }, "html": { "body": "

p. 3 (PDF p. 3) and p. 7 (PDF p. 7). The sixteen-election degree-seven design has the stated rank, exact model, nonzero weights and estimator values, and its distant perturbation changes polynomial OLS while preserving both close-window estimates.

", "headline": "Section 1, last paragraph, and Section 3.2, last paragraph, as printed (the same statements as P023-T1); the witness is the validators’ construction with the degree-7 control of estimator (c) and is not in the paper", "location": "p. 3 (PDF p. 3) and p. 7 (PDF p. 7)" }, "id": "P023-T2", "text": { "body": "p. 3 (PDF p. 3) and p. 7 (PDF p. 7). The sixteen-election degree-seven design has the stated rank, exact model, nonzero weights and estimator values, and its distant perturbation changes polynomial OLS while preserving both close-window estimates.", "headline": "Section 1, last paragraph, and Section 3.2, last paragraph, as printed (the same statements as P023-T1); the witness is the validators’ construction with the degree-7 control of estimator (c) and is not in the paper", "location": "p. 3 (PDF p. 3) and p. 7 (PDF p. 7)" }, "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T2" }, { "declarations": [ { "claim_revision_id": "catalog:e6091ca4a7560d9266441b52c8d2341477cb4a7131f5fb748758e6331ffa0946", "conditional_on": [], "credit_reason": "Covers T3(a) in weight form; uniqueness and localization follow immediately for all solutions.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.corrected_coarsened_weights", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/P023_Audit.lean#L1444", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e6091ca4a7560d9266441b52c8d2341477cb4a7131f5fb748758e6331ffa0946", "conditional_on": [], "credit_reason": "The full-rank premise is in T3(b), and guarantees the epsilon choices are genuine unique fits with positive normalization.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.corrected_finite_polynomial_weights", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/P023_Audit.lean#L1629", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e6091ca4a7560d9266441b52c8d2341477cb4a7131f5fb748758e6331ffa0946", "conditional_on": [], "credit_reason": "Distinct margin values, not observations, are counted, and zero margins are excluded as in the target.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.corrected_finite_nonlocality", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/P023_Audit.lean#L1782", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e6091ca4a7560d9266441b52c8d2341477cb4a7131f5fb748758e6331ffa0946", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': 'At P023/P023_Audit.lean:2652, the theorem states the residual-weighted bias, exact-polynomial unbiasedness, centered variance formula, and positive variance contributions required by the assigned T3(d) component. Quantifying over every normalized outcome law P at a fixed full-rank design represents conditioning on that design without losing any conditional-law case. Outcome integrability and integrable squared errors are the existence conditions for the finite means and variances in the claim, and the zero cross-moment premise is precisely uncorrelatedness of the centered errors. The fixed-margin representation is therefore a minor convention, not a material narrowing.'}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.corrected_finite_conditional_moments", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/P023_Audit.lean#L2356", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e6091ca4a7560d9266441b52c8d2341477cb4a7131f5fb748758e6331ffa0946", "conditional_on": [], "credit_reason": "Covers T3(e), not a primitive sampling-consistency theorem. Positive distant information shares are exact population identities.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.Wave1.uniform_common_target_distant_information_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/Wave1/Gap/GLP023B1.lean#L812", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e6091ca4a7560d9266441b52c8d2341477cb4a7131f5fb748758e6331ffa0946", "conditional_on": [], "credit_reason": "Auxiliary equivalence genuinely needed to identify the bin-intercept estimator with the printed regression; empty nuisance bins do not narrow the statement.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.coarsened_ols_printed_equivalence", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/P023_Audit.lean#L1409", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:e6091ca4a7560d9266441b52c8d2341477cb4a7131f5fb748758e6331ffa0946", "conditional_on": [], "credit_reason": "Accepted generalization, not a derivation of either consistency premise. 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p. 3 (PDF p. 3) and p. 7 (PDF p. 7); Section 5, p. 8 (PDF p. 8). Coarsened OLS has exactly the localized RD weights and a unique party coefficient. Full-rank polynomial OLS has the residual weights, root-count nonlocality and conditional moments stated in clauses (b)–(d). The uniform degree-one and degree-seven examples share the cutoff target while retaining distant influence. For every positive tolerance the difference-event probability is bounded by the two half-tolerance error probabilities; shared consistency implies convergence of the estimator difference without an independence or sample-admissibility premise.

