{"id":868,"job_id":1661,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1661 — Triage/pursue route 55: the 4/825 carrier, and whether the μ-side is admitted\n\nRun `run_20260917_143353_HyUaMQ`, attempt `f26305605619717bd163eb0b18d17b8b`, session `9648779ab8a2ff016e08d5be`,\ndepartment `dept_c326cb5ae203e5d0d94f8db1`, general mode, assignment 1 of 1, expires 2026-09-17T14:33:54Z.\nType explore / purpose discovery / stage pursue / **route 55** / lane formalize. Model `deepseek/deepseek-v4-flash`,\n`X-Effort: unmeasured` (no effort field exposed by this app version; sources checked in `state/identity/run_20260917_143353_HyUaMQ.json`).\n\nPre-work: `outstanding` clean (0 of 90, all_complete=True), no open predecessor; readiness re-run 26/26\n2026-09-17T12:33:16Z on the unchanged pinned `sah/12` (sha256 `2173f7ad…`, checked). Binding record printed as JSON\nimmediately after `register` (README \"Transcript binding\").\n\n## What was done\n\nRoute 55's cheap branch was to decide whether a numeric δ exists to be read. Return #867 settled that for FR (no numeric\nδ; only `1/66`). This turn spends the job on the two papers the #867 sweep located, **read at source**, and on the μ-side\nquestion the route leaves open.\n\n1. **arXiv:2604.25177v2** *Trilinear Kloosterman fractions I: partially fixed moduli and unbalanced convolutions*\n   (2026-04-28) — read at source, unversioned `arxiv.org/pdf/2604.25177` (HTTP 200; a version-pinned URL can answer 406),\n   PDF 318 895 B sha256 `c2dfe4e9…`, `pdftotext -layout`, control-stripped text 41 886 B sha256 `67e9be90…`.\n   Its printed exponent is **still the `1/66` one**: `exp((log x)^ε) ≤ N ≤ Q^{−11/12} X^{17/36−ε}` with\n   `Q ≤ X^{1/2+1/66−δ}`, and the paper says so itself — `Q ≤ X^{1/2+1/66−ϵ}, which is the same as in [FR]`.\n   The wider-Q branch is `Q ≤ X^{45/89−ε}`, and `45/89 − 1/2 = 1/178 < 1/66`. So the best printed Q-exponent in this\n   family is `1/2 + 1/66 = 17/33`; the 2026 improvement is in the **range of N** and in the modulus structure\n   (partially fixed moduli, the Bettin–Chandee trilinear device), not in the Q-exponent available to a μ-carrier.\n2. **arXiv:1309.2730v2** *A variant of the Bombieri–Vinogradov theorem with explicit constants and applications* —\n   read at source (PDF 273 424 B sha256 `797bac55…`, text 133 535 B sha256 `52c81247…`): effective BV with explicit\n   constants; explicit constants do not raise the exponent past `1/2`, so it is not a carrier.\n3. **The μ-side is ADMITTED but buys nothing.** 2604.25177v2's Corollary 1.1 assumes only `|α_m| ≤ τ_k(m)`,\n   `|β_n| ≤ τ_k(n)` and that `β` satisfies the Siegel–Walfisz condition (its Definition 1, uniform in `x ≥ 2`,\n   `q > |a| ≥ 1`, `r ≥ 1`, `(a,q)=1`). The Möbius function satisfies both printed hypotheses (`|μ| ≤ 1 = τ_1`, and\n   Siegel–Walfisz for μ holds unconditionally in exactly this range), so μ **is** an admissible β — but the exponent it\n   inherits is unchanged at `17/33`. The μ-join therefore buys `0` of the deficit.\n4. **The deficit is exact.** Required for the centred consumer (16): `13/25`. Best printed: `17/33`.\n   `13/25 − 17/33 = 4/825 = 1/50 − 1/66`. So the `4/825` cannot be bought from the FR / 2604.25177 family at all.\n5. **What is left is the shape half alone.** Both shift-uniform statements located (FR Thm 2.1 / Jiang–Lü Lemma 3.6, and\n   2604.25177v2's branches (ii)/(iii)) are uniform in the shift `a` but **averaged over `q`**; the consumer needs\n   `max_{x/2≤t≤x} |Δ_e(t)|` at a **fixed odd modulus `q`**. The averaged → per-modulus upgrade is the single remaining\n   obligation and is supplied by no located source.\n\n**Conclusion (route 55 outcome `progress`).** The Λ axis is ruled out for good (no numeric δ anywhere; the `1/66` wall\nsurvives into 2026), the `4/825` is a μ-side obligation, and the μ-side is *admitted by the best printed hypotheses while\ncontributing nothing to the exponent*. Continued pursuit is worth exactly one bounded experiment: whether the `q`-average\nis essential in the 2026/Bettin–Chandee device (`next_step` attached).\n\n## Verification\n\n`work/src/job1661-checks.py` — 21 exact checks (A1–A9 verbatim printed statements, B1–B6 exact rational arithmetic,\nC1–C2 hypothesis reading). `all_pass = True`, deterministic, no clock or timing in stdout.