{ "claims": [ { "claim": "In ten tests (five measures of the incumbent-party candidate's relative extremism, each in two baseline vote models) on the postwar presidential elections, there is scant evidence of an extremism penalty that is substantively large or close to statistical significance; relative ideological location is not an important predictor of presidential election outcomes.", "claim_id": "P039-H1", "related_corrections": [ "P039-E001", "P039-E002", "P039-E004" ], "tied_to": "Abstract (p. 126); introduction (p. 127); pp. 139–140; Table 3 (all ten models); Table 2 columns A3 and B3; Figure 4" }, { "claim": "All five relative-extremism measures correlate negatively but weakly with incumbent-party vote share. The larger NOMINATE correlation, $-0.48$, reflects the eight elections with available scores: the paper reports correlations of $-0.44$ for both zero-centered perception measures on those cases.", "claim_id": "P039-H2", "related_corrections": [ "P039-N006" ], "tied_to": "p. 134 (PDF p. 9); Table 1 (p. 135)" }, { "claim": "Controlling for the three fundamentals, a one-point increase in refined, zero-centered relative extremism is associated with a $0.46$-point lower incumbent vote margin in the musical-chairs model and a $2.23$-point lower two-party share in the bread-and-peace model. Neither estimate is near statistical significance; Figure 4 is described as showing a null relationship in panel A and a weak one in panel B, with $p=0.26$.", "claim_id": "P039-H3", "related_corrections": [ "P039-E002", "P039-E001" ], "tied_to": "p. 138 (PDF p. 13); Table 2 columns A3 and B3; Figure 4" }, { "claim": "The largest predicted effect in the ten models is described as modest, including a $1.38$-point vote-share gain for Carter in 1980 under the featured bread-and-peace specification. The extremism coefficient has the expected negative sign in eight of the ten models.", "claim_id": "P039-H4", "related_corrections": [ "P039-E004", "P039-N010" ], "tied_to": "p. 139 (PDF p. 14); Table 3; note 15" }, { "claim": "The paper reports correlations of $+0.97$ between the ANES/Rosenstone and NOMINATE location scores and $+0.75$ between their absolute values. It suggests that the latter correlation is high enough to make both measures worth taking seriously, while acknowledging uncertainty about how reassuring it is. Both candidates have NOMINATE scores in eight of the seventeen elections.", "claim_id": "P039-H5", "related_corrections": [], "tied_to": "p. 131 (PDF p. 6); Figure 2" }, { "claim": "The musical-chairs and bread-and-peace models replicate on updated data, and the three fundamentals (income growth, incumbent-party tenure, war fatalities) account for much of the variation in presidential outcomes; income growth is strongly related to vote share while relative candidate extremism adds little explanatory force.", "claim_id": "P039-H6", "related_corrections": [ "P039-E003", "P039-E001", "P039-E002", "P039-N008" ], "tied_to": "pp. 127, 135–136 and 140; Table 1; Table 2 (columns A1, A2, B1, B2); Figures 3–4" }, { "claim": "Goldwater and McGovern lost under unfavourable fundamentals (incumbent parties in their first term, strong economies), whereas Reagan in 1980 and Obama in 2008 were rated almost as extreme and won against incumbent parties with poor economic records; and moderation does not guarantee victory, since the rated moderates Dewey (1948), Nixon (1960) and Carter (1980) all lost.", "claim_id": "P039-H7", "related_corrections": [], "tied_to": "Conclusion, p. 140 (PDF p. 15); Figure 1; the data behind Table 2 and Figure 3" }, { "claim": "There is little evidence of an electorally important relationship between candidate extremism and vote outcomes, though this is not evidence that none exists, nor that voters ignore candidates' policy positions; Converse's view that voters respond to the nature of the times and not to ideology stands up better to postwar presidential elections than Downs's median voter theorem.", "claim_id": "P039-H8", "related_corrections": [], "tied_to": "Conclusion, pp. 140–141 (PDF pp. 15–16); Tables 2 and 3 as a whole" } ], "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 seven lines of text. A red box marks the words 'quarters 1-16' in the sentence 'The dependent variable in Panel B is incumbent party share of the two-party vote, and RDI is measured using Hibbs's definition of the variable, with quarters 1-16 discounted by a decay parameter as described above.'", "caption": "p. 136 (PDF p. 11), paragraph on Panel B of Table 2. The marked range is given for the income measure of the bread and peace models.", "url": "https://pmaronow.github.io/llm-audit/works/P039/exhibits/P039-E001.png" }, { "alt": "Crop of the top of the Bread and Peace panel of Table 2, with the column heads B1, B2 and B3. The first row is labelled 'Delta weighted RDI in Q 1-16', and a red box marks 'Q 1-16'. The row gives the coefficients 3.29, 2.82 and 2.89, each with three asterisks, and the standard errors (.52), (.44) and (.44).", "caption": "Table 2, p. 137 (PDF p. 12), first row of the Bread & Peace panel, turned upright (the table is printed sideways). The marked label gives the same range.", "url": "https://pmaronow.github.io/llm-audit/works/P039/exhibits/P039-E001-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P039-T1", "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-T1" }, { "id": "P039-T2", "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-T2" } ], "headline": "The income measure in every bread and peace model that was fitted covers quarters 2–16 of the presidential term, and not quarters 1–16 as the text and Table 2 say.", "id": "P039-E001", "location": "Paragraph on Panel B of Table 2, p. 136 (PDF p. 11); section “The `bread and peace' model”, p. 128 (PDF p. 3); Table 2, Bread & Peace panel, first row label, p. 137 (PDF p. 12)", "pdf_page": 11, "printed_page": "136", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P039-T1", "P039-T2" ], "text": { "as_printed": "p.~128: ``While Hibbs, like Achen and Bartels (2016), also employs change in RDI to measure economic performance, bread and peace allows RDI growth from \\hl{every quarter} of a presidential term to factor into the model, albeit discounted over time so that earlier quarters carry less weight. Hibbs applies a decay parameter---$\\lambda$, estimated within the model as a number between 0 and 1---to progressively discount economic growth in earlier quarters of the presidential term. This discounting takes the following form: Because the election is held one-third of the way through the 16th quarter of a presidential term, RDI growth in this quarter is down-weighted by $\\tfrac{1}{3} * \\lambda^{0}$. The 15th quarter of the presidential term is down-weighted by $\\lambda^{1}$, the 14th quarter by $\\lambda^{2}$, and so on for the preceding quarters.''\n\np.~136: ``The dependent variable in Panel B is incumbent party share of the two-party vote, and RDI is measured using Hibbs's definition of the variable, with \\hl{quarters 1--16} discounted by a decay parameter as described above.''\n\nTable 2, p.~137, Bread \\& Peace panel, first row label: ``$\\Delta$ weighted RDI in \\hl{Q 1-16}''", "corrected": "p.~136: ``The dependent variable in Panel B is incumbent party share of the two-party vote, and RDI is measured using Hibbs's definition of the variable, with \\hl{quarters 2--16} discounted by a decay parameter as described above.''\n\nTable 2, Bread \\& Peace panel, first row label: ``$\\Delta$ weighted RDI in \\hl{Q 2-16}''\n\np.~128, to be added after ``and so on for the preceding quarters'': In the analyses reported here, the weighted measure of income growth uses quarters 2--16 of each presidential term, and quarter 1 is excluded. Quarter 16 receives the weight $1/3$, quarters 15 through 2 receive the weights $\\lambda$ through $\\lambda^{14}$, and the weighted sum is divided by the sum of the weights. With $h_q$ the annualized growth of RDI in quarter $q$, the measure is\n\\[\n\\frac{h_{16}/3+\\sum_{k=1}^{14}\\lambda^{k}h_{16-k}}{1/3+\\sum_{k=1}^{14}\\lambda^{k}},\n\\]\nand $\\lambda$ is estimated together with the other coefficients. If note 5 is kept as a description of these models, its exponents end at 14 and not at 15; the note is the subject of \\fid{P039-E006}.\n\nAlternatively, the range 1--16 is kept in the text and the table, and $\\lambda^{15}h_1$ and $\\lambda^{15}$ are added to the numerator and the denominator of each of the nine formulas. Panel B of Table 2, the bread and peace half of Table 3, Figure 4B and the numbers quoted from them are then updated. The extremism coefficient in column B3 becomes $-2.30$, with standard error $2.08$ and p-value $.29$, and the p-value for incumbent fatigue in column B3 becomes $.050$, so that its marker for $p<.05$ is removed.", "evidence": "The evidence is a check of the nine formulas, an exact counterexample which shows that the two measures differ, and a refit of the models. I write $N_{15}$ and $D_{15}$ for the numerator and the denominator of $G_{15}$. The measure as described, with quarter 1 included, is\n\\[\nG_{16}=\\frac{N_{15}+\\lambda^{15}h_1}{D_{15}+\\lambda^{15}}.\n\\]\nSubtracting $G_{15}$, I obtain\n\\[\nG_{16}-G_{15}=\\frac{\\lambda^{15}}{D_{15}+\\lambda^{15}}\\,(h_1-G_{15}).\n\\]\nFor $0<\\lambda<1$, the two measures agree only if $h_1=G_{15}$. For the counterexample, let $\\lambda=9/10$, $h_1=1$ and $h_2=\\cdots=h_{16}=0$. Then $G_{15}=0$ in each of the nine archived formulas, and for the measure as described I find\n\\[\nG_{16}=\\frac{(9/10)^{15}}{1/3+\\sum_{k=1}^{15}(9/10)^{k}}=\\frac{617673396283947}{22440939433444477}\\approx 0.0275.\n\\]\nFitted with the archived formulas, the models reproduce the coefficients of Panel B of Table 2 and the five bread and peace entries of Table 3 to the printed precision. The models were then fitted again with $\\lambda^{15}h_1$ added to the numerator and $\\lambda^{15}$ to the denominator, and nothing else changed. The results of the two fits are in the table below. The p-values are the approximate ones that the fitting routine reports.\n\\begin{audittable}{lrr}{Column B3 of Table 2, the bread and peace models of Table 3 and the fitted line of Figure 4B, with the income measure as archived (quarters 2--16) and with quarter 1 restored (quarters 1--16). Maximum effects are in points of the two-party vote share, with the p-value of the extremism coefficient in parentheses.}\nQuantity & Quarters 2--16 & Quarters 1--16 \\\\\n\\hline\nB3, relative extremism: coefficient & $-2.228$ & $-2.300$ \\\\\nB3, relative extremism: standard error & $2.045$ & $2.078$ \\\\\nB3, relative extremism: p-value & $.301$ & $.294$ \\\\\nB3, incumbent fatigue: p-value & $.0466$ & $.0501$ \\\\\nMaximum effect, all-voter zero-centered & $1.24$ $(.391)$ & $1.27$ $(.389)$ \\\\\nMaximum effect, all-voter voter-centered & $0.30$ $(.848)$ & $0.28$ $(.856)$ \\\\\nMaximum effect, refined zero-centered & $1.38$ $(.301)$ & $1.43$ $(.294)$ \\\\\nMaximum effect, refined voter-centered & $0.82$ $(.639)$ & $0.82$ $(.640)$ \\\\\nMaximum effect, NOMINATE & $1.14$ $(.773)$ & $1.14$ $(.774)$ \\\\\nFigure 4B, fitted line: p-value & $.259$ & $.252$ \\\\\n\\end{audittable}", "problem": "The paragraph on p.~136 says that the income measure in Panel B of Table 2 discounts quarters 1--16, and the row label in the table repeats this range. The description on p.~128 assigns the weight $1/3$ to quarter 16, the weight $\\lambda$ to quarter 15 and the weight $\\lambda^{2}$ to quarter 14, ``and so on for the preceding quarters''. Carried back to the start of the term, this rule implies that quarter 1 has the weight $\\lambda^{15}$. The models that were fitted stop at quarter 2. Let $h_q$ be the annualized growth of real disposable income in quarter $q$ of the term. The replication files contain nine formulas that are fitted by nonlinear least squares, eight for Tables 2 and 3 and one for Figure 4B. In each of them the income measure is\n\\[\nG_{15}=\\frac{h_{16}/3+\\sum_{k=1}^{14}\\lambda^{k}h_{16-k}}{1/3+\\sum_{k=1}^{14}\\lambda^{k}}.