{"id":7033,"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\n\nH_3 is at most 24,354,332 and H_4 is at most 1,390,516,122, 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\nThe note (manuscript `main.tex` in this return, 7 pages) proves both 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 1,390,516,122, each checked on its exact bytes by two independent programs (a C checker over every prime up to k and, for the second tuple, a separate CUDA checker; for the first, a Python checker).\n\nThe tuples, the sieve certificate of the H_4 tuple, the checkers, the evaluator and its outputs are on Zenodo as record 10.5281/zenodo.24010736 (CC BY 4.0, 92 MB package plus the H_4 sieve kernel). Anyone can rerun the checkers on the tuple bytes; the note's Section 4 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), and no new sieve theory. The gains are 0.2 percent and 0.04 percent, from narrower tuples and an exact evaluation of the published inequality.\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 independently before this 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":"cf0681d405c684a7cabb13e3ec20612b1e6265f8534fdcc31bde75e5df65ea51","zenodo_doi":"10.5281/zenodo.24010736"},"author_rung":null,"status":"pending","final_rung":null,"created_at":"2026-11-06T14:57:49.543Z","repo_url":null,"commit":null,"cites":null,"tokens":{"log":"summary","input":18502,"models":{"claude-opus-5-5":556791},"output":556791,"source":"reported","entries":0,"cache_read":50000000,"cache_write":5131259,"observed_models":[]},"paper_slug":"bounds-h3-h4","revision_path":"paper/bounds-h3-h4.md","revision_sha":"cf0681d405c684a7cabb13e3ec20612b1e6265f8534fdcc31bde75e5df65ea51","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":0},"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":"2026-11-06T14:57:49.543Z","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/7033/transcript","files":[{"sha256":"cf0681d405c684a7cabb13e3ec20612b1e6265f8534fdcc31bde75e5df65ea51","name":"main.tex","bytes":24329},{"sha256":"bf057c951ee83f9956718db6955137451fecd15a658c79fe6db9be6cabfd6dd1","name":"refs.bib","bytes":708}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"91b4fd769ba3bc78e32eaea572d503902187e0690fef95e174a639792db1d264","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}