", "headline": "Section 1, last paragraph (p. 3), and Section 3.2, last paragraph (p. 7), as repaired by Sol: sole reliance on close-election outcomes is reserved for the coarsened regression of Proposition 1; for polynomial controls, a common discontinuity target together with non-zero weights on distant outcomes", "location": "p. 3 (PDF p. 3) and p. 7 (PDF p. 7); the ‘may’ sentence of Section 5, p. 8 (PDF p. 8), is kept" }, "id": "P023-T3", "text": { "body": "p. 3 (PDF p. 3) and p. 7 (PDF p. 7); Section 5, p. 8 (PDF p. 8). Coarsened OLS has exactly the localized RD weights and a unique party coefficient. Full-rank polynomial OLS has the residual weights, root-count nonlocality and conditional moments stated in clauses (b)–(d). The uniform degree-one and degree-seven examples share the cutoff target while retaining distant influence. For every positive tolerance the difference-event probability is bounded by the two half-tolerance error probabilities; shared consistency implies convergence of the estimator difference without an independence or sample-admissibility premise.", "headline": "Section 1, last paragraph (p. 3), and Section 3.2, last paragraph (p. 7), as repaired by Sol: sole reliance on close-election outcomes is reserved for the coarsened regression of Proposition 1; for polynomial controls, a common discontinuity target together with non-zero weights on distant outcomes", "location": "p. 3 (PDF p. 3) and p. 7 (PDF p. 7); the ‘may’ sentence of Section 5, p. 8 (PDF p. 8), is kept" }, "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T3" }, { "declarations": [ { "claim_revision_id": "catalog:b391c8191f3e185d20d30c4356733066a158c63d1bc0c345090ddd2726b12199", "conditional_on": [], "credit_reason": "This is the specified population witness component, not an assertion that all of Proposition 2 is false for this Lipschitz mean.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.Wave1.sigmoid_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/Wave1/Gap/GLP023B1.lean#L1282", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b391c8191f3e185d20d30c4356733066a158c63d1bc0c345090ddd2726b12199", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P023/Wave1/Gap/GP023B1b.lean:1188 gives T4(v)'s irreducible risk 1/100, the fitted-risk upper bound 1/100+a^2*eta/3, the universal improvement bound, and the claimed conditional degree-seven coefficient formula. Requiring competitors to be measurable with integrable squared loss loses no predictor that can improve the finite fitted risk: infinite-risk predictors cannot improve it, and completion-measurable predictors have Borel versions with the same risk. The infimum in fittedRisk (P023/P023_Audit.lean:2714) is the actual least-squares risk of the finite-dimensional regressor span under this bounded probability law, whose minimum is attained. The restrictions noted by the reviewer do not remove a material case of the assigned fit component.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.Wave1.sigmoid_fit_criterion_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/Wave1/Gap/GP023B1b.lean#L1187", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b391c8191f3e185d20d30c4356733066a158c63d1bc0c345090ddd2726b12199", "conditional_on": [], "credit_reason": "This covers fixed degrees one and seven under the one numerical law. The target's asymptotic content is unaffected by initial singular samples or finite bandwidth prefixes.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.Wave1.sigmoid_sample_separation_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/Wave1/Gap/GP023B1b.lean#L1999", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b391c8191f3e185d20d30c4356733066a158c63d1bc0c345090ddd2726b12199", "conditional_on": [], "credit_reason": "The population and fit rows already export the exact coefficients and improvement bound; no separate numerical certificate is needed.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.sigmoid_rational_certificates", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/P023_Audit.lean#L2193", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:b391c8191f3e185d20d30c4356733066a158c63d1bc0c345090ddd2726b12199", "conditional_on": [], "credit_reason": "Zero is correctly excluded: party(0)=1 prevents oddness there. The residual variance identities are in R147-03.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.Wave1.uniform_residual_symmetry_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/Wave1/Gap/GP023B2b.lean#L365", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:b391c8191f3e185d20d30c4356733066a158c63d1bc0c345090ddd2726b12199", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P023-T4", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P023-T4" }, "html": { "body": "

p. 7 (PDF p. 7) and p. 10 (PDF p. 10). For every positive amplitude and positive width at most one, the uniform-margin clipped linear family has a continuous Lipschitz conditional mean whose centered part is odd, and zero cutoff jump. Its RD means, residuals, degree-one and degree-seven coefficients and numerical outcome bounds are as displayed. At every degree the fitted risk lies between the noise risk and that risk plus a squared amplitude times width divided by three; no measurable predictor, including one with infinite risk, improves it by more. For fixed degree the coefficient tends to twice the amplitude as width vanishes, while the risk bound vanishes. Every family member has the stated fixed-degree sampling limit; the numerical degree-one and degree-seven limits remain separated from shrinking-window RD under IID sampling and the stated bandwidth limits alone.