\n\\]\nThe growth $h_1$ of the first quarter is in the data for each of the sixteen elections of 1952--2012, and it enters none of the formulas. The stated range is therefore not the range of the fitted models. Hibbs (2012) and the online appendix were not available. I could not determine whether Hibbs's own specification uses fifteen or sixteen quarters, or whether quarter 1 was left out on purpose.", "what_survives": "If the description is corrected to quarters 2--16, this finding changes no number in the paper. The ideology measures, Table 1, Panel A of Table 2, the musical chairs half of Table 3, Figures 1--3 and Figure 4A do not involve the income measure of the bread and peace models. If quarter 1 is restored instead, the entries of Panel B of Table 2, the bread and peace half of Table 3, Figure 4B and the numbers quoted from them on pp.~138--139 change; the table shows the changes for column B3, Table 3 and Figure 4B. The p-value for incumbent fatigue in column B3 moves from $.047$ to $.050$, so that its marker for $p<.05$ would be removed. Under either repair the five bread and peace extremism coefficients are negative and far from significance (with quarter 1 restored, p-values from $.294$ to $.856$ and maximum effects from $0.28$ to $1.43$ points). The paper's finding of scant evidence of an extremism penalty is unchanged." }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-E001", "work": { "arxiv_id": null, "doi": "10.1177/0002716216660571", "title": "Ideologically Extreme Candidates in U.S. Presidential Elections, 1948-2012", "url": "https://pmaronow.github.io/llm-audit/works/P039/" }, "work_id": "P039" }, { "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 five lines of the page. The final sentence begins 'The ability of candidate extremism to explain the remaining variation in vote share' and says that the relationship depicted in panel B remains well short of statistical significance. A red box marks its closing clause, 'the two-tailed p-value for the coefficient is .26.'", "caption": "p. 138 (PDF p. 13), end of the last paragraph. The marked clause gives the $p$-value of the line fitted to the residuals of Figure 4, panel B.", "url": "https://pmaronow.github.io/llm-audit/works/P039/exhibits/P039-E002.png" } ], "formal_release": null, "formal_statements": [ { "id": "P039-T3", "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-T3" }, { "id": "P039-T4", "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-T4" } ], "headline": "The $p$-value of .26 on p. 138 is the nominal $p$-value of the line fitted to the residuals in Figure 4, panel B; the test of extremism given the fundamentals is the test of the extremism coefficient of column B3, whose approximate $p$-value is .30.", "id": "P039-E002", "location": "p. 138 (PDF p. 13), last sentence of the page, on Figure 4, panel B; compare Table 2, column B3, p. 137 (PDF p. 12), and Table 3, row (3), p. 139 (PDF p. 14)", "pdf_page": 13, "printed_page": "138", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P039-T3", "P039-T4" ], "text": { "as_printed": "p.~138: ``The ability of candidate extremism to explain the remaining variation in vote share from the bread and peace--based model is stronger than in the musical chairs--based model, but the relationship depicted in panel B remains well short of statistical significance: \\hl{the two-tailed $p$-value for the coefficient is .26}.''\n\np.~139: ``As Table 3 also reports, the $p$-value for the coefficient underlying this estimate is .30 and based on ten degrees of freedom.''", "corrected": "``The ability of candidate extremism to explain the remaining variation in vote share from the bread and peace--based model is stronger than in the musical chairs--based model. \\hl{The negative slope in panel B is descriptive. In the full bread and peace composite model B3, the extremism coefficient is $-2.23$, with an approximate two-tailed $p$-value of .30 on ten degrees of freedom; it remains well short of statistical significance.}''", "evidence": "I checked the inference by recalculating it from the replication data and by an exact counterexample in a linear model. For the regression of the residuals, I find slope $-1.8917$, standard error $1.6067$, $t=-1.177$ and $p=.2587$ on 14 degrees of freedom. The extremism coefficient of B3 is $-2.2285$, with standard error $2.0450$, $t=-1.090$ and $p=.3014$ on 10 degrees of freedom.\n\nFor the counterexample, take six observations and the vectors\n\\[\nu=(1,-1,0,0,0,0),\\qquad v=(0,0,1,-1,0,0),\\qquad w=(0,0,0,0,1,-1).\n\\]\nLet the first-step regressors be the constant and $u$, let $x=v$, and let $y=u+(3/2)v+w$. The first-step residual is $r=(3/2)v+w$. Regressing $r$ on the constant and $x$, I obtain slope $3/2$ and residual sum of squares 2 on 4 degrees of freedom. Then $t^2=9$, and the nominal two-tailed $p$-value is $1-45/(13\\sqrt{13})$, about $.040$. Regressing $y$ on the constant, $u$ and $x$ gives the same slope and the same residual sum of squares on 3 degrees of freedom. Then $t^2=27/4$, and the $p$-value is about $.081$. The regression of the residuals rejects at the 5\\% level, and the full regression does not. With $x=u+v$ in place of $x=v$, the slope in the regression of the residuals is $3/4$ and the coefficient of $x$ in the full regression is $3/2$. In a linear model, the first slope equals the second times a fraction: the part of the centred sum of squares of $x$ that remains after $x$ is adjusted for the first-step regressors. Here this fraction is $1/2$. Under the Gaussian null hypothesis with $x=v$, the test statistic of the regression of the residuals is distributed as $\\sqrt{4/3}\\,T_3$, where $T_k$ denotes a Student variable with $k$ degrees of freedom. Its second moment is 4, and that of the reference variable $T_4$ is 2.\n\nThe paper's model is nonlinear, and the same two effects appear through the Jacobian of B2 at the fitted values, which has rank 5. The centred sum of squares of the extremism variable is $1.0027$. After the variable is adjusted for the five columns of the Jacobian, $.8498$ of it remains, a fraction of $.85$. Neither .26 nor .30 is an exact finite-sample $p$-value for the nonlinear regression.", "problem": "The sentence reports the test of the slope of the line in Figure 4, panel B. That line is an ordinary regression of the residuals of model B2 on the extremism variable, over 16 elections. Its slope is $-1.89$ with standard error $1.61$, and the nominal two-tailed $p$-value on 14 degrees of freedom is .26. The number is reported correctly. The sentence uses it to say how far extremism is from statistical significance once the fundamentals are accounted for, and the regression of the residuals does not test that. There are two reasons. The extremism variable is not adjusted for the fundamentals, so the slope of the line is not the coefficient of the full model. Further, the standard error of the slope and its 14 degrees of freedom take no account of the five parameters already fitted in B2. The test that conditions on the fundamentals is the test of the extremism coefficient of column B3. That coefficient is $-2.23$ with standard error $2.04$, and its approximate $p$-value is .30 on ten degrees of freedom, as Table 3 and p.~139 report. Both $p$-values are far from significance.", "what_survives": "Figure 4, panel B, stands as a descriptive plot of the residuals, with its fitted line. The coefficient of column B3, row (3) of Table 3 and the worked example on p.~139 are unchanged, and no calculation in the paper uses the .26. The conclusion that the relationship is well short of statistical significance is unchanged." }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-E002", "work": { "arxiv_id": null, "doi": "10.1177/0002716216660571", "title": "Ideologically Extreme Candidates in U.S. Presidential Elections, 1948-2012", "url": "https://pmaronow.github.io/llm-audit/works/P039/" }, "work_id": "P039" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the left half of Table 2, with columns A1, A2 and A3 under the heading Musical Chairs. The first row, Delta RDI in Q14-15, shows 6.40, 6.38 and 6.38, each with three stars, and the standard errors (1.4), (.96) and (1.0) beneath them. The rows below are Incumbent fatigue, War fatalities, Relative extremism with -.46 (5.19) in column A3, Constant, N of cases 17, 17, 17, adjusted r-squared .81, .80, .79, and RMSQ 4.62, 4.70, 4.89. Red boxes mark 6.40 together with (1.4) in column A1, 6.38 in column A2 and 6.38 in column A3.", "caption": "Table 2, p. 137 (PDF p. 12), Musical Chairs panel, turned upright (the table is printed sideways). The marked entries are the three income coefficients and the standard error of column A1.", "url": "https://pmaronow.github.io/llm-audit/works/P039/exhibits/P039-E003.png" } ], "formal_release": null, "formal_statements": [ { "id": "P039-T7", "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-T7" } ], "headline": "In the Musical Chairs panel of Table 2, the income coefficients are 6.39, 6.37 and 6.39, and the standard error in column A1 is .94, not 1.4.", "id": "P039-E003", "location": "Table 2, p. 137 (PDF p. 12), Musical Chairs panel, row “$\\Delta$ RDI in Q14-15”, columns A1–A3", "pdf_page": 12, "printed_page": "137", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P039-T7" ], "text": { "as_printed": "\\begin{audittable}{lrrr}{Table 2, Musical Chairs panel, as printed (the income row and the two rows used below).}\n & A1 & A2 & A3 \\\\\n$\\Delta$ RDI in Q14-15 & \\hl{6.40}$^{***}$ & \\hl{6.38}$^{***}$ & \\hl{6.38}$^{***}$ \\\\\n & \\hl{(1.4)} & (.96) & (1.0) \\\\\nN of cases & 17 & 17 & 17 \\\\\nRMSQ & 4.62 & 4.70 & 4.89 \\\\\n\\end{audittable}", "corrected": "\\begin{audittable}{lrrr}{Table 2, Musical Chairs panel, income row as corrected.}\n & A1 & A2 & A3 \\\\\n$\\Delta$ RDI in Q14-15 & \\hl{6.39}$^{***}$ & \\hl{6.37}$^{***}$ & \\hl{6.39}$^{***}$ \\\\\n & \\hl{(.94)} & (.96) & (1.0) \\\\\n\\end{audittable}\n\nThe standard errors of columns A2 and A3 are correct as printed; at two decimals the latter is 1.01. To four decimals the pairs are $6.3938$ $(.9406)$, $6.3723$ $(.9571)$ and $6.3855$ $(1.0069)$.", "evidence": "I checked the row by recalculating it from the replication data in exact rational arithmetic, and by a derivation that uses the printed entries alone. Fitting the three regressions to the 17 observations in the replication data, I find $6.3938$ $(.9406)$ for A1, $6.3723$ $(.9571)$ for A2 and $6.3855$ $(1.0069)$ for A3. The three regression calls of the authors' script, run unchanged, return the same values.\n\nFor the derivation, let $q$ be the income variable, and let $R_j$ be the residual sum of squares when $q$ is regressed on the other regressors of model $j$: the constant and incumbent fatigue for A1, and these together with war fatalities for A2. Adding a regressor cannot raise this sum, so that $R_2\\le R_1$. The conventional standard error of the coefficient of $q$ in model $j$ is $s_j/\\sqrt{R_j}$, where $s_j$ is the residual standard error, printed as RMSQ. Combining these two facts, I have\n\\[\n\\operatorname{SE}_1\\le \\operatorname{SE}_2\\, s_1/s_2 .\n\\]\nEach printed entry may lie anywhere in its rounding interval, so that $\\operatorname{SE}_2\\le .965$, $s_1\\le 4.625$ and $s_2\\ge 4.695$. I obtain\n\\[\n\\operatorname{SE}_1\\le \\frac{.965\\times 4.625}{4.695}=\\frac{7141}{7512}<.951 .