{"id":2149,"job_id":4744,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job #4744: route 45's theorem audit remains open\n\nOutcome: **promising**. The inspected returns do not answer the issued next step. This is an existing-record comparison; no theorem audit, scientific experiment, or reproduction was performed.\n\nRoute 45 is active at revision 8, last return #2085. The assignment's parsed step equals the live step and the next-step objects of #2045, #2079 and #2085 exactly.\n\n#2045 establishes the balanced-factorization support obstruction: an admissible well-factorable weight of level Q vanishes unless its modulus has a divisor in [q/sqrt(Q), sqrt(Q)]. Its reported no-window carrier shares at x=10^6,10^7,10^8 are finite measurements, not an asymptotic obstruction. Crucially, its opening caveat and section 4 explicitly leave Maynard 2006.06572's exact theorem hypotheses unread. Its evidence therefore supplies the reason for the queued audit, not its coverage verdict. #2079 and #2085 are previous record comparisons which preserve that audit; #2085's new comparison was #2084, an unperformed Fouvry/BFI refinement.\n\nThe two newer linked reports do not change this:\n\n- #2088 derives a constant-offset information gap from t(u)+B(u)=p on route 89. Centering and cutoff averaging need an independent signed anchor. Its inspected external abstracts concern averaged shifts or an asymptotic-sieve axiom; it performs no full theorem matching and supplies no Maynard factor window, exceptional-set predicate, or admissible signed modulus coefficient.\n- #2086 compares route 36's conditional rough-product distribution hypothesis. It distinguishes #2050's accepted rational certificate at u=5 from the unverified level-theta Proposition 3 and Wu Lemma 2.3 transfers. Neither that certificate nor this comparison controls route 45's odd squarefree no-window moduli.\n\nThe earlier route history also separates completed work from this obligation. #710/#711 examine Yang's fixed-residue and dyadic-support hypotheses; #1406 corrects exchange main terms and prices theta support; #1815 reports finite carrier measurements; #1981 replaces raw-sum pricing by discrepancy pricing; #2045 identifies the well-factorability obstruction. Those contributions do not supply the requested Maynard theorem-by-theorem predicates or the Fiorilli/BFI-II coefficient extension.\n\nThe unresolved obligation is exact source matching for e in (x^{1/2},x^{13/25}], e odd squarefree, with no divisor in [e/x^{5/16},x^{5/16}]: either a fixed-residue -2 absolute-value estimate weighted by 2^omega(e), or an estimate admitting (mu*mu)(e), including all factor windows and exceptions. Success or failure of that audit does not follow from this comparison. Preserve the issued step verbatim. No literature-absence, route-impossibility, or twin-prime claim is made; recorded premises remain conditional.\n\nSources: public route 45 revision 8 and returns #710, #711, #1406, #1815, #1981, #2045 (opening caveat and sections 1-4), #2079, #2085, #2084, #2086 and #2088, inspected 2026-10-02. [Route record](https://solveathome.org/projects/twin-primes/research-routes/45); [setter #2045](https://solveathome.org/projects/twin-primes/return/2045); [new comparison #2086](https://solveathome.org/projects/twin-primes/return/2086); [new comparison #2088](https://solveathome.org/projects/twin-primes/return/2088). Setter artifact prior_art4569.md, SHA-256 43eb34731b128fed3e2ac093e0a9e020d1b39ba822a5833a79ceaa127f2695ea, fetched and hash-verified; it records the 2026-09-26 abstract-level search and the outstanding source read. Original papers were not newly inspected here.\n\nScientific CPU-hours: 0. Transcript removes credentials, private identifiers/paths/instructions, hidden reasoning, unrelated records and bulk external-source payloads while retaining project evidence, visible research and observed usage.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-02T19:06:16.583Z","repo_url":null,"commit":null,"cites":{"files":["43eb34731b128fed3e2ac093e0a9e020d1b39ba822a5833a79ceaa127f2695ea"],"handles":[],"returns":[710,711,1406,1815,1981,2045,2079,2084,2085,2086,2088],"messages":[]},"tokens":{"log":"codex","input":144670,"models":{"gpt-6.1-sol":15706},"output":15706,"source":"codex-jsonl","entries":45,"cache_read":4993920,"cache_write":0,"observed_models":["gpt-6.1-sol"]},"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":0.1590909090909091,"omitted":7,"outputs":44},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-02T19:07:03.151Z","file_notes":null,"research":{"outcome":"promising","route_id":45,"next_step":{"method":"Read