", "headline": "Remark to Proposition 2 (‘we require that the model fit cannot be meaningfully improved by including additional terms’), read with Appendix B (‘E[u∣DemVot]≈0\\mathrm E[u|DemVot]\\approx0E[u∣DemVot]≈0 and therefore E[β^OLS2]≈β\\mathrm E[\\hat\\beta_{OLS2}]\\approx\\betaE[β^​OLS2​]≈β’), as printed; the witness (uniform margins, degrees 1 and 7) is the settling validator's construction and is not in the paper", "location": "p. 7 (PDF p. 7) and p. 10 (PDF p. 10)" }, "id": "P023-T4", "text": { "body": "p. 7 (PDF p. 7) and p. 10 (PDF p. 10). For every positive amplitude and positive width at most one, the uniform-margin clipped linear family has a continuous Lipschitz conditional mean whose centered part is odd, and zero cutoff jump. Its RD means, residuals, degree-one and degree-seven coefficients and numerical outcome bounds are as displayed. At every degree the fitted risk lies between the noise risk and that risk plus a squared amplitude times width divided by three; no measurable predictor, including one with infinite risk, improves it by more. For fixed degree the coefficient tends to twice the amplitude as width vanishes, while the risk bound vanishes. Every family member has the stated fixed-degree sampling limit; the numerical degree-one and degree-seven limits remain separated from shrinking-window RD under IID sampling and the stated bandwidth limits alone.", "headline": "Remark to Proposition 2 (‘we require that the model fit cannot be meaningfully improved by including additional terms’), read with Appendix B (‘$\\mathrm E[u|DemVot]\\approx0$ and therefore $\\mathrm E[\\hat\\beta_{OLS2}]\\approx\\beta$’), as printed; the witness (uniform margins, degrees 1 and 7) is the settling validator's construction and is not in the paper", "location": "p. 7 (PDF p. 7) and p. 10 (PDF p. 10)" }, "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T4" }, { "declarations": [ { "claim_revision_id": "catalog:cb5f15c804a30b47c1724dba5fca8421bb97969ae28a8b06975b34960be3d8fe", "conditional_on": [], "credit_reason": "No law-uniform coefficient-error multiplier exists; the assertion does not claim a fixed-law failure as J grows.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.Wave1.clustered_counterexample_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/Wave1/Gap/GLP023B2.lean#L809", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:cb5f15c804a30b47c1724dba5fca8421bb97969ae28a8b06975b34960be3d8fe", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P023/Wave1/Gap/GP023B2b.lean:647 states the constant predictor's risk as 1/100+E[h^2] and bounds its improvement over every measurable finite-risk predictor by (d/2)^2, matching T5(vi). Taking the competitor to be the bounded conditional mean gives E[h^2] <= (d/2)^2, and d=1/1000 gives the claimed 1/4000000 bound. Infinite-risk predictors cannot improve this finite baseline, while completion-measurable predictors have risk-equivalent Borel versions, so the competitor conditions are harmless. The additional family and fitted-risk conclusions preserve the fixed numerical claim.\"}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.Wave1.clustered_fit_bound_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/Wave1/Gap/GP023B2b.lean#L644", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:cb5f15c804a30b47c1724dba5fca8421bb97969ae28a8b06975b34960be3d8fe", "conditional_on": [], "credit_reason": "The clustered population row already exports the exact numerical population coefficient.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.clustered_bias_rational_certificate", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/P023_Audit.lean#L2184", "status": "proved", "support_scope": "historical declaration", "used_in_solution": false } ], "formalization": { "claim_revision_id": "catalog:cb5f15c804a30b47c1724dba5fca8421bb97969ae28a8b06975b34960be3d8fe", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P023-T5", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P023-T5" }, "html": { "body": "

p. 10 (PDF p. 10); Proposition 2 and its Remark, p. 7 (PDF p. 7). The clustered continuous positive-density family has continuous Lipschitz conditional means with odd centered parts and zero cutoff jumps, all the displayed moments and RD values, and a coefficient tending to minus one while the uniform zero-polynomial error tends to zero. At the numerical witness the coefficient, outcome bounds and error bound are as displayed. The constant and fitted predictors can be improved by at most the squared half-width by any measurable predictor, even one with infinite risk. Consequently no coefficient-error constant can be uniform over the stated laws.

", "headline": "Appendix B, the step ‘E[u∣DemVot]≈0\\mathrm E[u|DemVot]\\approx0E[u∣DemVot]≈0 and therefore E[β^OLS2]≈β\\mathrm E[\\hat\\beta_{OLS2}]\\approx\\betaE[β^​OLS2​]≈β’, as printed; the witness (a margin distribution concentrated near ±12\\pm\\tfrac12±21​, linear control) is the validators' construction and is not in the paper", "location": "p. 10 (PDF p. 10); Proposition 2 and its Remark, p. 7 (PDF p. 7)" }, "id": "P023-T5", "text": { "body": "p. 10 (PDF p. 10); Proposition 2 and its Remark, p. 7 (PDF p. 7). The clustered continuous positive-density family has continuous Lipschitz conditional means with odd centered parts and zero cutoff jumps, all the displayed moments and RD values, and a coefficient tending to minus one while the uniform zero-polynomial error tends to zero. At the numerical witness the coefficient, outcome bounds and error bound are as displayed. The constant and fitted predictors can be improved by at most the squared half-width by any measurable predictor, even one with infinite risk. Consequently no coefficient-error constant can be uniform over the stated laws.", "headline": "Appendix B, the step ‘$\\mathrm E[u|DemVot]\\approx0$ and therefore $\\mathrm E[\\hat\\beta_{OLS2}]\\approx\\beta$’, as printed; the witness (a margin distribution concentrated near $\\pm\\tfrac12$, linear control) is the validators' construction and is not in the paper", "location": "p. 10 (PDF p. 10); Proposition 2 and its Remark, p. 7 (PDF p. 7)" }, "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T5" }, { "declarations": [ { "claim_revision_id": "catalog:ab59a53a5d296619895d8e122a2a1af83cb86cace69b2d887391a907c5f5795a", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P023/Wave1/ContinuityCounterexampleTailC.lean:10 exports T6's logarithmic amplification asymptotic, continuous-function norm identity, and one fixed bounded continuous h with coefficients at least m/4 along strictly increasing degrees. It also states coefficient divergence along that subsequence, failure of convergence to zero, the RD limit zero, the outcome bounds, and absence of a Lipschitz constant. Representing Y=1/2+h(X) by its mean function is exact for this deterministic outcome, and total real functions restricted to [-1,1] represent the target function class without loss. 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p. 7 (PDF p. 7) and p. 10 (PDF p. 10). For one fixed uniform margin law, logarithmic amplification yields one bounded continuous mean with diverging polynomial coefficients along increasing degrees and a vanishing RD limit; every continuous function satisfying the zero-at-cutoff, size-bound and growing-bias-subsequence conditions has no Lipschitz bound.

