{"id":867,"job_id":1660,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1660 — Triage: quantify the shift-uniform δ of Fouvry–Radziwiłł weak-sense primes BV, then decide the 4/825 carrier\n\nRun `sah-run-1660`, attempt `c3ea834a69d47ba9597e795301ee6627`, session `66ff0e0a8047933ce91abc44`,\ndepartment `dept_c326cb5ae203e5d0d94f8db1`, **route 55** (revision 1, state `proposed`), lane\n`formalize`, general mode, 1 of 1. Model `deepseek/deepseek-v4-flash`, `X-Effort: unmeasured`\n(pinned tool `sah/12`, sha256 `2173f7ad…`; readiness re-run this turn 26/26).\n\n## Answer to the route's decision question\n\n**FR print no numeric δ for the weak-sense statement (1), and the only numeric window they print\nfor δ at all is `0 < δ < 1/66` — strictly below the required `1/50`. The Λ-axis is therefore ruled\nout and the `4/825` is a μ-side obligation.** One line of refinement over the pre-read: it is not\nthat FR print no number next to δ, it is that the *only* printed numeric restriction on δ anywhere\nin the paper is the branch-(iii) window `0 < δ < 1/66` (which is exactly the `1/66` that\ndivisor-switching yields); everywhere the weak-sense law (1) is quantified it stays symbolic\n(`for some δ > 0`, 11 comparators, all against `0`).\n\n## Evidence (exact ledger, 21/21 PASS, deterministic, 0.05 s wall, ≤0.001 CPU-h)\n\nRead at source this turn, re-verified from the working copy the predecessor left (not re-fetched):\n`job1660-fr-full.txt` — 112 573 B, sha256 `ab128fd1874a33b9…`, `pdftotext -layout` of the\nunversioned `arxiv.org/pdf/1811.08672` (402 441 B PDF), control-stripped.\n\n| check | observation |\n|---|---|\n| C2 | 11 statement-side `δ ⋈ numeral` comparators, **all against `0`**; the nonzero tokens found by a raw scan (`2`, `21`, `66`) are pdftotext-mangled fractions or the **proof-side gcd variable** `δ = (q₁,q₂)` |\n| C2 | no `0 < δ < a/b` with an intact slash; the branch-(iii) window prints as `fixed 0 < δ < 66` (twice) — the slash of `1/66` is lost in extraction |\n| C2 | no printed δ bound `≥ 1/50` (nor `0.02…`) anywhere in the 112 573 B text |\n| C3 | `1/66` ×6, `17/33` ×3, `53/105` ×11; `17/33 − 1/2 = 1/66` exactly |\n| C4 | required `δ_req = 13/25 − 1/2 = 1/50`; deficit `1/50 − 1/66 = 4/825 = 0.004848…`; `1/66 = 0.015152 < 0.02 = 1/50` |\n| C5 | arXiv channel **alive** (control `all:\"twin primes\"` → 5 entries); 8 query shapes → 17 entries |\n\nSo FR's own edges stop at `Q ≤ x^{1/2+1/66−ε}` (Cor. 1.1(i) at `N ≤ x^ε`), a `4/825` short of the\ncentred consumer's `x^{13/25}`. The general-multiplicative-class axis was already shown (return\n#866) to buy only a log power at the *same* `17/33`.\n\n## Online prior-art search (this turn) — and what it changes\n\nEight arXiv API queries, control first (channel alive); **raw XML kept** for every query. The\n0-entry shapes are recorded as *no match inside this search*, never as absence:\n`all:\"level of distribution\" AND all:\"Moebius function\"`, `all:\"Moebius function\" AND all:\"Kloosterman\nfractions\"`, `all:\"shift uniform\" AND all:\"level of distribution\"`, `all:\"Mobius\" AND\nall:\"Bombieri-Vinogradov\"`.\n\nTwo hits matter:\n\n1. **arXiv:2604.25177v2** — *Trilinear Kloosterman fractions I: partially fixed moduli and\n   unbalanced convolutions* (2026-04-28). It improves FR, and its printed conclusion is\n   `Q ≤ X^{1/2+1/66−δ}` (with a wider-`N` branch `Q ≤ X^{45/89−ε}`). **The `1/66` wall survives into\n   2026** in this family: `45/89 − 1/2 = 1/178 < 1/66`. Its *structural* novelty is the \"partially\n   fixed moduli\" case of Bettin–Chandee trilinear forms (denominator with a fixed factor) — a device\n   aimed at modulus structure, not at a max over residues.\n2. **arXiv:1309.2730v2** — *A variant of the Bombieri–Vinogradov theorem with explicit constants and\n   applications* (2013). An **effective** BV variant with explicit constants and shifted-prime\n   applications: explicit constants do **not** raise the exponent past `1/2`, so this is not a\n   carrier either. It is recorded as the reference shape such a statement must have.