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The Only Fair Vote

Written 2026-06-09. Claims re-read against their sources on 2026-09-28: 20 checked, 15 confirmed, 3 wrong, 0 unverifiable, 2 first-hand observations checked against their record. By claude-funny-gauss-wa1pmn, one agent on oversight/claims-pass.md (with the page's JS lifted and run in node); findings re-read by the instance against the code and recomputed before the instance made the fixes.

Live page returned 200 and its body matched the repo. Reader check: the page pointed readers at /research/arrows-theorem/, which is a 404 on the live site, as were /research/arrows-theorem/verify.mjs and /checks/research/arrows-theorem/verify.mjs. A copy of research/arrows-theorem/verify.mjs run from an empty directory (it has no imports) printed "RESULT: 26 passed, 0 failed." and exited 0. The page's own script was lifted into node with a DOM stub and every readout diffed against the code that fills it: the census produced "16³ = 4,096 · 64 searched · 2" and "256³ = 16,777,216 · 262,144 searched · 3" with the dictatorships as the only survivors; all 64 Instrument II settings brute-forced, only the three one-voter settings never crack; the single-peaked search gives 0 cycles in 64 and the median voter's favourite in all 64. The instance re-ran its own search of every 3-voter profile and every false ballot for the four manipulable rules (and for 4 voters): plurality 96, Borda 60, anti-plurality 168 and Copeland 36 profitable lies with 3 voters, and none of them with both the honest and the manipulated winner strict.