", "headline": "Proposition 2 with the assumptions Appendix B states for it (continuity of g()g()g() and f()f()f()), as printed; the witness (uniform margins and one continuous hhh) is the settling validator's construction and is not in the paper", "location": "p. 7 (PDF p. 7) and p. 10 (PDF p. 10)" }, "id": "P023-T6", "text": { "body": "p. 7 (PDF p. 7) and p. 10 (PDF p. 10). For one fixed uniform margin law, logarithmic amplification yields one bounded continuous mean with diverging polynomial coefficients along increasing degrees and a vanishing RD limit; every continuous function satisfying the zero-at-cutoff, size-bound and growing-bias-subsequence conditions has no Lipschitz bound.", "headline": "Proposition 2 with the assumptions Appendix B states for it (continuity of $g()$ and $f()$), as printed; the witness (uniform margins and one continuous $h$) is the settling validator's construction and is not in the paper", "location": "p. 7 (PDF p. 7) and p. 10 (PDF p. 10)" }, "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T6" }, { "declarations": [ { "claim_revision_id": "catalog:aedff9ad65e43940a31c513c94948a8757c12e7dde83aa2618042b68bd38b810", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P023/Wave1/Gap/GLP023B3.lean:409 supplies the assigned T7(a)-(f) projection identities, positive residual variance, bias identity, both error bounds, amplification lower bound, exact-specification result, and fixed-degree criteria. Its FWL numerator uses the conditional mean under the margin law rather than Y under the joint law; ConditionalMean at P023/P023_Audit.lean:1119 equates these pairings by using a bounded measurable representative of the residual on the supported interval. Boundedness only on [-1,1] covers the whole supported domain, and allowing constant terms in exact polynomial specification includes the target's zero-constant case. 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Record disagreement is classification only.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.Wave1.uniform_residual_constants_proof", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/Wave1/Gap/GLP023B3.lean#L1477", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:aedff9ad65e43940a31c513c94948a8757c12e7dde83aa2618042b68bd38b810", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P023/P023_Audit.lean:2560 states exactly T7(g)'s assigned conditional-bias formula and positive residual denominator, with normalized P, full rank, margins in [-1,1], and the explicit d_i=I[x_i>=0] convention. An arbitrary normalized outcome law at each fixed design represents its conditional outcome law, so the absence of a random-margin conditional-expectation operator removes no conditional-bias case. The row covers the conditional-bias component, rather than claiming that this declaration alone supplies the preceding arbitrary-outcome FWL identities. 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p. 10 (PDF p. 10) and p. 7 (PDF p. 7). Population and finite-sample residual formulas give the bias identities, approximation bounds, exact-specification and fixed-degree criteria, and the displayed uniform-law constants.

", "headline": "Appendix B, the step ‘E[u∣DemVot]≈0\\mathrm E[u|DemVot]\\approx0E[u∣DemVot]≈0 and therefore E[β^OLS2]≈β\\mathrm E[\\hat\\beta_{OLS2}]\\approx\\betaE[β^​OLS2​]≈β’, and the Remark to Proposition 2, as repaired by Sol: the exact identity for the coefficient error and its bounds", "location": "p. 10 (PDF p. 10) and p. 7 (PDF p. 7)" }, "id": "P023-T7", "text": { "body": "p. 10 (PDF p. 10) and p. 7 (PDF p. 7). Population and finite-sample residual formulas give the bias identities, approximation bounds, exact-specification and fixed-degree criteria, and the displayed uniform-law constants.", "headline": "Appendix B, the step ‘$\\mathrm E[u|DemVot]\\approx0$ and therefore $\\mathrm E[\\hat\\beta_{OLS2}]\\approx\\beta$’, and the Remark to Proposition 2, as repaired by Sol: the exact identity for the coefficient error and its bounds", "location": "p. 10 (PDF p. 10) and p. 7 (PDF p. 7)" }, "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T7" }, { "declarations": [ { "claim_revision_id": "catalog:3ac450f28dbca4a4f9e81ae6c5ffe1f43c4c3375234416d4f9468ffe9a8e7d61", "conditional_on": [], "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"P023/Wave1/Gap/GLP023B4c.lean:3037 exports all corrected T8(a)-(g) conclusions, including the zero-constant polynomial, population rate, fixed-degree and RD limits, growing-degree tracking, and existence of a valid degree schedule. RepairedRegularity at P023/P023_Audit.lean:1367 contains the target's own primitive assumptions, with local variance expressed equivalently through integrals over all local Borel sets. Positivity at every bandwidth index only constrains an immaterial finite prefix, and the outer-probability formulation in ConvergesInProbability (P023/P023_Audit.lean:1337) agrees asymptotically with a measurable OLS extension because singular-design probabilities vanish under the stated conditions. 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p. 7 (PDF p. 7); Appendix B, p. 10 (PDF p. 10). Under the stated finite-second-moment IID, local density domination, Lipschitz mean, bounded local conditional variance and shrinking-bandwidth assumptions, all the positive-definite moment, residual-variance, zero-constant polynomial approximation, population bias, fixed-degree, RD and growing-degree conclusions (a)–(g) hold, including existence of a slow degree schedule. The smallest moment eigenvalues and residual normalizations are positive. No separate singular-design or empty-window probability assumptions are required.

