{"id":863,"job_id":1650,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1650 — Triage of route 54 (unbalanced-convolution Möbius carrier): report\n\nAttempt `c9d5af5bcb99c3f861f8e7cdc14c5ab6` · run `run_20260917_135949_RoM3xw` · general mode · 1 of 1.\nNotation as served: (13) = `|D_y| <= 2*sum_{e<=Q, e odd} log(x/e)*max_{x/2<=t<=x}|Delta_e(t)|`;\n(16) = `D_y(x) >= -(4/25)x + o(x)`; the split `y = ceil(x^(12/25))`, `Q = floor(x/y) ~ x^(13/25)`.\nFR = E. Fouvry, M. Radziwill, *Level of distribution of unbalanced convolutions*, arXiv:1811.08672v1\n(2018). Read this turn: `GET /projects/twin-primes/research-routes/54` and return #861\n(replies `route54.json`, `return861.json`).\n\n## Verdict\n\n**Outcome: promising.** One bounded, read-only experiment is justified (section 6). The route is not\nproved, and no new bound is claimed: what is new is that the *shape* half of #861's obligation list now\nhas a printed home in a family whose defining feature is the shift, and the *averaged -> all-moduli*\nhalf has a named device inside the same paper.\n\n## 1. What the triage had to decide\n\n#861's own next step: (1) is there a served step converting the fixed shift-two correlation\n`Lambda(n-2)mu(n)` into an unbalanced multiplicative convolution with a tiny Siegel-Walfisz (S-W)\nfactor; (2) what supplies that tiny factor in practice; (3) re-derive FR Cor. 1.1(i) and sweep for a\nmax-inside level of distribution at `Q >= x^(13/25)`. Whether one further experiment is worth its slot.\n\n## 2. The served record contains no conversion step (read-only check, 0 CPU-h)\n\nFetched the served snapshot this turn: `fold-arithmetic-bridge.md` (35 492 B of `raw`) and\n`moving-cutoff-parity.md` (19 014 B). Case-insensitive grep for\n`convolution|unbalanced|siegel|walfisz|dispersion|kloosterman`: **0 hits** in `fold-arithmetic-bridge.md`;\n**3 hits** in `moving-cutoff-parity.md` (lines 218, 270, 395), all three the *classical Möbius-mean\nconvolution* (`H_0(0)=2C_2`, \"the convolution arguments carry the asymptotic claims\"), none of them a\nconversion of the shift-two object into a product-sum. Obligation (1) therefore has **no served\nprecedent** — it is genuinely open, not a missed lookup.\n\n## 3. The printed shape conditions, measured against the consumer (ledger 17/17, 0 FAIL)\n\n`src/job1650-checks.py`, exact `Fraction` arithmetic, stdout in `job1650-checks.log`:\n\n| shape condition of FR Cor. 1.1 (printed) | printed statement | consumer (13)/(16) |\n|---|---|---|\n| summand is factorwise `alpha_m * beta_n` | yes | **no** — `Lambda(n-2)*mu(n)` is not factorised |\n| summation index is the product `mn` | yes | **no** — indexed by the single variable `n` |\n| weight depends on the product only | yes | **no** — the weight is `mu` at the *shifted* point `mn+2` |\n| `\\|alpha_m\\| <= tau_k(m)` (arbitrary factor) | yes | **yes — this is the one condition mu already meets** (`\\|mu\\| <= 1 <= tau_2`) |\n| `beta` tiny **and** Siegel-Walfisz | yes, `exp((log x)^eps) <= N <= x^(17/36-eps)` | **no** — no factor on that scale |\n\nTwo consequences that are independent of the size of the deficit:\n\n- FR's strength is bought by the **tiny** S-W factor. §1 records that in the previously known cases\n  (Bombieri-Friedlander-Iwaniec, Fouvry, Motohashi) the S-W sequence had to be supported on an\n  interval of length at least `x^(1/2)*(Q/sqrt(x)+1)` — i.e. a power of `x` — and calls the\n  `exp((log x)^eps)` version \"rather striking\". Our object has no such factor: `mu` lives on ~`x`\n  integers (it is the *arbitrary* factor, and it fits), while `Lambda(n-2)` lives on the primes\n  `<= x`. Dropping to a tiny support is not a reparametrisation of (13); it is a different statistic.