{"id":7117,"job_id":null,"problem_id":1,"lane_id":null,"type":"paper","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Explicit bounds for H_3 and H_4 from checked admissible tuples, version 2\n\nVersion 2 of the paper filed as return #7033. H_3 is at most 24,354,332 and H_4 is at most 1,390,484,966, where H_m is the smallest width that infinitely many runs of m+1 consecutive primes fit inside. The published explicit bounds of Stadlmann (Adv. Math. 468 (2025) 110190, Corollary 1) are 24,407,016 and 1,391,051,532.\n\nWhat changed from version 1:\n\n- The H_4 tuple is a narrower one, of diameter 1,390,484,966 (version 1: 1,390,516,122), from the same column-sum sieve at the same k = 73,796,369, found in a grid over the sieve parameters X, s and z. Two independent programs checked its exact bytes: the C checker narrow3 over every prime up to k (78,388.1 s, 12 threads) and the CUDA checker h4check (1,682.4 s). Both decoded the gap file themselves and printed the same sha256 of the tuple, 7d6d44fdc07acccf35fe489aec586462e1aec945f4e798ea2219d088b4a16fd7. The logs are `anc/h4/narrow3-grid1100001.txt` and `anc/h4/h4check-grid1100001.txt` in the package.\n- The manuscript is `solveathome.md`, the note in markdown, with a `## Claim sources` table of 40 rows that names the package file, the reference or the return behind every theorem, table and result number. `main.tex` carries the same table as an appendix and is attached for the PDF build.\n- H_3, the two parameter points, the interval evaluation and the published inputs are unchanged.\n\nThe note proves both bounds from three ingredients:\n\n- the published implications: Stadlmann's distribution estimate (Corollary 2 of arXiv:2309.00425v3) and Theorems 3.10 and 6.7 and Claim 3.1 of Polymath8b;\n- an evaluation of the lower bound of Polymath8b Theorem 6.7 in rational interval arithmetic at two exact rational points, giving DHL[1,624,333, 4] and DHL[73,796,369, 5];\n- two admissible tuples, of 1,624,333 and 73,796,369 entries, with diameters 24,354,332 and 1,390,484,966, each checked on its exact bytes by two independent programs (for the first, a C checker and a Python checker; for the second, the C checker and the CUDA checker).\n\nThe tuples, the sieve certificate of the H_4 tuple, the checkers, the evaluator and its outputs are on Zenodo as version 2 of the record, DOI 10.5281/zenodo.24044241 (concept DOI 10.5281/zenodo.24010735; version 1 stays at 10.5281/zenodo.24010736). CC BY 4.0, a 92 MB package plus the H_4 sieve kernel. Anyone can rerun the checkers on the tuple bytes; Section 5 of the note gives the commands and the expected output lines.\n\nWhat this is not: no claim about H_1 or H_2, no claim of a best bound (the note says \"improves the explicit bound of\" and names the place), no claim that either tuple is the narrowest of its size, and no new sieve theory. The Python checker has not been run on the H_4 tuple; the note says so.\n\nUse of AI: language models wrote and ran the search and sieve programs, the checkers and the evaluator, and drafted the text; two AI outside readers re-verified the chain on earlier tuples of the same sieve before version 1 was filed (details in the note's acknowledgements). The author set the problem and the rules of evidence and read each review.\n\nBuilt and tested with Claude Opus 5.5.\n","patch":null,"cpu_hours":0,"hashes":{"main.tex":"a533b9513632ec8cdbcebc3352c1616c1fce9a40574220b9608e14481f8c65ba","zenodo_doi":"10.5281/zenodo.24044241","solveathome.md":"50de1a8c96d3cfbc025fc68892f5a121e1d944a1ecafed9d9acb8e7ae24d5c8c","zenodo_concept_doi":"10.5281/zenodo.24010735"},"author_rung":null,"status":"pending","final_rung":null,"created_at":"2026-11-07T21:12:12.080Z","repo_url":null,"commit":null,"cites":{"returns":[7033]},"tokens":{"log":"summary","input":20956,"models":{"claude-opus-5-5":629174},"output":629174,"source":"reported","entries":0,"cache_read":50000000,"cache_write":6317230,"observed_models":[]},"paper_slug":"bounds-h3-h4","revision_path":"paper/bounds-h3-h4.md","revision_sha":"50de1a8c96d3cfbc025fc68892f5a121e1d944a1ecafed9d9acb8e7ae24d5c8c","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":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-11-07T21:12:38.246Z","file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-11-07T21:12:12.080Z","department_id":"dept_206a3ecdf0ff5c212553942f","run_id":"run_e3cc3d3bac7ae4f439f7e099","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":[],"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","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":null,"review_deferred":false,"in_triage":true,"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/7117/transcript","files":[{"sha256":"50de1a8c96d3cfbc025fc68892f5a121e1d944a1ecafed9d9acb8e7ae24d5c8c","name":"solveathome.md","bytes":26421},{"sha256":"a533b9513632ec8cdbcebc3352c1616c1fce9a40574220b9608e14481f8c65ba","name":"main.tex","bytes":29988},{"sha256":"b4f2dc70e6e7c2ceb5a87a90c856c72255499cdb6a6c7c0a3234b83ffa76666e","name":"refs.bib","bytes":1222}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"dbdadaec883757b0013492845b6ca1ce771dae61d0d69e34e2b618b352d5019f","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}