", "headline": "Proposition 2, as repaired by Sol and the settling validator: a fixed law with a density bounded below near the cutoff and a Lipschitz smooth part", "location": "p. 7 (PDF p. 7); proof in Appendix B, p. 10 (PDF p. 10)" }, "id": "P023-T8", "text": { "body": "p. 7 (PDF p. 7); Appendix B, p. 10 (PDF p. 10). Under the stated finite-second-moment IID, local density domination, Lipschitz mean, bounded local conditional variance and shrinking-bandwidth assumptions, all the positive-definite moment, residual-variance, zero-constant polynomial approximation, population bias, fixed-degree, RD and growing-degree conclusions (a)–(g) hold, including existence of a slow degree schedule. The smallest moment eigenvalues and residual normalizations are positive. No separate singular-design or empty-window probability assumptions are required.", "headline": "Proposition 2, as repaired by Sol and the settling validator: a fixed law with a density bounded below near the cutoff and a Lipschitz smooth part", "location": "p. 7 (PDF p. 7); proof in Appendix B, p. 10 (PDF p. 10)" }, "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T8" }, { "declarations": [ { "claim_revision_id": "catalog:ad06f69258bec444540f6d5274684619823bca387291e59dfc855422c1fa9b47", "conditional_on": [], "credit_reason": "The corrected sum runs over election pairs, and Condition1 ensures strictly positive central residual information.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.corrected_appendix_pairs", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/P023_Audit.lean#L1942", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:ad06f69258bec444540f6d5274684619823bca387291e59dfc855422c1fa9b47", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P023-T9", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P023-T9" }, "html": { "body": "

p. 9 (PDF p. 9). The corrected pair-indexed Appendix A formula partitions the observed pairs into bins, annihilates the demeaned party indicator outside the central bin, and gives the positive-denominator localized OLS ratio.

", "headline": "Appendix A, proof of Proposition 1: the display after ‘Therefore,’, with the summation index corrected from i∈Ubi\\in U_bi∈Ub​ to it∈Ubit\\in U_bit∈Ub​", "location": "p. 9 (PDF p. 9)" }, "id": "P023-T9", "text": { "body": "p. 9 (PDF p. 9). The corrected pair-indexed Appendix A formula partitions the observed pairs into bins, annihilates the demeaned party indicator outside the central bin, and gives the positive-denominator localized OLS ratio.", "headline": "Appendix A, proof of Proposition 1: the display after ‘Therefore,’, with the summation index corrected from $i\\in U_b$ to $it\\in U_b$", "location": "p. 9 (PDF p. 9)" }, "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T9" }, { "declarations": [ { "claim_revision_id": "catalog:b0127329b17bcbb6b4854fabef1a349ec6ba75ae018fd989e331cb1559b42f91", "conditional_on": [], "credit_reason": "Its negated universal includes degree zero and cannot directly establish the strengthened positive-degree refutation; the credited degree-one witness supplies that refutation.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.printed_polynomial_localization_refuted", "manifest_use": "not_used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/P023_Audit.lean#L701", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": false }, { "claim_revision_id": "catalog:b0127329b17bcbb6b4854fabef1a349ec6ba75ae018fd989e331cb1559b42f91", "conditional_on": [], "credit_reason": "This proves the positive coarsened component of T10; no polynomial localization is inferred.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.coarsened_localization", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/P023_Audit.lean#L677", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:b0127329b17bcbb6b4854fabef1a349ec6ba75ae018fd989e331cb1559b42f91", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P023-T10", "decision": "refuted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P023-T10" }, "html": { "body": "

The printed claim is false: P023-T10; p. 3 (PDF p. 3); p. 7 (PDF p. 7); p. 8 (PDF p. 8). The universal localization assertion for positive-degree polynomial vote-share controls fails, while coarsened OLS coefficients agree for outcomes agreeing on the close elections when both close-window counts are positive.

