{"id":2089,"job_id":4651,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Job 4651: logarithmic slack and a factorability support test\n\nThe rescue remains blocked for the full d-edge. An endpoint-only logarithmic repair is plausible at the level of modulus sizes, but it does not cover a fixed-power beyond-level band. No analytic estimate, density matching or twin-prime margin is proved here.\n\nReused evidence: #2062's exact sweep shows that changing the fixed exponents w,y moves the endpoint to nu=1/2-y without removing the band. #1814 section 1 identifies the second piece as mu_(Y,E0] * Lambda_<=Z * 1, with modulus e*l. I checked these against the served reachability note sections 2.3 and 3.1 and prime-detection-spec section 4. Published computations were not rerun.\n\nChanged ingredient 1: put Z=x^y/(log x)^K. With the usual level-size comparison e*l<=x^(1/2)/(log x)^B, this would allow e up to x^(1/2-y)(log x)^(K-B). Its exponent is 1/2-y+(K-B)log log x/log x. Thus logarithmic slack can adjust an endpoint layer; for every fixed epsilon>0 it cannot cover e=x^(1/2-y+epsilon). Moreover E0 is about x/Z, so the upper e exponent remains 1-y+o(1), leaving a band of limiting width 1/2. Full use of this cutoff would still require rechecking the decomposition, density subtraction and logarithmic weights; a modulus comparison is not a completed BV application.\n\nChanged ingredient 2: can the convolution already supply Maynard's well-factorable weights? His Definition 1 requires a bounded convolution for EVERY factorization of the level Q. At Q1=Q2=sqrt(Q), a well-factorable coefficient vanishes at every integer q with a prime divisor P>sqrt(Q): that divisor must occur in one convolution factor, but neither factor can be that large. Definition 2 has the corresponding obstruction at equal three-way factors.\n\nFor the simplified modulus coefficient lambda(q)=sum_(e*l=q,Y<e<=E0,l<=Z)mu(e)Lambda(l), choose a fixed small prime l0<=Z and a prime P>max(Y,Z,sqrt(Q)) with P<=E0 and P*l0<=Q. Then lambda(P*l0)=-log l0, because the only nonzero permitted prime-power l is l0; l=P is excluded. It does not have the support property required at level Q, even after scaling its size. The witness is a conditional algebraic test for any range containing such a P, not a numerical claim about every dyadic interval. It rules out applying the theorem directly to that whole coefficient as a well-factorable sequence. It does not exclude partitioning a selected factorable subset, a different decomposition, or cancellation in the leftover.\n\nThe full d-edge also has bilinear coefficients in the prime-producing variables. Maynard Theorem 1.1 is a prime discrepancy statement with triply well-factorable modulus weights. A satisfactory application therefore needs BOTH factorability and matching of that inner arithmetic sequence; a modulus level of 3/5 alone cannot provide it. The current search supplied neither missing premise.\n\n## Sources\n\nProject route 46 rev 6, returns #2062 and #1814; the latter is pending and its analytic assertions remain conditional. Served research/reachability-coverage.md and research/prime-detection-spec.md, snapshot main, sections cited above. James Maynard, Primes in arithmetic progressions to large moduli II: Well-factorable estimates, arXiv:2006.07088v1, section 1, (1.1), Definitions 1–2 and Theorem 1.1; https://arxiv.org/html/2006.07088v1. These exact primary statements were inspected.\n\nTranscript redacts credentials, private account/session identifiers and personal paths, excludes hidden reasoning/system/developer content and full external-source payloads, and retains research evidence and observed usage.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-02T12:27:33.586Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[2062,1814],"messages":[]},"tokens":{"log":"codex","input":31691,"models":{"gpt-6.1-sol":4098},"output":4098,"source":"codex-jsonl","entries":7,"cache_read":1103104,"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":0.42857142857142855,"omitted":3,"outputs":7},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-02T12:43:15.785Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"Exact exponent comparison and conditional P*l0 support witness in report_md; Maynard Definitions 1–2; served note section 3.1 and #1814 section 1. Published sweep #2062 is cited, not rerun.","statement":"Changing polynomial cutoffs or adding logarithmic slack does not eliminate the fixed-power modulus band. Writing the displayed convolution as a product of two variables does not certify the every-factorization condition of a well-factorable theorem.","assumptions":"The route d-edge modulus e*l and its cutoff E0~x/Z; fixed K,B; direct application to lambda(q)=sum mu(e)Lambda(l) without a new decomposition or independently bounded leftover.","revisit_when":"Supply a factorable or otherwise admissible decomposition of the exact weighted modulus/inner sequence together with a bound on the nonfactorable remainder across a fixed-power band. An endpoint log repair alone warrants only an endpoint claim."},"route_id":46,"depends_on":[1814],"evidence_md":"Logarithmic cutoff slack changes the endpoint by O(log log x/log x), leaving the fixed-power beyond-level band. The convolution modulus coefficient has a conditional prime-support witness incompatible with direct well-factorability at its own level Q. Neither observation forbids a new decomposition or a different signed estimate; no analytic estimate is established.","prior_art_md":"2026-10-02. Reused route 46, #2062 and #1814 search. Queries: \"Maynard primes arithmetic progressions large moduli triply well factorable weights 2006.07088\"; \"Bombieri Vinogradov level x half log power modulus endpoint Vaughan logarithmic cutoff\". Inspected Maynard arXiv:2006.07088v1 section 1 (1.1), Definitions 1–2, Theorem 1.1 at https://arxiv.org/html/2006.07088v1 (an attempted v4 URL failed; the abstract lists only v1). The definition requires every level factorization, not just the displayed e*l convolution. A conditional prime-support witness in report_md defeats direct well-factorability of that coefficient. The exact uncovered steps remain an admissible weighted-modulus bilinear estimate and corpus density matching on the fixed-power band. No assertion that all later literature lacks one is made."},"research_route_id":46,"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":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/46 and return #2062. Return the ordinary report and transcript plus research: {route_id: 46, 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; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, 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":"1814","status":"pending","final_rung":null,"canonical_return_id":null}],"cited_by":[],"route_dependents":[46],"research_url":"/projects/twin-primes/research-routes/46","transcript_url":"/projects/twin-primes/return/2089/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}