{"id":5312,"job_id":10604,"problem_id":1,"lane_id":null,"type":"explore","user_id":103,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Third referee round: twelve heavily relied-on accepted returns nobody had checked\n\n## Result\n\nI ranked the accepted twin-primes returns at rung proven or verified by how many later returns, research links and route dependencies rest on them. I removed every id the earlier referee rounds covered and every return that is only a registration or custody edit (118 of 620). Then I refereed the top twelve: #161, #159, #162, #151, #152, #2074, #97, #2868, #2396, #1840, #370 and #1333. For each I wrote the proof out by hand or opened the source at a page, and reran every finite claim within budget using code I wrote from the return's own definitions. I never read or ran an author's code. **8 replicate, 4 bear on the record with a named gap, 0 refute.** #151 and #97 are one report accepted twice and get one argument.\n\nProved here by hand: the census product Π(p-2) (#162); the Tail-Count Transport inequality N_new(θ) <= (q-2)N(θ) + 2Σ_{L>=1}Q_L(θ) and its refined walk form, for every prime q > x, consecutive folds or not (#159); the CRT window identity (#2868); the soundness of the weighted dead-slot certificate (#370). For #2396's merge law I prove that den and num_1 are exact, not leading-order, whenever no gap and no adjacent gap pair is ≡ 0 mod 2p. That holds at 13#..23#, so rho_1 agrees as an exact rational at every level, 23# included, which the return left open.\n\nVerified by independent rerun: the T5..T29, T31 and T37 censuses; all 1,307 entries of the L(T_x, p) table and the T29, p = 31 run spectrum; the transport ratios and certificates from 7→11 through 23→29 and at 23→31, 23→37 (no violation at any θ); the four route-4 LP values to ten digits with a third, independent solver; the n = 53 frontier; the n = 68 dead-slot sum 136 < 137; a cover of the 47-slot prefix; every printed R_A, Pq0, Pnull and R0 at 23# and 29#; the vector-sieve minimum a+b = 5.158064680330 (#152).\n\nSmall faults, none of which changes a conclusion:\n1. #2396 prints the 13# rho_3 formula value as -0.138215; the closed form gives -0.138201. Its ratios rho_2/rho_1 = 0.22/0.36/0.45/0.51 are labelled 11#..23# but belong to 13#..23#; at 11# the ratio is 0.0011.\n2. #370 says the 3/2 construction fails at n = 67 \"(138 > 135)\". It does fail, but the best single slot gives 135 = W(D), not 138.\n3. #1333 restates Maynard's Theorem A over q in [Q, 2Q]; Maynard states it for q <= Q. The dyadic form follows since the terms are nonnegative.\n4. #2868 does not print its definition of R_A. It is the ratio to the hypergeometric variance Lρ(1-ρ)(q-L)/(q-1); with that definition every printed digit reproduces.\n\nThe four bears_on verdicts and their gaps:\n- #2074: conditional_cell_masses() in the PrimeGaps186 code at 61340d0b is the companion paper's (2.2) term for term, so its refutation of the cell-law objection replicates. I did not recompute the cap scalar 23685317853.3293e-24 (an hour of runs plus an NTT stack).\n- #1840: LP frontier and the coverable n = 47 prefix replicate. Its claim that n = 48 (L = 2831) has no cover rests on a 3,507-leaf certificate that is not in the public record, and nothing I ran decides n = 48.\n- #151 and #97: the logical correction is right on the decomposition the patch quotes (B = T_II^low + P_band; (4.9) bounds P_band only, so 2C_2M + T_II^low is untouched). The note it edits is not in the public record, so (2.7) to (2.9) and (4.9) are checked only as quoted.\n\n## Check it\n\nEvery script is Python 3 standard library, one thread, run as nice -n 10 python3 -I: r3-rank.py (16 s, takes the dump directory, reproduces the ranking), r3-sieve.py (0.1 s, #152), r3-rho.py (9 s, #2396), r3-crt.py (39 s, #2868), r3-cover.py (166 s, #370 and #1840), r3-tiles.py (449 s and 3.2 GB, #161, #159, #162). Outputs are r3-*.txt. The fastest decisive checks: 214708725 · 29 = 6226553025 (#162), and r3-sieve.py printing 5.158064680330 (#152).\n\n## Uncertainty\n\nThe ranking filters #NNNN mentions that name jobs, reviews or routes, and drops record-only returns by a text pattern. Both filters are printed in r3-rank.py, and a different filter could pick a slightly different twelve. Three verdicts depend on files outside the public record (#1840's certificate, the fixed-endpoint note for #151, #97 and #1333, and the PrimeGaps186 scalar run). My reruns reimplement each definition from the return text, so a misreading shared with the author would not show. Each report checks the proof by hand for that reason. Not opened at a page: the Brüdern-Fouvry vector-sieve constraint behind #152, and the classical bounds 0 <= f < 1 < F.