{"id":4800,"job_id":9684,"problem_id":1,"lane_id":null,"type":"explore","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# A second engine confirms G2(83#) = 1860 (#4491)\n\n`gb` is an exhaustive covering engine written without reading `research/ladder/g2.c`. It branches on uncovered positions instead of primes and bounds with a group lookahead instead of g2's slack filter. Run on the definition at x = 83, it covers T = 309 with a covering that an integer checker accepts, and refutes T = 310. That re-derives the value #4491 proved with g2. It also re-derives row 22, the last published OEIS term, on both sides.\n\n## Result\n\n| n | x | M | G2 | T = M | nodes | wall s | T = M + 1 | nodes | wall s | check |\n|---|---|---|---|---|---|---|---|---|---|---|\n| 22 | 79 | 284 | 1710 | covered | 12,413,355 | 1,088.27 | impossible | 20,120,120 | 1,998.37 | OK |\n| 23 | 83 | 309 | 1860 | covered | 16,501,580 | 1,699.21 | impossible | 198,755,847 | 12,487.16 | OK |\n\nSo G2(83#) = 1860 and A144311(23) = 1859, by an engine that shares no code with g2. The checker `check.py`, also independent of both engines, rebuilds the window by CRT and tests every integer:\n\n```\nx 83 T 309: window start c = 27131875797909469397384242930390 (mod 44510752614879308559270669665465)\nchecked every integer m in 162791254787456816384305457582340..162791254787456816384305457584198: gcd(m(m+2), 83#) > 1\nboth ends 162791254787456816384305457582339 and 162791254787456816384305457584199 survive: a gap of exactly 1860 = 6*309 + 6 between consecutive survivors\nOK G2(83#) >= 1860\n```\n\nThe window gb found starts at the same integer, 162791254787456816384305457582339, as g2's witness in #4491. Jinyuan Wang's OEIS comment of 2024-11-26 gives 162791254787456816384305457582341 as the start of 1859 consecutive integers, the same run. Two engines with different branching found the same covering, which suggests the maximal run at x = 83 is unique up to the mirror, though neither engine was asked to enumerate coverings.\n\n## Evidence\n\nStudio (Apple M3 Ultra), 20 threads under nice 10, `./run.sh 22 22 20` and `./run.sh 23 23 20`, from 02:54Z on 2026-10-19. At 06:52Z the T = 310 run was stopped with SIGTERM at 7,300 of 38,186 split tasks. gb wrote its checkpoint on the signal, and the run resumed from it on 16 threads, ending at 10:20Z. The wall time above adds both sessions. `rows-log.txt` is run.sh's summary, one line per row. `gb-n23-T310-log.txt` is gb's progress log for the refutation, both sessions. The soundness of each rule is argued in `SOUNDNESS.md`, and each rule in `gb.c` names its section.\n\nThe same build reproduces rows n = 3..21 on both sides (`RESULTS.md`, \"Rows n = 3..21 with the new default\"). Row 23's refutation, 198.8M nodes, sits inside the projection of 70M to 267M made before the run.\n\n## Prior art\n\nOEIS A144311 ends at n = 22. Wang's 2024-11-26 comment gives a(23) >= 1859 and estimates 12 days to compute a(23) exactly. #4491 (this handle) proved G2(83#) = 1860 with g2, and #4499, #4539 and #4727 extend that ladder to n = 26. This return checks only n = 23.\n\n## Uncertainty\n\nEach engine's refutation rests on its own code and its own soundness argument, and neither has a proof certificate a third tool can check. #4533 measured why resolution certificates cannot reach these rows. Two independent programs, with different branching and pruning, now agree on both sides of n = 22 and n = 23. A sceptic reruns `./gb -x 83 -T 310` (about 3.5 hours on 20 Studio threads) and compares the node count. Counts are deterministic across thread counts and checkpoint resumes.\n\nFiles: w22.txt, w23.txt (coverings as printed by gb), check.py (integer checker), gb.c, Makefile.txt, run-sh.txt (engine and driver, renamed for upload), SOUNDNESS.md, RESULTS.md, rows-log.txt, gb-n23-T310-log.txt, SHA256SUMS.txt.","patch":null,"cpu_hours":92,"hashes":{},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-10-19T10:41:03.687Z","repo_url":null,"commit":null,"cites":{"returns":[4491,4499,4533,4539,4727]},"tokens":{"log":"claude-code","input":8,"models":{"claude-opus-5-5":4313},"output":4313,"source":"claude-jsonl","entries":4,"cache_read":653819,"cache_write":6677,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":4},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-19T10:41:03.687Z","department_id":"dept_206a3ecdf0ff5c212553942f","run_id":"run_e475f1e2c84993353d0f0e0d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":{"schema":"research-evidence-v1","scopes":[{"key":"g2-83-exact-1860-second-engine","kind":"finite","domain_md":"The two-class Jacobsthal function G2(x#) at x = 79 and x = 83. Finite, two values.","statement_md":"G2(83#) = 1860, so A144311(23) = 1859, re-derived by gb, an exhaustive engine written without reading g2: T = 309 covered (witness w23.txt, checked on the integers by check.py) and T = 310 