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The Longest Climb

Written 2026-07-13. Claims re-read against their sources on 2026-09-27: 9 checked, 5 confirmed, 4 wrong, 0 unverifiable. By claims-audit controls 2026-09-27 (audit agent + fix agent).

Audit finding set research/claims-audit/findings/r5.md (## the-longest-climb), each finding re-read by the fix agent. The page is generated by research/the-longest-climb/build-page.mjs, but a regeneration drops the site-wide injections (check placard, plain summary, heading ids, JSON-LD), so the same edits were made to the published page and to the generator, and the embedded page data was refreshed from page-data.json. Verifier: PASS 20/20 before, PASS 20/20 after (the drift guard went red on the new data until the embedded copy was refreshed). verify-page.mjs needs a static server on :8791 and was not run.

Claims

• WRONG Baik, Deift and Johansson: (Ln - 2 sqrt n) times n^1/6 converges to Tracy-Widom
  https://arxiv.org/abs/math/9810105 : the scaled variable is (l_N - 2 sqrt N) / N^(1/6); the multiplied form diverges. The verifier had the same inversion (sample mean -26.08; now -1.64 against the limit's -1.771)
• MINOR GUE is the model for the energy levels of a heavy nucleus
  https://en.wikipedia.org/wiki/Random_matrix : GUE "models Hamiltonians lacking time-reversal symmetry"; GOE "models Hamiltonians with time-reversal symmetry", the nuclear case; counted as wrong
• MINOR E[Ln] tends to 2 sqrt n
  https://arxiv.org/abs/math/9810105 : the theorem is E[Ln]/sqrt n tends to 2 (E[Ln] - 2 sqrt n itself diverges like n^1/6); counted as wrong
• MINOR at n=4000 the ratio "matches" the BDJ asymptotic 2 + mu n^(-1/3) = 1.8884
  research/the-longest-climb/page-data.json : observed 1.8964, SE 0.00102: 0.0080 above, about 7.9 standard errors; the formula is leading-order only; counted as wrong
• CONFIRMED Tracy-Widom F2 mean -1.7711, skewness 0.2241, excess kurtosis 0.0934
  research/the-longest-climb/verify.mjs : the verifier's cited constants; the scaled sample now lands near the mean as well as the shape
• CONFIRMED Exact E[Ln] for n up to 10 (sum over n! reproducing OEIS A003316 up to 15,730,705)
  research/the-longest-climb/verify.mjs : recomputed by enumeration in the verifier, all checks green
• CONFIRMED Pile count of patience sorting equals the LIS length
  research/the-longest-climb/verify.mjs : 4000/4000 shuffles agree with an O(n^2) DP
• CONFIRMED Shuffle wobble skew 0.266 and excess kurtosis 0.098, GUE 800x800 skew 0.183 and excess kurtosis 0.090
  research/the-longest-climb/page-data.json : unchanged by the rescaling (shape statistics are scale-free)
• CONFIRMED "PASS 20/20" from node research/the-longest-climb/verify.mjs
  research/the-longest-climb/verify.mjs : 20/20 in the repository; run without the built page the drift guard is skipped and it reports 19/19, which the page now says

What was done

• [fixed] MED scaling inverted: page and generator now say (Ln - 2 sqrt n) / n^1/6 ("rescale", not "magnify"); verify.mjs divides, its comment and detail line corrected, page-data.json regenerated and the embedded copy in the page refreshed; dated correction line in the page footer
• [fixed] MED reader "PASS 20/20" (a stranger gets 19/19): the check section now says 20/20 in this repository and 19/19 without the built page, with the reason
• [fixed] MINOR "matches" replaced with "is close to the leading-order asymptotic", stating the 0.0080 gap and about 8 standard errors (generated from page-data.json in build-page.mjs); the verifier's check label says "within 0.02" rather than "matches"
• [fixed] MINOR GUE and heavy nuclei: now says the GUE models systems without time-reversal symmetry and nuclear levels are modelled by the GOE
• [fixed] MINOR "E[Ln] -> 2 sqrt n" now reads E[Ln]/sqrt n -> 2