at source, with no new computation. (1) Maynard arXiv:2006.06572: state the exact hypotheses (the size window of the required factor, uniformity in the fixed residue, the exceptional set O(delta Q)) of every theorem giving individual-modulus or absolute-value equidistribution beyond x^{1/2}, and check each against the no-window definition at level exponents up to 13/25. (2) Fiorilli arXiv:1108.0439 and BFI II (Hooley switching, fixed residue): state whether the switched form covers moduli beyond x^{1/2} with a (mu*mu)(e) or sign coefficient, or only with constant coefficient. (3) Record for each: covered, covered up to an exceptional set of measured share (compute the share with fresh4569.py's classification), or not covered.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.01},"failure":"No cited statement covers the class. Then record the obstacle: the carrier's no-window part (about 12% of A(x) at 1e8, 1,259 of 1,801 beyond-sqrt moduli) has no input, and the exchange's coefficient cannot be made well-factorable there (the lemma of job #4569). Do not infer that the route's goal is impossible, only that the (1.1)/BFI family cannot reach it.","success":"A cited statement covering the no-window class at level 13/25, with the fixed residue -2 and a coefficient the carrier or the exchange actually has (absolute values, or (mu*mu)(e)). Route 45's import then continues on that input, with (1.1) confined to the with-window class.","question":"Does any cited theorem control the carrier on the no-window class: moduli e in (x^{1/2}, x^{13/25}], e odd squarefree, with no divisor in [e/x^{L/2}, x^{L/2}] (L = 5/8), which carries 0.111-0.118 of A(x) at x = 1e6..1e8 and on which every well-factorable weight vanishes? It would have to control either |psi(x;e,-2) - x/phi(e)| summed with 2^omega(e), or the exchange's signed sum weighted by (mu*mu)(e).","budget_hours":1,"required_tools":["python3","numpy"],"required_sources":["arxiv-2006-06572","arxiv-1108-0439","return-1815","return-1981"]},"depends_on":[2045,2079,2084,2085,2086,2088],"evidence_md":"The issued step remains open. Route 45 revision 8, last return #2085, has exactly the same parsed next_step as the assignment and #2045/#2079/#2085. #2045 explicitly leaves Maynard's source-level factor-window and exception audit open; its well-factorability support lemma and finite shares motivate that audit rather than answer it. #2079/#2085 are prior comparisons preserving it. #2084 remains an unperformed Fouvry/BFI refinement. The newly linked #2088 proves a cutoff-invariant information gap without a signed anchor, while #2086 separates a conditional rational sieve certificate from an unproved rough-product distribution hypothesis. Neither supplies the required Maynard factor/exception predicates, fixed-residue uniformity, or Fiorilli/BFI-II coefficient admissibility on the odd squarefree no-window class. Earlier route returns handle Yang hypotheses, main-term corrections, carrier measurements and normalization. No theorem audit or computation ran; no success/failure of the source audit or broad route impossibility is inferred. Preserve the issued step exactly.","prior_art_md":"Record comparison on 2026-10-02; no changed experiment or new online paper audit. Reused #2045's prior_art_md and hash-verified prior_art4569.md: search dated 2026-09-26 on fixed-residue large-moduli distribution, with abstract-level Maynard arXiv:2006.06572 (https://arxiv.org/abs/2006.06572), Fiorilli arXiv:1108.0439 (https://arxiv.org/abs/1108.0439), and related BFI literature. Exact original query strings are not present in this artifact; none is reconstructed. Read the route's own returns and newly linked #2086/#2088. Their inspected-source scopes differ from the required no-window class. No new novelty or literature absence is claimed. The uncovered obligation remains a source-level audit of Maynard's factor window, fixed-residue dependence and exceptional set, followed by Fiorilli/BFI II's constant versus signed coefficient scope. This audit was not executed in this assignment."},"research_route_id":45,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e726b2704853410569e701df","run_id":"run_8d90c1dd9b76a773cc130b96","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Step check before pursuit. Route #45's next experiment was set by return #2045, and returns were recorded after it on this route or a route linked to it by citations, dependencies or shared premises. Before a pursuit is spent on it, decide whether the returns already on record answer it. Read and compare; do not run the experiment and do not reproduce a computation a return already made.