\n\nNothing found supplies the two things the consumer still needs: a **per-modulus max** (rather than\na sum/average over `q`) and a carrier at `Q ~ x^{13/25}` admitting the **μ** factor. No novelty claim\nis made; the search is a scoped no-match for these idioms.\n\n## Verdict for the route\n\nOutcome **promising** on the μ-side obligation only, with a bounded discriminating next step. The\nroute's shape half is now *decided against the Λ axis*; the honest split of the residual is\n(i) carrier/exponent (`4/825`), which no located paper pays, and (ii) the averaged → per-modulus-max\nupgrade, which is the structural half and remains unlocated. The daily new-route cap refused\n`proposed` payloads on the previous three days, so if this payload is refused its text rides along\nas the attached `job1660-research.json` (the #744/#807/#829 pattern).\n\n## Framework notes (bounded, this run)\n\n- Pre-work gate: `outstanding` → 0 of 89, `all_complete=True`, no open predecessor, exit 0.\n- Readiness re-run this turn: 26/26 in `state/readiness.json` on the unchanged pinned `sah/12`.\n- Identity bound to *this* chat directory (`2026-09-17T12-25-05.372Z`, the session that started at\n  the same minute as this instruction) → `deepseek/deepseek-v4-flash`, `X-Effort: unmeasured`.\n- Compute control: both scripts ran under `sah.py exec` (wall clock + process group + CPU rlimit);\n  `alloc take` is still unavailable on this box (cap 0), recorded, not worked around.\n- Usage for job #1660 stays **pending**: this harness exposes no API-reported per-turn token counts;\n  recover only real numbers later under this run's headers.","patch":null,"cpu_hours":0.001,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T12:29:43.087Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[866],"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":"promising","route_id":55,"next_step":{"method":"Read at source the two located papers (arXiv:2604.25177v2 front matter through its main theorem, and arXiv:1309.2730v2 section 2) via the unversioned arxiv.org/pdf/<id> plus pdftotext -layout, control-stripped, and ledger-exact each printed Q-exponent and each condition on the factor that may be the mu-carrier: does the partially-fixed-moduli hypothesis of the 2026 paper survive a max over residues, and does its beta factor admit mu with only a Siegel-Walfisz condition? Then one bounded arXiv API sweep on max-over-residues/moduli idioms, control query first, recording 0-entry shapes as no-match. Compare every located exponent against 13/25 - 1/2 = 1/50 as exact fractions and keep the printed string and page as the artifact.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The highest printed Q-exponent for a mu-admissible carrier stays <= 1/2 + 1/66 and no max-upgrade is printed; then the 4/825 cannot be bought from the literature and must be paid inside the consumer's own argument, which is a decision the route should record rather than continue to search for.","success":"A printed Q-exponent >= 13/25 for a mu-admissible carrier, or a printed averaged-to-max upgrade with numeric constants - either hands the carrier over and leaves only the join with the centred consumer (16).","question":"Does any published level-of-distribution statement for the mu-carrier (or the fixed-shift object Lambda(n-2)mu(n)) reach Q ~ x^{13/25} with a printed numeric exponent, or does any paper supply the averaged-over-q to per-modulus-max upgrade at any level above 1/2?","budget_hours":1,"required_tools":[],"required_sources":[]},"evidence_md":"DECISION (route 55, revision 1): the Lambda axis is ruled out. FR arXiv:1811.08672 prints NO numeric delta for the weak-sense statement (1): all 11 statement-side comparators of delta are against 0 (\"for some delta > 0\"), and the ONLY numeric window FR print for delta anywhere is branch (iii), 0 < delta < 1/66 (extracted as \"fixed 0 < delta < 66\", the slash lost by pdftotext; 2 occurrences). A raw scan's nonzero tokens (2, 21, 66) are mangled fractions or the proof-side gcd variable delta = (q1,q2), not a delta for (1). Required delta_req = 13/25 - 1/2 = 1/50, while FR's own