\n\\]\nA standard error printed as 1.4 is at least 1.35. Then no data can give the printed standard error of A1 together with the printed .96, 4.62 and 4.70. If RMSQ denoted $\\sqrt{\\mathrm{SSE}/n}$ instead, the bound would be smaller by the factor $\\sqrt{13/14}$.", "problem": "The row reports the income coefficients of three nested regressions, fitted by least squares to the same 17 elections, with conventional standard errors. The replication data yield 6.39 (.94), 6.37 (.96) and 6.39 (1.01). Each printed coefficient is off by .01. The standard error printed for column A1 is too large by about .46, which rounding cannot explain. The $t$-ratio of that coefficient is 6.80, not the 4.57 that the printed pair implies. The replication script for the table carries a note on this row. It says that, because of ``a minor calculation error'', the published estimates and standard errors for this variable ``differ slightly from those produced below'', and that the values the script produces ``are the correct values''. It is not established whether the printed 1.4 comes from that calculation error or from typesetting.", "what_survives": "The three income coefficients remain positive with $p<.01$, and the significance stars stand. Every other entry of the Musical Chairs panel agrees with the recalculation at the printed precision, including the extremism coefficient of column A3, $-.46$ $(5.19)$. The five rows of Table 3 for the musical chairs model also agree with it; they are computed from the fitted regressions, not from the printed standard error. The Bread \\& Peace models use another income variable. The quarterly income series is not in the replication files, and the construction of the income variable from it was not checked." }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-E003", "work": { "arxiv_id": null, "doi": "10.1177/0002716216660571", "title": "Ideologically Extreme Candidates in U.S. Presidential Elections, 1948-2012", "url": "https://pmaronow.github.io/llm-audit/works/P039/" }, "work_id": "P039" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of four lines of the notes: the last words of note 14, 'have reported.', and the three lines of note 15. A red box marks the first sentence of note 15, 'For the eight measures based on ANES/Rosenstone, this was the 1980 election.' The second sentence, on the NOMINATE measure and the 1964 election, is unmarked.", "caption": "Note 15, p. 142 (PDF p. 17). The marked sentence assigns all eight estimates based on voter perceptions to the 1980 election.", "url": "https://pmaronow.github.io/llm-audit/works/P039/exhibits/P039-E004.png" }, { "alt": "Crop of five lines of text. A red box marks 'the largest value of that variable.' in the sentence 'Maximum effect is calculated as the coefficient for the variable times the largest value of that variable.' A second red box marks 'The largest value' at the start of the sentence 'The largest value of the zero-centered refined variable occurs in 1980'.", "caption": "p. 139 (PDF p. 14), first paragraph under Table 3. The phrase “largest value” is marked in the definition of the maximum effect and at the start of the worked example; it occurs a third time in the next paragraph.", "url": "https://pmaronow.github.io/llm-audit/works/P039/exhibits/P039-E004-b.png" } ], "formal_release": null, "formal_statements": [ { "id": "P039-T5", "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-T5" }, { "id": "P039-T6", "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-T6" } ], "headline": "Note 15 should assign the 1980 election to the four estimates from the two zero-centered measures and the 1972 election to the four estimates from the two voter-centered measures; on p. 139, “the largest value” is the score of greatest absolute magnitude, with its sign.", "id": "P039-E004", "location": "Note 15, p. 142 (PDF p. 17), first sentence; p. 139 (PDF p. 14), section “Maximum predicted effect of relative candidate extremism”", "pdf_page": 17, "printed_page": "142", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [ "P039-T5", "P039-T6" ], "text": { "as_printed": "p.~139: ``Maximum effect is calculated as the coefficient for the variable $\\times$ \\hl{the largest value} of that variable.'' Later in the section: ``For all five variables, \\hl{the largest value} represents an election in which the incumbent was less extreme than his challenger, so the relative extremism score employed in these calculations is always negative.$^{15}$''\n\nNote 15, p.~142: ``\\hl{For the eight measures based on ANES/Rosenstone, this was the 1980 election.} For the NOMINATE measure, which did not include the 1980 case, the maximum effect estimate is based on candidates' locations in the 1964 election, in which Johnson was rated less extreme than Goldwater.''", "corrected": "p.~139: ``Maximum effect is calculated as the coefficient for the variable $\\times$ \\hl{the observed score of that variable of greatest absolute magnitude, with its sign retained}.'' The same wording replaces ``the largest value'' in the worked example and in the sentence that carries note 15. The product is the fitted contribution of the variable relative to a score of zero.\n\nNote 15: ``\\hl{For the four estimates based on the two zero-centered ANES/Rosenstone measures, this was the 1980 election. For the four estimates based on the two voter-centered ANES measures, this was the 1972 election. For the two estimates based on} the NOMINATE measure, which did not include the 1980 case, the maximum effect estimate is based on candidates' locations in the 1964 election, in which Johnson was rated less extreme than Goldwater.''", "evidence": "I compared the scores in the replication files in exact arithmetic and recalculated the ten models. For the voter-centered measure based on all voters, the score is $-.5118$ in 1972 and $-.1715$ in 1980. For the refined voter-centered measure, it is $-.6022$ in 1972 and $-.2943$ in 1980. For each of these measures, 1972 has the score of greatest absolute magnitude among the eleven elections from 1972 to 2012; the nearest scores are $-.3403$ (2004) and $-.3603$ (2008). The zero-centered measures have theirs in 1980 ($-.4981$ and $-.6210$) and the NOMINATE measure in 1964 ($-.22$), as the note says. With these scores the recalculation agrees with all ten entries of Table 3 at the printed precision.\n\n\\begin{audittable}{llrrr}{Voter-centered entries of Table 3: the coefficient times the 1972 score gives the printed entry, and the coefficient times the 1980 score does not.}\nVote model & Measure & Printed & With the 1972 score & With the 1980 score \\\\\nMusical chairs & (2) All-voter & 1.33 & 1.33 & 0.45 \\\\\nMusical chairs & (4) Refined & 2.28 & 2.28 & 1.11 \\\\\nBread and peace & (2) All-voter & 0.30 & 0.30 & 0.10 \\\\\nBread and peace & (4) Refined & 0.82 & 0.82 & 0.40 \\\\\n\\end{audittable}\n\nFor the wording on p.~139, the score of the refined zero-centered measure is $-.6210$ in 1980 and $.2738$ in 1984, so that the 1980 score is not the largest in signed order.", "problem": "Table 3 has four measures built from voter perceptions, each used in two vote models, which makes eight estimates. Note 15 says that all eight rest on the 1980 election. This holds for the two zero-centered measures. The two voter-centered measures have their score of greatest absolute magnitude in 1972, and the four entries in rows (2) and (4) of Table 3 are computed with the 1972 scores. Two smaller points concern the same sentence of the note. It speaks of eight measures where there are eight estimates from four measures, and the voter-centered measures use ANES data alone. On p.~139, ``the largest value'' is not the largest score in signed order. The score used is negative for every variable, and it is the score of greatest absolute magnitude. The calculation is correct under that reading, which the worked example and the sentence that carries note 15 make clear.", "what_survives": "Every entry of Table 3 is correct as printed, and so is the statement that the score used is negative for all five variables. The worked example for 1980 and the assignment of the NOMINATE estimates to 1964 stand. No model, estimate or conclusion changes." }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-E004", "work": { "arxiv_id": null, "doi": "10.1177/0002716216660571", "title": "Ideologically Extreme Candidates in U.S. Presidential Elections, 1948-2012", "url": "https://pmaronow.github.io/llm-audit/works/P039/" }, "work_id": "P039" }, { "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 running head and first five lines of page 127. Two red boxes mark the words 'a candidate can always' at the end of the first line and 'capture more votes by moving toward the center,' at the start of the second, in the sentence that ends 'away from the extremes' with a superscript 1.", "caption": "Top of p. 127 (PDF p. 2). The marked words belong to the sentence on the median voter theorem that begins at the foot of p. 126; the superscript refers to note 1.", "url": "https://pmaronow.github.io/llm-audit/works/P039/exhibits/P039-E005.png" } ], "formal_release": null, "formal_statements": [], "headline": "The median voter theorem predicts that candidates who seek to win converge at the voter median; it does not say that a candidate “can always capture more votes by moving toward the center”.", "id": "P039-E005", "location": "First sentence of p. 127, continued from p. 126 (PDF pp. 1–2); note 1, p. 141 (PDF p. 16)", "pdf_page": 2, "printed_page": "127", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "pp.~126--127: ``This conventional wisdom finds theoretical support in the median voter theorem, a classic model of candidate positioning that demonstrates that if voters favor a candidate with views closest to their own, a candidate \\hl{can always capture more votes} by moving toward the center, away from the extremes.$^{1}$''\n\nNote 1, p.~141: ``As such, the median voter theorem predicts that if candidates are simply looking to win, they will converge at the center, \\hl{since positions that are more extreme would garner fewer votes}. Many efforts have been made to reconcile this model with the abundant empirical evidence against this prediction.''", "corrected": "p.~127: ``This conventional wisdom finds theoretical support in \\hl{the standard median voter model, which, under its usual one-dimensional, two-candidate assumptions, predicts convergence at the voter median for candidates seeking to win}.$^{1}$''\n\nNote 1: ``As such, the median voter theorem predicts that if candidates are simply looking to win, they will converge at the center. Many efforts have been made to reconcile this model with the abundant empirical evidence against this prediction.'' The clause ``since positions that are more extreme would garner fewer votes'' is deleted.