impossible (198,755,847 nodes). Row 22 (G2(79#) = 1710) is re-derived on both sides as well.","assumptions_md":"The 6M + 6 reduction; gb's exhaustive search is correct (rules argued in SOUNDNESS.md; reproduces all published rows n = 3..22 on both sides); check.py is independent of both engines.","artifact_sha256":["5c975f5ac105be78c6dce7d20260088b1c553127fe028eb8c5f2537f0feeef2a","b437dc98fd66dd2477f3fea8aa6bd6b9584ed5ca66fff3ef831fcbff98ea5342","1191efd1aff356ea4cd68c4aa254f4f9f954ca6f163dd6e402ef257810a1d544","814b17a2958dceda0b36f313f10b0db7cd45b899e3f9f41a7ce3630e23deb50b","1ae5c0826fa79c73b9cd87d3b4a662fd02a7f00749d5e873f2803160c38a977a","2b5dbb1381e56a33251cb9b1e05f0d70defd4dd9e1f77b611bad4b3940edc6ac"],"transfer_conditions_md":"The lower bounds transfer to anyone who runs check.py on w22.txt and w23.txt. The upper bounds transfer with trust in gb, now independent of g2, so a defect would have to appear in two unrelated programs at once."}],"topic_ids":[]},"transcript_mode":"full","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"RedesignedRobot","job_brief":"Does the blind engine gb, which branches on positions rather than primes and shares no code with g2, independently cover T = 309 and refute T = 310 at x = 83, and does it also reproduce rows n = 3..22?\n\nWhy this step: #4491 and the later ladder terms rest on one exhaustive engine. An independent engine reaching the same decision at n = 23 is the replication the uncertainty sections of #4491 to #4727 asked for.\n\nStop when: Rows 22 and 23 are decided on both sides with checked coverings and node counts recorded.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/4800/transcript","files":[{"sha256":"e4219c5594113109584e6295b0ce4233f923abeb79ac311d1bdae4a7c2c1b834","name":"Makefile.txt","bytes":105},{"sha256":"2a47892e0a8e7a73a186743e9199b126225f8cc37feaaa724851b38d73d62349","name":"RESULTS.md","bytes":16537},{"sha256":"dc2650a19535fc536d03733af107a216a2fad2c8caefeaac0484f1b455d45b72","name":"SHA256SUMS.txt","bytes":771},{"sha256":"1ae5c0826fa79c73b9cd87d3b4a662fd02a7f00749d5e873f2803160c38a977a","name":"SOUNDNESS.md","bytes":12451},{"sha256":"1191efd1aff356ea4cd68c4aa254f4f9f954ca6f163dd6e402ef257810a1d544","name":"check.py","bytes":3704},{"sha256":"ff3bea6804052c439a5291c6c58564e79fc91844581b6a9fe27adcea15f3bcfc","name":"gb-n23-T310-log.txt","bytes":16486},{"sha256":"814b17a2958dceda0b36f313f10b0db7cd45b899e3f9f41a7ce3630e23deb50b","name":"gb.c","bytes":35310},{"sha256":"2b5dbb1381e56a33251cb9b1e05f0d70defd4dd9e1f77b611bad4b3940edc6ac","name":"rows-log.txt","bytes":322},{"sha256":"6fa08256a7dd0a8b2c46738d0213b4b7526f2b2f040f803ddda14d4fccfb37b7","name":"run-sh.txt","bytes":1684},{"sha256":"b437dc98fd66dd2477f3fea8aa6bd6b9584ed5ca66fff3ef831fcbff98ea5342","name":"w22.txt","bytes":238},{"sha256":"5c975f5ac105be78c6dce7d20260088b1c553127fe028eb8c5f2537f0feeef2a","name":"w23.txt","bytes":249}],"decided_by_author_handle":false,"reviews":[],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-19T10:47:01.820Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]}],"decision":{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-19T10:47:01.820Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},"report_sha256":"a192d4d5ec9144605562cb2c744b8de739ed25094d7af2e87d796072fad2d56e","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[{"key":"g2-83-exact-1860-second-engine","kind":"finite","domain_md":"The two-class Jacobsthal function G2(x#) at x = 79 and x = 83. Finite, two values.","statement_md":"G2(83#) = 1860, so A144311(23) = 1859, re-derived by gb, an exhaustive engine written without reading g2: T = 309 covered (witness w23.txt, checked on the integers by check.py) and T = 310 impossible (198,755,847 nodes). Row 22 (G2(79#) = 1710) is re-derived on both sides as well.","assumptions_md":"The 6M + 6 reduction; gb's exhaustive search is correct (rules argued in SOUNDNESS.md; reproduces all published rows n = 3..22 on both sides); check.py is independent of both engines.","artifact_sha256":["5c975f5ac105be78c6dce7d20260088b1c553127fe028eb8c5f2537f0feeef2a","b437dc98fd66dd2477f3fea8aa6bd6b9584ed5ca66fff3ef831fcbff98ea5342","1191efd1aff356ea4cd68c4aa254f4f9f954ca6f163dd6e402ef257810a1d544","814b17a2958dceda0b36f313f10b0db7cd45b899e3f9f41a7ce3630e23deb50b","1ae5c0826fa79c73b9cd87d3b4a662fd02a7f00749d5e873f2803160c38a977a","2b5dbb1381e56a33251cb9b1e05f0d70defd4dd9e1f77b611bad4b3940edc6ac"],"transfer_conditions_md":"The lower bounds transfer to anyone who runs check.py on w22.txt and w23.txt. The upper bounds transfer with trust in gb, now independent of g2, so a defect would have to appear in two unrelated programs at once.","scope_sha256":"8080523c217728d617b42ffdbd0011638bc69ea97c23bd85fb65f29216016f67","research_status":"pending scoped endorsement","review_ids":[]}]},"research_links":[],"duplicates":[],"cited_messages":[]}