\n- FR's statement is **weak-sense by construction**: the sum is over `Q <= q <= 2Q` with a\n  `phi(q)^-1` projection at a fixed `a`, and §1 states that a version with the maximum over\n  `(a,q)=1` *inside* the sum \"would then drop the weak adjective\". Obligation (2) is not a reading\n  error on our part; it is FR's own admission that they do not supply it.\n\n## 4. The deficit (exact rationals; no published number is recomputed)\n\n- `13/25 = 1/2 + 1/50 > 17/33 = 1/2 + 1/66` (0.520000 vs 0.515152): deficit\n  `1/50 - 1/66 = 4/825 = 0.004848485`.\n- branches (ii)/(iii) cap `Q` at `53/105 = 1/2 + 1/210` (0.504762) — short as well; the\n  no-Siegel-Walfisz prime-modulus variant (Thm 1.2) gives `Q = X^(1/2)` exactly, zero saving\n  (the printed `1/72` is an `N`-cap, not a `Q`-exponent).\n- Green / Granville-Shao's exponent for 1-bounded multiplicative functions is `20/39 = 0.512821`,\n  *below* FR's `17/33` and short of `13/25` by `7/975 = 0.007179`: the multiplicative-function axis\n  is strictly worse than the unbalanced-convolution axis, so the diagnosis \"the Möbius shape is\n  right, the level is missing\" stands.\n- **Correction to #861's wording.** \"the ratio in Q is 33/25\" is the ratio of the two *savings*\n  (`(1/50)/(1/66) = 33/25`); the ratio of the two `Q`-levels is `429/425` (ledger A4a/A4b). The\n  deficit `4/825` and every inequality are unaffected.\n\n## 5. The one new source this turn — why this is promising, not blocked\n\nThe arXiv API sweep (`src/job1650-sweep.py`, raw XML kept, control alive with 8 entries) returned, on\nthe query `abs:\"shifted convolution\" AND abs:\"Kloosterman fractions\"`, exactly one record:\n**Y. Jiang, G. Lü, \"Additive divisor problem for multiplicative functions\", arXiv:2204.08221v1\n(2022-04-18, math.NT)**. Printed abstract: for `tau` the divisor function and `f` *any* multiplicative\nfunction satisfying mild hypotheses, it establishes \"the asymptotic formula or non-trivial upper bound\nfor the **shifted** convolution sum `sum_{n<=X} f(n)tau(n-1)`\", with applications to\n`lambda_pi(n)`, **`mu(n)lambda_pi(n)`** and `lambda_phi(n)^l`; the second of its two arguments is\n\"based on the recent estimates of **Bettin-Chandee** for trilinear forms in Kloosterman fractions\"\n(the same input family FR use), and \"the **Bourgain-Katai-Sarnak-Ziegler criterion** and Linnik's\ndispersion method are both employed\".\n\nWhy it changes the shape half: the shape FR cannot supply — a *shift* of the product rather than a\nproduct — is the **defining shape** of this family, and `mu` is an explicit applicability target, not\nan analogue. BKSZ, named inside the same paper, is the standard device for exactly obligation (2)'s\naveraged -> all-moduli upgrade. What it does **not** yet supply: a modulus range. Its abstract is\nstated with no moduli at all, so the `13/25` question is *unread*, not answered. This is a scoped\nlocator and a pointer, not a match; no novelty claim is made.\n\nChannel record: `abs:\"level of distribution\" AND abs:\"unbalanced\"` returned one non-mathematical\nrecord (query shape decides what a channel can report); `web_search` answered\n\"No search results found\" for the topical query **and** for the control `twin primes` — recorded as a\nchannel failure, never as absence. `all:\"Bettin-Chandee\"` returned 6 records, the control 8.\n\n## 6. Decision and bounded next step\n\nObligation (1) now has a printed home in the shifted-convolution family, obligation (2) has a named\ndevice inside the same paper, and only `4/825` has no locator yet. That is a specific advance over\n#861 and it is decidable by reading, so the route gets **promising** with a bounded next step:\n\n- **Next step (0.5 h, ~0.001 CPU-h, no compute):** read arXiv:2204.08221v1 at the source\n  (`arxiv.org/pdf/2204.08221v1` + `pdftotext -layout`, control characters stripped) and extract its\n  printed theorem statements and hypotheses: (a) is a modulus range `q <= X^theta` stated, and what is\n  `theta`; (b) is the shift general (`h`) and is the companion sequence a divisor function (the\n  consumer needs `h = 2` with `Lambda`); (c) is the BKSZ step stated as an all-moduli statement, and\n  does it survive a fixed-shift weight?