{
  "title": "At least one of beta(4), beta(6), ..., beta(32) is irrational",
  "author": "twinheterodox",
  "main_tex": "fifteen-beta-values.tex",
  "compiler": "pdfLaTeX",
  "expected_pages": 15,
  "canonical_repository_directory": "https://github.com/twinheterodox/beta4/tree/main/fifteen-beta-values",
  "publication_state_at_packaging": "Prepared working tree; not yet committed or pushed.",
  "commands_from_extracted_archive_root": [
    "pdflatex -interaction=nonstopmode -halt-on-error fifteen-beta-values.tex",
    "pdflatex -interaction=nonstopmode -halt-on-error fifteen-beta-values.tex",
    "python3 -B anc/checks/run_hexagon_checks.py"
  ],
  "optional_product_replay": {
    "dependency": "python-flint==0.9.0",
    "command": "python3 -B anc/checks/run_hexagon_checks.py --with-product"
  },
  "checks": "The default runner uses only Python 3.10 or later and its standard library. It reconstructs the arithmetic partition, certifies the real-phase maximum and recomputes all seed coefficients at both primes.",
  "additional_github_checks": "The repository also includes the separate C++ direct-floor checker and all independent-audit implementations. C++ source is not an accepted ancillary extension in the Hexagon schema checked for this package. Those implementations and their dependent runners are omitted from this ZIP; their numerical reports are retained.",
  "audit_scope": "The independent audit identifies the 14-page predecessor by hash. The current 15-page manuscript includes later explanatory revisions; the audit response and exposition update record their scope. This is not formal verification.",
  "paths_in_historical_records": "Preserved reviews, source ledgers and check reports refer to the canonical repository layout. The source-copy mapping is in anc/MANIFEST.json.",
  "canonical_repository_guide": "AT LEAST ONE OF beta(4), beta(6), ..., beta(32) IS IRRATIONAL\ntwinheterodox; prepared with Codex; 11 November 2026\n\nCanonical PDF: output/pdf/fifteen-beta-values.pdf\nLaTeX source: paper/fifteen-beta-values.tex\n\nThe paper proves an irrational member of a set of fifteen even Dirichlet\nbeta values, improving Krisman's seventeen-value set through beta(36).\nThe new step is a denominator refinement: a pole alone in its residue\nclass has integral principal parts, and a zero identity avoids its\npotential contribution to the rational constant's denominator. Only\nrepeated pole classes enter the large-prime cost.\n\nThe paper gives one complete proof with parameters r=3/4 and base index\nN=96. It includes an isolated-pole example, the exact arithmetic and\nphase bounds, and a nonzero seed transferred to infinitely many indices\nby prime dilation. Appendix A supplies the small-prime estimates.\n\nBUILD\n\nFrom the repository root, with LaTeX and Poppler installed:\n\n    python3 -B fifteen-beta-values/tools/build_paper.py\n\nThe build writes the canonical PDF and paper/build-record.json. It\nrejects unresolved references and overfull boxes. Temporary files go\nunder build/ and are ignored by Git.\n\nCHECK\n\n    python3 -B fifteen-beta-values/checks/run_checks.py\n\nSee checks/README.txt for scope, optional dependencies and the command\nthat also regenerates the truncated-product seed calculation.\n\nPREPARATION AND REVIEW\n\nThe denominator insight and fifteen-value conclusion were obtained\nseparately in two Codex research sessions using GPT-6-Astra max and\npublic sources, without access to each other's unpublished work. This\nmanuscript uses one parameter choice and incorporates the simpler\npolynomial maximum argument developed in the second session.\n\nAn independent audit reported on 11 November 2026 found no mathematical\nerror or gap affecting the theorem and independently reproduced all\nthree computational inputs. Its report and the hashes of the audited\n14-page version are in review/independent-audit-2026-11-11.txt and\nreview/independent-audit-2026-11-11.json. The auditor's unchanged programs\nand numerical results are in checks/independent-audit/.\n\nThe current PDF adds an audit note and explanations of the pole-block\nresidue counts and the two odd-product lengths, and explains the integer\ncontradiction and the arithmetic integral before their machinery. These\nlater edits and their validation are recorded in\nreview/audit-response-2026-11-11.json and\nreview/exposition-update-2026-11-11.json.\nFinal source and PDF consistency checks are recorded in\nreview/publication-preparation-2026-11-11.json.\nThe original internal review remains in review/assembly-review.json\nwith its historical scope. The independent audit accepts the stated\nclassical inputs; it is not formal verification or a priority assessment.\n\nSources, versions, checksums and actual reading scopes are recorded in\nsources/reading-record.json. Third-party PDFs are not redistributed.\n",
