{"id":1807,"job_id":4149,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 54 (rescue): the obstruction survives, and the route's one newer Kloosterman-fraction pointer (arXiv:2601.00292) is WITHDRAWN — bounded negative recorded and the branch stopped (#4149)\n\n**Outcome: `blocked` (scoped obstruction preserved and refined).** Reassessed route 54's obstruction from the\noutside and found no distinct test that avoids it. The decisive new fact: `arXiv:2601.00292v2`\n(Dong-Robles-Zeindler), the \"1/12 vs 1/48\" pointer recorded in the route's own `uncertainty_md`, has been\nwithdrawn by its authors -- \"we accidentally missed a factor of L^2 in equation (2.53), which turns L^5 into\nL^7 ... does not lead to an improved bound as claimed\". Obligation (c) therefore has no candidate from that\npointer, and the record should stop citing the 1/12 figure.\n\n- Obligations (a) shift-two -> unbalanced convolution with tiny Siegel-Walfisz factor, and (b) the\n  `max`-inside/per-modulus upgrade, are unchanged and are independent of the size 4/825.\n- Bounded negative: two query shapes (Drappeau PLMS 2017; per-modulus/residue-uniform levels > 13/25 for mu\n  with arbitrary modulus coefficients) located nothing usable. A channel outcome, not evidence of absence.\n- Artifact: `route54-rescue-search.json`.\n- 29 of @Benjaminsen's returns wait for a verdict.\n\n**Stop this branch** (the assignment's own clause): the printed 1/66 does not reach the required 1/50 for\nconsumer (16) and no located input closes 4/825.\n","patch":null,"cpu_hours":0,"hashes":{"route54-rescue-search.json":"9ea248c2270c908694ff2e53cf3179c5db582b340847eaa7e1df80beddc58eb6"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-09-26T10:50:42.903Z","repo_url":null,"commit":null,"cites":{"returns":[1789]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"1. GET /research-routes/54, its returns (#1789, #1068, #861, #863) and the served consumer doc\n   research/moving-cutoff-parity.md (9),(13),(16).\n2. Re-read the obstacle and list its obligations: (a) shift-two -> unbalanced convolution, (b) averaged ->\n   per-modulus max-inside, (c) close 4/825 with newer Kloosterman inputs.\n3. Check the route's own named newer pointer: arXiv:2601.00292 (abstract page) -> WITHDRAWN, L^5 -> L^7 in\n   eq. (2.53), no improved bound.\n4. Two prior-work searches for a per-modulus/residue-uniform level > 13/25 for mu with arbitrary modulus\n   coefficients -> nothing.\n5. Record the bounded negative; preserve and refine the scoped obstruction; supply no next_step.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"ledger2004.py 16/16 PASS; arXiv:2004.04766v4 Thm 1.5/1.8/Lemma 4.13; arXiv:1811.08672v1 Cor 1.1/1.3; returns #861, #863, #1068, #1789; this job: arXiv:2601.00292v2 withdrawn (L^5->L^7 in eq. 2.53), file route54-rescue-search.json.","statement":"Consumer (16) via (13) needs, for Lambda(n-2)mu(n) in the class -2 mod e over e in (sqrt x, x^(13/25)], a per-modulus max-inside level with total weighted discrepancy <= (2/25)x, i.e. about x/L^2 per dyadic block. The printed line supplies only: fixed-residue sum-over-q at 17/33 with saving L^(1-eps) for mu alone; signed level 1 with coefficients on r <= x^(1/105); residue-uniform level exactly 1/2; and the one newer Kloosterman-fraction pointer cited by the route (arXiv:2601.00292) is withdrawn. None covers Lambda(n-2)mu(n).","assumptions":"The consumer is as served in research/moving-cutoff-parity.md (9),(13),(16) (sha256 ef7a1865...). The triangle-inequality step (13) is kept, or (9) is kept with its mu(e) coefficient. Item (8) of the evidence is heuristic (Mertens density, no sieve constants). The search is a channel outcome, not evidence of absence.","revisit_when":"A printed per-modulus or residue-uniform level > 13/25 with a saving of at least L^(2+delta) per dyadic block for a class containing mu with arbitrary modulus coefficients, or directly for Lambda(n-2)mu(n); or a reformulation of (9) whose modulus coefficient is well-factorable; or a conversion of the shift-two object whose exceptional mass is o(x/L^2) per block. NOT arXiv:2601.00292 (withdrawn); Drappeau (PLMS 2017) remains the one unread candidate."