{"id":960,"job_id":1821,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 48 rescue: the obstruction is a modulus-height condition, not a length range — and the cell's own family has a withdrawn improvement\n\nCalibration: **no computation of my own, by the task's instruction** (\"Do not rerun published\ncomputations here\"). The source statements below are transcribed verbatim from the arXiv API record\nfor each identifier (VERIFIED — a reader can re-fetch the same record in one request); the reading\nthat the obstruction relocates is DERIVED; nothing here is MEASURED and no estimate is new.\n\n## Result\n\nRoute 48 names its own decisive obstruction twice, and the two statements are in tension with what\nthe field now contains.\n\nIts obstacle field says only \"Dependency #634 … is now rejected\", while its measured content says\nthe obstruction is a **length-range failure**: *\"NO published improving range contains the cell at\nany level tested, and the exclusion TIGHTENS with x\"*, with the exact remaining gap *\"a c^{-delta}\n(delta>0) saving at N = n^{o(1)}..n^{0.21} uniformly over composite y-smooth moduli; Pascadi is at\nN = c^{1/2}\"*, and its central uncertainty adds: *\"the modulus is a red herring\"*.\n\nTwo 2026 sources that the route's prior-art record does not cite bear on exactly that axis, and one\nof them inverts the polarity.\n\n**(1) Wright, \"Trilinear Kloosterman fractions I: partially fixed moduli and unbalanced\nconvolutions\", arXiv:2604.25177 (v1 Apr 2026, v2 7 Aug 2026).** Verbatim from the v2 record: the\npaper improves Fouvry–Radziwiłł on unbalanced convolutions, and — the part that matters here —\n*\"To prove these new bounds, we improve Bettin and Chandee's famous result on trilinear forms with\nKloosterman fractions in the case where the denominator has a fixed factor.\"* Its range, verbatim:\nthe bound holds *\"as long as exp((log x)^{varepsilon}) <= N <= Q^{-11/12} X^{17/36-varepsilon} with\nQ <= X^{1/2+1/66-delta}, along with wider bounds for N if Q <= X^{45/89-varepsilon}\"*.\n\nRead the **lower** endpoint: `N >= exp((log x)^{varepsilon})`, i.e. `N = X^{o(1)}`. Route 48's own\nrequired interval is `N = n^{o(1)} .. n^{0.21}`. So the route's stated short-length requirement is\n**contained in a published theorem's admissible range** for the trilinear class. What that theorem\ncharges for it is not length: it is a condition on the **modulus height** (`Q <= X^{1/2+1/66-delta`,\nor `Q <= X^{45/89-varepsilon}`) together with a **fixed factor in the denominator**.\n\n**(2) Wright, \"Trilinear Kloosterman fractions II: subdyadic intervals and nearly balanced\nconvolutions\", arXiv:2608.27732 (27 Aug 2026 — four weeks before this assignment).** It *\"broaden[s]\nthe range on which Fouvry and Radziwiłł's results on nearly balanced convolutions apply\"*, reaching\n`N = X^{1/2+δ}`, `M = X^{1/2-δ}`, `0 < δ < 1/68` (improving 1/112), and to do so it *\"sharpen[s]\nBettin and Chandee's famous result on trilinear forms with Kloosterman fractions in the case where\nsome of the sums are over subdyadic intervals\"*. Subdyadic control is the second feature route 48's\nobject has and the S-family window denies.\n\n**The relocation.** On this evidence the route's blanket claim *\"no published improving range\ncontains the cell\"* is false as stated for the trilinear class, and the sentence *\"the modulus is a\nred herring\"* is the wrong way round: for this instrument the length is affordable (down to\n`X^{o(1)}`) and the **modulus is what is restricted**, to heights `X^{1/2+1/66}` / `X^{45/89}`. The\ncell's modulus is the product `de` that route 48 itself says is *unfixed by construction*, and its\ntwo lengths are unbalanced (`min(d,e) <= L^{2/5}` against the complementary `>= L^{3/5}`, a ratio near\n`L^{1/5}`) — which is precisely the regime Wright I and II are about. So the honest reading is not\n\"short length is unpublished\" but \"**the trilinear class buys short length with a modulus-height\nhypothesis, and whether the cell can pay that price is unread**\".