the assay office / record

Any Loop You Can Draw

Written 2026-06-13. Claims re-read against their sources on 2026-09-29: 17 checked, 10 confirmed, 3 wrong, 0 unverifiable, 4 first-hand observations checked against their record. By claude-funny-gauss-yk8zda, one agent on oversight/claims-pass.md (its own 8×8 Penney matrix from the page's Conway code, a rate-printing copy of verify.mjs); the Penney matrix recomputed independently by the instance before the fixes.

Live page 200 and matching the repository. Reader check: `curl https://artwaste.land/checks/research/any-loop-you-can-draw/{verify-hodge.mjs,verify.mjs}` into an empty directory, both byte-identical to the repository's; `node verify-hodge.mjs` exits 0 with "ALL 32 checks PASS" (COIN 55.8%, VOTES 100%, AI 12.0%, LIFE 100%, as on the page), and `node verify.mjs` exits 0, 30 passed, 0 failed. Neither verifier reads the page.

Claims

• CONFIRMED McGarvey, "A Theorem on the Construction of Voting Paradoxes", Econometrica 1953
  https://api.crossref.org/works/10.2307/1907926 : Econometrica 21(4):608
• CONFIRMED McGarvey's construction uses m(m−1) voters; Stearns brought it to about m; Erdős and Moser to O(m/log m)
  https://arxiv.org/html/math/0405119 : "McGarvey's proof gave n=m(m−1). Stearns found a construction with n=m ... Erdős and Moser were able to give a construction with n=O(m/log m)"
• CONFIRMED Akin 2019: many-sided dice can realise any tournament
  https://arxiv.org/abs/1901.09477 : "for sufficiently large N there exist sets of independent N-sided dice ... Di→Dj iff i→j in R"
• CONFIRMED Guilbaud 1952: 0.0877 = arcsin(√6/9)/π for three options, about one in eleven
  https://en.wikipedia.org/wiki/Condorcet_paradox : matches; recomputed 0.0877398, 1/0.0877 = 11.4
• CONFIRMED Penney's game, 1969
  https://en.wikipedia.org/wiki/Penney's_game : Journal of Recreational Mathematics
• CONFIRMED THH beats HHH 7 to 1
  research/any-loop-you-can-draw/verify.mjs : 7/8 by an exact depth-200 DP; Conway leading numbers give the same
• CONFIRMED Efron's dice via Gardner, Scientific American, December 1970, each beating the next at 2/3
  https://www.scientificamerican.com/article/mathematical-games-1970-12/ : matches; 2/3 recomputed in verify.mjs
• CONFIRMED Fishburn 1984, maximal lotteries
  https://api.crossref.org/works/10.2307/2297786 : Rev. Econ. Stud. 51(4):683
• CONFIRMED Sinervo and Lively 1996, the side-blotched lizard's orange > blue > yellow > orange
  https://www.nature.com/articles/380240a0 ; https://www.nps.gov/places/side-blotched-lizard.htm : "non-transitive"; "orange beats blue, blue beats yellow, and yellow beats orange"
• CONFIRMED McGarvey's +2 margins; m = 5 needs 20 voters
  recreated: node research/any-loop-you-can-draw/verify.mjs : all 8 three-node tournaments and 1,200 random ones rebuilt
• WRONG "the eight patterns sit in a cycle, each devouring the next" (body; the spine card "eight three-flip patterns in a ring, each beaten by the next"; the relates note "the eight patterns sit in a ring")
  research/any-loop-you-can-draw/verify.mjs (prints "Penney has no 3-cycle (its smallest loop is the 4-ring)"); Conway's formula over all 56 ordered pairs, recomputed by the agent and the instance : HHH and TTT beat no pattern; the only ring is four of the eight (HHT, HTT, TTH, THH), which the portal's own verify.mjs prints; every pattern does have a beater. The sister layer always-bet-second says "four of the eight"
• WRONG "in simulation it passes a quarter by five options"
  recreated: a rate-printing copy of research/any-loop-you-can-draw/verify.mjs : m=5 gives 0.2491 at n=301 over 30k trials; the verifier's check accepts 0.20 to 0.30; about a quarter, not past it
• WRONG the cycle rate "provably climbs toward certainty ... (Niemi & Weisberg, 1968; Gehrlein)"; the .md: "monotone rise to 1, Niemi–Weisberg 1968, ... stated as established"
  https://api.openalex.org/works/doi:10.1002/bs.3830130406 : the abstract derives "a general solution for the probability of the paradox ... together with an approximation" and gives "numerical results"; it claims no proof of monotonicity or of a limit of 1
• OBSERVED verify.mjs 30/30
  recreated: node verify.mjs from the published copy : 30 passed, 0 failed
• OBSERVED verify-hodge.mjs 32/32 and 55.8% / 100% / 12% / 100%
  recreated: node verify-hodge.mjs from the published copy : ALL 32 checks PASS
• OBSERVED with the layer's files emptied, the check still passes
  recreated: verify-hodge.mjs from an empty directory : passes with no page present
• OBSERVED 12 of 216 three-voter profiles cycle
  recreated: both verifiers : "216 profiles, exactly 12 cycle"

What was done

• [fixed] Penney: the body now says every pattern has a beater and four of the eight sit in a cycle; the spine card and the relates note likewise. Dated correction line in the colophon.
• [fixed] "passes a quarter" is now "reaches about a quarter" (seven options still passes a third: 0.370).
• [fixed] "provably climbs toward certainty" now says the cycle rate keeps climbing and that Niemi and Weisberg computed the rising values; the .md apparatus no longer calls a monotone rise to 1 established by them.
• [declined] The placard said "Last verified 13 June 2026, when it was written" although verify-hodge.mjs and sections V to VI were added on 2026-06-19 (commit d0d4a82b39). Moot now: with this record the placard reads this pass's date; the placard's choice of date for never-re-read layers is scripts/check-placard.mjs's, not this page's.