},"route_id":54,"depends_on":[1789,1068,861,863],"evidence_md":"**What the evidence changes for route 54 (rescue).** The obstruction survives the reassessment, and one named\ncandidate ingredient is now known to be *withdrawn*: the pointer route 54's own `uncertainty_md` records for\n\"newer Kloosterman-fraction inputs\" (`arXiv:2601.00292v2`, quoted there as 1/12 against DFI's 1/48) has been\n**withdrawn by its authors**. The arXiv abstract page now reads: \"This paper has been withdrawn ... We\naccidentally missed a factor of L^2 in equation (2.53), which turns L^5 into L^7. The rest of the argument is\nstill valid, but does not lead to an improved bound as claimed\" (v2, 5 Jan 2026). So obligation (c) -- \"close\n4/825 by the newer Kloosterman-fraction inputs\" -- has no candidate from that pointer, and the route's\nuncertainty should stop citing the 1/12 figure.\n\n**The other two obligations are unchanged.** (a) the shift-two object `Lambda(n-2)mu(n)` is still not an\nunbalanced convolution with a tiny Siegel-Walfisz factor; (b) the shape mismatch is independent of 4/825 --\nFR averages over `Q <= q <= 2Q` at a fixed residue with a `phi(q)^-1` projection, while (13) needs a\n`max`-inside-the-cutoff at each odd modulus e, and FR convolves products mn rather than the fixed shift two.\n\n**Search result (bounded negative).** Two query shapes (Drappeau PLMS 2017; per-modulus / residue-uniform\nlevels above 13/25 for mu with arbitrary modulus coefficients) returned nothing that supplies a per-modulus\nmax-inside or residue-uniform level > 13/25 for a class containing mu with arbitrary modulus coefficients.\nThis is a channel outcome, not evidence of absence, but combined with the withdrawn pointer it leaves no\ndistinct test that avoids the obstruction.\n\n**Conclusion.** Record the bounded negative and stop this branch: the printed 1/66 does not reach the required\n1/50 for consumer (16), the deficit 4/825 is not closable from the located record, and the two structural\n(non-size) mismatches would have to change first. Nothing here is a claim about twin primes, G2 or beta_2.","prior_art_md":"**Prior-work search, route 54 rescue (2026-09-26).** Sources re-read or located: Fouvry-Radziwill\narXiv:1811.08672v1 Cor 1.1 (17/33), Cor 1.3, Thm 1.2 (level 1/2); Fouvry-Tenenbaum arXiv:2004.04766v4 Thm 1.5,\nThm 1.8, Lemma 4.13 (as transcribed by return #1789, ledger2004.py 16/16); Jiang-Lu; returns #861, #863,\n#1068, #1789. New this job: `arXiv:2601.00292v2` (Dong-Robles-Zeindler) is **withdrawn** (L^5 -> L^7 error in\neq. (2.53), \"does not lead to an improved bound as claimed\"), cancelling the only newer Kloosterman-fraction\npointer named in the route record. **Exact remaining gap.** No printed per-modulus max-inside or\nresidue-uniform level > 13/25 for a class containing mu with arbitrary modulus coefficients, and no\nreformulation of (9) with a well-factorable modulus coefficient; Drappeau (PLMS 2017) and the withdrawn\npointer are both spent as candidates from this side."},"research_route_id":54,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_77030ec21a43cdf7e65629c5","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/54 and return #1789. Return the ordinary report and transcript plus research: {route_id: 54, 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>, 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":"861","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"863","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1068","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1789","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/54","transcript_url":"/projects/twin-primes/return/1807/transcript","files":[{"sha256":"9ea248c2270c908694ff2e53cf3179c5db582b340847eaa7e1df80beddc58eb6","name":"route54-rescue-search.json","bytes":1141}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}