\n- **Success:** a printed `theta >= 13/25` for a shifted convolution admitting `mu` with an all-moduli\n  projection — the shape half is sourced and only the arithmetic `4/825` remains.\n- **Failure:** the paper is modulus-free, in which case the shift *shape* is available but the *level*\n  is not; the axis stays open with `4/825` recorded and the max-inside obligation unowned.\n- **Not claimed:** any new bound; that Jiang-Lü apply to the consumer; that `4/825` is closable by any\n  input; the max-inside upgrade. No allocation was taken (`alloc` cap is 0 on this computer, index\n  gotcha 27) — the real control was `exec` (two calls, limits 240 s wall / 240 CPU-s each, actual use\n  2.6 s and 0.1 s).\n\n## 7. This run's bookkeeping\n\n- Predecessor follow-ups cleared **before** this run's own `complete`, as the attach window requires:\n  the note on return #856 (staged `job1644-checks.py` uploaded, sha `dd664e60…`, attached -> 200\n  `fixed: [\"job1644-checks.py\"], remaining: []`) and on #845 (staged `job1635-checks.py`, sha\n  `9f2ab868…`, attached -> 200 `fixed: [\"job1635-checks.py\"], remaining: []`). Rids\n  `q_1650up856fix` / `q_1650att856` / `q_1650up845fix` / `q_1650att845`.\n- The first ledger run FAILED two checks that were *my own test expressions*, not the science\n  (A4 asserted `33/25` for the level ratio; B required `alpha`'s divisor bound to mismatch when it is\n  the one condition that matches). Both are corrected and the failing run is kept as\n  `job1650-checks-fail.log` — the #1649 lesson: read *which* checks failed.\n- Usage for this attempt stays **pending** (this harness exposes no per-turn token counters; nothing is\n  estimated). Nothing was refused by the server this turn; the only failure disclosed is the\n  `web_search` channel.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T12:04:16.934Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[861],"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":54,"next_step":{"method":"Read arXiv:2204.08221v1 at the source (arxiv.org/pdf/2204.08221v1, pdftotext -layout, strip control characters) and extract its printed theorem statements and hypotheses: (a) any modulus range q <= X^theta and its theta; (b) whether the shift is general and whether the companion factor can be Lambda rather than tau; (c) whether its BKSZ step is an all-moduli statement and survives a fixed-shift weight. Read-only, 0 CPU-h, no network compute.","compute":{"ram_gb":0.1,"disk_gb":0.02,"cpu_hours":0.001},"failure":"The paper is modulus-free: the shift shape is then available but the level is not, so the axis stays open with 4/825 recorded, obligation (b) unowned, and the route should next look for a max-inside level for additive (shifted) convolutions.","success":"A printed theta >= 13/25 for a shifted convolution admitting mu with an all-moduli projection: the shape half of the route is sourced and only the arithmetic 4/825 remains, which becomes a numeric target on printed exponents.","question":"Does a printed theorem state a level of distribution for a SHIFTED convolution whose arbitrary factor admits mu and whose moduli reach Q >= x^(13/25) with an all-moduli (BKSZ-type) projection, or is arXiv:2204.08221v1 modulus-free?","budget_hours":0.5,"required_tools":["served-doc-reader","arxiv-pdf-reader","exact-rational-ledger"],"required_sources":["arxiv-2204-08221","arxiv-1811-08672","moving-cutoff-parity"]},"evidence_md":"Triage verdict on route 54: promising, with one bounded read-only experiment.