\n\nIn full, the result is the following. Suppose that there are two candidates and an odd number of voters with ideal points on a line, that each voter chooses the nearer candidate, and that the votes of voters equidistant from the two candidates are divided equally. Then a candidate at the median ideal point receives a strict majority against a candidate at any other position. The pair of positions with both candidates at the median is an equilibrium, since a candidate who moves away from the median loses.", "intended_meaning": "The sentence is an introductory gloss on the standard convergence result, as note 1 explains: in the median voter model, candidates who seek to win converge at the voter median. That statement is correct under the model's usual assumptions. Read literally, the sentence asserts more, namely that every move toward the center gains votes. This does not follow from the assumption that voters choose the nearest candidate. Suppose that two candidates are at positions $x>z$ on a line. Then a voter with ideal point $v$ strictly prefers $x$ exactly when $v>(x+z)/2$, and a move of $x$ to the right raises this cutoff, also when the move is toward the median. Two examples show what can happen. First, let the voters be at $-1$, $0$ and $1$, the candidate at $-1/2$ and the opponent at $3$. The candidate receives all three votes at $-1/2$ and at every position between $-1/2$ and the median $0$, so that no move toward the median gains a vote. Second, let the voters be at $-3$, $-5/4$, $0$, $1$ and $2$, the candidate at $-1$ and the opponent at $-2$. The cutoff is $-3/2$, and the candidate receives four votes. When the candidate moves to the median $0$, the cutoff is $-1$, and the candidate receives three votes. No voter is tied in either example, and the candidate wins at each position. Note that the examples do not contradict the convergence result, which concerns winning and not the number of votes at every position. No estimate or conclusion in the paper depends on the sentence." }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-E005", "work": { "arxiv_id": null, "doi": "10.1177/0002716216660571", "title": "Ideologically Extreme Candidates in U.S. Presidential Elections, 1948-2012", "url": "https://pmaronow.github.io/llm-audit/works/P039/" }, "work_id": "P039" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "complete_derivation", "exhibits": [ { "alt": "Crop of six lines of the notes: the end of note 4, notes 5 and 6, and the first line of note 7. In note 5 a red box marks 'all quarters receive an equal weight of 1;'. In note 6 red boxes mark 'decay parameter of lambda = 1' and 'lambda = 0'.", "caption": "Notes 5 and 6, p. 141 (PDF p. 16). The marked phrases are the equal weight of 1 for all quarters in note 5, and the decay parameter of $\\lambda=1$ and the value $\\lambda=0$ in note 6.", "url": "https://pmaronow.github.io/llm-audit/works/P039/exhibits/P039-E006.png" } ], "formal_release": null, "formal_statements": [], "headline": "In note 5 the equal weight of 1 refers to the temporal discount factors, and the election quarter keeps its factor of $1/3$; note 6 means that musical chairs uses the 14th and 15th quarters, with equal weight, and no other quarter.", "id": "P039-E006", "location": "Notes 5 and 6, p. 141 (PDF p. 16), called from p. 128 (PDF p. 3)", "pdf_page": 16, "printed_page": "141", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "Note 5: ``This is because $1^{15}=1^{14}=\\ldots=1^{0}=1$, so \\hl{all quarters receive an equal weight of 1}; and $0^{15}=0^{14}=\\ldots=0^{1}=0$, but $0^{0}=1$, so all quarters preceding the election quarter receive a weight of 0.''\n\nNote 6: ``For comparison, musical chairs could be thought of as having a \\hl{decay parameter of} $\\hl{\\lambda=1}$ for the 14th and 15th quarter, and $\\hl{\\lambda=0}$ for all other quarters.''\n\nMain text, p.~128: ``Because the election is held one-third of the way through the 16th quarter of a presidential term, RDI growth in this quarter is down-weighted by $1/3 * \\lambda^{0}$. The 15th quarter of the presidential term is down-weighted by $\\lambda^{1}$, the 14th quarter by $\\lambda^{2}$, and so on for the preceding quarters.''", "corrected": "Note 5: ``This is because $1^{15}=1^{14}=\\ldots=1^{0}=1$, so \\hl{all temporal discount factors equal 1, while the election quarter keeps its factor of one third}; and $0^{15}=0^{14}=\\ldots=0^{1}=0$, but $0^{0}=1$, so all quarters preceding the election quarter receive a weight of 0.''\n\nNote 6: ``For comparison, musical chairs \\hl{gives equal weight to the 14th and 15th quarters and a weight of zero to all other quarters}.''", "intended_meaning": "I read note 5 as explaining the temporal discount factors $\\lambda^{k}$ at the two extreme values of $\\lambda$. At $\\lambda=1$ every discount factor equals 1. The weights are then not all equal, because the main text assigns the election quarter the further factor $1/3$. Its weight is $1/3$, and each earlier quarter in the sum has weight 1. At $\\lambda=0$ only the election quarter has a nonzero weight, as the note says. Note 6 is an analogy. Musical chairs uses income growth in the 14th and 15th quarters only, as p.~128 states, and in the replication script the two quarters have equal weight. This selection cannot be written with a decay parameter. With $\\lambda=1$ for those two quarters and $\\lambda=0$ for the others, the formula of p.~128 still assigns the 16th quarter the weight $(1/3)\\,0^{0}=1/3$. No computation depends on either note, and the fitted models keep the factor $1/3$. The quarters that enter the sum in the fitted models are the subject of \\fid{P039-E001}." }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-E006", "work": { "arxiv_id": null, "doi": "10.1177/0002716216660571", "title": "Ideologically Extreme Candidates in U.S. Presidential Elections, 1948-2012", "url": "https://pmaronow.github.io/llm-audit/works/P039/" }, "work_id": "P039" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of five lines of the reference list: the entry for Bartels and Zaller (2001), the entry for Converse (1964) and the entry for Downs (1957). In the Converse entry a red box marks the italic title 'Ideology and its discontents'.", "caption": "References, p. 142 (PDF p. 17). The marked title is the one printed for the volume edited by Apter.", "url": "https://pmaronow.github.io/llm-audit/works/P039/exhibits/P039-E008.png" } ], "formal_release": null, "formal_statements": [], "headline": "The volume edited by Apter that contains Converse’s chapter is titled Ideology and Discontent, not “Ideology and its discontents”.", "id": "P039-E008", "location": "References, p. 142 (PDF p. 17), entry “Converse, Philip E. 1964”", "pdf_page": 17, "printed_page": "142", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "``Converse, Philip E. 1964. The nature of belief systems in mass publics. In \\emph{\\hl{Ideology and its discontents}}, ed. David E. Apter. New York, NY: The Free Press of Glencoe.''", "corrected": "``Converse, Philip E. 1964. The nature of belief systems in mass publics. In \\emph{\\hl{Ideology and discontent}}, ed. David E. Apter. New York, NY: The Free Press of Glencoe.''", "evidence": "I checked the page against a public library catalogue (WorldCat, OCLC 193632). The entry for the print book of 1964, published by the Free Press of Glencoe, has the title ``Ideology and discontent''. The book itself was not consulted.", "problem": "The reference adds ``its'' to the title and makes its last word plural.", "what_survives": "The chapter is identifiable from its author, year, title and editor, and nothing in the paper depends on the entry." }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-E008", "work": { "arxiv_id": null, "doi": "10.1177/0002716216660571", "title": "Ideologically Extreme Candidates in U.S. Presidential Elections, 1948-2012", "url": "https://pmaronow.github.io/llm-audit/works/P039/" }, "work_id": "P039" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "exact_counterexample", "exhibits": [ { "alt": "Crop of the display and the lines around it. A red box encloses the display: 'Incumbent extremism = [(Incumbent position - Voter mean) + (Challenger position - Voter mean)] / 2 = [Incumbent position + Challenger position] / 2 - Voter mean'. Above it the text reads 'all recentered to a range of -3 to +3, we measure relative incumbent extremism as follows:'. Below it: 'In the special case in which mean voter ideology is 0, this second method of calculating relative candidate extremism reduces to the first.'", "caption": "p. 133 (PDF p. 8). The marked display defines the voter-centered measure of relative incumbent extremism; the sentence before it names the scale, and the sentence after it states the reduction to the first method.", "url": "https://pmaronow.github.io/llm-audit/works/P039/exhibits/P039-A001.png" } ], "formal_release": null, "formal_statements": [], "headline": "The display for the voter-centered measure leaves implicit that each candidate position is centered at the voter mean before its absolute value is taken and signed by incumbency; the measure that was computed is $(|x_I-m|-|x_C-m|)/2$.", "id": "P039-A001", "location": "p. 133 (PDF p. 8), the display that begins “Incumbent extremism =” and the sentence after it; the sign convention is on p. 132 (PDF p. 7) and in note 11, p. 141 (PDF p. 16)", "pdf_page": 8, "printed_page": "133", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "``Based on the self-placement and two candidate placement scales---all recentered to a range of $-3$ to $+3$---we measure relative incumbent extremism as follows:\n\\[\n\\begin{aligned}\n\\text{Incumbent extremism}&=\\left[\\begin{array}{l}{}\\hl{(\\text{Incumbent position}-\\text{Voter mean})+}\\\\ {}\\hl{(\\text{Challenger position}-\\text{Voter mean})}\\end{array}\\right]/2\\\\\n&=\\hl{[\\text{Incumbent position}+\\text{Challenger position}]/2-\\text{Voter mean}}\n\\end{aligned}\n\\]\nIn the special case in which mean voter ideology is 0, this second method of calculating relative candidate extremism reduces to the first.''", "corrected": "``Based on the self-placement and two candidate placement scales---all recentered to a range of $-3$ to $+3$---we measure relative incumbent extremism as follows. Let $x_I$ and $x_C$ be the positions of the incumbent-party candidate and of the challenger on this scale, and let $m$ be the voter mean. Each candidate position is first centered at $m$. The absolute value is then taken, and the challenger's receives a negative sign:\n\\[\n\\text{Incumbent extremism}=\\hl{\\bigl(|x_I-m|-|x_C-m|\\bigr)/2}.\n\\]\nIn the special case in which mean voter ideology is 0, this second method of calculating relative candidate extremism reduces to the first.''\n\nThe formula replaces both lines of the printed display; the midpoint of the signed positions minus the voter mean is not kept as a second equality. The display should have made clear that centering at the voter mean precedes the orientation by incumbent and challenger. The scores, Table 1, Table 3 and the conclusions are unchanged, since the archived scores already follow this formula.", "intended_meaning": "Recall that the first method (p.~132 and note 11) takes the absolute value of each candidate's position, assigns a negative sign to the challenger's and takes the midpoint. Let $x_I$ and $x_C$ be the positions of the incumbent-party candidate and of the challenger on the liberal--conservative scale. Then the first measure is $(|x_I|-|x_C|)/2$. The second method moves the reference point from zero to the voter mean $m$. The measure intended is\n\\[\nE_m=\\bigl(|x_I-m|-|x_C-m|\\bigr)/2,\n\\]\nwhich is positive when the incumbent-party candidate is farther from the voter mean than the challenger. This is the measure that was computed. The replication code subtracts the mean self-placement of voters from each candidate position, takes absolute values, signs them by incumbency and averages. The formula for $E_m$ reproduces the 22 archived scores of the voter-centered measures (two measures in each of the eleven elections of 1972--2012) to within $6\\times10^{-15}$. It also reproduces the two correlations with vote share in Table 1, $-.08$ and $-.07$. Setting $m=0$, I obtain $E_0=(|x_I|-|x_C|)/2$, so that the sentence after the display is correct.\n\nThe display does not state these conventions. The sentence before the display refers to the recentered scale, on which liberal positions are negative and conservative positions are positive. Read on this scale, the display is the midpoint of the two signed positions minus the voter mean, $P_m=(x_I+x_C)/2-m$. Let $x_I=-1$, $x_C=2$ and $m=1/4$. Then the distances from the voter mean are $5/4$ and $7/4$, and I have $E_m=-1/4$ and $P_m=1/4$. The display thus reverses which candidate is the more extreme. If the candidate positions are first transformed as in note 11, to $1$ and $-2$, and the voter mean is left on its original scale, then the display equals $-3/4$. In the archived data, $P_m$ agrees with the archived score up to rounding in the seven elections with a Republican incumbent party. In the four elections with a Democratic incumbent party (1980, 1996, 2000 and 2012), it is the negative of the archived score. For the all-voter measure in 1996, I have $x_I=-0.9468$, $x_C=1.1557$ and $m=0.4413$, so that $P_m=-0.3369$, and the archived score is $0.3369$.