{
  "source_repository_path": "shifted-even-beta/review/independent-audit-2026-11-05.txt",
  "source_sha256": "97ac889b58cdb07014d1e72830054235adaaecdb5b294d1c948343fb2a0fe588",
  "report_text": "INDEPENDENT AUDIT REPORTED 5 NOVEMBER 2026\n\nReport supplied by the user; auditor identity and model were not specified.\nLine wrapping, table layout and local filesystem references are normalized.\nThe audited plain-text proof has SHA-256\nc7197fc5cd341c3cb71ab30515be836f2cea0156b1252284229bd8d059cdf258.\nThe subsequently assembled PDF has a separate assembly review.\n\nI found no substantive mathematical error or gap in the underlying plain-text proof.\nThe general shifted-interval theorem survives the audit, including:\n\n  For every positive odd m, at least one of\n  β(m+1),β(m+3),…,β(67m-1) is irrational.\n\nI checked the arguments for arbitrary allowed parameters, particularly\nthe new steps:\n\n• Differentiated Gram matrix: the kernel formula, exact pole orders,\n  polynomial independence, contour sign (-1)^{DL/2+(m+1)/2}, convergence\n  and positivity all check out.\n• Denominators: m-fold zeros preserve the grouped cancellation after\n  differentiation. The clearing exponent is correctly increased to S+k.\n• Derivative decay: the fixed-radius Cauchy estimate works even near the\n  last zero, and the infinite tail is controlled. Parameters remain fixed\n  while n→∞; uniform convergence in m is unnecessary.\n• Prime dilation: local coefficient extraction is justified. The weaker\n  nonselected-pole bound vₚ(A_{ν,h})≥ν-k is sufficient because τ>k. The\n  surviving constant has the correct factor χ(p)(ℓ+1)^{-S}.\n• Final contradiction and endpoints: the cancellation weights retain a\n  nonzero seed constant, and the uniform estimate pays the additional\n  derivative cost for every odd m and every B.\n\nUsing independently written code, without running or importing the\nsupplied checkers, I reproduced:\n\n  Exact check                                                    Passed\n  Differentiated-kernel evaluations                                 120\n  Gram entries, comparing moments with differentiated\n    partial fractions                                               90\n  Positive principal minors / integer seed vectors              30 / 12\n  Selected-pole congruences / other-pole bounds                800 / 6,800\n  Constant congruences                                               14\n  Grouped integrality checks, seed and prime indices              6,564\n\nI also tested seed index ℓ=3, prime 17, hence n=51 beyond the supplied\ncases. Its coefficient and constant congruences passed.\n\nAll four displayed interval certificates pass, as do the rational\ninequalities supporting the universal endpoint.\n\nOne minor wording correction: at line 490 of the audited plain-text proof,\nreplace “its numerator or denominator” with “the numerator or denominator\nof q_{w}(1).” The intended exclusion is clear from the next equation, but\nthe pronoun is ambiguous.\n\nAudited version: SHA-256 prefix c7197fc5cd341c3c. This is an independent\nmathematical audit, not formal verification or certification of priority.\nNo files were changed.\n"
}