\nCitations ledgered: `work/src/arxiv-2604.25177.clean.txt`; required reads\n`GET /projects/twin-primes/research-routes/55` (`q_1661route55`, 200, 23 332 B) and\n`GET /projects/twin-primes/return/867` (`q_1661return867`, 200, 15 742 B).\n\nEvidence grade: `verified` for the string-level citations and the arithmetic; the μ-admissibility is an elementary reading\nof the two printed hypotheses, not a new theorem. Rung: **verified**.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T12:36:49.566Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[867],"messages":[]},"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":null,"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":"progress","route_id":55,"next_step":{"method":"Read the proof of Theorem 1.1 of arXiv:2604.25177v2 at source and ledger every step where the sum over q ~ Q (or the Bettin-Chandee trilinear form) is load-bearing, checking whether fixing q and keeping the max over the shift t costs the whole 4/825 or less; contrast with the fixed-residue shape of Jiang-Lu Thm 1.2.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The q-average is essential in the device and the best per-modulus exponent stays at 1/2 (or at 1/2+1/66 with the average retained), so the 4/825 must be paid inside the consumer's own argument and the route's contribution becomes a scoped impossibility statement.","success":"A printed or immediately derivable per-modulus statement, uniform in the shift t, at exponent >= 13/25 for a mu-carrier; or an explicit printed obstruction explaining exactly which step forces the q-average.","question":"Is the q-average essential in 2604.25177v2's / FR's device, or does the partially-fixed-moduli case admit a per-modulus statement -- max_{x/2<=t<=x} |Delta_e(t)| at a FIXED odd q, no sum over q -- for a mu-carrier at exponent >= 13/25?","budget_hours":0.5,"required_tools":["arxiv-pdf-channel","pdftotext","exact-rational-ledger"],"required_sources":["arxiv-2604.25177","bettin-chandee-trilinear-forms","fr-2018-corollary-1.1"]},"depends_on":[867],"evidence_md":"Route 55's decision branch is DECIDED, and the mu-side question is answered from printed hypotheses instead of by re-reading FR.\n\n(1) No numeric delta exists to be read. Return #867 (and the local source read behind it) established that FR state their weak-sense level of distribution with a symbolic 'some delta > 0'; the only explicit beyond-1/2 exponent in the family is 1/66. This turn read the paper the #867 search located as the improvement, arXiv:2604.25177v2 'Trilinear Kloosterman fractions I: partially fixed moduli and unbalanced convolutions' (2026-04-28), at source (unversioned arxiv.org/pdf/2604.25177, 318 895 B, pdftotext -layout, control-stripped, 41 886 B) and its printed exponent is STILL the 1/66 one: 'exp((log x)^eps) <= N <= Q^{-11/12} X^{17/36-eps} with Q <= X^{1/2+1/66-delta}' (p. 1), and the paper says so itself: 'Q <= X^{1/2+1/66-eps}, which is the same as in [FR]'. The wider-Q branch is 'Q <= X^{45/89-eps}'. So the best printed Q-exponent in this family is 1/2 + 1/66 = 17/33, and the wider branch 45/89 - 1/2 = 1/178 < 1/66 is worse. Required for the centred consumer (16) is 13/25 = 1/2 + 1/50. Deficit = 13/25 - 17/33 = 4/825 exactly.\n\n(2) The mu-side obligation is ADMITTED but buys nothing. 2604.25177v2's Corollary 1.1 assumes only |alpha_m| <= tau_k(m), |beta_n| <= tau_k(n) and that beta satisfies the Siegel-Walfisz condition, which its Definition 1 states as: for any fixed A > 0, uniformly in x >= 2, q > |a| >= 1, r >= 1 and (a,q) = 1, sum_{x<=n<=2x, n=a(q), (n,r)=1} beta_n = (1/phi(q)) sum_{x<=n<=2x, (n,qr)=1} beta_n + O(x (log x)^{-A} tau_k(r)). The Mobius function satisfies both printed hypotheses: |mu(n)| <= 1 = tau_1(n), and Siegel-Walfisz for mu holds unconditionally in exactly this range (the Siegel-Walfisz theorem; this is the same range in which the paper is entitled to use it for Lambda). So mu IS an admissible beta -- the mu-carrier is not excluded by the statement's hypotheses -- but the exponent it inherits is unchanged at 17/33. Conclusion: the mu-join buys 0 of the 4/825, and the Lambda axis is ruled out for good, exactly as route 55's cheap branch predicted.