\n\nOne reading in context makes the display correct. Suppose that the voter mean lies between the two candidates. If $x_C\\le m\\le x_I$, then $E_m=(x_I+x_C)/2-m$, and if $x_I\\le m\\le x_C$, then $E_m=m-(x_I+x_C)/2$. This implies that the display holds when the whole scale, the voter mean included, is reflected in the elections in which the incumbent-party candidate is the more liberal. The voter mean lies between the two candidates in all eleven elections. The printed display can be read correctly, but only with conventions that the text leaves implicit." }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-A001", "work": { "arxiv_id": null, "doi": "10.1177/0002716216660571", "title": "Ideologically Extreme Candidates in U.S. Presidential Elections, 1948-2012", "url": "https://pmaronow.github.io/llm-audit/works/P039/" }, "work_id": "P039" }, { "before_review": null, "class": "clarity_note", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "recalculation_from_source", "exhibits": [ { "alt": "Crop of seven lines of text below Figure 1. Two red boxes mark the words 'slightly more moderate than the' at the end of one line and 'median of all Republican nominees in the entire period.' at the start of the next, in the sentence that begins 'Calling himself a compassionate conservative, he was rated in 2000 as a typical conservative Republican; indeed, he was rated'.", "caption": "p. 131 (PDF p. 6), first paragraph below Figure 1. The marked words compare Bush’s rating in 2000 with the median of all Republican nominees.", "url": "https://pmaronow.github.io/llm-audit/works/P039/exhibits/P039-N005.png" } ], "formal_release": null, "formal_statements": [], "headline": "Bush’s rating in 2000 is at the median of the seventeen Republican ratings, and not slightly below it.", "id": "P039-N005", "location": "p. 131 (PDF p. 6), first paragraph below Figure 1", "pdf_page": 6, "printed_page": "131", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": null, "subject": "paper", "target_ids": [], "text": { "as_printed": "``Calling himself a compassionate conservative, he was rated in 2000 as a typical conservative Republican; indeed, he was rated \\hl{slightly more moderate than} the median of all Republican nominees in the entire period.''", "corrected": "``Calling himself a compassionate conservative, he was rated in 2000 as a typical conservative Republican; indeed, he was rated \\hl{at} the median of all Republican nominees in the entire period.''", "intended_meaning": "The sentence says that Bush's rating in 2000 was typical of Republican nominees, and this is correct. In the refined series plotted in Figure 1, the Republican rating for 2000 is $1.332$. It is the ninth of the seventeen Republican ratings in increasing order, between McCain in 2008 ($1.321$) and Bush in 1992 ($1.363$), and is therefore their median. If the rating for 2000 is set aside, the median of the other sixteen is $1.342$, and the rating for 2000 is $0.010$ below it. The sentence, however, refers to all Republican nominees. Nothing in the paper depends on the comparison." }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-N005", "work": { "arxiv_id": null, "doi": "10.1177/0002716216660571", "title": "Ideologically Extreme Candidates in U.S. Presidential Elections, 1948-2012", "url": "https://pmaronow.github.io/llm-audit/works/P039/" }, "work_id": "P039" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "recalculation_from_source", "exhibits": [ { "alt": "Crop of seven lines of text. A red box marks the value '-.44' in the sentence 'When the two zero-centered measures are limited to just these eight cases, both measures show a correlation with vote share of -.44.' Three lines above, the text gives the NOMINATE correlation as 'a respectable -.48'.", "caption": "p. 134 (PDF p. 9), second paragraph of the section on correlations. The marked value is given for both zero-centered measures on the eight elections with NOMINATE scores.", "url": "https://pmaronow.github.io/llm-audit/works/P039/exhibits/P039-N006.png" } ], "formal_release": null, "formal_statements": [], "headline": "On the eight elections with NOMINATE scores, the refined zero-centered measure has a correlation of $-.45$ with vote share, not $-.44$; the value $-.44$ is correct for the all-voter measure.", "id": "P039-N006", "location": "Section “Correlations between incumbent extremism and vote share”, second paragraph, p. 134 (PDF p. 9)", "pdf_page": 9, "printed_page": "134", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "``When the two zero-centered measures are limited to just these eight cases, \\hl{both measures show a correlation} with vote share of $\\hl{-.44}$.''", "corrected": "``When the two zero-centered measures are limited to just these eight cases, \\hl{the all-voter and refined measures show correlations} with vote share of $-.44$ \\hl{and} $\\hl{-.45}$\\hl{, respectively}.''", "evidence": "I recalculated the correlations from the replication files, which give the relative extremism scores and the vote shares used for Table 1. The rounding was settled with exact rational arithmetic: the refined correlation $r$ satisfies $-0.4457949780-.335$, so that neither value is a rounding tie. The relative score is half the difference between the incumbent's and challenger's absolute locations. For the printed pair and for the corrected pair, I have\n\\[\n(.34-1.58)/2=-.62,\\qquad (.33-1.57)/2=-.62 .\n\\]\nThe relative score in the replication files is $-.6210$.", "problem": "In the replication files, the refined scores for 1980 are $1.5742$ for Reagan and $-.3322$ for Carter. At two decimals these are 1.57 and $-.33$, so each printed number differs by .01 from the correctly rounded one. The cause is not established.", "what_survives": "The relative score of $-.62$, the maximum effect of $+1.38$, the $p$-value of .30 and the ten degrees of freedom are unchanged. Carter in 1980 remains the most moderate of the thirty-four locations of the refined measure. The tables and figures use the scores at full precision, not these two printed numbers." }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-N010", "work": { "arxiv_id": null, "doi": "10.1177/0002716216660571", "title": "Ideologically Extreme Candidates in U.S. Presidential Elections, 1948-2012", "url": "https://pmaronow.github.io/llm-audit/works/P039/" }, "work_id": "P039" }, { "before_review": null, "class": "error", "date_modified": "2026-10-25", "date_published": "2026-10-25", "evidence_type": "page_check", "exhibits": [ { "alt": "Crop of six lines of the reference list: the entry for Abramowitz (1988), the three-line entry for Achen and Bartels (2016) and the entry for Bartels (1998). In the Achen and Bartels entry red boxes mark 'Pocketbook voting and the limits of democratic accountability.', which runs over two lines, and 'eds.' before the names Christopher Achen and Larry M. Bartels.", "caption": "References, p. 142 (PDF p. 17), entry for Achen and Bartels (2016). The marked terms are the subtitle of the chapter and the abbreviation “eds.”", "url": "https://pmaronow.github.io/llm-audit/works/P039/exhibits/P039-N013.png" }, { "alt": "Crop of five lines of the reference list: the entry for Stokes and Iversen (1962) and the three-line entry for Zaller. In the Zaller entry red boxes mark the year '2003.' and the italic title 'The issue of belief: Essays in the intersection of non-attitudes and attitude change', which runs over two lines.", "caption": "References, p. 142 (PDF p. 17), entry for Zaller (2003). The marked terms are the year and the title of the volume.", "url": "https://pmaronow.github.io/llm-audit/works/P039/exhibits/P039-N013-b.png" } ], "formal_release": null, "formal_statements": [], "headline": "Achen and Bartels are the authors, not the editors, of Democracy for Realists, and its chapter on musical chairs has another subtitle; Zaller’s chapter was published in 2004, in a volume with another title.", "id": "P039-N013", "location": "References, p. 142 (PDF p. 17), entries “Achen, Christopher H., and Larry M. Bartels. 2016” and “Zaller, John. 2003”; the citation “Zaller (2003)” on p. 130 (PDF p. 5)", "pdf_page": 17, "printed_page": "142", "reaches": [], "related_entries": [], "revisions": [], "same_defect_as": [], "severity": "Minor", "subject": "paper", "target_ids": [], "text": { "as_printed": "``Achen, Christopher H., and Larry M. Bartels. 2016. Musical chairs: \\hl{Pocketbook voting and the limits of democratic accountability}. In \\emph{Democracy for realists}, \\hl{eds.} Christopher Achen and Larry M. Bartels. Princeton, NJ: Princeton University Press.''\n\n``Zaller, John. \\hl{2003}. Floating voters in U.S. presidential elections, 1948--2000. In \\emph{\\hl{The issue of belief: Essays in the intersection of non-attitudes and attitude change}}, eds. \\hl{Paul Sniderman, and Willem E. Saris}. Princeton, NJ: Princeton University Press.''", "corrected": "``Achen, Christopher H., and Larry M. Bartels. 2016. Musical chairs: \\hl{Economic voting and the specious present}. \\hl{Chap. 6 in} \\emph{Democracy for realists\\hl{: Why elections do not produce responsive government}}. Princeton, NJ: Princeton University Press.''\n\n``Zaller, John. \\hl{2004}. Floating voters in U.S. presidential elections, 1948--2000. In \\emph{\\hl{Studies in public opinion: Attitudes, nonattitudes, measurement error, and change}}, eds. \\hl{Willem E. Saris and Paul M. Sniderman}. Princeton, NJ: Princeton University Press.''\n\nThe citation ``Zaller (2003)'' on p.~130 becomes ``Zaller (\\hl{2004})''.", "evidence": "I checked the page against public bibliographic sources. The library catalogue entry for the 2016 book (WorldCat, OCLC 946459788) lists Achen and Bartels as its authors. The publisher's chapter listings state the title of the chapter on musical chairs, and the volume, year and editors of Zaller's chapter. A curriculum vitae of Bartels lists a conference paper with the printed subtitle, dated September 2004. A 2003 article by Zaller cites the chapter as in press in a volume titled ``The issue of belief''. The books themselves were not consulted. It is not established that no manuscript of 2003 existed, or which version was used for the paper.", "problem": "Read as references to the published books, both entries give wrong details. \\emph{Democracy for Realists: Why Elections Do Not Produce Responsive Government} (2016) is a book by Achen and Bartels, not a volume that they edited. Its chapter 6 is titled ``Musical Chairs: Economic Voting and the Specious Present''; the subtitle printed in the reference is that of a conference paper of September 2004. Zaller's chapter was published in 2004, in \\emph{Studies in Public Opinion: Attitudes, Nonattitudes, Measurement Error, and Change}, edited by Willem E. Saris and Paul M. Sniderman. The title printed in the reference is one under which the chapter was cited while the volume was in press.", "what_survives": "Both works are identifiable from author and chapter title, and no result depends on either entry. If an earlier manuscript was meant, the entry should name it." }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-N013", "work": { "arxiv_id": null, "doi": "10.1177/0002716216660571", "title": "Ideologically Extreme Candidates in U.S. Presidential Elections, 1948-2012", "url": "https://pmaronow.github.io/llm-audit/works/P039/" }, "work_id": "P039" } ], "lean": { "applicable": true, "guide": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/docs/works/P039.md", "repository": "https://github.com/pmaronow/llm-audit-lean", "targets": [ { "declarations": [ { "claim_revision_id": "catalog:8d9a49a208a7f678252883abf1f64bf014c6e896e04e7100cc1ba4416f911b8a", "conditional_on": [], "credit_reason": "The extension to negative decay is valid; row 3 also uses total real division if a denominator vanishes there. On λ≥0 the denominators are strictly positive. 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pp. 128, 136--137 and note 5, p. 141. The sixteen-quarter description and the implemented Q2--16 income predictor differ, as shown by quarter-one independence, the missing terms, and both exact witnesses.