  "canonical_check_guide": "FINITE CERTIFICATES FOR THE FIFTEEN-VALUE PAPER\n\nFrom the repository root:\n\n    python3 -B fifteen-beta-values/checks/run_checks.py\n\nThis requires Python 3.10 or later and a C++17 compiler named g++.\nThe Python calculations use the standard library. The command regenerates\nthe arithmetic partition, checks its 38,371 midpoint maxima by two\nalgorithms, checks the phase by polynomial and concavity arguments, and\nrecomputes all 2,077 seed coefficients at each of 10007 and 10009 by a\nlogarithmic-derivative recurrence. The saved truncated-product rows are\nread only after that calculation, for comparison.\n\nTo regenerate those rows as well, install the optional pinned dependency\nin a virtual environment and run:\n\n    python3 -m pip install -r fifteen-beta-values/checks/requirements-optional.txt\n    python3 -B fifteen-beta-values/checks/run_checks.py --with-product\n\nThe complete command above was run for the manuscript. The report is\ncheck-report.json; it records whether the product calculation was replayed\nand hashes the scripts and certificate data. Scratch executables and logs\ngo under ../build/checks/ and are not publication files.\n\nWHAT EACH CHECK ESTABLISHES\n\ncertify_fifteen.py\n  Enumerates affine-offset intersections and integer events on [1,400].\n  Sweeps the jumps in the residue variable to find the maxima. Integrates\n  the resulting step function using rational arithmetic with upward\n  64-bit rounding, adds an explicit infinite-tail bound, and proves the\n  arithmetic cost is below 41/5. Also certifies max Phi < -18.83 using\n  concavity on an initial interval and a tail derivative bound.\n\nverify_fifteen_arithmetic.cpp\n  Recomputes the maximum at every supplied midpoint by evaluating the\n  floor counts directly on each residue cell. It checks the maxima, not\n  a separately enumerated event partition. The general completeness\n  argument for that partition is in Section 4.2 of the manuscript.\n\ncertify_phase_polynomial.py\n  Reconstructs the degree-33 critical polynomial, checks its coefficient\n  signs, and brackets the unique positive root. Rational logarithm\n  enclosures prove max Phi <= -18.835112315041. This script imports no\n  other checker.\n\nseed_fifteen.py\n  Uses python-flint to multiply truncated finite-field series at all\n  poles. Checks reflection and three rational-function evaluations per\n  prime. Produces the coefficient rows and nonzero constant residues\n  3738 modulo 10007 and 5333 modulo 10009.\n\nverify_fifteen_seed.py\n  Reconstructs the seed coefficients by logarithmic differentiation and\n  a Taylor recurrence, using only Python's standard library. Checks both\n  prime moduli by trial division, odd-index cancellation, and agreement\n  with all 4,154 saved coefficients. It does not import seed_fifteen.py.\n\nThe finite calculations establish the numerical inputs specified above.\nThe denominator lemma, uniform analytic estimates, prime-dilation\nargument and final irrationality contradiction are general proofs in\nthe manuscript. These replays are not a separate mathematical audit or\nformal verification of those arguments.\n\nINDEPENDENT AUDIT\n\nThe separate audit reported on 11 November 2026 checked the complete\nargument and independently reproduced the arithmetic, phase and seed\ninputs. Its original programs and results are in independent-audit/;\nthe report and audited-file hashes are in ../review/. To replay those\nprograms in a scratch directory, with the optional dependency installed:\n\n    python3 -B fifteen-beta-values/checks/replay_independent_audit.py\n\nThe independent-audit/README.txt explains the algorithms and coverage.\n"
}