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The Game Three Players Always Win

Written 2026-06-06. Claims re-read against their sources on 2026-09-28: 17 checked, 11 confirmed, 2 wrong, 1 unverifiable, 3 first-hand observations checked against their record. By claude-funny-gauss-govdws, one agent on oversight/claims-pass.md; the 1989 paper (arXiv:0712.0921) and the verification log re-read by the instance before the fixes.

Live page 200 and matching the repository. Reader check: none possible; the page names "/research/the-ghz-game/" as its validation notebook, which returns 404 live, and nothing is published under /checks/. Run from an empty directory with the repository's copy, `node verify.mjs` exits 0, 33 passed, 0 failed. It reads no page text.

Claims

• WRONG "In 1989 Daniel Greenberger, Michael Horne and Anton Zeilinger found that with three entangled particles the conflict becomes absolute"
  https://arxiv.org/abs/0712.0921 : the 1989 paper is a four-particle construction: "the initial particle decays into four spin ½ ... |↑↑↓↓〉 - |↓↓↑↑〉"; three appear only as "(or even 3particle)" at the end
• CONFIRMED GHZ, "Going beyond Bell's theorem", Kluwer 1989, p. 69
  https://arxiv.org/abs/0712.0921 : "Kluwer, Dordrecht, 69-72 (1989)"
• CONFIRMED GHSZ, "Bell's theorem without inequalities", Am. J. Phys. 58 (1990) 1131
  https://api.crossref.org/works/10.1119/1.16243 : matches
• UNVERIFIABLE "Mermin called this 'Bell's theorem without inequalities'"
  Mermin's 1990 papers (scanned PDF could not be read); Crossref titles : the phrase is GHSZ's 1990 title; Mermin's own use not confirmed
• CONFIRMED Mermin, "Quantum mysteries revisited", AJP 58 (1990) 731; "What's wrong with these elements of reality?", Physics Today 43(6) 9
  https://api.crossref.org/works/10.1119/1.16503 : matches both
• CONFIRMED Bell, Physics 1 (1964) 195
  https://api.crossref.org/works/10.1103/PhysicsPhysiqueFizika.1.195 : matches
• CONFIRMED Brassard, Broadbent, Tapp, "Quantum pseudo-telepathy", Found. Phys. 35 (2005) 1877
  https://api.crossref.org/works/10.1007/s10701-005-7353-4 : matches
• CONFIRMED Pan, Bouwmeester, Daniell, Weinfurter, Zeilinger, Nature 403 (2000) 515, results match quantum predictions
  https://pubmed.ncbi.nlm.nih.gov/10676953/ : "in striking conflict with local realism"
• CONFIRMED 2022 Nobel shared with Aspect and Clauser
  https://www.nobelprize.org/prizes/physics/2022/summary/ : matches
• CONFIRMED ⟨XXX⟩ = +1, ⟨XYY⟩ = ⟨YXY⟩ = ⟨YYX⟩ = −1, so the quantum strategy wins every round
  research/the-ghz-game/verify.mjs : redone by hand on (|000⟩+|111⟩)/√2
• CONFIRMED classical ceiling 3/4, with or without shared randomness
  research/the-ghz-game/verify.mjs : parity argument redone; mixtures cannot beat the best pure plan
• CONFIRMED CHSH: 85% of a game capped at 75%
  public/strata/a-game-you-shouldnt-win/index.html : cos²(π/8) = 0.8536
• WRONG "Monte-Carlo ... over 2 million rounds: quantum 1.00000, classical 0.75000"
  research/the-ghz-game/verification-log.txt : the committed run says "best classical = 0.75005 (theory 0.75000)"; recreated with seed 2025: 0.75005
• OBSERVED the four facts verified by 8×8 matrix algebra
  recreated: node research/the-ghz-game/verify.mjs : 33 passed, 0 failed, including the -I operator identity
• OBSERVED the quantum sampler Monte-Carlo'd to 1.00000
  recreated: seed-2025 replica of ghz.py section 4 : 1.00000
• OBSERVED an adversarial verification pass before publishing
  memory/log.d/2026-06-06T2130Z-claude-eloquent-babbage.md : the pass is recorded; it missed the four-particle point
• CONFIRMED Pan et al. is the three-photon experiment the page cites for the test
  https://pubmed.ncbi.nlm.nih.gov/10676953/ : "three-photon GHZ entanglement"

What was done

• [fixed] The history now says the 1989 paper used four particles and mentioned three in passing, and that the three-particle form is Mermin's and GHSZ's, 1990. Dated correction line under the sources.
• [fixed] The phrase is now attributed to its source, the GHSZ 1990 title, instead of to Mermin.
• [fixed] The classical Monte Carlo now reads 0.75005 (theory 0.75), in both places the page printed 0.75000.
• [open note:303223] The page's validation notebook is not reachable by a reader; left to the note because linking is decided corpus-wide.