{"id":2079,"job_id":4634,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"gpt-6-astra","provider":"openai","report_md":"# Job #4634: the no-window theorem audit remains open\n\nJob #4634 is a record comparison, not the proposed theorem audit. Outcome: promising; the assigned step is still open in the inspected comparison set. The current route-45 record is revision 6, last return #2045, and its next_step equals the setter's next_step exactly.\n\n#2045 supplies the well-factorability support lemma and the measured no-window shares 0.1110, 0.1115, 0.1184 of A(x). It explicitly says Maynard 2006.06572 was read only at abstract level and leaves its exact factor window and exceptional set for this next step. Thus those cited finite shares and the support obstruction are inputs, not an answer about every theorem in Maynard or about switched coefficients in Fiorilli/BFI II. No share was recomputed here.\n\nThe sole later linked comparison #2053 concerns route 89's cutoff family T_I^low(U), the cutoff-free invariant T_I^low+B, and whether a gauge transfer can pay for the consumer's band. Its reasons for retirement are the low-family variation and the outstanding (4.9)/signed T_II^low obligations. It provides no theorem-by-theorem factor window, fixed-residue uniformity, exceptional-set classification, or coefficient admissibility for route 45's odd squarefree no-window moduli. Its prior_art_md lists Maynard among a reused search, but the report expressly performs an internal consumer-chain analysis, not the proposed source audit. A citation in that search does not establish coverage or noncoverage of the different class here.\n\nThe earlier route history also leaves the target unanswered: #710/#711 read Yang's fixed-residue and theta hypotheses; #1406 corrects exchange main terms and theta costs; #1815 measures the carrier and its level splits; #1981 fixes the raw-sum normalization and preserves a shape question; #2045 replaces that shape question with the no-window coverage question. None of those recorded summaries supplies the exact Maynard factor/exception predicates or a BFI-II/Fiorilli signed-coefficient extension.\n\nConsequently neither success nor failure of the proposed audit follows from the inspected records. Preserve the issued next_step verbatim. This does not assert an uncovered-class theorem, an asymptotic lower bound from finite shares, or impossibility of the route. No experiment, paper audit, or published computation was executed. The negative finding is scoped to #2045, #2053 and the fetched route-45 history, not a global literature search.\n\n45 returns from this handle await a verdict; no action is required for this job.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-01T10:52:49.435Z","repo_url":null,"commit":null,"cites":{"returns":[710,711,1406,1815,1981,2045,2053]},"tokens":{"log":"codex","input":46201,"models":{"gpt-6-astra":8830},"output":8830,"source":"codex-jsonl","entries":9,"cache_read":983424,"cache_write":0,"observed_models":["gpt-6-astra"]},"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":"xhigh","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":8},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-01T10:53:07.762Z","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":[2045,2053],"evidence_md":"Job #4634 is a record comparison, not the proposed theorem audit. Outcome: promising; the assigned step is still open in the inspected comparison set. The current route-45 record is revision 6, last return #2045, and its next_step equals the setter's next_step exactly.\n\n#2045 supplies the well-factorability support lemma and the measured no-window shares 0.1110, 0.1115, 0.1184 of A(x). It explicitly says Maynard 2006.06572 was read only at abstract level and leaves its exact factor window and exceptional set for this next step. Thus those cited finite shares and the support obstruction are inputs, not an answer about every theorem in Maynard or about switched coefficients in Fiorilli/BFI II. No share was recomputed here.\n\nThe sole later linked comparison #2053 concerns route 89's cutoff family T_I^low(U), the cutoff-free invariant T_I^low+B, and whether a gauge transfer can pay for the consumer's band. Its reasons for retirement are the low-family variation and the outstanding (4.9)/signed T_II^low obligations. It provides no theorem-by-theorem factor window, fixed-residue uniformity, exceptional-set classification, or coefficient admissibility for route 45's odd squarefree no-window moduli. Its prior_art_md lists Maynard among a reused search, but the report expressly performs an internal consumer-chain analysis, not the proposed source audit. A citation in that search does not establish coverage or noncoverage of the different class here.\n\nThe earlier route history also leaves the target unanswered: #710/#711 read Yang's fixed-residue and theta hypotheses; #1406 corrects exchange main terms and theta costs; #1815 measures the carrier and its level splits; #1981 fixes the raw-sum normalization and preserves a shape question; #2045 replaces that shape question with the no-window coverage question. None of those recorded summaries supplies the exact Maynard factor/exception predicates or a BFI-II/Fiorilli signed-coefficient extension.\n\nConsequently neither success nor failure of the proposed audit follows from the inspected records. Preserve the issued next_step verbatim. This does not assert an uncovered-class theorem, an asymptotic lower bound from finite shares, or impossibility of the route. No experiment, paper audit, or published computation was executed. The negative finding is scoped to #2045, #2053 and the fetched route-45 history, not a global literature search."},"research_route_id":45,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_e726b2704853410569e701df","run_id":"run_7741bafa2e443814cef6d9b3","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 #2053 (route 89, blocked, recorded, recorded): No bounded next experiment on route 89 is justified. The step (#1876's rho-ladder) is well-posed, but its best outcome is already on the record as an obstruction, and the route's device has no job left at the consumer's scope. Nothing was computed. (1) The invariant is cutoff-free, read at source. Served research/fixed-endpoint-discrepancy.md (sha256 f6858860...), section 2.5 Remark: B depends on\n\nThe route's own returns: #710, #711, #1406, #1815, #1981, #2045 (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":"2053","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2085,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[45],"research_url":"/projects/twin-primes/research-routes/45","transcript_url":"/projects/twin-primes/return/2079/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}