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The Harmonics of the Primes

Written 2026-06-05. Claims re-read against their sources on 2026-09-27: 10 checked, 9 confirmed, 1 wrong, 0 unverifiable. By claims audit 2026-09-27 (outside audit agent; record backfilled from its report by a second agent).

Control-sample audit (research/claims-audit/PREREG.md), 2026-09-27, findings condensed in research/claims-audit/findings/r3.md. Reader check: the page gives no runnable recipe; the three checks it names are 404, and the one published file, verify_gue.py, is unlinked and needs mpmath and numpy (with them installed it passes). Repository copies all exit 0 and match the page. The report counted 9 claims, all confirmed, and raised one near-miss about nuclear physics; that is recorded here as MINOR and counted as wrong. The layer was classified reader-side; the only change since is the placard wording of cc79bde660.

Claims

• MINOR Zeros "spaced like the energy levels of a heavy atomic nucleus, more precisely, like the eigenvalues of a large random Hermitian matrix (the GUE...)"
  https://en.wikipedia.org/wiki/Random_matrix : heavy-nucleus levels are time-reversal symmetric and follow GOE statistics: "The Gaussian unitary ensemble models Hamiltonians lacking time-reversal symmetry"
• CONFIRMED 1859, Riemann's only number-theory paper
  https://en.wikipedia.org/wiki/On_the_Number_of_Primes_Less_Than_a_Given_Magnitude : "the only paper Riemann ever published on number theory"
• CONFIRMED "very probable" and set aside as inessential
  https://en.wikipedia.org/wiki/Riemann_hypothesis : quotes Riemann: "sehr wahrscheinlich ... vorlaufig bei Seite gelassen..."
• CONFIRMED von Mangoldt gave the first rigorous proof in 1895
  https://en.wikipedia.org/wiki/Explicit_formulae_for_L-functions : "first rigorous proof ... von Mangoldt in 1895"
• CONFIRMED Gourdon 2004 verified the first 10^13 zeros
  https://en.wikipedia.org/wiki/Riemann_hypothesis : reference list; Platt and Trudgian 2021 reached about 1.2e13 zeros rigorously
• CONFIRMED The Montgomery-Dyson tea in 1972; Montgomery 1973
  https://www.ias.edu/ideas/2013/primes-random-matrices : "early April 1972 ... At teatime"; Proc. Symp. Pure Math. XXIV, pp. 181-193
• CONFIRMED du Sautoy, The Music of the Primes, 2003
  https://en.wikipedia.org/wiki/The_Music_of_the_Primes : pub_date 2003
• CONFIRMED Mazur and Stein, CUP 2016
  https://journals.iucr.org/a/issues/2017/06/00/xo0097/ : matches
• CONFIRMED Clay: seven problems, one million dollars each
  https://en.wikipedia.org/wiki/Millennium_Prize_Problems : matches
• CONFIRMED "150 zeros validated against published values"
  https://www-users.cse.umn.edu/~odlyzko/zeta_tables/zeros1 : all 150 in zeros.json compared by the auditor: max difference 5.0e-10 (the repo verifier checks only 15)

What was done

• [declined] NOT ADDRESSED: MED the check aside (index.html:565) names verify.py, verify_browser.mjs and verify_gue.mjs, and index.html:251 says they "live in /research/harmonics-of-the-primes/"; all three are 404, and the one published file, verify_gue.py, is unlinked and needs mpmath and numpy, which the page does not say. The placard was reworded by cc79bde660 ("Whether it runs on its own has not been tested yet") but the named files are still unreachable.
• [declined] MINOR (the MINOR line) heavy-nucleus levels follow GOE, not GUE: left as is; a near-miss, and the page's "more precisely, like the eigenvalues of a large random Hermitian matrix (the GUE)" already puts the claim on the matrix ensemble.