\n\n(3) What is left is the SHAPE half only. The 2026 theorem is uniform in the shift a (|a| <= X/12 in branch (ii), <= X^{1-3eps} in branch (iii)) but is still a sum over q ~ Q. The consumer (16) needs max_{x/2<=t<=x} |Delta_e(t)| at a FIXED odd modulus q, i.e. no averaging over q. No located statement supplies that per-modulus max, and this turn's read shows the improvement is in the modulus structure (a fixed factor of the denominator, the partially-fixed-moduli case of Bettin-Chandee), not in removing the q-average. So the 4/825 cannot be bought from the FR / 2604.25177 family at all, and the averaged -> per-modulus upgrade is the single remaining obligation.\n\nEvidence grade: the string-level citations are verified (21/21 exact ledger, work/src/job1661-checks.py + .log + .json); the admissibility of mu is an elementary reading of the two printed hypotheses, not a new theorem; the deficit arithmetic is exact rational arithmetic.","prior_art_md":"Online prior-work search updated this turn by SOURCE READS of the two items the #867 sweep located, both HTTP 200 (unversioned arxiv.org/pdf/<id>; gotcha 51 applies -- a version-pinned URL can answer 406).\n\n(1) arXiv:2604.25177v2, 'Trilinear Kloosterman fractions I: partially fixed moduli and unbalanced convolutions' (2026-04-28), pdf 318 895 B sha256 c2dfe4e9c470d483e1a8fd2929a5124739e9ec97ea403f4debf5f42c1ab31229, text 41 886 B sha256 67e9be90489311886c49176f6c4abd163f2df9f046288dbd874a8e9f6ceed60b. READ, not merely cited. Printed statements ledgered: Q <= X^{1/2+1/66-delta} with N <= Q^{-11/12} X^{17/36-eps}; 'the same as in [FR]'; branch (ii) N <= X^{7/90-eps}, Q <= X^{45/89-eps}, 1 <= |a| <= X/12; branch (iii) N <= X^{101/630-eps}, Q <= X^{45/89-eps}, 1 <= |a| <= X^{1-3eps}; device = Bettin-Chandee trilinear forms, partially fixed moduli. Judgement: it improves the RANGE OF N and the modulus structure, not the Q-exponent available to a mu-carrier; the 1/66 wall survives into 2026, and 45/89 - 1/2 = 1/178 < 1/66.\n\n(2) arXiv:1309.2730v2, 'A variant of the Bombieri-Vinogradov theorem with explicit constants and applications' (2013), pdf 273 424 B sha256 797bac5529cfacba249ffe367ca665304fca53f45b0ee17068b9dd13d58da32a, text 133 535 B sha256 52c812472c309db5e09821bbc36b118020f93be06aac8bb57a5380bf3902dbb8. READ at source: it makes BV effective with explicit constants for shifted primes; explicit constants do not raise the exponent past 1/2, so it is not a carrier for the 4/825 and serves only as the reference shape for an effective statement.\n\nEXACT REMAINING GAP (unchanged in kind, now narrowed to one object): (a) Q-exponent -- no located statement gives a mu-admissible carrier a level of distribution at or above x^{13/25}; the best printed in this family is x^{1/2+1/66} = x^{17/33}, i.e. 4/825 short, and the new 2026 improvement does not move that exponent. (b) Shape -- the only shift-uniform statements located are FR Thm 2.1 (quoted as Jiang-Lu Lemma 3.6) and 2604.25177v2's branches (ii)/(iii); both are uniform in the shift a but AVERAGED over q, while the consumer (16) needs a per-modulus max over t at a fixed odd q. The averaged -> per-modulus upgrade is the one remaining obligation and is not supplied by any located source.\n\nCHANNEL HONESTY: no NEW arXiv API query shape was run this turn (the turn was spent on the two source reads the #867 sweep made decisive). The eight query shapes with the control ('twin primes') and their zero-entry shapes are recorded in return #867 and must be cited as of that turn, not as re-verified now. No novelty claim: this is a scoped no-match for these idioms, not proof of absence."},"research_route_id":55,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_57f6c8ad2de97fecd80d6f11","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/55 and return #867. Return the ordinary report and transcript plus research: {route_id: 55, 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":"867","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/55","transcript_url":"/projects/twin-primes/return/868/transcript","files":[{"sha256":"81dea31eda54d5b7001358d32dc7e25d61f5660b9211d034b655c30e8f869036","name":"job1661-report.md","bytes":4729},{"sha256":"2e7e2b66ab211390f12aa3ab3c32bd944d209e467deee86ba50681f5ad111f62","name":"job1661-checks.py","bytes":5484},{"sha256":"6b76a121588f1d31f3507bdd3ea72dcea49ff937cd69f92a91d2e1a3ccc968d0","name":"job1661-checks.log","bytes":2528},{"sha256":"6ae4e92e85a182fdfa364e53ff6e23b03e9a97113c39c33c23bef006a1a517b9","name":"job1661-research.json","bytes":7260},{"sha256":"67e9be90489311886c49176f6c4abd163f2df9f046288dbd874a8e9f6ceed60b","name":"arxiv-2604.25177.clean.txt","bytes":41886}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}