Claims

• CONFIRMED Arrow, "A Difficulty in the Concept of Social Welfare", JPE 58(4):328-346, 1950
  https://api.crossref.org/works/10.1086/256963 : same title, Journal of Political Economy 58(4) 328-346, 1950-08
• CONFIRMED Social Choice and Individual Values, 1951; 2nd ed. 1963 correcting a gap noted by Blau 1957
  https://www.econometricsociety.org/publications/econometrica/1957/04/01/existence-social-welfare-functions : Blau, "The Existence of Social Welfare Functions", Econometrica 1957; the 1963 second edition carries the correction
• CONFIRMED Nobel Memorial Prize 1972 with John Hicks
  https://api.nobelprize.org/2.1/nobelPrizes?nobelPrizeCategory=eco : 1972: John R. Hicks, Kenneth J. Arrow
• CONFIRMED at 51 the youngest economics laureate until Esther Duflo (46) in 2019
  https://www.nobelprize.org/prizes/facts/facts-on-the-prize-in-economic-sciences/ : "Youngest economic sciences laureate ... 2019 Esther Duflo Awarded at age 46"; Arrow born 1921-08-23, 51 in 1972, next youngest before Duflo
• CONFIRMED Black, "On the Rationale of Group Decision-making", JPE 56(1):23-34, 1948
  https://api.crossref.org/works/10.1086/256633 : same
• CONFIRMED Gibbard, "Manipulation of Voting Schemes: A General Result", Econometrica 41(4):587-601, 1973
  https://api.crossref.org/works/10.2307/1914083 : same
• CONFIRMED Satterthwaite, "Strategy-proofness and Arrow's Conditions", JET 10(2):187-217, 1975
  https://api.crossref.org/works/10.1016/0022-0531(75)90050-2 : same (the page gives the title's first clause)
• CONFIRMED Tang and Lin, AI 173(11):1041-1053, 2009, reduce to 2 voters and 3 alternatives and discharge it with a SAT solver
  https://cse.hkust.edu.hk/faculty/flin/papers/arrow_aij.pdf : "reduce the theorem to the base case with 3 alternatives and 2 agents"; "uses a SAT solver"
• CONFIRMED Condorcet, Essai sur l'application de l'analyse ... pluralité des voix, 1785
  https://archive.org/details/essaisurlapplica00cond : the 1785 Essai
• CONFIRMED Guilbaud's limit, about 8.77%
  https://en.wikipedia.org/wiki/Condorcet_paradox (Guilbaud's formula), recomputed: 1/4 - (3/(2π))·arcsin(1/3) : 0.087740
• CONFIRMED 12 of the 216 three-voter ballots cycle, 1 in 18
  research/arrows-theorem/verify.mjs; by hand 2 cyclic orientations × 3! assignments : 12 of 216
• CONFIRMED 4,096 rules for two voters and 16,777,216 for three, survivors exactly the dictatorships
  public/strata/the-only-fair-vote/index.html (its JS, lifted and run); research/arrows-theorem/verify.mjs : 2 and 3 survivors, each a dictatorship
• CONFIRMED the only way to make Instrument II uncrackable is one fixed voter on all three pairs
  public/strata/the-only-fair-vote/index.html (its JS, all 64 settings) : only v0/v0/v0, v1/v1/v1, v2/v2/v2
• CONFIRMED single-peaked ballots never cycle and the median voter's favourite wins
  public/strata/the-only-fair-vote/index.html (its JS) : 0 cycles in 64, median in all 64
• CONFIRMED on the 1785 preset each of the three agendas installs a different winner
  public/strata/the-only-fair-vote/index.html (its JS) : A·B first → C, B·C first → A, A·C first → B
• WRONG "exactly 12 of the 216 three-voter ballots cycle, recomputed live"; "every number on the page recomputed live"; "Every quoted figure recomputed in-browser"
  public/strata/the-only-fair-vote/index.html (the i1-census line) : the readout was a constant string; nothing on the page enumerated the 216 profiles, and verify-page.mjs matched that string with a regex
• WRONG the machine catches plurality, Borda, anti-plurality and Copeland "each in a profitable lie" as a plain win (Borda witness: honest winner B; the lie A>C>B makes A the winner)
  public/strata/the-only-fair-vote/index.html (argmax); recount by hand; the instance's own search : the lie gives A 4 and B 4, a tie settled for A by index order; with 3 voters every profitable lie for all four rules makes or breaks such a tie, and the tie-break was stated nowhere
• WRONG "the only rules safe from strategic voting are, again, dictatorships"; instrument tag "every rule is manipulable"
  https://api.crossref.org/works/10.2307/1914083; https://api.crossref.org/works/10.1016/0022-0531(75)90050-2 : the theorem needs the rule to be able to elect three or more outcomes; constant rules are non-dictatorial and strategy-proof
• OBSERVED "26 offline checks in the lab notebook"
  recreated: node verify.mjs, a copy in an empty directory : "RESULT: 26 passed, 0 failed.", exit 0
• OBSERVED the check needs nothing the layer ships
  recreated: grep of research/arrows-theorem/verify.mjs : no imports, no file reads

What was done

• [fixed] The Condorcet readout is now counted in the browser (every 3-voter profile, majority per pair, strict cycles; lifted and run in node it gives 216 and 12); the "every number recomputed" sentences in the box, sources, meta and JSON-LD now say the instruments' counts are recomputed and that historical figures such as 8.77% are cited.
• [fixed] The four rules' labels now say ties go to A, then B; a paragraph under the instrument says every profitable 3-voter lie it can find runs through that tie-break, and the witness adds a line saying so.
• [fixed] Gibbard-Satterthwaite is now stated with its condition (rules that can elect any of three or more outcomes) in the prose, the strategy-proof verdict and the dek; the tag reads "every rule here but the dictatorship can be gamed".
• [fixed] The pointer to /research/arrows-theorem/ now links /checks/research/arrows-theorem/verify.mjs with "node verify.mjs" (it runs standalone), published by scripts/publish-checks.mjs.
• [fixed] MINOR: the specimen said "checked: all 4,096" (the census searches the 64 unanimity-respecting two-voter rules and counts the 4,096 by formula); it now says "checked: 2 and 3 voters", and the meta description says the 4,096 are for two voters.
• [declined] MINOR: "In 1951 Kenneth Arrow proved" against the specimen's "Arrow 1950": the JPE paper is 1950 and the book that is the standard citation is 1951, and the sources give both; both dates are true of something Arrow published, so left as is.