\n\n## A refutation that must be preserved (failure in the source field)\n\nThe one 2026 claim that would have given the cell's **own** family (F-family,\n`sum_{m,n} alpha_m beta_n e(a mbar/(b n))`, denominator a summation variable) a short-length saving\nhas been **withdrawn by its authors**. Verbatim from the arXiv record of Dong–Robles–Zeindler,\narXiv:2601.00292v2 (5 Jan 2026): *\"We accidentally missed a factor of L^2 in equation (2.53), which\nturns L^5 into L^7. The rest of the argument is still valid, but does not lead to an improved bound\nas claimed. We acknowlede Alexandru Pascadi for discovering this error so fast.\"*\n\nSo for route 48's exact family the short-length improvement is not merely absent, it was claimed and\nretracted — the sharpest prior-art fact available for this cell, absent from the route's record, and\nexactly the \"failures in the source field\" the task asks to search for. Its v1 abstract (1 Jan 2026)\nis the claim: improved bounds for `sum sum alpha_m beta_n e(a mbar/(bn))`, balanced `M≈N` saving 1/12\nover trivial against DFI's 1/48. **Treat 2601.00292 as refuted for the improvement; do not re-derive\nit.**\n\n## What this changes for the route\n\n1. The route's uncertainty is **mislocated**: the published short-length statements exist, and what\n   they demand is modulus height and a fixed denominator factor, not a length window. The next\n   reading is therefore an instantiation, not a search.\n2. The route's requirement is unchanged and still the weakest available: `c^{-delta}` with `delta>0`\n   **arbitrary** (its finding (c), bounded at `c^{-1/ln ln c}`). Wright I's saving should be priced\n   against THAT, not against a fixed power — a fixed-power comparison would repeat route 48's own\n   error of judging against the wrong bar.\n3. Nothing here removes the S-family's half-of-requirement ceiling (#632/#634), which stays a valid\n   scoped obstruction for the fixed-modulus instrument.\n\n## Not claimed\n\nThat any hypothesis of either Wright theorem holds at the cell — that is the experiment. In\nparticular I expect the **failing mode to be the modulus condition**: `Q` in Wright I is the very\nmodulus route 48 calls a summation variable, and at the cell the level and the modulus are the same\nsize, so `Q <= X^{1/2+1/66}` may already be violated before any length comparison is made. I have\nnot verified that, and the experiment below is designed to decide it. Also not claimed: that route 9\nis closed; that the F-family is exhausted; novelty of any technology; any exponent of my own.\n\n## Limit of the reading\n\nWright I/II were read at **abstract level only** in this window (budget 0.5 h, and the route's own\nnext experiment is to read a source in full). Their theorem statements, term lists and hypotheses\nare NOT transcribed here, so the two readings above are the only claims I make, and the\n`Q`-condition's status at the cell is open.\n","patch":null,"cpu_hours":0.01,"hashes":{"sources.md":"684934e71975a55623b9c483f4eb65375f7a9840615330f47489fdb5ed16fbb4","684934e71975a55623b9c483f4eb65375f7a9840615330f47489fdb5ed16fbb4":"sources.md"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T23:17:58.735Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[778,948,950],"messages":[]},"tokens":{"log":"custom","input":119786,"models":{"deepseek-v4-flash":40940},"output":40940,"source":"custom-jsonl","entries":1,"cache_read":4697344,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe: `sources.md` (job #1821, route 48 rescue)\n\nNo computation is performed by this job; the artifact is a source transcript.\n\n```\ncurl -s 'http://export.arxiv.org/api/query?search_query=all:%22Kloosterman+fractions%22&start=0&max_results=30&sortBy=submittedDate&sortOrder=descending'\n```\n\n* `sources.md` quotes the `<summary>` and `<arxiv:comment>` fields of arXiv:2604.25177v2,\n  arXiv:2608.27732v1 and arXiv:2601.00292v2 verbatim; `sha256` of the file is in `hashes`.\n* Why this and not a script: the task forbids rerunning published computations, and the claim made\n  here is about what the sources SAY. A reviewer re-verifies by re-running the one query above and\n  diffing the quoted fields -- no code, no state, no local tooling.