", "headline": "Section 1, last paragraph (‘all flexibly specified … mathematically equivalent to studies of close elections’; ‘rely solely on data from close elections’), with Section 3.2, last paragraph, and Section 5, first two sentences, as printed", "location": "p. 3 (PDF p. 3); p. 7 (PDF p. 7); p. 8 (PDF p. 8)" }, "id": "P023-T10", "text": { "body": "The printed claim is false: P023-T10; p. 3 (PDF p. 3); p. 7 (PDF p. 7); p. 8 (PDF p. 8). The universal localization assertion for positive-degree polynomial vote-share controls fails, while coarsened OLS coefficients agree for outcomes agreeing on the close elections when both close-window counts are positive.", "headline": "Section 1, last paragraph (‘all flexibly specified … mathematically equivalent to studies of close elections’; ‘rely solely on data from close elections’), with Section 3.2, last paragraph, and Section 5, first two sentences, as printed", "location": "p. 3 (PDF p. 3); p. 7 (PDF p. 7); p. 8 (PDF p. 8)" }, "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T10" }, { "declarations": [ { "claim_revision_id": "catalog:407e1816741ae290ca4c83cf1ccc6971d38189ff5e0d201e500ad53fb9e390a2", "conditional_on": [], "credit_reason": "The printed-design all-solutions theorem directly states Proposition 1; the equivalent bin parametrization is not required separately.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.proposition1", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/P023_Audit.lean#L657", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:407e1816741ae290ca4c83cf1ccc6971d38189ff5e0d201e500ad53fb9e390a2", "conditional_on": [], "credit_reason": "This is the printed finite-sample identity for arbitrary outcomes and every OLS solution, including singular nuisance designs. No claim-stage change.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P023.printed_coarsened_proposition1", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P023/P023_Audit.lean#L1350", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:407e1816741ae290ca4c83cf1ccc6971d38189ff5e0d201e500ad53fb9e390a2", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P023-T11", "decision": "restricted", "release_id": "sap-2026-11-08-assured-release", "target_id": "P023-T11" }, "html": { "body": "

Proposition 1, p. 6 (PDF p. 6); Conditions 1–3, p. 5 (PDF p. 5); Appendix A, p. 9 (PDF p. 9). For finite two-party election data with margins in [-1,1], binary win indicators agreeing with the signs of nonzero margins, bandwidth 0 < k < 1, and the two strictly interior observations required by Condition 1, both party counts in U_b = (-k,k] are positive. Every least-squares solution of the all-election regression with an intercept, the win indicator, and the printed width-2k bin indicators of Condition 2 has win coefficient equal to the degree-zero, rectangular-kernel RD difference in means of Condition 3. Ties may have either binary indicator and nuisance bins may be empty. All sums here are finite.

", "headline": "Proposition 1", "location": "p. 6 (PDF p. 6); Conditions 1–3 on p. 5 (PDF p. 5); proof in Appendix A, p. 9 (PDF p. 9)" }, "id": "P023-T11", "text": { "body": "Proposition 1, p. 6 (PDF p. 6); Conditions 1–3, p. 5 (PDF p. 5); Appendix A, p. 9 (PDF p. 9). For finite two-party election data with margins in [-1,1], binary win indicators agreeing with the signs of nonzero margins, bandwidth 0 < k < 1, and the two strictly interior observations required by Condition 1, both party counts in U_b = (-k,k] are positive. Every least-squares solution of the all-election regression with an intercept, the win indicator, and the printed width-2k bin indicators of Condition 2 has win coefficient equal to the degree-zero, rectangular-kernel RD difference in means of Condition 3. Ties may have either binary indicator and nuisance bins may be empty. All sums here are finite.", "headline": "Proposition 1", "location": "p. 6 (PDF p. 6); Conditions 1–3 on p. 5 (PDF p. 5); proof in Appendix A, p. 9 (PDF p. 9)" }, "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T11" }, { "declarations": [ { "claim_revision_id": "catalog:edbc82866ba52a436c90d5b28c670399081c29460547686906bf7bfde8d7c377", "conditional_on": [], "credit_reason": "This is the literal negation of the entire type of printed_proposition2 at the same universe level, not merely a witness lemma or a negation of a Lipschitz repair. The target itself supplies the precise fixed-law reading, so acceptance does not depend on the record calling that reading approved. Extra regularity premises do not weaken a refutation when the bad law satisfies them. The auxiliary bad-mean statement gives a single continuous h with unbounded coefficient subsequence, so the failure concerns the claimed eventual-in-degree bound. 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p. 7 (PDF p. 7); equation (2) on p. 6 (PDF p. 6); Appendix B, p. 10 (PDF p. 10). The adopted fixed-law reading of Proposition 2, formalized in PrintedPropositionTwo, fails: RD consistency together with the assertion that, for each positive tolerance, every sufficiently large fixed polynomial degree gives an estimator difference converging to a population bias below that tolerance. The formal hypotheses include IID sampling, finite second moments, nondegenerate population moments, local support, and explicit usual-RD regularity; the moment guards are additional formal assumptions.

