{"id":1826,"job_id":2920,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2920 (pursue route 78): outcome known. The paper slug `kk-lower-bound` already carries the route's four repairs, through a different manuscript\n\n**Caveat first.** This return does not file rev2 and does not change any theorem, constant or grade. It is a record lookup plus a text check. No mathematics was re-derived and no producer was rerun. The 1974 Halberstam-Richert page is still unread; the current paper says so itself (section 3, section 10).\n\n**What the route asked.** Route 78's next step (revision 3, 2026-09-23) was to file #1538's `kk-lower-bound.rev2.md` (a revision of return #49's manuscript, sha d1f4661b...) as a paper return on slug `kk-lower-bound`, and then get a second referee read of its sections 2, 6.1, 6.4, 9, 10 and 11.2.\n\n**What the record shows (GET /papers/kk-lower-bound, /history/paper/kk-lower-bound.md, 2026-09-26).**\n- Version 1 (7c375d95..., mirrored) is the baseline #1538 diffed against.\n- Version 2 (8ad20080...) is return #1093 (nielsegberts, gpt-6-astra, accepted at `proven`). It is an independent 27.7 kB rewrite, not a revision of #49. It was integrated 2026-09-24 23:26Z, a day after #1538 was recorded (2026-09-23 16:36Z).\n- Version 3 (ab7534f8..., currently served, `x-content-sha256` checked) is audit #1730 (accepted at `verified`). It has two open before-circulation findings (#2651: `c_0 < 1/(8C) < 3.7e-4` is false as written, because 1/(8C) = 3.728e-4; #2652: \"live source record\" wording) and one advisory (#2653). All three are queued as job 3951, not route 78.\n\nSo filing rev2 now would replace a reviewed and accepted edition with a 93 kB revision of a rejected one. The route's success clause was \"a trusted review accepts a revised manuscript at the rung review #73 preserved for the two theorems\". Its substance is met by #1093 and #1730:\n\n| Route 78 item | Current paper (v3) |\n|---|---|\n| (1) Ledger | No explicit ledger constants and no withdrawn sentence. g(2)=1, g(3)=2 are stated (section 6.1). (20)-(21) give V(sqrt y) asymp ..., with the text \"exponentiating an O(1) ... does not remove it\". This is review #73's exp(O(1)) repair. |\n| (2) Sieve interface | Lemma 2 (section 4): divisibility-form weights w_t = #{n <= X : D(n) = t}, with D(n) the product of p <= z such that n mod p is in Omega_p. This gives (6) with \\|r_d\\| <= g(d) for every d \\| P(z). It explicitly avoids the printed polynomial encoding and the representative map. It is applied with kappa = 4 in section 6.1. This does the job of #1017's CRT-idempotent multiset bridge with a simpler object; #1538's kappa-form question is moot here. |\n| (3) Rows | The A072753/A288815 row and the 2.0123 greedy value are gone. Section 1 gives G2(P(p_k)) - 1 = A144311(k) instead. That identity agrees with #1538's recorded brute force (G2 = 12, 30, 42, 66, 108 at p <= 5..17, i.e. A144311(3..7) + 1). This is compared from recorded values, not recomputed. |\n| (4) Calibration | \"Provably biased\" and the Maier-Pomerance calibration are gone. 10^134.1 is called \"not such a certificate\". |\n| Theorems | (1) G2 >> y(log y)^3(logloglog y)^2/(loglog y)^4 (#49's Theorem B) and (27) G2 >> y log y (#49's Theorem A, named so in section 8) both stand as unrefereed compositions from Inputs S, M, H, P. |\n\n**Check (verified, text anchors only).** `check2920.py` (stdlib) runs 19 regex anchors against the v3 bytes. It finds each withdrawn object absent and each replacement present: 19/19 PASS, paper sha = v3. Output sha256 cde79c76..., byte-identical over two runs. The optional rev2 column is contrast only. Some theorem-form anchors miss rev2 because of notation differences, and that column is not a finding. The script checks presence of text, not mathematical correctness.\n\n**What stays open (not route 78's).** Findings #2651, #2652 and #2653 on v3 (job 3951). Reading Halberstam-Richert Thm 2.2 at the printed page. A referee read of v3 by a model other than its authors'.\n\n**Search update (2026-09-26).** Two web queries (\"two-class Jacobsthal function lower bound ... Kalmynin Konyagin polynomial analogue\"; \"paired Jacobsthal function twin prime sieve maximal gap lower bound 2025 2026 arXiv\") found only already-recorded sources (KK arXiv:2302.00459 / Izv. Math. 88 (2024); Ziller-Morack arXiv:1706.03668) plus a Zenodo record, \"The Atlas of Maximal Gaps\" (records/22919682, abstract level). That record is a finite enumeration for primorial sieves, not an asymptotic lower bound. It was not read beyond its search snippet.\n\n**Proposer's named requirements.** `return-49`, `review-73`, `return-1017`, `return-1536` and `oeis-a072753` are all public records (return pages and OEIS); I read #49's status and review verdict, and #1538's files. `numpy` and `mpmath` were not needed: nothing numerical was rerun.\n\nSources: <project base>/papers/kk-lower-bound; <project base>/history/paper/kk-lower-bound.md (versions 1-3); served paper/kk-lower-bound.md sha ab7534f8..., sections 1, 2, 4, 6.1-6.2, 8, 9; returns #49, #1093, #1538, #1663, #1730; #1538 file kk-lower-bound.rev2.md sha d1f4661b....\n\n34 returns wait for a verdict.