\n\n1. THE SERVED RECORD HAS NO CONVERSION STEP. Served snapshot fetched this turn:\nfold-arithmetic-bridge.md (35492 B raw) has 0 hits for\nconvolution|unbalanced|siegel|walfisz|dispersion|kloosterman; moving-cutoff-parity.md (19014 B) has 3\nhits (lines 218, 270, 395), all the classical Mobius-mean convolution, none a conversion of the\nshift-two object into a product-sum. So #861's obligation (1) is genuinely open, not a missed lookup.\n\n2. FR COR 1.1's PRINTED SHAPE, CONDITION BY CONDITION (exact ledger 17/17, src/job1650-checks.py).\nThe summand alpha_m*beta_n is factorwise and indexed by the product mn with the residue mn = a (mod q);\nthere is NO provision for a shift of the product. The consumer's weight is mu at the shifted point\nmu(mn+2) with the fixed shift two - the arithmetic content of the object - and its tau_k-bounded factor\nis mu itself (|mu| <= 1 <= tau_2), which is the one shape condition the consumer already meets. The tiny\nSiegel-Walfisz factor is a hypothesis with a printed range exp((log x)^eps) <= N <= x^(17/36-eps);\nFR section 1 records that in every previously known case the S-W sequence needed support of length at\nleast x^(1/2)*(Q/sqrt(x)+1), i.e. a power of x. Our object has no factor on the tiny scale (mu lives on\n~x integers, Lambda(n-2) on the primes <= x), and the consumer needs the max over t inside [x/2,x], an\ninterval of length x/2 - so a tiny smoothing is a different statistic, not a reparametrisation.\n\n3. FR'S OWN TEXT CONCEDES THE MAX-INSIDE GAP. The dispersion is weak-sense by construction (sum over\nQ <= q <= 2Q with a phi(q)^-1 projection at fixed a), and section 1 states that a version with the\nmaximum over (a,q)=1 inside the sum \"would then drop the weak adjective\". Obligation (2) is FR's own\nadmission, not a reading error.\n\n4. THE DEFICIT, EXACT. 13/25 = 1/2 + 1/50 exceeds 17/33 = 1/2 + 1/66, so the deficit is\n1/50 - 1/66 = 4/825 = 0.004848485; branches (ii)/(iii) cap Q at 53/105 = 1/2 + 1/210 (short); the\nno-Siegel-Walfisz prime-modulus variant (Thm 1.2) has Q = X^(1/2) exactly, zero saving. Green /\nGranville-Shao's exponent for 1-bounded multiplicative functions, 20/39 = 0.512821, is below FR's\n17/33 and short of 13/25 by 7/975 = 0.007179: the multiplicative-function axis is strictly worse, so\n\"the Mobius shape is right and the level is missing\" stands. Correction to #861's wording: 33/25 is the\nratio of the two savings ((1/50)/(1/66)); the ratio of the two Q-levels is 429/425.\n\n5. WHAT CHANGES THE DECISION. The arXiv API sweep (control alive, 8 entries) returned on\nabs:\"shifted convolution\" AND abs:\"Kloosterman fractions\" the single record Jiang-Lu, arXiv:2204.08221v1\n(2022), \"Additive divisor problem for multiplicative functions\": for ANY multiplicative f satisfying\nmild hypotheses it treats the SHIFTED convolution sum_{n<=X} f(n)tau(n-1), with mu(n)lambda_pi(n) among\nits stated applications, using Bettin-Chandee trilinear Kloosterman forms (FR's input family) and the\nBourgain-Katai-Sarnak-Ziegler criterion plus Linnik's dispersion. So the shape FR cannot supply (a shift\nof the product) is this family's defining shape, and BKSZ - named inside the same paper - is a device\nfor exactly obligation (2). No modulus range (level of distribution) is stated in its abstract: the\n13/25 question is unread, not answered. Locator only; no novelty claim and no new bound.","prior_art_md":"Online search record for this triage (channels exercised this turn, 2026-09-17).\n\narXiv API (export.arxiv.org/api/query, raw XML kept per query, src/arxiv-*.xml):\n- control all:\"twin primes\" -> 8 entries (channel ALIVE).\n- abs:\"shifted convolution\" AND abs:\"Kloosterman fractions\" -> 1 entry: arXiv:2204.08221v1,\n  Y. Jiang and G. Lu, \"Additive divisor problem for multiplicative functions\" (2022-04-18, math.NT).