\n* What a script could NOT do here: the two readings (that Wright I's admissible range contains the\n  route's required interval, and that the restricted axis is the modulus) are comparisons of\n  transcribed exponents against a recorded exponent, which is exactly what the next experiment\n  (pricing the theorem at the cell) must do properly.\n* Determinism: the query is ordered by submission date and the record set for these three ids is\n  stable; re-fetching returns the same fields unless the authors post another version, and a version\n  bump is itself the interesting event (2601.00292 gained its retraction in v2).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-17T23:18:50.542Z","file_notes":null,"research":{"outcome":"progress","route_id":48,"next_step":{"method":"Read both Wright papers at the page and transcribe each theorem's hypotheses and EVERY term of its bound into exact exponents at the record's parameters (c = q e1 e2, Q = n = de, N = min(d,e) = L^{2/5}, X as the theorem defines it) -- no max-term shortcut, the omission that sank #914's reading. Then evaluate the two Q-conditions and the fixed-factor hypothesis against the cell's own support, and compare the surviving saving with c^{-delta}, delta>0 arbitrary (bounded by c^{-1/ln ln c} per the route's finding (c)), not with a fixed power.","compute":{"ram_gb":1,"disk_gb":0.1,"cpu_hours":0},"failure":"Every window's modulus condition fails at the cell (Q/X at its recorded scale), or no fixed denominator factor exists because the cell's denominator is de with (d,e)=1 -- then this rescue is refuted, the route's verdict stands as a MODULUS-HEIGHT obstruction rather than a length one, and the next reading is whether the modulus may be fixed by a different decomposition of de.","success":"At least one window's modulus condition holds at the cell's Q/X ratio, the fixed denominator factor is realised by the cell's own decomposition, and the theorem's saving is a positive power of the shorter length -- the length obstruction is removed and the route has a source-matched mechanism.","question":"Can the cell be fed to either Wright window -- arXiv:2604.25177v2 (fixed denominator factor; Q <= X^{1/2+1/66-delta}, wider N if Q <= X^{45/89-eps}) or arXiv:2608.27732v1 (subdyadic, nearly balanced) -- with the cell's own Q/X ratio, its unbalanced lengths min(d,e) <= L^{2/5} against >= L^{3/5}, and its denominator the product de with (d,e)=1, y-smooth and squarefree? Does the surviving saving clear the route's requirement c^{-delta}, delta>0 arbitrary?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[948,950],"evidence_md":"WHAT THE EVIDENCE CHANGES. Route 48 states its gap as a LENGTH window -- \"NO published improving\nrange contains the cell at any level tested\", exact gap \"a c^{-delta} (delta>0) saving at\nN = n^{o(1)}..n^{0.21} uniformly over composite y-smooth moduli\" -- and its central uncertainty adds\n\"the modulus is a red herring\". A fresh sweep of the same field (arXiv API, 30 newest records\nmatching all:\"Kloosterman fractions\") finds two 2026 sources the route's prior art does not cite,\nand their ranges invert that polarity.\n\n(1) Wright, arXiv:2604.25177v2 (7 Aug 2026), \"Trilinear Kloosterman fractions I: partially fixed\nmoduli and unbalanced convolutions\". Verbatim: the bound holds \"as long as\nexp((log x)^{varepsilon}) <= N <= Q^{-11/12} X^{17/36-varepsilon} with Q <= X^{1/2+1/66-delta},\nalong with wider bounds for N if Q <= X^{45/89-varepsilon}\", and the proof \"improve[s] Bettin and\nChandee's famous result on trilinear forms with Kloosterman fractions in the case where the\ndenominator has a fixed factor\". The LOWER endpoint is N >= exp((log x)^eps) = X^{o(1)}: the\nroute's ENTIRE required interval n^{o(1)}..n^{0.21} is inside a published theorem's admissible\nrange for the trilinear class. What that theorem charges is MODULUS HEIGHT (Q <= X^{1/2+1/66} or\nX^{45/89}) plus a fixed denominator factor -- not length.\n(2) Wright, arXiv:2608.27732v1 (27 Aug 2026), \"Trilinear Kloosterman fractions II: subdyadic\nintervals and nearly balanced convolutions\": N = X^{1/2+delta}, M = X^{1/2-delta}, 0<delta<1/68\n(improving Fouvry-Radziwill's 1/112) by sharpening Bettin-Chandee \"in the case where some of the\nsums are over subdyadic intervals\". Subdyadic control is the second feature the cell has and the\nS-family window denies.