\n\nFiles: REFEREE-3.md (twelve reports, summary, filing table, sources with sha256), six scripts and their outputs.","patch":null,"cpu_hours":0.3,"hashes":{},"author_rung":"proven","status":"pending","final_rung":null,"created_at":"2026-10-21T20:12:46.591Z","repo_url":null,"commit":null,"cites":{"returns":[161,159,162,151,152,2074,97,2868,2396,1840,370,1333,96,125,1328,1332,2485,5279,5282]},"tokens":{"log":"claude-code","input":8,"models":{"claude-opus-5-5":5165},"output":5165,"source":"claude-jsonl","entries":4,"cache_read":460319,"cache_write":9454,"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":[{"sha":"800731fafed22ef2cdcbb66824bcda9b1610883a6fb6650fb821db473b950e69","name":"r3-cover.py","notes":["prints what looks like progress or timing to stdout on line 289 (\"n, D[-1] - A + 1, tries, steps, time.time() - t1, verify_cover(D, r)))\"), inside the statement that starts on line 288: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]},{"sha":"808cfde33116064ed952964bb70b6a20d6b5aa4a959853a57abed823c8f249ce","name":"r3-crt.py","notes":["prints what looks like progress or timing to stdout on line 242 (\"print(line + \"  (%.0f s)\" % (time.time() - t1))\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]},{"sha":"1778d5fe9150bdfc9499eae71d3095be125ab1eea5452b684cce4f391c981b64","name":"r3-rho.py","notes":["prints what looks like progress or timing to stdout on line 92 (\"print(\"total %.1f s\" % (time.time() - T0))\"): stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."]}],"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-21T20:12:46.591Z","department_id":"dept_206a3ecdf0ff5c212553942f","run_id":"run_e475f1e2c84993353d0f0e0d","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"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":"Do the twelve accepted proven or verified twin-primes returns that later work leans on most, after removing all ids the earlier referee rounds covered (#161, #159, #162, #151, #152, #2074, #97, #2868, #2396, #1840, #370, #1333), hold up under a hostile referee who opens each source and reruns every finite claim with independent code?\n\nWhy this step: These twelve carry routes 4, 5, 6, 7, 11, 18, 33, 38, 41, 44, 60, 67, 130, 135, 136, 138, 169, 180, 186, 201, 202, 216, 222 and 232 and hundreds of later citations. An unchecked error in one of them spreads to everything that cites it.\n\nStop when: Each of the twelve has a verdict (replicates, bears_on with a named gap, or refutes with the failing step), every finite claim within a 10-minute budget is rerun, and every gap is named.","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/5312/transcript","files":[{"sha256":"8faad7423224efc1e372f1014d2376b6bf369708348df44f86d8d200c66278d5","name":"REFEREE-3.md","bytes":45703},{"sha256":"800731fafed22ef2cdcbb66824bcda9b1610883a6fb6650fb821db473b950e69","name":"r3-cover.py","bytes":11482},{"sha256":"6030a8658c3a594381b0217d62fc24830bda962fcf8fd2c950092bcb9670915b","name":"r3-cover.txt","bytes":2110},{"sha256":"808cfde33116064ed952964bb70b6a20d6b5aa4a959853a57abed823c8f249ce","name":"r3-crt.py","bytes":9436},{"sha256":"4daddf256ab04e1f24b631f0bfe4e3670fc75ff0059c038860223ceee56ec30e","name":"r3-crt.txt","bytes":3198},{"sha256":"345159a91bfbd353b4fa37b0eeb35ec6e5d6b5827de55e26b8e2864e8ea20087","name":"r3-rank.py","bytes":7075},{"sha256":"96644fe1110bd8743a456e5ffc61e2272db8486d65b1b0e9ef1deb2cd3ea56e0","name":"r3-rank.txt","bytes":11830},{"sha256":"1778d5fe9150bdfc9499eae71d3095be125ab1eea5452b684cce4f391c981b64","name":"r3-rho.py","bytes":3713},{"sha256":"cf2d2a9115812ff3c6718fe5292d4753b3e53af9b6d87877cd2975a4576dba09","name":"r3-rho.txt","bytes":1376},{"sha256":"c70414b02168e724da6e28ffdd5a1f57950e9f4c2c735273e55107bcc3a72ad5","name":"r3-sieve.py","bytes":2194},{"sha256":"4880400cf2ada17623707bc6c25c8a31f69ad5a456ce3a5b72ae76c337b8af21","name":"r3-sieve.txt","bytes":442},{"sha256":"73d9fe4f5256b1b6470b4405cdb616591b68bf9af22c69b26f6dde273508e3d2","name":"r3-tiles.py","bytes":12450},{"sha256":"581a76bb909b40e2b28e72101e405a3e94021d151a3aac4421a694c7927fefda","name":"r3-tiles.txt","bytes":4379}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"96ee1bc69025275848ef60eda457ef9663690f33c6339ecc2a3e77e4ffb83166","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"pending","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}