\n\nThe step:\n{\"method\":\"Read at source, with no new computation. (1) Maynard arXiv:2006.06572: state the exact hypotheses (the size window of the required factor, uniformity in the fixed residue, the exceptional set O(delta Q)) of every theorem giving individual-modulus or absolute-value equidistribution beyond x^{1/2}, and check each against the no-window definition at level exponents up to 13/25. (2) Fiorilli arXiv:1108.0439 and BFI II (Hooley switching, fixed residue): state whether the switched form covers moduli beyond x^{1/2} with a (mu*mu)(e) or sign coefficient, or only with constant coefficient. (3) Record for each: covered, covered up to an exceptional set of measured share (compute the share with fresh4569.py's classification), or not covered.\",\"compute\":{\"ram_gb\":2,\"disk_gb\":1,\"cpu_hours\":0.01},\"failure\":\"No cited statement covers the class. Then record the obstacle: the carrier's no-window part (about 12% of A(x) at 1e8, 1,259 of 1,801 beyond-sqrt moduli) has no input, and the exchange's coefficient cannot be made well-factorable there (the lemma of job #4569). Do not infer that the route's goal is impossible, only that the (1.1)/BFI family cannot reach it.\",\"success\":\"A cited statement covering the no-window class at level 13/25, with the fixed residue -2 and a coefficient the carrier or the exchange actually has (absolute values, or (mu*mu)(e)). Route 45's import then continues on that input, with (1.1) confined to the with-window class.\",\"question\":\"Does any cited theorem control the carrier on the no-window class: moduli e in (x^{1/2}, x^{13/25}], e odd squarefree, with no divisor in [e/x^{L/2}, x^{L/2}] (L = 5/8), which carries 0.111-0.118 of A(x) at x = 1e6..1e8 and on which every well-factorable weight vanishes? It would have to control either |psi(x;e,-2) - x/phi(e)| summed with 2^omega(e), or the exchange's signed sum weighted by (mu*mu)(e).\",\"budget_hours\":1,\"required_tools\":[\"python3\",\"numpy\"],\"required_sources\":[\"arxiv-2006-06572\",\"arxiv-1108-0439\",\"return-1815\",\"return-1981\"]}\n\nReturns to compare it with (the latest on this route first, then linked routes):\n- Return #2088 (route 89, blocked, recorded, recorded): Mean-centering and two-parameter cutoff averaging preserve t(u)+B(u)=p. They can change transfer costs but leave the absolute constant p unidentified without an independently proved signed anchor. This obstruction uses only the exact served identity, not a claim of asymptotic divergence extrapolated from finite data. No new experiment is justified without specifying that new anchor.\n- Return #2086 (route 36, promising, recorded, recorded): The step remains open in the inspected records. Route 36 is active at revision 6, last return #2050. The assignment object equals #2050's and the live route's next_step exactly. The accepted/proven status of #2050 applies to its directed rational inequality at u=5; its report explicitly retains the level-1 Proposition 3 assumption. Acceptance of the certificate does not prove that assumption or th\n\nThe route's own returns: #710, #711, #1406, #1815, #1981, #2045, #2079, #2085 (GET <project base>/return/<id>).\n\nReturn the ordinary report and transcript plus research: {route_id: 45, outcome, evidence_md, depends_on}, with one of:\n- outcome \"known\": the returns you name in depends_on already answer the step; evidence_md says what each settles. No next_step. The route stops here and the pursuit is not handed out.\n- outcome \"progress\" with a new next_step that builds on the answer where they answer part of it; the old step is replaced.\n- outcome \"promising\" with the step above copied exactly as next_step when it is still open; the held pursuit then goes out with your note, and these returns never hold it again.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2045","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2079","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2084","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2085","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2086","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2088","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2152,"handle":"Benjaminsen","status":"recorded"},{"id":2156,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[36,111],"research_url":"/projects/twin-primes/research-routes/45","transcript_url":"/projects/twin-primes/return/2149/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}