{
  "title": "Irrational even beta values in shifted intervals",
  "version": 2,
  "contributor": "twinheterodox",
  "source_git_commit": "6524b5f153fa8c18d200ade0f059ae955b6d9fb3",
  "source_git_tag": "shifted-even-beta-v2",
  "source_url": "https://github.com/twinheterodox/beta4/tree/shifted-even-beta-v2/shifted-even-beta",
  "existing_hexagon_version": "https://hexagonmath.org/2611.00160v1",
  "main_tex_file": "shifted-even-beta.tex",
  "compiler": "pdfLaTeX",
  "expected_pages": 20,
  "compile_from_archive_root": [
    "pdflatex -interaction=nonstopmode -halt-on-error shifted-even-beta.tex (three passes)"
  ],
  "exact_checks_from_archive_root": [
    "python3 -B anc/checks.py",
    "python3 -B anc/endpoints.py",
    "python3 -B anc/worked_example.py"
  ],
  "optional_numerical_exploration": "python3 -B anc/optimization.py",
  "requirements": "The checkers use only Python's standard library. Do not use -O: assertions perform the checks.",
  "result_files": "Each checker writes the JSON file beside its source. Packaged JSON files are the reference results.",
  "package_layout": "TeX inputs are at the archive root and in sections/. Supporting scripts and records are in anc/. The eleven main-file input paths omit the repository's paper/ prefix; all other TeX bytes are unchanged.",
  "review_paths": "Paths within preserved review records refer to the tagged GitHub repository layout, not the flattened ZIP. The package manifest beside the ZIP records the mapping.",
  "check_guide": "EXACT CHECKS FOR SHIFTED EVEN-BETA INTERVALS, VERSION 2\n\nRun from the shifted-even-beta directory:\n\n  python3 -B checks/checks.py\n  python3 -B checks/endpoints.py\n  python3 -B checks/worked_example.py\n\nThese programs use only the Python standard library. Do not use -O:\ntheir assertions perform the checks. Each writes the corresponding\nJSON result beside its source.\n\nThe parameters now agree directly with the manuscript: m is positive\nodd, B is the count, r is the integer zero-range parameter, and a is\nthe even pole order. The endpoint is a+m-1. The v1 checker's notation\nand numerical records belong to the earlier version.\n\nchecks.py\n  Reproduces 8,909 structural checks with exact rational arithmetic:\n    57 differentiated-kernel evaluations;\n    49 Gram entries, comparing residues with moments;\n    19 positive leading principal minors;\n     9 integer seed-vector identities;\n    54 rational-function reconstructions;\n  1308 reflection identities;\n   332 differentiated grid zeros;\n  4516 grouped-integrality checks;\n   404 selected-pole congruences;\n  1940 other-pole valuation bounds;\n    19 constant congruences;\n   202 beta-coefficient valuation bounds.\n\n  The nine seed cases are (B,m,r):\n    (1,1,2), (1,3,2), (1,5,1), (2,1,1), (2,3,1),\n    (2,1,2), (3,1,1), (3,3,1), (4,1,1).\n  All use a=mDL+1. Complete prime cases are recorded in checks.json.\n  These small parameters check identities, not endpoint decay.\n\nendpoints.py\n  Reproduces all 21 rational endpoint certificates, including the 12\n  displayed in Section 8.2; checks the uniform inequalities for every\n  m and B; certifies the six listed numerical values of a_B using\n  rational exponential bounds; and proves the seven integer-r\n  comparisons in Appendix B.\n\n  Floating point locates trial u and candidate ceilings. Every claimed\n  inequality and ceiling is then certified with rational arithmetic.\n  U_approx and f_approx are illustrative floating-point displays.\n  The general proof supplies the all-parameter arguments.\n\nworked_example.py\n  Checks B=2, m=1, r=2, a=70 from Section 3.1. It compares the displayed\n  moment matrix with the seed partial fractions, verifies the\n  determinant formula, and checks the integer multiplier's seed vector.\n  This example was added after the independent audit and is a self-check.\n\noptimization.py\n  Optional numerical exploration:\n    python3 -B checks/optimization.py\n\n  Reproduces the limiting profile transition for B=1 and scans integer\n  r=1,...,15 for B=1,2,3,4,5,10. Its JSON is numerical evidence, not\n  an exact certificate. The manuscript's 18.58 bound and integer-r\n  comparisons are certified by endpoints.py.\n\nThe checkers are supplied self-checks, not an independent audit.\nThe independent auditor's separately written calculations and general\nproof review are recorded in review/independent-audit-2026-11-06.json.\nNo proof-assistant verification is claimed.\n",
  "audit_scope": "The independent report concerns the general plain-text proof and independently reproduced certificates. The typeset PDF and added worked example were checked separately. These records do not claim formal verification.",
  "license_note": "The Hexagon deposit retains the contributor-approved CC BY 4.0 setting. The repository's Apache-2.0 notice is preserved in LICENSE.json."
}