{"taskid":"59f8deca115aa901df74db204be5a398ecf47e17878fe23cbcb07e4df3ae59e0i0","node":"find364","result":{"type":"negative","from":1,"to":10000000000000000000,"script":"search_ep364.py","log":"out/run_odd_1e19.log","runtime_env":"full runs: CPython 3.12.7 + numpy 2.5.4 (Windows x86-64, Intel i5-10210U, single process); verifier replay / crosscheck / brute: python3 standard library only (no third-party packages needed; numpy auto-accelerates when present)","method":"mod-4 lemma: any triple of consecutive powerful numbers has odd endpoints at distance 2 and a powerful middle. Complete (not sampled) enumeration of ALL odd powerful numbers <= 1e19 via the unique representation m = a^2*b^3 (a,b odd, b squarefree; Golomb 1970), adaptive blocks with cross-block carry; sorted gap scan finds every distance-2 pair; every candidate middle tested by exact integer factorization (trial division <= 1e6, deterministic Miller-Rabin, Pollard rho, perfect-power analysis). A second fully independent pipeline (all powerful numbers, no parity reduction) covers [1, 1e17]. Per-1e15-chunk fingerprints (count/xor/sum64/exact-sum/min/max) are replayable from the attached package with the standard library only.","result_summary":{"claim":"no triple of consecutive powerful numbers in [1, 10^19]","odd_powerful_count":2537872835,"gap2_pairs_count":9,"gap2_pairs":["25;27","70225;70227","130576327;130576329","189750625;189750627","512706121225;512706121227","13837575261123;13837575261125","99612037019889;99612037019891","1385331749802025;1385331749802027","3743165875258953025;3743165875258953027"],"gap2_pairs_equal_oeis_A076445_in_range":true,"triples_found":[],"middle_nonpowerful_witnesses":[["26",[[2,1],[13,1]],[2,1]],["70226",[[2,1],[13,1],[37,1],[73,1]],[2,1]],["130576328",[[2,3],[29,1],[197,1],[2857,1]],[29,1]],["189750626",[[2,1],[13,1],[61,1],[181,1],[661,1]],[2,1]],["512706121226",[[2,1],[13,1],[757,1],[2521,1],[10333,1]],[2,1]],["13837575261124",[[2,2],[19,1],[61,1],[199,1],[1381,1],[10861,1]],[19,1]],["99612037019890",[[2,1],[5,1],[89,1],[103093,1],[1085657,1]],[2,1]],["1385331749802026",[[2,1],[13,1],[37,1],[73,1],[109,1],[1297,1],[139537,1]],[2,1]],["3743165875258953026",[[2,1],[13,1],[150217,1],[489061,1],[1959673,1]],[2,1]]],"fingerprints":{"xor_hex":"14cdff534290e471","sum64_hex":"84fc8bc5ffc57f87","sum_exact":"8460235187835511201459437447","min":"1","max":"9999999992610520281"},"chunk_manifest":{"width":1000000000000000,"chunks":10001,"file":"out/odd_manifest_1e19.csv","per_chunk_fields":["count","xor_hex","sum64_hex","sum_exact","min","max","d2_count"]}},"cross_method_validation_1e17":{"all_powerful_count":686552743,"even_count":432856760,"odd_subset_equals_odd_pipeline_count":253695983,"consecutive_pairs":29,"consecutive_pairs_equal_oeis_A060355_in_range":true,"gap2_pairs":8,"triples":0,"quadruples":0},"verification_guide":{"spec_verifier":"materialize find.json from this result (chain pin content), extract the attachment, then: python3 spec-ep364.py <dir>","replay_any_chunk":"python3 search_ep364.py --from <c*10^15> --to <(c+1)*10^15-1> --engine stdlib  # diff vs out/odd_manifest_1e19.csv row c","crosscheck_suite":"python3 crosscheck_ep364.py [--heavy]  # final line must read '== crosscheck PASS =='","brute_definition":"python3 brute_ep364.py --to 300000","spot_checked_chunks":["0","1","3743","9999"],"spot_check_result":"4 chunks x 2 engines, 8/8 fingerprints MATCH vs manifest (out/spotcheck.log)","full_rerun":"python3 search_ep364.py --from 1 --to 10000000000000000000 --engine numpy --manifest out/odd_manifest_1e19.csv --summary-out out/run_odd_1e19_summary.json"},"anchors":{"literature":["erdosproblems.com/364 (accessed 2026-10-12): status VERIFIABLE; no quadruples; page states no triple n<7.38e28 via OEIS A076445","arXiv:2609.25011: kernel-certified (Lean 4) exclusion below 1e14; cites larger uncertified computations (exhaustive to 1e22; conditional ~7.38e28)","OEIS A076445 b-file (2026-10-12): 9 in-range terms matched exactly; OEIS A060355 b-file: in-range terms matched exactly"],"prior_onchain_records":["a760d1b443750b9de1ecf4538e9d4e11376b2305d2fa69f33e4a51870b41f1c4i0 negative [1,1e13]+Pell (fail verdict 192477)","0367db2b202171943e3beb6bba60ad458696a9746dde47fce5fc3f4bcca4b17ci0 negative [2,1e15]","7df13aa919f839589823da9c970d23216fa57f99ad6397faf83f8112ce5d8dc2i0 negative [1,2e15]","fe787977db78f1e3d1101c95f359279c901438ba01586db03e3a5450730748f6i0 negative [1,1e18] (superseded)","4ddfb6e967af63e5a0bc49dcf398882d36052deeb4ac0749ab164721f612d7d0i0 negative [1,1e18]"]},"proof_archive":{"contents":"README + 3 scripts + full logs (odd 1e19, all 1e17) + 10001-chunk manifest + 100-chunk manifest + crosscheck/spotcheck/brute logs + OEIS anchor refs + sha256sums; find.json is derived from the chain pin (not inside the pack, to avoid self-hash circularity)","pack_sha256":"94cd1ff92ad0498ca3a1cb17f8ea13a60f72ed7fe462e8fa768835fac5670f2e"},"disclosure":"Honest scope: this record does not claim to exceed the literature's uncertified exhaustive bound 1e22 nor the conditional 7.38e28 statement; it is a fully replayable, chain-anchored exclusion record: 10^5 x beyond the kernel-certified 1e14 boundary and 10 x beyond the previous best strict replayable on-chain record [1, 1e18]. Verifier replay path is python3-stdlib-only; every claim reproduces from the attached package.","hash":"dc3b15f3abb2baf32d2583be46e8517e2a75b1e2f6b8d9dcde4c789b0a0e588d"},"hash":"4916a3d65a2f8c61bdd1b4767760d025c4a9687187b9620ecb8db8971849a593","contentType":"application/json;utf-8","attachment":"metafile://c4caab875cf0535a838a1c1713e2f046879ffd556dba2ea44d27d616ee5d896ei0","childids":[]}