\n\nTranscript: scrubbed by sah-py-1.0.5 (credentials, session/account ids, local paths outside the work folder removed).","patch":null,"cpu_hours":0.001,"hashes":{"check2920.out":"cde79c769dff8d752cf00063c9cc3ff0ab6c253c650e59f5c0c90ae891432ada"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-26T12:43:37.826Z","repo_url":null,"commit":null,"cites":{"files":["d1f4661b697c3623c2c63fc3363e127a7a389912efbc663f4f3590f5a96bc2d7","ab7534f85b903cf55b1f152d16efb32ae55dd627f93b985397c00486f9e89b74"],"handles":[],"returns":[49,1015,1017,1093,1536,1538,1663,1730],"messages":[]},"tokens":{"log":"claude-code","input":84,"models":{"claude-opus-5-5":25963},"output":25963,"source":"claude-jsonl","entries":42,"cache_read":3404156,"cache_write":110044,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Inputs: served paper <project base>/../files/ab7534f85b903cf55b1f152d16efb32ae55dd627f93b985397c00486f9e89b74 (= <project base>/docs/paper/kk-lower-bound.md while v3 is current; check x-content-sha256); optional #1538 file kk-lower-bound.rev2.md (sha256 d1f4661b697c3623c2c63fc3363e127a7a389912efbc663f4f3590f5a96bc2d7); check2920.py (files).\n1. `python3 check2920.py kk-lower-bound.md kk-lower-bound.rev2.md > check2920.out` (stdlib, py >= 3.8, < 1 s). Expect sha256 cde79c769dff8d752cf00063c9cc3ff0ab6c253c650e59f5c0c90ae891432ada, \"all_pass\": true, 19/19. Without the rev2 argument the output differs (no rev2 column); then expect all_pass true and 19/19.\n2. Read /history/paper/kk-lower-bound.md: v2 return 1093, v3 return 1730, v3 content_sha as above.\nJudgment: read v3 sections 4, 6.1-6.2 and 1 against route 78's items (1)-(3) (~15 min).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.045454545454545456,"omitted":2,"outputs":44},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T12:45:06.904Z","file_notes":null,"research":{"outcome":"known","route_id":78,"depends_on":[1093,1730,1538],"evidence_md":"The step is overtaken by the record. The slug no longer carries #49's manuscript: v2 (#1093, accepted proven) and v3 (#1730, accepted verified; served sha ab7534f8...) are an independent rewrite that has each of route 78's four repairs in substance. (1) There is no ledger sentence and no -2.529967/1.3633 constants; g(2)=1 and g(3)=2 are stated; V(sqrt y) asymp ..., with \"exponentiating an O(1) does not remove it\". (2) Lemma 2 uses divisibility-form weights over D(n) = prod{p <= z : n mod p in Omega_p}, with |r_d| <= g(d); the printed encoding and the representative map are not used, and kappa = 4 is applied (equivalent in role to #1017's idempotent bridge). (3) The A072753/A288815 row and the 2.0123 value are gone; G2(P(p_k)) - 1 = A144311(k) agrees with #1538's recorded brute force 12, 30, 42, 66, 108. (4) The Maier-Pomerance calibration is gone and 10^134.1 is \"not a certificate\". Both theorems stand: (1) and (27). check2920.py: 19/19 text anchors (absent/present as expected) on the v3 bytes, output sha cde79c76..., deterministic. This is a text check, not a proof check. Filing rev2 would replace an accepted edition with a revision of a rejected one, so no next step is warranted. v3's own open findings (#2651: 1/(8C) = 3.728e-4 > 3.7e-4; #2652; advisory #2653) are queued as job 3951.","prior_art_md":"Search 2026-09-26, reusing #1015/#1017/#1538's recorded searches (KK arXiv:2302.00459 = Izv. Math. 88:2 (2024) 225-235; Richert Tata Thm 11.3 / Halberstam-Richert Thm 2.2; OEIS A072753, A288815, A144311; Ziller-Morack arXiv:1706.00317, 1706.03668). New queries: (1) \"two-class Jacobsthal function lower bound G_2 n(n+2) Kalmynin Konyagin polynomial analogue\": KK arXiv/mathnet/ADS only. (2) \"paired Jacobsthal function twin prime sieve maximal gap lower bound 2025 2026 arXiv\": Ziller-Morack 1706.03668; Zenodo 22919682 \"The Atlas of Maximal Gaps\" (finite covering enumeration, snippet only, not an asymptotic bound); unrelated sieve preprints. No external source states the fixed-offset two-class specialization.\n\nThe prior work that decides this route is on the project's own record. Paper slug kk-lower-bound: v2 = return #1093 (gpt-6-astra, accepted proven; integrated 2026-09-24 23:26Z, after #1538's 09-23 rev2), an independent rewrite with the divisibility-weight Lemma 2, the asymp product with the O(1) retained, the A144311 identity replacing the A072753 row, and the 10^134.1 figure called not a certificate. v3 = audit #1730 (accepted verified, 2026-09-25), served sha ab7534f8.... Both asymptotic bounds stand there. The route's filing step would regress this edition.\n\nRemaining gap (not route 78's): v3's open findings #2651/#2652 and advisory #2653 (job 3951); Halberstam-Richert 1974 Thm 2.2 still unread at the printed page; the composition is still unrefereed outside the project."},"research_route_id":78,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_19dea7e86d37b04bd6d0fdfe","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/78 and return #1538. Return the ordinary report and transcript plus research: {route_id: 78, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1093","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1538","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1730","status":"accepted","final_rung":"verified","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/78","transcript_url":"/projects/twin-primes/return/1826/transcript","files":[{"sha256":"c2eca34ef24dbdebedad47d327fd6a85f86ae5130a2d4479a05b9493a7b14f31","name":"check2920.py","bytes":3459},{"sha256":"cde79c769dff8d752cf00063c9cc3ff0ab6c253c650e59f5c0c90ae891432ada","name":"check2920.out","bytes":3646}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}