", "headline": "The description of the bread and peace income variable of Table 2, Panel B: ‘with quarters 1–16 discounted by a decay parameter as described above’ (p. 136), the row label ‘∆ weighted RDI in Q 1-16’ (Table 2) and the weighting rule of p. 128, as printed; the witnesses are the validators' constructions and are not in the paper", "location": "p. 136 (PDF p. 11), second paragraph of text; Table 2, p. 137 (PDF p. 12); the weighting rule is on p. 128 (PDF p. 3) and note 5 on p. 141 (PDF p. 16)" }, "id": "P039-T1", "text": { "body": "pp. 128, 136--137 and note 5, p. 141. The sixteen-quarter description and the implemented Q2--16 income predictor differ, as shown by quarter-one independence, the missing terms, and both exact witnesses.", "headline": "The description of the bread and peace income variable of Table 2, Panel B: ‘with quarters 1–16 discounted by a decay parameter as described above’ (p. 136), the row label ‘∆ weighted RDI in Q 1-16’ (Table 2) and the weighting rule of p. 128, as printed; the witnesses are the validators' constructions and are not in the paper", "location": "p. 136 (PDF p. 11), second paragraph of text; Table 2, p. 137 (PDF p. 12); the weighting rule is on p. 128 (PDF p. 3) and note 5 on p. 141 (PDF p. 16)" }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-T1" }, { "declarations": [ { "claim_revision_id": "catalog:28d65528e235778b86f5d033b5bc6c8a730ed2a55d78c195a655539f232f795f", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.income_denominators_positive", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Income.lean#L78", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:28d65528e235778b86f5d033b5bc6c8a730ed2a55d78c195a655539f232f795f", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.income_growth_difference", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Income.lean#L93", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:28d65528e235778b86f5d033b5bc6c8a730ed2a55d78c195a655539f232f795f", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.income_growth_equality_iff", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Income.lean#L104", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:28d65528e235778b86f5d033b5bc6c8a730ed2a55d78c195a655539f232f795f", "conditional_on": [], "credit_reason": "The powers are asserted for all natural exponents, which includes the target exponents 0–14. Quarter 16 retains its factor 1/3 at unit decay, as the corrected target requires.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.income_endpoint_values", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Income.lean#L115", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:28d65528e235778b86f5d033b5bc6c8a730ed2a55d78c195a655539f232f795f", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P039-T2", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P039-T2" }, "html": { "body": "

pp. 128, 136--137 and note 5, p. 141. The corrected Q2--16 predictor has positive normalization, ignores quarter one, and satisfies the exact difference, equality characterization and endpoint formulas.

", "headline": "The description of the bread and peace income variable (p. 128, p. 136, the Table 2 row label and note 5), corrected: quarters 2–16, with exponents 0 to 14", "location": "p. 136 (PDF p. 11); Table 2, p. 137 (PDF p. 12); p. 128 (PDF p. 3); note 5, p. 141 (PDF p. 16)" }, "id": "P039-T2", "text": { "body": "pp. 128, 136--137 and note 5, p. 141. The corrected Q2--16 predictor has positive normalization, ignores quarter one, and satisfies the exact difference, equality characterization and endpoint formulas.", "headline": "The description of the bread and peace income variable (p. 128, p. 136, the Table 2 row label and note 5), corrected: quarters 2–16, with exponents 0 to 14", "location": "p. 136 (PDF p. 11); Table 2, p. 137 (PDF p. 12); p. 128 (PDF p. 3); note 5, p. 141 (PDF p. 16)" }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-T2" }, { "declarations": [ { "claim_revision_id": "catalog:933978dd7a738add431b6776e8ab4be1535c8357d5c1a7523acb8a3aff19e93b", "conditional_on": [], "credit_reason": "Auxiliary nonnegativity is used inside the existing fit proofs and is not a claim component.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.squaredLength_nonnegative", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Regression.lean#L74", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:933978dd7a738add431b6776e8ab4be1535c8357d5c1a7523acb8a3aff19e93b", "conditional_on": [], "credit_reason": "Auxiliary zero-criterion lemma is used inside existing uniqueness proofs.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.squaredLength_zero_iff", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Regression.lean#L81", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:933978dd7a738add431b6776e8ab4be1535c8357d5c1a7523acb8a3aff19e93b", "conditional_on": [], "credit_reason": "Auxiliary SSE expansion is already used inside the exported least-squares fits.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.normal_equation_expansion", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Regression.lean#L87", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:933978dd7a738add431b6776e8ab4be1535c8357d5c1a7523acb8a3aff19e93b", "conditional_on": [], "credit_reason": "Auxiliary normal-equation uniqueness is already included in each concrete witness fit.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.normal_equation_unique", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Regression.lean#L109", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:933978dd7a738add431b6776e8ab4be1535c8357d5c1a7523acb8a3aff19e93b", "conditional_on": [], "credit_reason": "Auxiliary Gram-certificate lemma is already used inside the exported concrete fits; no separate helper assertion is included.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.gram_certificate_full_rank", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Regression.lean#L131", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:933978dd7a738add431b6776e8ab4be1535c8357d5c1a7523acb8a3aff19e93b", "conditional_on": [], "credit_reason": "Baseline component of target (i). 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No claim-stage verdict changed after reading the record.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.regression_five_percent_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Regression.lean#L646", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:933978dd7a738add431b6776e8ab4be1535c8357d5c1a7523acb8a3aff19e93b", "conditional_on": [], "credit_reason": "Positive σ in ℝ≥0 represents every positive real scale in the target. Total division extends the statistic by zero on zero-SSE outcomes, a Gaussian null event, so the law and moments retain their usual meaning. All three asserted second moments are nonzero, hence cannot arise from the Bochner integral’s nonintegrable-value-zero convention. No claim-stage verdict changed after reading the record.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.residual_gaussian_reference_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_GaussianCompletion.lean#L463", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:933978dd7a738add431b6776e8ab4be1535c8357d5c1a7523acb8a3aff19e93b", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P039-T3", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P039-T3" }, "html": { "body": "

Figure 4 and p. 138, compared with Table 2, p. 137 and Table 3, p. 139. The six-observation OLS witnesses give different tests at 5%, different slopes for the correlated regressor, and the stated Gaussian-null scaled-t3 law and moments. For every nonnegative threshold the two analytic formulas are the actual t4 and t3 two-sided probability tails. The intermediate square and arctangent comparisons and four-decimal p-value enclosures hold. The usual measure-theoretic conditions are assumed; statistics are totalized on the Gaussian null set of zero SSE.

