{"id":2123,"job_id":4674,"problem_id":1,"lane_id":1,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Return #1694: a failed revision, with its mathematical input preserved\n\nSource-audit outcome, rung **heuristic**; no distinct research proposal. Review #458 refuted #1694 as a document revision, not Möbius Bombieri–Vinogradov or the endpoint-transfer method. I verified the rejected artifact's SHA-256 and its five-cell §2 row at line 55. The accepted repair #1769/review #523 is now served exactly (f6a966ed…): its §2 rows have three cells, and the Forum Math. Sigma title, dates and attribution are corrected. Those repairs stand.\n\nThe surviving scope error is already owned by open finding #3437: current `mobius-bv-derivation.md` lines 9 and 64 call the unlocated clause “exact class-sup,” although the missing displayed clause is the **maximum over y≤T**. Finding #3438 separately distinguishes pointwise prime restriction from Dirichlet convolution. The current OUTCOMES entry, lines 2298–2310, still repeats the older five-channel fixed-endpoint access negative. None of this is a counterexample to the theorem.\n\nParent inspection of #1694’s eight-record original transcript found one web-search call whose query mixed the two Granville–Shao titles; the report disclosed that it did not freshly open the three carried readings. I read the original search record #157 and review #333. Its eleven channels included access failures, not eleven theorem refutations. The later Granville–Shao source came through #1183/review #420 on 2026-09-25, separately from that sweep. #1694 explicitly carried those readings rather than freshly opening all three sources; I freshly inspected the decisive primary statements here.\n\nGranville–Shao, *When does the Bombieri–Vinogradov Theorem hold for a given multiplicative function?*, arXiv:1706.05710v1, §1 equation (1.1), Theorem 1.1(b), and §2 Theorem 2.1, supply fixed-endpoint class-maximal distribution. For f=μ, its pointwise restriction to primes is −1_P; classical prime BV and the Möbius Siegel–Walfisz input give the required hypothesis. Theorem 2.1's varying X is an input range; its conclusion is at x. This source match preserves the holding portion of reviews #420 and #458. [Primary text](https://arxiv.org/html/1706.05710v1).\n\nThe distinction is bibliographic, not a new analytic obstruction. For a bounded sequence, the usual mesh transfers sufficiently strong fixed-endpoint bounds to a maximum over endpoints at an extra logarithmic price. Write L=log T, h=T/L^K and use O(L^K) grid points. Between adjacent points, the progression sum changes by O(h/q+1), and its coprime-average correction by O(h/φ(q)); summing over q≤Q gives O(h log T+Q). Endpoints below T/L^C cost O(T/L^(C−1)+Q) trivially. At all remaining grid points t, taking B>B_0+C/2 ensures Q≤sqrt(t)/(log t)^B_0; take the fixed-endpoint saving A_0>A+K and C,K>A+2. The grid sum and both interpolation errors are then O_A(T/L^A). This is a standard consequence with adjustable logarithmic losses, not a located verbatim published statement of (M), nor a new route. For the raw Möbius sum, removing the coprime-average correction uses the existing K1 import (Corollary 13.4 with progression modulus 1 and excluded primes dividing q); that source theorem was not independently revalidated here.\n\nThe current `shifted-prime-decomposition.md` §2, lines 82–90, already contains this mesh mechanism for intervals in [T/2,T] below a fixed power margin from 1/2. Shao, arXiv:1607.01814 §1, freshly inspected, supplies a fixed-endpoint exceptional-set assertion and a stronger Gowers-norm formulation with its stated restrictions; it does not furnish a distinct endpoint rescue. Le Boudec's Fall 2014 EPFL Sheet III, Exercise 4, p. 2, freshly inspected, already states raw fixed-endpoint μ-BV as an exercise and is already cited in the project note. Vaughan 1980 p. 113 remains a carried reading from #157/review #333: fresh publisher/matwbn attempts did not yield the paper. [Shao](https://ar5iv.labs.arxiv.org/html/1607.01814), [Le Boudec](https://wiki.epfl.ch/tan-tnt/documents/TNT%202014-2015/Sheet%203.pdf).\n\nNo experiment or scientific computation was run. The exact remaining obligation is to repair the existing qualifier/provenance wording against the served head, and, if a direct published citation of (M) is required, locate a statement with its y-maximum or cite the fixed-endpoint theorem together with an explicitly checked mesh argument. This pass neither closes unread textbook pages nor supplies distribution beyond 1/2, the coupled two-Möbius residual, or a twin-prime lower bound.\n\nSources: #1694/review #458 and served revision 46a06ae8ce792e7a3a24a94c3f677cc189a547ef36f3de474851892190bd86d6; #157/review #333; #1183/review #420; #1769/review #523; current snapshot `main` fetched 2026-10-02: `research/mobius-bv-derivation.md` §§1–3 (SHA-256 f6a966ed725fc9cd38bc610d54b568a3bb44516f80eaf85f51e430569df9b904), `research/OUTCOMES.md` owning entry and Closed routes (49364d8848f14f6b4f4692ccf27606e5407a155073fece37139c5873b7511f5a), `research/shifted-prime-decomposition.md` §2 (244b964c99966fef7c2891098f174cbd5d54eb2a7f9009d149232239542c6a84), and `research/SEARCH-CONVENTIONS.md` §1 (790e2429b7cd083381f9551b86886b45cd0c5f87268010cddf5977b9664c48cb). Full third-party sources were consulted without republication; no numerical outputs were reproduced.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-02T15:57:40.792Z","repo_url":null,"commit":null,"cites":{"files":["46a06ae8ce792e7a3a24a94c3f677cc189a547ef36f3de474851892190bd86d6","f6a966ed725fc9cd38bc610d54b568a3bb44516f80eaf85f51e430569df9b904"],"handles":[],"returns":[1694,157,1183,1769],"messages":[]},"tokens":{"log":"codex","input":150518,"models":{"gpt-6.1-sol":18023},"output":18023,"source":"codex-jsonl","entries":44,"cache_read":3906688,"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":0.23255813953488372,"omitted":10,"outputs":43},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-02T15:58:22.794Z","file_notes":null,"research":null,"research_route_id":null,"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":"Read return #1694 and its search record, then search online for the method and changed alternatives before testing them. Check whether its negative conclusion closes only a statement or attempt. Use published numerical results with citations, reserving reproduction for later validation. Inspect the decisive evidence, then seek a concrete alternative. Preserve valid refutations. A promising alternative should return research.proposal with parent evidence in cites.returns, a prior-art comparison and the cheapest next experiment. If nothing changes, record the scoped obstacle and stop. This is a bounded sample; do not reproduce the whole investigation.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[],"cited_by":[],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/2123/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}