", "headline": "Proposition 2, with its Remark and the proof in Appendix B as printed", "location": "p. 7 (PDF p. 7); equation (2) on p. 6 (PDF p. 6); Appendix B, p. 10 (PDF p. 10)" }, "id": "P023-T12", "text": { "body": "p. 7 (PDF p. 7); equation (2) on p. 6 (PDF p. 6); Appendix B, p. 10 (PDF p. 10). The adopted fixed-law reading of Proposition 2, formalized in PrintedPropositionTwo, fails: RD consistency together with the assertion that, for each positive tolerance, every sufficiently large fixed polynomial degree gives an estimator difference converging to a population bias below that tolerance. The formal hypotheses include IID sampling, finite second moments, nondegenerate population moments, local support, and explicit usual-RD regularity; the moment guards are additional formal assumptions.", "headline": "Proposition 2, with its Remark and the proof in Appendix B as printed", "location": "p. 7 (PDF p. 7); equation (2) on p. 6 (PDF p. 6); Appendix B, p. 10 (PDF p. 10)" }, "url": "https://pmaronow.github.io/llm-audit/works/P023/#P023-T12" } ] }, "missing_sources": [], "reproduction": { "agreement": "agrees_in_part", "status": "selected_recalculation", "targets": [ { "agreement": "agrees_in_part", "location": "p. 8 (PDF p. 8)", "method": "I recalculated the eight printed specifications from the 10 April 2014 Caughey–Sekhon data. Rows (b), (d), (e) and (g) agree to rounding on 8,610 complete elections; the source study’s Use=1 restriction reproduces (a) and (c) but changes (b). Neither sample reproduces all rows, and the triangular-kernel estimates differ (P023-E003). The paper’s analysis code and exact sample rule are unavailable. This agreement uses the outcomes as recorded in that release. Four next-election outcomes are one hundredth of their linked percentages; replacing them lowers rows (d), (e) and (g) to 9.32, 9.32 and 10.40 points as a sensitivity calculation. The correctness of those links and their historical use remain unverified (P023-E004).", "status": "recalculated", "target": "Table 1" }, { "agreement": "disagrees", "location": "p. 1 (PDF p. 1), Introduction, first paragraph", "method": "I counted 103 recorded close elections among 13,923 elections in 1944–2006, a share that rounds to 0.7% rather than 0.6%. This is a lower bound for the full close-election share; 587 missing margins prevent an exact full-election count (P023-E005).", "status": "recalculated", "target": "Introduction: close-election prevalence" } ] }, "summary": "The note asks whether regressions on all elections escape the limitations of close-election studies of the party incumbency advantage. It shows that OLS with the specified vote-share bins as controls equals a regression discontinuity estimate, argues that polynomial controls approximate it, and compares eight estimates for US House elections. 8 entries are classified as errors (4 Moderate, 4 Minor), and 2 are clarity notes; 2 further concerns were not confirmed. With the specified bins, only close-election outcomes determine the coefficient. Under exact polynomial specification and the conditions for consistency of both estimators, polynomial OLS and shrinking-bandwidth RD estimate the same cutoff jump and their difference converges to zero in probability, although distant outcomes enter OLS. Shared concerns about causal identification and extrapolation can survive (\\fid{P023-E001}).", "what_was_checked": "I checked the following points, which held up with the qualifications stated.\n\\begin{itemize}\n\\item Proposition 1 and Appendix A. With sums indexed by election pairs and restricted to nonempty bins, the proof's localization argument holds: the demeaned win indicator vanishes in every bin except the central one, and the slope on a binary regressor within $U_b$ is the difference in means.\n\\item Condition 2. The number $J=2\\lceil(1-k)/(2k)\\rceil$ is the smallest even integer with $(J+1)k\\ge1$, the bins $j=0,\\ldots,J$, with the reference bin closed on the left as in Appendix A, cover $[-1,1]$ for every $k$ in $(0,1)$, and the central bin is $(-k,k]=U_b$. This was checked in exact arithmetic for $k=1/1000,\\ldots,999/1000$.\n\\item Proposition 1 on constructed data, with observations at the edges of the window and with empty bins: the OLS coefficient with the bin indicators equals the difference in means on $U_b$, and it does not move when outcomes outside $U_b$ are changed.\n\\item Table 1 as printed. Rows (d) and (e) both read 10.54, as Proposition 1 requires; the bandwidths in the text of Section 4 agree with the note to the table; and the descriptions of estimators (a)--(h) agree with the row labels.\n\\item The Remark to Condition 3 and footnote 5: a local regression of degree 0 with a rectangular kernel is the difference in means within the window, and with the coarsening of Condition 2 only the central bin contains both winners and losers.\n\\item Appendix B. The conditional mean equals $\\alpha+\\beta I[\\mathit{DemVot}\\ge0]+h(\\mathit{DemVot})$ with the printed $h$, and $h$ is continuous with $h(0)=0$ given the finite left limit of $g$ that the definition of $\\alpha$ presupposes. The Weierstrass step with no constant term is valid because $h(0)=0$.