", "headline": "p. 138, last sentence, with Figure 4, panel B: the two-tailed p-value of the line fitted to the B2 residuals (.26), read as a test of the statistical significance of candidate extremism once the fundamentals are accounted for; the witness is the validator's construction and is not in the paper", "location": "p. 138 (PDF p. 13), last sentence, and Figure 4, panel B; compare Table 2, column B3 (p. 137, PDF p. 12) and Table 3, row (3) (p. 139, PDF p. 14)" }, "id": "P039-T3", "text": { "body": "Figure 4 and p. 138, compared with Table 2, p. 137 and Table 3, p. 139. The six-observation OLS witnesses give different tests at 5%, different slopes for the correlated regressor, and the stated Gaussian-null scaled-t3 law and moments. For every nonnegative threshold the two analytic formulas are the actual t4 and t3 two-sided probability tails. The intermediate square and arctangent comparisons and four-decimal p-value enclosures hold. The usual measure-theoretic conditions are assumed; statistics are totalized on the Gaussian null set of zero SSE.", "headline": "p. 138, last sentence, with Figure 4, panel B: the two-tailed p-value of the line fitted to the B2 residuals (.26), read as a test of the statistical significance of candidate extremism once the fundamentals are accounted for; the witness is the validator's construction and is not in the paper", "location": "p. 138 (PDF p. 13), last sentence, and Figure 4, panel B; compare Table 2, column B3 (p. 137, PDF p. 12) and Table 3, row (3) (p. 139, PDF p. 14)" }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-T3" }, { "declarations": [ { "claim_revision_id": "CT-02:P039-T4:accepted", "conditional_on": [], "credit_reason": "Auxiliary Gram invertibility is already incorporated in corrected_residual_regression.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.full_rank_gram_unit", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Regression.lean#L169", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P039-T4:accepted", "conditional_on": [], "credit_reason": "The projection identities are already part of corrected_residual_regression.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.ols_projection_identities", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Regression.lean#L181", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P039-T4:accepted", "conditional_on": [], "credit_reason": "The needed uniqueness is already part of corrected_residual_regression.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.full_rank_ols_unique", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Regression.lean#L225", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P039-T4:accepted", "conditional_on": [], "credit_reason": "Its attenuation and SSE conclusions are already included in corrected_residual_regression.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.residual_attenuation_algebra", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Regression.lean#L671", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P039-T4:accepted", "conditional_on": [], "credit_reason": "Its guarded squared-t conclusions are already included in corrected_residual_regression.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.residual_t_squared_algebra", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Regression.lean#L694", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P039-T4:accepted", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.corrected_residual_regression", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Regression.lean#L1478", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "CT-02:P039-T4:accepted", "conditional_on": [], "credit_reason": "{'source': 'adjudication+second review', 'adjudicator': 'P039/P039_PaperInstances.lean:61 adds the outer hypothesis hσ : 0 < σ, so the residual and covariance identities and the covariance rank/non-identity conclusions at lines 73–76 all require positive noise, whereas the first sentence of T4(f) asserts them from Eε = 0 and Cov(ε) = σ²I without that restriction. For n = 6, k = 2, Z = [1, (1,−1,0,0,0,0)], x = (0,0,1,−1,0,0), γ = 0 and ε = 0 on a one-point probability space, σ = 0 satisfies the target assumptions, but Cov(r) = 0 has rank 0 rather than 4 and equals σ²I; the excluded case falsifies substantive target conclusions, rather than merely leaving an easy true boundary case unproved. The additional negation of LiteralCovarianceClaim at P039/P039_PaperInstances.lean:80 correctly identifies this false target subclaim but does not establish it or supply the valid zero-noise identities from the credited declaration. The Student-law premise at P039/P039_PaperInstances.lean:77 is explicitly permitted by record.json’s T4 note, so the reviewer’s withdrawn objection to that premise is rejected. The necessary correction concerns the audit’s formal T4(f), whose unguarded rank/non-identity assertion is absent from the supplied empirical paper correction.', 'suggested_fix': 'Add σ > 0 to the first covariance premise of the formal target T4(f), and record this as an additional correction to the validator’s formal claim; this makes the credited positive-noise assessment faithful without changing the empirical replacement sentence. If the target is to retain its valid covariance identities for σ = 0, leave those identities unconditional, place the positive-noise guard only on covariance rank/non-identity, and use P039/P039_Regression.lean:1744 (corrected_residual_covariance_and_law) as the corresponding credited declaration.', 'second_review': 'T4(f) says Cov(r) = σ²M is “a matrix of rank n − k and not σ²I_n” without a positive-noise premise, whereas vectorCovariance is the ordinary centered-integral covariance matrix (P039/P039_Regression.lean:1545). At σ = 0 with zero noise, n = 6, k = 2, Z = [1, (1,−1,0,0,0,0)] and x = (0,0,1,−1,0,0), the target assumptions hold but Cov(r) = 0 has rank 0 rather than 4 and equals σ²I_n; P039/P039_PaperInstances.lean:10 defines precisely this unguarded rank/non-identity claim, refuted at line 37. The credited theorem explicitly requires hσ : 0 < σ (P039/P039_PaperInstances.lean:61), so the substantive error is in the audit’s formal correction, not in an empirical covariance assertion from the paper. Correct T4(f) by guarding rank/non-identity with σ > 0; if its valid zero-noise identities are retained, use P039/P039_Regression.lean:1744, which exports those identities unconditionally and guards only rank/non-identity.', 'severity': None}", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.PaperDomains.corrected_residual_covariance_assessment", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_PaperInstances.lean#L58", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "CT-02:P039-T4:accepted", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P039-T4", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P039-T4" }, "html": { "body": "

p. 138, final sentence, with the exact linear-OLS interpretation. For every full-rank control design containing an intercept and every additional regressor outside its span, clauses (a)--(e) give the projection, attenuation, SE and t identities. Under positive isotropic noise the residual covariance has rank n-k. Under the Gaussian null the full statistic has the central t(n-k-1) law; an orthogonal centered regressor gives the stated scaled residual-line law, which differs from t(n-2) when k is at least two, including residual df one or two. The usual measure-theoretic conditions are assumed. The empirical nonlinear application in (g) is expressly outside the formal claim.

", "headline": "p. 138, last sentence, as repaired by Sol: the line of Figure 4, panel B is descriptive, and the test of extremism given the fundamentals is the test of its coefficient in the full model B3 (formal content: the algebra that relates the two regressions)", "location": "p. 138 (PDF p. 13), last sentence" }, "id": "P039-T4", "text": { "body": "p. 138, final sentence, with the exact linear-OLS interpretation. For every full-rank control design containing an intercept and every additional regressor outside its span, clauses (a)--(e) give the projection, attenuation, SE and t identities. Under positive isotropic noise the residual covariance has rank n-k. Under the Gaussian null the full statistic has the central t(n-k-1) law; an orthogonal centered regressor gives the stated scaled residual-line law, which differs from t(n-2) when k is at least two, including residual df one or two. The usual measure-theoretic conditions are assumed. The empirical nonlinear application in (g) is expressly outside the formal claim.", "headline": "p. 138, last sentence, as repaired by Sol: the line of Figure 4, panel B is descriptive, and the test of extremism given the fundamentals is the test of its coefficient in the full model B3 (formal content: the algebra that relates the two regressions)", "location": "p. 138 (PDF p. 13), last sentence" }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-T4" }, { "declarations": [ { "claim_revision_id": "catalog:f40b5131711adf361ac27f5199da77b0b7087ed1677fdc06f94ffdcddf5a9b23", "conditional_on": [], "credit_reason": "This covers the literal signed-order clause T5(ii). As the target note clarifies, the printed worked example indicates a largest-magnitude intention and its positive 1.38 is not shown numerically erroneous by this lemma; the result diagnoses the wording of the rule.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.signed_extremism_maxima", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Extremism.lean#L165", "status": "proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f40b5131711adf361ac27f5199da77b0b7087ed1677fdc06f94ffdcddf5a9b23", "conditional_on": [], "credit_reason": "Fidelity sweep: ok", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.voter_centered_1980_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Extremism.lean#L175", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true }, { "claim_revision_id": "catalog:f40b5131711adf361ac27f5199da77b0b7087ed1677fdc06f94ffdcddf5a9b23", "conditional_on": [], "credit_reason": "This covers the literal signed-order clause T5(ii). As the target note clarifies, the printed worked example indicates a largest-magnitude intention and its positive 1.38 is not shown numerically erroneous by this lemma; the result diagnoses the wording of the rule.", "credit_status": "ok", "fidelity_status_baseline": "ok", "lean_name": "AronowAudit.P039.literal_largest_value_counterexample", "manifest_use": "used", "named_axioms": [], "says": null, "source": "https://github.com/pmaronow/llm-audit-lean/blob/b9e9df44831afed4f2b8f63be5c0cc9169a5fda4/AronowAudit/P039/P039_Extremism.lean#L189", "status": "counterexample_proved", "support_scope": "historical declaration", "used_in_solution": true } ], "formalization": { "claim_revision_id": "catalog:f40b5131711adf361ac27f5199da77b0b7087ed1677fdc06f94ffdcddf5a9b23", "crosswalk_url": "../../data/lean-crosswalk.json#/targets/P039-T5", "decision": "whole", "release_id": "sap-2026-11-08-assured-release", "target_id": "P039-T5" }, "html": { "body": "

p. 139 and note 15, p. 142. Both voter-centered score comparisons refute the 1980 selection, and all five signed maxima are positive, with the worked signed-maximum product negative.

", "headline": "p. 139: ‘Maximum effect is calculated as the coefficient for the variable × the largest value of that variable’ and ‘The largest value of the zero-centered refined variable occurs in 1980’; note 15: ‘For the eight measures based on ANES/Rosenstone, this was the 1980 election’; as printed. The witnesses are the validators' calculations on the deposited scores and are not in the paper", "location": "p. 139 (PDF p. 14), the two paragraphs under the heading ‘Maximum predicted effect of relative candidate extremism’; note 15, p. 142 (PDF p. 17)" }, "id": "P039-T5", "text": { "body": "p. 139 and note 15, p. 142. Both voter-centered score comparisons refute the 1980 selection, and all five signed maxima are positive, with the worked signed-maximum product negative.", "headline": "p. 139: ‘Maximum effect is calculated as the coefficient for the variable × the largest value of that variable’ and ‘The largest value of the zero-centered refined variable occurs in 1980’; note 15: ‘For the eight measures based on ANES/Rosenstone, this was the 1980 election’; as printed. The witnesses are the validators' calculations on the deposited scores and are not in the paper", "location": "p. 139 (PDF p. 14), the two paragraphs under the heading ‘Maximum predicted effect of relative candidate extremism’; note 15, p. 142 (PDF p. 17)" }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-T5" }, { "declarations": [ { "claim_revision_id": "catalog:a00397745bd38f21fc6d86baebae1246d4906e21076ccda37075d9071a156ab6", "conditional_on": [], "credit_reason": "Covers the selection, uniqueness and score-sign portions of T6(i)–(ii). The five exact selected-score equalities are also exposed by unfolding the score definition; the description does not claim that signed-order maxima occur in these years. 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p. 139 and corrected note 15, p. 142. The corrected magnitude rule uniquely selects the stated elections with positive attained margins in both samples, gives the signed fitted contribution and its maximum magnitude, and preserves the rounded worked example.

