{"id":1392,"job_id":2769,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job 2769: triage of route 125 (one-class Jacobsthal exponent), outcome: known\n\nCaveat first: no bound on g(x#), G_2 or twin primes is claimed. This is a prior-art triage. The Granville transfer to x# at exponent exactly 2 was not checked.\n\n**Question.** Is a bounded next experiment on route 125 (return #1380) justified? Success criterion: find a step that prior work does not cover.\n\n**Finding (cited).** Both halves of the proposed next step are covered.\n1. M3's \"enumerate exact optima for n = 6..10\": Ziller & Morack (arXiv:1611.03310v2) publish every maximal sequence for n = 2..54 (p <= 251) as an ancillary file. I downloaded and parsed its header lines, e.g. n=16: run 52 in reduced form, 240 maximal sequences.\n2. M1's \"is the switching step tight\": Iwaniec's x^2 is the linear-sieve sifting limit (f(s)=0 for s<=2). Granville (arXiv:2010.01211) shows, assuming Siegel zeros, that the Jurkat-Richert linear-sieve bounds cannot be improved, and discusses Iwaniec's Jacobsthal bound there. Sharpening the sieve error term therefore meets a known conditional obstruction.\n3. The \"one log saving\" target and its value are in the literature: Integers 18 (2018) #A26 links sub-quadratic bounds on h(n) to an elementary proof of Dirichlet's theorem. No improvement on Iwaniec was found.\n\n**Verified locally (rung: verified for arithmetic only).** The route's figures gamma(64)=1.2217, 87.1x and the 1.609 shape exponent reproduce from the cited A048670(64)=1110, p=311. Corrections: Sum 1/p >= 1 holds from x=5, not x=17. The brief's inequality cap(R) >= pi(x)+R Sum 1/p is false for all R in [2,1200] at n=64; the derivation file's version is correct, so Lemma 3 stands.\n\n**Not covered.** A provable structural constraint drawn from the published n <= 54 maximal configurations. That would need its own proposal with a pattern and falsifier stated in advance. It is not recommended as automatic pursuit here.\n\nMethod: web and arXiv search, reading of sources, one deterministic arithmetic script (check.mjs; output SHA-256 c4a42df33518e1ab8b9cb5cac6d5e87211b5cd6c5382c8315381a4cc480ffc2e). No new computation of published values.\n\n44 returns wait for a verdict.\n\nTranscript: scrubbed of the credential, local absolute paths, env values, session/account identifiers and third-party PDF payloads; kept otherwise.\n\n## Sources\n- Ziller & Morack, arXiv:1611.03310v2, anc/remainders.txt (SHA-256 c615edcac626c152d17e7ab2756cce1cdaa5ed0e133f8329c64d1c3d9be00ade).\n- Granville, arXiv:2010.01211v1, section 1, pp. 1-5.\n- Integers 18 (2018) #A26, arXiv:1708.05415 (abstract).\n- Costello & Watts, arXiv:1208.5342 (abstract); arXiv:1209.3464 (withdrawn v2).\n- FGKMT, JAMS 31 (2018) 65-105 (via Granville and search summaries).\n- Return #1380 and its 0019-DERIVATION.md (sha256 dbc06b54...).","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-09-22T20:06:34.991Z","repo_url":null,"commit":null,"cites":{"files":["dbc06b54e6b9f82c81db01721ccd932559ff408b2f81640d884843b4b1bfe582"],"handles":[],"returns":[1380],"messages":[]},"tokens":{"log":"claude-code","input":106,"models":{"claude-opus-5-5":35461},"output":35461,"source":"claude-jsonl","entries":53,"cache_read":4406888,"cache_write":118714,"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":57},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-22T20:07:14.558Z","file_notes":null,"research":{"outcome":"known","route_id":125,"depends_on":[1380],"evidence_md":"Route 125's contribution and its next step are already covered by prior work.\n\n(1) Next step (i), M3: \"enumerate exact optima for n = 6..10\". Ziller & Morack (arXiv:1611.03310v2) publish ancillary file anc/remainders.txt with EVERY maximal sequence for n = 2..54 (p_n <= 251), with reverse-pair indices. Inspected: header lines give run length and n_seq, e.g. n=6: 10, 2; n=9: 19, 12; n=16: 52, 240; n=54: 428, 4 (odd-prime reduced form; h(n) = 2*omega(n)+2, e.g. n=6 gives 22). So the proposed enumeration already exists, to p = 251 rather than n = 10. Any structure study should start from these published lists, not from a new search.\n\n(2) Next step (ii), M1: \"is Iwaniec's switching step proven tight?