{
  "title": "Irrational even beta values from powered hypergeometric kernels",
  "author": "twinheterodox",
  "main_tex": "powered-beta-kernels.tex",
  "compiler": "pdfLaTeX",
  "expected_pages": 13,
  "canonical_repository_directory": "https://github.com/twinheterodox/beta4/tree/main/powered-beta-kernels",
  "repository_link_status": "Prepared locally; the new directory becomes available after GitHub publication.",
  "guide": "IRRATIONAL EVEN BETA VALUES FROM POWERED HYPERGEOMETRIC KERNELS\n11 November 2026\n\nEach of the following sets contains an irrational number:\n\n  beta(6), beta(8), ..., beta(74);\n  beta(8), beta(10), ..., beta(110).\n\nThe earlier shifted-interval paper gives endpoints 92 and 128 for these\nstarting indices. The new intervals contain 35 and 52 candidate values,\nrespectively; each removes nine values from the earlier interval.\nThe two assertions need not identify distinct irrational numbers.\n\nThe proof raises Krisman's hypergeometric kernel to a fixed power and\ndifferentiates to remove the early beta values. The denominator cost and\nreal decay scale together. A prime-dilation argument and a finite nonzero\nseed then give infinitely many nonzero integers tending to zero under\nthe assumption that all values in the chosen interval are rational.\n\nCanonical files\n\n  paper/powered-beta-kernels.tex\n  output/pdf/powered-beta-kernels.pdf\n\nThis directory is the canonical working version. The original research\nworkspace is retained privately as a historical record.\n\nBuild from this directory\n\n  python3 -B tools/build_paper.py\n\nThe build requires pdfLaTeX with the usual AMS and Latin Modern packages,\nand Poppler's pdfinfo. Intermediate files go in the Git-ignored build/\ndirectory. The source has no external figures or bibliography files.\n\nComputational checks\n\nSee checks/README.txt. The seed checker and arithmetic-budget checker\nuse only the Python standard library. The primary seed generator and\nrational dilation checker also need python-flint 0.9.0. Two original\nhelper functions are retained verbatim; unrelated determinant pilots\nare not included. The complete finite certificate data are supplied.\n\nThe auditor's separate implementations and numerical records are also\navailable in checks/audit/. They are distinct from the manuscript's\nsupplied checkers. The auxiliary arithmetic-budget program needs a\nC++17 compiler. No executable binary is distributed.\n\nAttribution and inputs\n\nKrisman's kernel and quantitative arithmetic/decay estimates are imported\ninputs. The prime-dilation strategy follows the earlier shifted-interval\npaper. Rivoal, Lai--Sprang and Zudilin are cited directly for the relevant\nmethodological precedents; Sorokin gives earlier shifted-interval context.\nsources/reading-record.json identifies all six sources, their hashes and\nthe passages actually read. Third-party PDFs are not redistributed.\n\nReview scope\n\nreview/independent-audit-2026-11-11.txt preserves the separate AI audit,\nwith local paths normalized for public use. Its companion JSON identifies\nthe audited source and PDF hashes. The audit found no substantive error\nor gap and independently reconstructed the finite certificates. It used\nfresh implementations after preliminary inspection and replay of the\nsupplied checkers; it was not a blind review. No Lean project was rebuilt,\nand this is not formal verification.\n\nThe current manuscript is a subsequent exposition and attribution\nrevision. review/assembly-review.json records the transfer into this\nrepository, preservation checks, portable replays and PDF comparison.\nThe general power criterion remains conditional on a nonzero seed;\nthe manuscript verifies that condition for powers 1, 2 and 3.\n\nThe original material follows the repository's Apache-2.0 license.\nPrepared with Codex. Author: twinheterodox.\n",
  "guide_path_note": "The guide above describes the canonical repository layout, whose reproduction guide is README.txt. In this source ZIP the main TeX file is at the root, and the manuscript checkers and data are under anc/checks/.",
  "python_dependency_for_optional_checks": "python-flint==0.9.0",
  "run_from_extracted_archive_root": [
    "python3 -B anc/checks/check_power_seed.py anc/checks/results/power_seed_k1_p10007.json anc/checks/results/power_seed_k2_p10007.json anc/checks/results/power_seed_k2_p10009.json anc/checks/results/power_seed_k3_p10007.json anc/checks/results/power_seed_k3_p10009.json --output anc/checks/POWER_SEED_VALIDATION.json",
    "python3 -B anc/checks/check_krisman_budget.py",
    "python3 -B anc/checks/check_prime_dilation.py",
    "python3 -B anc/checks/power_kernel_seed.py --power 2 --prime 10007 --output anc/checks/results/power_seed_k2_p10007.json"
  ],
  "audit_implementation_note": "The full independent-audit implementations, including the C++ helper, are in the repository. This ZIP includes their audit report and numerical outputs; its runnable scripts are the manuscript checkers.",
  "scope": "Exact finite checks support the identities and seed conditions. The general mathematical arguments are in the paper. The independent audit is not formal verification."
}