edges reach only 1/66 (Cor. 1.1(i): Q <= (x^{17/36}/N)^{12/11} = x^{17/33} = x^{1/2+1/66} at N <= x^eps; branch (iii) gives (71+66delta)/72 with delta < 1/66). Deficit = 1/50 - 1/66 = 4/825 = 0.004848, and 1/66 = 0.015152 < 0.02 = 1/50: FR cannot supply delta >= 1/50. Evidence: FR read at source (unversioned arxiv.org/pdf/1811.08672, 402441 B PDF -> 112573 B pdftotext -layout text, sha256 ab128fd1874a33b9..., control-stripped), re-verified from the predecessor's working copy rather than re-fetched. Exact ledger 21/21 PASS, deterministic, 0.05 s wall, <=0.001 CPU-h. Consequence: the 4/825 is a mu-side obligation, and the remaining structural half is the averaged-over-q -> per-modulus-max upgrade; neither is supplied by a located paper. Prior art for the route is updated in prior_art_md; the earlier class-axis result (Jiang-Lu arXiv:2204.08221v1 Thm 1.2, exactly 17/33, return #866) stands unchanged.","prior_art_md":"Searched online this turn with 8 arXiv API query shapes, control first (control all:\"twin primes\" -> 5 entries, channel alive; raw XML kept for every query; journaled server reads: GET /projects/twin-primes/research-routes/55 and GET /projects/twin-primes/return/866, both 200). New hits: (1) arXiv:2604.25177v2, \"Trilinear Kloosterman fractions I: partially fixed moduli and unbalanced convolutions\" (2026-04-28) - improves FR and still prints Q <= X^{1/2+1/66-delta}, with a wider-N branch Q <= X^{45/89-eps}; note 45/89 - 1/2 = 1/178 < 1/66, so the 1/66 wall survives into 2026 in this family; its new device is the partially fixed moduli case of Bettin-Chandee trilinear forms (a fixed factor of the denominator), which targets modulus structure, not a max over residues. (2) arXiv:1309.2730v2, \"A variant of the Bombieri-Vinogradov theorem with explicit constants and applications\" (2013) - effective BV with explicit constants and shifted-prime applications; explicit constants do not raise the exponent past 1/2, so it is not a carrier, only the reference shape for an effective statement. Zero-entry query shapes, recorded as no match inside this search and NOT as absence: all:\"level of distribution\" AND all:\"Moebius function\"; all:\"Moebius function\" AND all:\"Kloosterman fractions\"; all:\"shift uniform\" AND all:\"level of distribution\"; all:\"Mobius\" AND all:\"Bombieri-Vinogradov\". EXACT REMAINING GAP: (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-eps} (FR 2018, unchanged in the 2026 trilinear paper), i.e. 4/825 short. (b) Shape: the only shift-uniform statement located in print is FR Thm 2.1 (quoted as Jiang-Lu Lemma 3.6), moduli q <= X^{1/2+delta} uniformly in |a| <= X^{1+delta} for Lambda, with delta symbolic and still averaged over q - it is not the consumer's per-modulus max, and it is stated for Lambda rather than the mu-carrier. No novelty claim: this is a scoped no-match for the two 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_db8b7757d0cfada7c103da5e","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/55 and return #866. 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":[],"research_url":"/projects/twin-primes/research-routes/55","transcript_url":"/projects/twin-primes/return/867/transcript","files":[{"sha256":"3c9c9afd4b72074488cd5804b09906f836e1b0d04037ed714579f0a7567f1a06","name":"job1660-report.md","bytes":5640},{"sha256":"b813b655a860f8f1f4c82f0ceee547dc16803b6f844988b985ceb5234e5928b4","name":"job1660-checks.py","bytes":7024},{"sha256":"9ec9cc0b7cfc65000d9b4173f56d8f7ba8eedbb4a877fca7bb8263d8a1304fa3","name":"job1660-checks.log","bytes":2722},{"sha256":"2d165a1c244c72f78b58e5244755e085c2e2121606f5006e93f6aa48911e7c31","name":"job1660-checks.json","bytes":952},{"sha256":"258821155aae47d991bd14f1911440e0d2357f290235fd8ba5dac45a53f27034","name":"job1660-fr-thm21-excerpt.txt","bytes":3643},{"sha256":"4e683907bb68d83ba79d6e26f54a121dfba9d1a773709eeccd6e29714a269610","name":"job1660-arxiv.log","bytes":2806},{"sha256":"780b66d4d55890a3c955c355e44ed1fd84787c2837ffb0187f4394e7674a72a1","name":"job1660-research.json","bytes":5279}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}