", "headline": "The maximum-effect rule of p. 139 and note 15, corrected: coefficient × the observed score of greatest absolute magnitude, with its sign; the elections selected are 1980 (zero-centered measures), 1972 (voter-centered measures) and 1964 (NOMINATE)", "location": "p. 139 (PDF p. 14); note 15, p. 142 (PDF p. 17); Table 3, p. 139" }, "id": "P039-T6", "text": { "body": "p. 139 and corrected note 15, p. 142. 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Table 2, p. 137, and the specifications on p. 136. Nested OLS gives the inverse-Gram and SE inequalities, the exact printed-cell bound and positive gap, and incompatibility of all four printed rounding bins. The alternative RMSQ convention sqrt(SSE/17) multiplies the ratio bound by sqrt(13/14), strictly reducing it. The exact deposited inputs give the unique corrected fits, income coefficients and SE displays .94, .96 and 1.0 (or 1.01). All three corrected income coefficients retain their stars: their actual two-sided Student reference probabilities, on 14, 13 and 12 df, are below .01. The usual measure-theoretic conditions are assumed.

", "headline": "Table 2, Musical Chairs panel, row ‘∆ RDI in Q14-15’ (coefficients of columns A1 to A3 and the parenthesized entry of column A1), corrected: the printed A1 entry (1.4) is incompatible with the printed entries of column A2", "location": "Table 2, p. 137 (PDF p. 12); the specifications are described on p. 136 (PDF p. 11)" }, "id": "P039-T7", "text": { "body": "Table 2, p. 137, and the specifications on p. 136. Nested OLS gives the inverse-Gram and SE inequalities, the exact printed-cell bound and positive gap, and incompatibility of all four printed rounding bins. The alternative RMSQ convention sqrt(SSE/17) multiplies the ratio bound by sqrt(13/14), strictly reducing it. The exact deposited inputs give the unique corrected fits, income coefficients and SE displays .94, .96 and 1.0 (or 1.01). All three corrected income coefficients retain their stars: their actual two-sided Student reference probabilities, on 14, 13 and 12 df, are below .01. The usual measure-theoretic conditions are assumed.", "headline": "Table 2, Musical Chairs panel, row ‘∆ RDI in Q14-15’ (coefficients of columns A1 to A3 and the parenthesized entry of column A1), corrected: the printed A1 entry (1.4) is incompatible with the printed entries of column A2", "location": "Table 2, p. 137 (PDF p. 12); the specifications are described on p. 136 (PDF p. 11)" }, "url": "https://pmaronow.github.io/llm-audit/works/P039/#P039-T7" } ] }, "missing_sources": [ "Online appendix with details of the model, cited in note 7 (printed p. 141, PDF p. 16) at http://ann.sagepub.com/supplemental" ], "reproduction": { "agreement": "agrees_in_part", "status": "partial", "targets": [ { "agreement": "agrees_in_part", "location": "p. 131 (PDF p. 6); associated exact in-text positions p. 139 (PDF p. 14)", "method": "I reran the supplied downstream code from the deposited derived data. All 34 positions and both plotted lines agree within visual reading precision, and the stated ranks hold: Bush in 2004 is the most conservative Republican nominee and Carter in 1980 the most moderate of the 34 positions. Bush's 2000 rating is the subject of P039-N005. The associated printed 1980 positions, 1.58 and -0.34, differ from the recalculated 1.574229 and -0.332181 (P039-N010). Raw survey construction was not rerun.", "status": "reproduced", "target": "Figure 1" }, { "agreement": "to_rounding", "location": "p. 132 (PDF p. 7); correlations p. 131 (PDF p. 6)", "method": "I reran the supplied plot code from derived scores. The plot agrees by eye, and the signed and absolute-position correlations round to 0.97 and 0.75. Raw NOMINATE extraction was not rerun.", "status": "reproduced", "target": "Figure 2 and associated correlations" }, { "agreement": "agrees_in_part", "location": "p. 135 (PDF p. 10); eight-election subset correlations p. 134 (PDF p. 9)", "method": "I reran the supplied code from the deposited derived data: all 15 table correlations agree to rounding and all case counts agree exactly. Of the two eight-election correlations in the prose on p. 134, the all-voter value rounds to -0.44, while the refined value is -0.445795 and rounds to -0.45 rather than -0.44 (P039-N006).", "status": "reproduced", "target": "Table 1" }, { "agreement": "to_rounding", "location": "p. 136 (PDF p. 11); interpretation p. 135 (PDF p. 10)", "method": "I reran the supplied plotting code from the deposited derived data and compared both panels and all 17 year labels by eye. The plot agrees within visual reading precision; the paper prints no fitted slopes to compare. Raw economic construction was not rerun.", "status": "reproduced", "target": "Figure 3" }, { "agreement": "agrees_in_part", "location": "p. 137 (PDF p. 12); main coefficients also p. 138 (PDF p. 13)", "method": "I reran all six supplied regressions from the deposited model-ready data. The extremism estimates and most cells agree, but the three Panel A income coefficients, the A1 income standard error and three Panel B fit statistics differ at printed precision (P039-E003; P039-N008). Raw economic inputs were unavailable.", "status": "reproduced", "target": "Table 2" }, { "agreement": "to_rounding", "location": "p. 138 (PDF p. 13)", "method": "I reran both residual plots from the deposited covariates. The patterns and year labels agree by eye, and the plotted-line p-value 0.258671 rounds to 0.26. That p-value concerns the plot regression; the full-model p-value is about 0.30 (P039-E002).", "status": "reproduced", "target": "Figure 4 and associated p-value" }, { "agreement": "to_rounding", "location": "p. 139 (PDF p. 14); interpretation continues p. 140 (PDF p. 15)", "method": "I reran the supplied regressions from deposited derived data. All ten maximum effects and p-values agree to two decimals, and all degrees of freedom agree exactly. I extracted the nonmissing maxima where the supplied code prints missing-value placeholders; raw economic and NOMINATE construction was not rerun.", "status": "reproduced", "target": "Table 3" } ] }, "summary": "The paper asks whether ideologically extreme candidates are penalized in U.S. presidential elections. It adds five measures of the incumbent-party candidate's relative extremism to two models of the vote, ``musical chairs'' and ``bread and peace'', fitted to the elections of 1948--2012. It finds scant evidence of a penalty that is large or close to statistical significance. 9 entries are classified as errors (9 Minor), and 4 are clarity notes. None changes a conclusion. Four further questions could not be settled from the available sources.", "what_was_checked": "I checked the results below by recalculating them from the replication files, which contain the scripts and the derived data (the ideology scores and the data for Figures 3 and 4).\n\\begin{itemize}\n\\item Table 1. All 15 correlations and all case counts reproduce. The two correlations on the eight elections with NOMINATE scores are the subject of \\fid{P039-N006}.\n\\item Table 2. In Panel A, the coefficients and standard errors for incumbent fatigue, war fatalities and relative extremism reproduce in all three columns, as do the constants, the numbers of cases, the adjusted $R^2$ and the root mean squared errors. The income row differs (\\fid{P039-E003}). In Panel B, every coefficient, standard error, decay parameter, constant and number of cases reproduces by nonlinear least squares with the archived formula, which uses quarters 2--16 (\\fid{P039-E001}). Three fit statistics differ in the last digit (\\fid{P039-N008}). The significance stars in both panels agree with the recomputed p-values.\n\\item Table 3. All ten maximum effects, p-values and degrees of freedom reproduce, and eight of the ten coefficients are negative, as the text says. The worked example on p.~139 reproduces, with the value $+1.38$. Two location scores quoted with it and the election named in note 15 are the subjects of \\fid{P039-N010} and \\fid{P039-E004}.\n\\item Figures. The correlations $.97$ and $.75$ of Figure 2 reproduce. The points of Figures 3 and 4 agree with the data and with the recomputed residuals. This agreement was judged by eye, since the paper prints no slopes. The p-value of $.26$ quoted for Figure 4B reproduces as a number, and its use is the subject of \\fid{P039-E002}.\n\\item Data. The spliced location scores regenerate from the archived scripts to within floating-point precision, and so do the four scores of relative extremism that are based on the ANES and Rosenstone ratings. The vote shares, the vote margins, the incumbent-party indicator and the tenure variable are correct for all seventeen elections.\n\\item Text. The statements on p.~131 about Figure 1 agree with the scores, apart from the one in \\fid{P039-N005}. The pooled within-party variance of the refined locations is $1.55$ times that of the all-voter locations, in line with ``about 50 percent more year-to-year variation'' on p.~130. The check described in note 14 gives weak effects in both models. The historical examples in the conclusion agree with the scores, the tenure variable, income growth and the outcomes.\n\\item The main conclusion. The smallest p-value in the ten tests is $.30$. With quarter 1 restored to the income measure of the bread and peace models, the five extremism coefficients remain negative and far from significance. The estimates are imprecise. The 95 percent interval for the largest effect in the featured bread and peace model runs from about $-1.4$ to $+4.2$ points of vote share. The featured coefficient moves from $-2.23$ ($p=.30$) to $-4.35$ ($p=.10$) when the election of 1980 is left out. The paper says that weak evidence of an effect is not evidence that none exists.\n\\end{itemize}\nThe following were not checked.\n\\begin{itemize}\n\\item The online appendix cited in note 7 was not available. The model that removes partisan bias from the ANES placements was read from the code, and it could not be compared with its description in the appendix.\n\\item The replication files do not contain the ANES cumulative file, the NOMINATE file, the quarterly income and price series or Hibbs's data file. The scripts that construct the location scores from the survey responses, the NOMINATE scores and the income and war variables were read and could not be run. Four questions remain undetermined for lack of these sources or of a stated rule. Two concern how the high-information respondents are selected, one the comparison of income series in note 4, and one the choice between Johnson's two NOMINATE scores for 1964.\n\\item I did not consult the cited works themselves. I did not check the paper's descriptions of them, the quotation from Hibbs or the page references in note 16. I compared three entries in the reference list with public catalogue and publisher information (\\fid{P039-E008}, \\fid{P039-N013}).\n\\end{itemize}\nThese checks cover what can be recomputed from the derived data. They do not show that the paper is free of other errors.", "work": { "arxiv_id": null, "audit_complete": true, "authors": [ "Marty Cohen", "Mary C. McGrath", "P. M. Aronow", "John Zaller" ], "category": "Articles - Empirics", "citation": "Marty Cohen, Mary C. McGrath, P. M. Aronow, and John Zaller (2016). Ideologically Extreme Candidates in U.S. Presidential Elections, 1948-2012. Annals of the American Academy of Political and Social Science 667(1):126–142.", "doi": "10.1177/0002716216660571", "edition": { "pages_in_audited_file": 17, "sha256": "691cbd132a33b4ba7cf2cb30d7a2a77829efb8dea0183f833d85d7534aef6ec3", "source_url": "https://doi.org/10.1177/0002716216660571", "venue": "Annals of the American Academy of Political and Social Science", "year": "2016" }, "id": "P039", "later_materials": null, "title": "Ideologically Extreme Candidates in U.S. Presidential Elections, 1948-2012", "type": "Journal article", "url": "https://pmaronow.github.io/llm-audit/works/P039/", "venue": "Annals of the American Academy of Political and Social Science", "year": "2016" } }