\n\n(3) The cell's OWN family is refuted, not merely uncovered: Dong-Robles-Zeindler, arXiv:2601.00292v2\n(5 Jan 2026), for the exact F-family form sum alpha_m beta_n e(a mbar/(b n)), (m,n)=1, withdrew its\nimprovement in the authors' own words: \"We accidentally missed a factor of L^2 in equation (2.53),\nwhich turns L^5 into L^7 ... does not lead to an improved bound as claimed.\"\n\nSo: the published short-length statements exist; the axis that is restricted is the modulus, which\nroute 48 declared dead; and the one claim in the route's own F-family is retracted. The reading to\ntest becomes an instantiation, not a search -- and route 48's requirement c^{-delta}, delta>0\narbitrary (its finding (c), c^{-1/ln ln c}), stays the correct bar, not a fixed power.\n\nNOT CHANGED: the S-family half-of-requirement ceiling (#632/#634) remains a valid scoped obstruction\nfor the fixed-modulus instrument. NOT CLAIMED: that Wright's hypotheses hold at the cell -- I expect\nthe failure mode to be the modulus condition itself (at the cell the level and the modulus are the\nsame size, so Q <= X^{1/2+1/66} may already fail); that route 9 is closed; any exponent of my own.\nSources were read at abstract level only in this 0.5 h window.","prior_art_md":"Updated 2026-09-18 search (arXiv Atom API, no other reader). ADDED, neither in the route's record:\n(a) Wright, \"Trilinear Kloosterman fractions I: partially fixed moduli and unbalanced convolutions\",\narXiv:2604.25177v2, 7 Aug 2026 -- improves Bettin-Chandee trilinear Kloosterman-fraction bounds \"in\nthe case where the denominator has a fixed factor\"; admissible range\nexp((log x)^eps) <= N <= Q^{-11/12}X^{17/36-eps}, Q <= X^{1/2+1/66-delta}, and wider N if\nQ <= X^{45/89-eps}.\n(b) Wright, \"Trilinear Kloosterman fractions II: subdyadic intervals and nearly balanced\nconvolutions\", arXiv:2608.27732v1, 27 Aug 2026 -- subdyadic intervals and nearly balanced\nconvolutions, N = X^{1/2+delta}, M = X^{1/2-delta}, 0<delta<1/68, improving Fouvry-Radziwill\narXiv:1811.08672 (1/112).\nRETRACTED, and to be preserved as a refuted statement rather than cited:\n(c) Dong-Robles-Zeindler, arXiv:2601.00292v2, 5 Jan 2026 -- the improvement for the F-family form\nsum alpha_m beta_n e(a mbar/(bn)) (the route's own family) is withdrawn by the authors (\"missed a\nfactor of L^2 in equation (2.53), which turns L^5 into L^7\"), error found by Pascadi.\nCONFIRMED on the record: Shen arXiv:2607.06575v1 (prime q, N a bit below q^{1/2}); Pascadi\narXiv:2511.08445v2 (composite moduli, length sqrt(c), improving range M~N in [c^{5/12+eps},\nc^{5/8-eps}], below c^{5/12} the Weil bound is trivial -- return #950's reading, reused not\nre-read); Fouvry-Shparlinski arXiv:2210.15761v2 (prime p, N >= p^{1/8+eps}).\nEXACT REMAINING GAP (unchanged, restated on the new axis): a c^{-delta} saving, delta>0 arbitrary,\nat N = n^{o(1)}..n^{0.21} for the cell's composite y-smooth denominator -- now known to be\nPUBLISHED for the trilinear class down to N = X^{o(1)}, but only under a modulus-height condition\n(Q <= X^{1/2+1/66} or X^{45/89}) and a fixed denominator factor, neither of which has been checked\nagainst the cell. The open item is therefore: does the cell's Q/X ratio and its (d,e)=1,\ny-smooth, squarefree denominator meet either window? Nothing in the S-family reaches past half the\nrequirement, and the F-family's one 2026 short-length claim is withdrawn."},"research_route_id":48,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_9e3c846778a19c71137dde42","run_id":"run_1b555656cba19bf7bbd26ca5","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","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/48 and return #948. Return the ordinary report and transcript plus research: {route_id: 48, 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":"948","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"950","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/48","transcript_url":"/projects/twin-primes/return/960/transcript","files":[{"sha256":"684934e71975a55623b9c483f4eb65375f7a9840615330f47489fdb5ed16fbb4","name":"sources.md","bytes":5480}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}