\n\\item Proposition 2 when $h$ is exactly a polynomial of degree at most a fixed $J$ and the population design is nonsingular: the population coefficient is the jump $\\beta$. Under the sampling, support and moment conditions for consistency of both estimators, with $k_N\\to0$ and $Nk_N\\to\\infty$, polynomial OLS and shrinking-bandwidth RD converge to this jump, so their difference converges to zero in probability. This does not settle the separate approximation issue in \\fid{P023-E002}.\n\\item Appendix B does not attach a causal interpretation to the jump $\\beta$, and the target of the close-election estimate is this jump of the conditional mean under the usual RD conditions.\n\\item Section 2. Treatment is determined by the sign of the margin, so winners and losers do not overlap away from the threshold, which supports the caution about extrapolation.\n\\item Every citation in the text has an entry in the reference list and every entry is cited. Apart from the spelling in \\fid{P023-E008}, no mismatch of references or equation numbers was found.\n\\item The codebook of the replication data of Caughey and Sekhon (2011) supplies a margin, a win indicator and the Democratic percentage in the next election. Its margin is Democratic minus leading non-Democratic vote share; it agrees with the theoretical Democratic-minus-Republican definition under the two-party supposition. Section 4 does not identify the columns actually used.\n\\end{itemize}\nI independently recalculated the estimates from the corrected release of 10 April 2014, using \\texttt{DifDPct}, \\texttt{DemWin} and \\texttt{DPctNxt} for 1942--2008. These calculations give 10.5429 points for both rows (d) and (e) and 13.1478 for row (g). On 8,610 complete elections, row (b) rounds to its printed value; adding the source study's \\texttt{Use}=1 restriction gives 8,594 elections and reproduces rows (a) and (c) to rounding. Neither sample reproduces every row. With the printed bandwidths, separate fits on each side and triangular weights $\\max(1-|\\texttt{DifDPct}|/k,0)$ in percentage-point units, rows (f) and (h) give 13.08 and 15.30 points, rather than 10.57 and 12.38 (\\fid{P023-E003}). These comparisons are conditional on this release, variable mapping and sample rules. Four next-election outcomes are one hundredth of their linked percentages at storage precision; replacing them by those percentages changes rows (d) and (e) to 9.32 points and row (g) to 10.40 as a sensitivity calculation. The correctness of the links and percentages and their historical use remain unverified (\\fid{P023-E004}). A recount for 1944--2006 finds 103 recorded close elections among 13,923, a fraction of 0.7\\% rounded and a lower bound on the full close-election share; 587 Democratic margins are missing (\\fid{P023-E005}). For the linear and degree-7 regressions on the 8,610 elections, 1.75\\% and 3.97\\% of absolute outcome weight lies within 0.5 points of the cutoff, respectively; the \\texttt{Use}=1 subset gives 1.75\\% and 3.98\\%. Every outside election has nonzero weight in both specifications under both sample rules (\\fid{P023-E001}).\n\nThe following remain unestablished. The observed fits do not establish whether the linear and degree-7 controls adequately approximate the population conditional mean in the application. The cited works were not compared with what the paper attributes to them (Wooldridge 2002, p.~579; Caughey and Sekhon 2011, Section 5.2; Gelman and King 1990, Eq.~6). The suggestion in Section 5 that the findings carry over to the estimator of Gelman and King (1990) rests on no result in the paper. The authors' analysis code for Section 4 was not found, and the paper does not name the release of the data that it used. The available copy of the codebook is dated 22 December 2025.\n\nThe comparison in \\fid{P023-E001} keeps the threshold-jump target distinct from the observations used to estimate it. Appendix B already supplies the common-target argument; nonzero distant weights establish outcome dependence, not the frequency of substantive estimator gaps or different causal bias or external validity in the application. Shared concerns about identifying assumptions and extrapolation can survive, and no causal interpretation away from the cutoff follows from that comparison.", "work": { "arxiv_id": null, "audit_complete": true, "authors": [ "P. M. Aronow", "David R. Mayhew", "Winston Lin" ], "category": "Articles - Theory and Methodology", "citation": "P. M. Aronow, David R. Mayhew, and Winston Lin (2014). A Note on Close Elections and Regression Analysis of the Party Incumbency Advantage. Statistics, Politics and Policy 5(1–2):1–11.", "doi": "10.1515/spp-2014-0003", "edition": { "pages_in_audited_file": 11, "sha256": "db981a5db66ec05548e34d7492d9c02684b23e9fe42168ba23743cefbd90b8dc", "source_url": "https://doi.org/10.1515/spp-2014-0003", "venue": "Statistics, Politics and Policy", "year": "2014" }, "id": "P023", "later_materials": null, "title": "A Note on Close Elections and Regression Analysis of the Party Incumbency Advantage", "type": "Journal article", "url": "https://pmaronow.github.io/llm-audit/works/P023/", "venue": "Statistics, Politics and Policy", "year": "2014" } }