{
  "source_repository_path": "even-beta-counts/review/independent-audit-2026-11-05.txt",
  "source_url": "https://github.com/twinheterodox/beta4/blob/06bd732c81d48125d619ea2e9bf514878d774a86/even-beta-counts/review/independent-audit-2026-11-05.txt",
  "source_sha256": "53821929bf69694a5c93bef117ee9fc412dde1f1802d9ecc61a48084a38619d1",
  "report_text": "Independent mathematical audit\nReport supplied on 5 November 2026\n\nRecordkeeping note: the reviewer name, model and completion date were not\nspecified. The report below is reproduced with the local filesystem path\nreplaced by a path relative to the count-paper directory. The accompanying\nJSON record identifies the PDF snapshot captured before the exposition edit;\nits checksum was recorded locally, not supplied by the reviewer.\n\nReport as supplied (path normalized)\n\nindependent audit complete: • I found no substantive mathematical error or gap. The proof supports\n  \\[\n  I(s)\\ge \\left\\lfloor\\pi\\sqrt{\\frac{2}{135}}\\sqrt{\\frac{s}{\\log s}}\\right\\rfloor\n  \\]\n  for every even s≥160, and the bound without the floor for even s≥42 170. output/pdf/even-beta-counts.pdf\n\n  I checked the general arguments, including:\n\n  • Gram construction, pp. 5–7: kernel independence, the contour’s sign and normalization, positivity, convergence, and the integer homogeneous multiplier. This establishes rank for every B.\n  • Denominators, pp. 7–9: the progression lemma and the grouped cancellation needed for the rational constants.\n  • Decay, pp. 9–11: Stirling normalization, the unique maximum, uniform estimates and infinite-tail control.\n  • Prime dilation, pp. 11–12: factorial units, permutation of fraction classes, local Laurent-coefficient extraction, and the surviving factor χ₄(p)(m+1)^{-S}.\n  • Contradiction and inversion, pp. 13–17: the nonzero Vandermonde minor, infinitely many nonzero cleared integers, ceiling/floor handling, and coverage of every claimed endpoint.\n\n  Using independently written arithmetic—not running or importing the supplied checkers—I reproduced:\n\n   Check                                                           Result\n  ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━  ━━━━━━━━━━━━━━━━━━━\n   Rational Gram entries                                       385 passed\n  ───────────────────────────────────────────────────  ───────────────────\n   Positive leading principal minors                            55 passed\n  ───────────────────────────────────────────────────  ───────────────────\n   Integer seed-vector identities                               10 passed\n  ───────────────────────────────────────────────────  ───────────────────\n   Sharp-endpoint decay parameter choices                   All 10 passed\n  ───────────────────────────────────────────────────  ───────────────────\n   Finite interval comparisons                             All 253 passed\n  ───────────────────────────────────────────────────  ───────────────────\n   Selected-pole coefficient congruences                       272 passed\n  ───────────────────────────────────────────────────  ───────────────────\n   Other-pole valuation bounds                               2,312 passed\n  ───────────────────────────────────────────────────  ───────────────────\n   Constant congruences / grouped integrality checks    10 / 4,802 passed\n\n  The smallest interval margins agree with the manuscript: approximately 0.0002452910722 and 0.0001432530917. The rational constants supporting the infinite range also pass.\n\n  One small exposition improvement: on p. 10, spell out why the Stirling error remains uniform down to t=1/B, where t/n→0. The asserted estimate is valid; its justification is compressed.\n\n  This was an independent mathematical audit, not formal verification or a fresh audit of all cited literature. No files were changed.\n"
}