\". Iwaniec's x^2 is the linear-sieve sifting limit: f(s) = 0 for s <= 2, with level ~ y and z = x, gives y ~ x^2. Granville, \"Sieving intervals and Siegel zeros\" (arXiv:2010.01211): assuming infinitely many Siegel zeros, the (Rosser-)Jurkat-Richert linear-sieve bounds cannot be improved. The paper discusses Iwaniec's Jacobsthal bound in the same setting and shows J(m) >> omega(m)(log omega(m))^B for some m when 1-beta < (log q)^-B. So a sharper sieve-only error term (M1) faces a known conditional obstruction. What is still open is a non-sieve input. Scope: Granville's statements are for general m and for sieve bounds. I did not check that they transfer to J(x#) with exponent exactly 2.\n\n(3) Target arithmetic: \"o(x^2) needs one log\" is stated in the literature. arXiv:1708.05415 (Integers 18, 2018, #A26) shows that plausible bounds on h(n), weaker than p_n^(2-eps), would give an elementary proof of Dirichlet's theorem. Search summaries (not the paper) say it names O(p_n^2/log p_n). The best upper bound is still Iwaniec j(P(x)) << x^2 (FGKMT JAMS 2018; Granville 2020; erdosproblems forum). I found no improvement. Costello-Watts (arXiv:1208.5342, abstract only) give computational upper bounds for finite ranges.\n\n(4) Lemmas 1-3 are standard (CRT covering form; one-class covering is trivially a two-class covering; counting capacity). Local check (check.mjs, deterministic): gamma(64)=1.2217, g/(p log p)=0.622, g/(p log^2 p)=0.108, p log^2 p exponent 1.609, x^2/g=87.1 all match. Corrections: Sum_{p<=x}1/p >= 1 already at x = 5, not 17. The brief's short form 'cap(R) >= pi(x) + R*Sum 1/p' is false (it fails for all 1199 R in [2,1200] at n = 64). The DERIVATION.md form (with -pi(x)) is correct, and Lemma 3's conclusion stands.\n\nUncovered, not claimed here: a provable structural constraint read from the published n <= 54 maximal configurations. That would need a separate proposal that states the pattern and its falsifier in advance.","prior_art_md":"Search 2026-09-22 (web search + arXiv export API; abstracts inspected, Granville PDF text-extracted and relevant passages read, Ziller-Morack ancillary file downloaded and parsed).\nQueries: \"Jacobsthal function upper bound improvement Iwaniec (omega log omega)^2 primorial\"; \"Jacobsthal function linear sieve limit beta=2 x^2 barrier\"; \"Ziller Morack Jacobsthal extremal sequences\".\n- Iwaniec, On the problem of Jacobsthal, Demonstratio Math. 11 (1978) 225-231: J(m) << (omega log omega)^2, i.e. j(x#) << x^2 (cited via Granville's reference list; not opened).\n- Granville, Sieving intervals and Siegel zeros, arXiv:2010.01211v1 (2020), section 1: under Siegel zeros the Jurkat-Richert linear-sieve bounds are not improvable; the Jacobsthal discussion is on p. 4; conditional lower bound J(m) >> omega (log omega)^B.\n- Ziller & Morack, Algorithmic concepts for the computation of Jacobsthal's function, arXiv:1611.03310v2: h(n) for p <= 251; exhaustive maximal sequences for n=2..54 in anc/remainders.txt (SHA-256 c615edca...0ade), also anc/permutations.txt, psi_min.txt, moduli.txt.\n- Hagedorn, Math. Comp. 78 (2009) 1073-1087: h(n) for n < 50.\n- Dirichlet's theorem and Jacobsthal's function, Integers 18 (2018) #A26, arXiv:1708.05415: plausible sub-p_n^2 bounds imply an elementary proof of Dirichlet.\n- Costello & Watts, arXiv:1208.5342 (computational upper bounds); arXiv:1209.3464 (v2 withdrawn claim of a stronger bound).\n- Paseman, arXiv:1311.5944: hopes for a subquadratic bound, none proved.\n- Ford-Green-Konyagin-Maynard-Tao, JAMS 31 (2018): best upper bound x^2 (Iwaniec); Maier-Pomerance conjecture x(log x)^(2+o(1)).\n- Ignored: Zenodo record 22865056 ('Atlas of maximal gaps'), not assessed.\nAccess gaps: Iwaniec 1978 and 1971 not opened. Friedlander-Iwaniec, Opera de Cribro, not checked for an explicit Jacobsthal/sifting-limit statement.\nExact remaining gap: j(x#) = o(x^2) is open. The sieve route is blocked at exponent 2 by the sifting limit (conditionally sharp by Granville). The M3 data exists (n <= 54). The only uncovered item is a structure theorem drawn from that data, which has no stated falsifiable pattern yet."},"research_route_id":125,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_da55f23c995cabb5136f4e91","run_id":"run_f446aa64a928a6f1088dfdeb","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/125 and return #1380. Return the ordinary report and transcript plus research: {route_id: 125, 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":"1380","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/125","transcript_url":"/projects/twin-primes/return/1392/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}