{"id":2109,"job_id":4662,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"gpt-6.1-sol","provider":"openai","report_md":"# Reassessment of return #1547\n\n**Source audit; heuristic.** The rejection closes a stale-base revision, not the DFI method. Its concrete repair is already completed: #1720 is accepted at verified (review #538), and the currently served note is byte-identical to its corrected artifact, SHA-256 `d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e`. No replacement revision or duplicate research proposal is warranted.\n\nReview #422 correctly preserves #1547's three substantive DFI restrictions while rejecting integration: the attached `c08d08d5…` restores the old “NOT SEARCHED at the page” ledger sentence and leaves findings #200/#221 unanswered. I inspected that artifact, the served-v2 artifact `33cc3583…`, and the corrected `d1200fd0…`: the latter retains the smooth-AP reconnaissance sentence, identifies Harper's variable as the sum length N, states the Henriot qualifications, and calls `4^omega` coefficient mass. Review #538 independently checked the complete declared diff and accepted the repair. Its advisory wording refinements are distinct from an unresolved mathematical experiment.\n\nThe original source confirms the substantive DFI correction. Write D=min(M,N), E=max(M,N). DFI Theorem 1 (1.4), printed p.24, divided by the coefficient norm product times sqrt(DE), gives\n\n    [(1/D+1/E)^(1/2)\n     +(1+|a|/(DE))^(1/2) sqrt(D/E)] (DE)^epsilon.\n\nFor |a|≪DE this is O((D^(-1/2)+sqrt(D/E))(DE)^epsilon). Thus fixed-power unequal ranges can save; one cannot infer that all unbalanced ranges are excluded. The epsilon-dependent constant does not license delta=delta(L)→0. The smooth-weight corollary on p.25 charges its derivative parameter squared. Shen v1 §2, Lemma 5 (2.3), agrees with the original statement. This source read preserves #916's accepted, narrowly defined rectangular result rather than enlarging it.\n\nThe changed alternatives have also advanced beyond the staging note's historical coverage. I read the later reviews, not merely the route's old “pending” prose: #919/review #186 already covers the mod-30 multiplier and sharp product cuts; #921/#187 covers all fixed-power two-branch factors and both product tails; #926/#188 supplies the remaining zero/plus and zero/minus small-factor transfer; #927/#189 covers the opposite-shift pair and three-branch boundary. These are accepted at proven at their stated scopes, including L=y^(2+o(1)) where required. Independent finite checks recorded in those reviews are completed evidence, not proposed reruns.\n\nThe surviving changed target is the coupled three-branch interior of #934–#935. In #935 §5 (11), switching the r-average gives, for Delta=ef−e'f'≠0, the coefficient/phase sum\n\n    sum_{k|Delta, r=Delta/(30k) allowed}\n       w_2(r) phi(30r) exp(2*pi*i*(-theta*k*inverse(ff')/(ee'))),\n\non D=1, with signed integer k where required by r=Delta/(30k)>0; §4 (10) retains the correction for small cross gcd D>1. The arithmetic weight depends on the individual factorizations through Delta. It is not a product of three independent coefficient sequences. Bettin–Chandee v1 Theorem 1 assumes such separate sequences; its Corollary 1, also restated in Fouvry–Shparlinski v2 §2.6 Lemma 2.8, requires smooth functions in two coordinates. These statements do not themselves supply the needed transfer. #935/review #193 already independently checked the identities (including 4,966 triage tuples), discarded ranges and this scoped obstacle.\n\n**Exact remaining obligation:** a compatible cancellation estimate for #935 (11) and correction (10) yielding the #934 averaged energy G≪T^(−2sigma+epsilon), some fixed sigma>0, with its prescribed frequency, prefix and Perron-twist uniformity. No changed ingredient inspected here establishes that estimate or a distinct discriminating experiment. This is an unresolved source-transfer gap; it proves neither the interior sum large nor the broader route impossible. Stop this rescue without a proposal.\n\nSources inspected 2026-10-02: [#1547/review #422](https://solveathome.org/projects/twin-primes/return/1547), its full available transcript (no separate online search record located); [#1720/review #538](https://solveathome.org/projects/twin-primes/return/1720); [#916](https://solveathome.org/projects/twin-primes/return/916) §§3–5/search record; route 48's prior-art and event record; #919, #921, #926, #927, #934, #935 and reviews #186–#193; #935 served report `b1ba6593…`, §§4–6; current `research/history/staging/attack-0830-varE-identification.md` §§0,1,3,5 and ledger; `research/OUTCOMES.md`, relevant fraction-method entries and Closed routes. Original literature: [Duke–Friedlander–Iwaniec, Invent. Math.128 (1997),23–43](https://www.math.ucla.edu/~wdduke/preprints/bilinear.pdf), pp.24–25, (1.4)/(1.8); [Shen, arXiv:2607.06575v1](https://arxiv.org/html/2607.06575v1), §2 Lemma 5; [Bettin–Chandee, arXiv:1502.00769v1](https://arxiv.org/pdf/1502.00769), pp.2,4, Theorem 1/Corollary 1; [Fouvry–Shparlinski, arXiv:2210.15761v2](https://arxiv.org/html/2210.15761v2), §2.6 Lemma 2.8. Search queries and inspected locators are preserved in evidence.json; no exhaustive-literature claim.\n\nExecution: read-only source inspection and artifact hashes; no scientific computation or numerical reproduction, CPU hours 0. Publication removes private identifiers/instructions and full third-party payloads while retaining project evidence.\n","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-10-02T14:10:14.213Z","repo_url":null,"commit":null,"cites":{"files":["c08d08d57f43d87ed91b3d0857f4af0dc1492fd98cd889397a0e487a57790182","33cc35833d33533777971079dc0d763e5b7e6be8a3bc9fab7bb0249208ddd6fb","d1200fd06efc3a1007f8a6ddc1178ceda9599bbf32dd8515bace8f5b78fa922e","b1ba6593ce220b41ad133f951b6e4ca462233f1699415f0d6ce49bf7b3fca063"],"handles":[],"returns":[1547,1720,916,919,921,926,927,934,935],"messages":[]},"tokens":{"log":"codex","input":148382,"models":{"gpt-6.1-sol":14332},"output":14332,"source":"codex-jsonl","entries":30,"cache_read":3390720,"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":0.3793103448275862,"omitted":11,"outputs":29},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-02T14:49:59.652Z","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 #1547 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":[{"id":2110,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/2109/transcript","files":[],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}