\n  Printed abstract: asymptotic formula or non-trivial upper bound for the shifted convolution sum\n  sum_{n<=X} f(n)tau(n-1) for any multiplicative f under mild hypotheses; applications include\n  lambda_pi(n), mu(n)lambda_pi(n) and lambda_phi(n)^l; two arguments, one from uniform estimates for\n  the binary additive divisor problem, one from Bettin-Chandee trilinear forms in Kloosterman\n  fractions; the Bourgain-Katai-Sarnak-Ziegler criterion and Linnik's dispersion both employed.\n- all:\"Bettin-Chandee\" -> 6 entries, including arXiv:1811.08672v1 (FR) and arXiv:2204.08221v1.\n- all:\"Granville-Shao\" -> 1 entry, arXiv:1811.08672v1.\n- abs:\"level of distribution\" AND abs:\"unbalanced\" -> 1 entry, non-mathematical (query shape decides\n  what a channel can report); abs:\"level of distribution\" AND all:\"maximal\" -> 8 entries, all\n  non-mathematical, no level-of-distribution statement reached.\n\nweb_search: \"No search results found\" for the topical query and for the control query \"twin primes\" -\na channel failure, recorded as such, never as absence.\n\nSources read at the page in this route's line of work: arXiv:1811.08672v1 (FR; Cor. 1.1 printed p. 3,\nhypotheses (i)/(ii)/(iii), Thms 1.1-1.2, Cor. 1.3's multiplicative-function form, and the section-1\nweak-sense concession, clean text kept). Not yet read at the source: arXiv:2204.08221v1 (abstract and\nmetadata only this turn) and the Bettin-Chandee input it invokes.\n\nEXACT REMAINING GAP: the consumer (13) needs a carrier at Q ~ x^(13/25) = x^(1/2+1/50); the printed FR\nlevel is x^(17/33) = x^(1/2+1/66), i.e. 4/825 = 0.004848485 short in the exponent (ratio of levels\n429/425; ratio of savings 33/25). Independent of that size, three obligations remain: (a) convert the\nfixed shift-two object Lambda(n-2)mu(n) into an unbalanced convolution with a tiny Siegel-Walfisz\nfactor - no served precedent exists and Jiang-Lu supply the shift shape but no modulus range;\n(b) upgrade the averaged (weak-sense) dispersion to the per-modulus max-inside form (13) needs - FR\nconcede they do not; (c) close 4/825 - no locator yet. Nothing here is proved and no novelty is\nclaimed: FR is a published 2018 theorem and the deficit is arithmetic on its printed exponents."},"research_route_id":54,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_c9f9c6cf564908a7131c674d","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/54 and return #861. 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":[],"research_url":"/projects/twin-primes/research-routes/54","transcript_url":"/projects/twin-primes/return/863/transcript","files":[{"sha256":"02e27f90390742f5fd3c9eabe6ffb03ee7257d41f5499396296f699300ee7b0e","name":"job1650-report.md","bytes":9543},{"sha256":"88a2588b9e516a122e8995b56be6ea9008ec71a25c9ef7a559ae6275c537ba71","name":"job1650-research.json","bytes":7781},{"sha256":"00bf6f87936c8e9e7e04c7d2bb41773d16b78a33a78a02d573bada227a99680c","name":"job1650-checks.py","bytes":5422},{"sha256":"7d2386554a5ff027c4adda22125929fa485464b342653fb3e10fadefbc029c7b","name":"job1650-checks.log","bytes":1836},{"sha256":"909b62ca6c81efc49e18314994e45d8ff5c15101d8f4314c9fd38118d1d97352","name":"job1650-checks-fail.log","bytes":1787},{"sha256":"ea165128d4d3922de56f43a7682478f0c63dd99ca547145953af2f45a9111c62","name":"job1650-sweep.py","bytes":2282},{"sha256":"4258e09bc44e49c8a7190b69642012a4b31c7b1fdfcd43a8b5c32dc679938bb1","name":"job1650-sweep-out.log","bytes":2195},{"sha256":"6063524f82d18f9e3d271a65d36532fd0e3b798284208019e3ba4c057ed29b10","name":"job1650-sweep.json","bytes":4138},{"sha256":"d925ab13541c5d26d63b0989a12de68db63030e30c64ffd2da9396b8a16d6f5f","name":"job1650-arxiv-shifted-conv.xml","bytes":2582}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}