{"id":710,"job_id":1503,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1503 — route draft: a *weight-based* convolution transfer for the fixed-shift product\n\n**Type** explore (discovery), lane `formalize`, stage discover, no route, general mode.\n**Attempt** `3f6f3fb0e4cc4888158be1222ac92629`, session `58550ac87f68619b22fbe13a`, run\n`run_791e8b4580c51f71fc8b4284` (local run dir `run_20260916_164534_FUS2Vg`).\n**Rung of the central finding: SOURCE-READ + INFERRED.** Nothing re-derived, no twin-prime claim,\n**no novelty claim**; `cpu_hours: 0` (no compute of substance; all checks are milliseconds).\n\n## 1. Recovery and framework (before this assignment's first research step)\n\n`outstanding` = **0 of 33 attempts; all_complete=True** (exit 0), no live `sah.py` processes,\n`state/HANDOFF.md` read in full (every predecessor closed, including #1502/return #709 with\n`pending_ops=[]`); predecessor `PENDING.md` items all recorded closed/permanently blocked — none cheap\nwas left undone. Readiness re-exercised **this session** on this computer: `readiness` → 26/26\n(`sah/12`, sha256 `2173f7ad…c5fe4f`, 14:44:09Z); `tests/path_fixture.py` → 5/5 (an assignment with no\nsubmission is reported outstanding, exit 2). Identity was bound to **this turn's** chat directory\n(`…/chats/2026-09-16T14-43-35.493Z`, whose `chat-meta.json.firstPrompt` is this instruction verbatim\nand whose log starts 14:43:39Z = `fbctl` `last_task_sent`), not to the newest record in the shared\nfolder. `X-Model: deepseek/deepseek-v4-flash`, `X-Effort: unmeasured` (no effort/reasoning field is\nexposed by this app version; the checked sources are recorded in `state/identity/`).\n\n## 2. What the record already rules out (cited, not re-run)\n\n* #1502 / return #709 closed the **Motohashi 1976** channel: its transfer is for the **multiplicative\n convolution** `f·g`, its §2 engine is the large sieve over divisor-indexed characters, and\n `dispersion` occurs 0 times in the paper. No pair `(f,g)` has `f·g(n) = Λ(n−2)μ(n)`, and hypothesis\n (·) *for a shifted factor is already the fixed-shift prime character sum* the target needs.\n* #1501 / return #708 showed the sequence obstruction is **not algebraic**: the divisor-exchange\n identity is exact and its inner sum is the ordinary shifted class sum `ψ(N;d,−2)`, which ordinary\n Bombieri–Vinogradov already covers. What survives is the carrier's `d`-dependent **moving cutoff**.\n* #1497 / return #701 and the served note `moving-cutoff-parity.md` §5: the μ-only estimate (M) cannot\n supply target (16); the route is blocked by *sequence*, not by *level*.\n\nSo a **new** route must change the *input class*, not the bookkeeping.\n\n## 3. The finding\n\n**Zhiyuan Yang, “Convolution-type Bombieri–Vinogradov theorem with well-factorable weights, and its\napplications”, arXiv:2608.13299v2** (submitted 2026-08-13, updated 2026-09-03, 27 pp., math.NT) —\none author; read from the **arXiv API abstract** (saved reply\n`sources/arxiv-well-factorable.xml`, gotcha-free channel; the paper was a *title-only* hit in #1501,\nnow a full record).\n\nVerbatim, the abstract's key sentence:\n\n> The key to the proof is that **for a special class of convolution forms equipped with\n> well-factorable weights, we may use the level `x^{5/8−o(1)}`** for Pascadi's prime-distribution\n> result with triple-well-factorable weights. We also use Pascadi's estimation of incomplete\n> Kloosterman sums.\n\nIts two applications are `#\\{p ≤ x : P⁺(p−1) ≥ p^c\\}` and `#\\{n ≤ x : P⁺(n) < P⁺(n+1)\\}`.\n\n**Why this is a different input class from Motohashi's.** Motohashi's transfer is a property of the\n*multiplicative* convolution `f·g` and is proved by divisor decomposition; Yang's is a statement about\n**well-factorable weights acting on convolution forms**, i.e. the same *device* that Vaughan's identity\nproduces when it decomposes `Λ(n−2)` into `λ_d`-weighted pieces. The obstruction #1502 identified was\nthat Motohashi's hypothesis (·) is itself a fixed-shift character sum — a *sequence* obstruction, not a\nweight obstruction. A weight-based transfer need not presuppose the shifted sequence: it estimates the\nweighted convolution form directly, at a level (`5/8 − o(1)`) **far above** the `x^{1/5}` the route\nasks for (mechanical check C1c).\n\n**The decisive page was then read (this is the route's own “cheapest refuting check”, executed).**\nThe paper was obtained as a PDF (`https://arxiv.org/pdf/2608.13299v2`, 503 016 B, sha256\n`4e33b95475042711…8f0fe563`), converted with `pdftotext -layout`, and §1.3 read verbatim. Its key\ninput — the author's own words “**In our works, we encountered the following type of estimation\n(This is the key!)**” — is, with `L = x^ν`, `(γ_d)` and `(λ_q)` divisor-bounded, `d ∼ D = x^θ`,\n`q ≤ Q = x^{L(θ,ν)−ε}`, `λ_q` well-factorable of level `x^{L(θ,ν)−ε}` (Rosser–Iwaniec);\n\n```\nΣΣ_{d,q} γ_d λ_q [ Σ_{l∼L} Σ_{p θ ≥ 0`), so\n the exchanged class sum must be presented in that shape — i.e. `Λ(n−2)` decomposed so that the shift\n `−2` sits in the residue `a` of a product `lp ≡ −2 (mod dq)`. The `μ` factor of our product fits the\n `γ_d` slot: `(μ*μ)(d)` is 1-bounded, hence divisor-bounded, exactly as (1.1) requires.\n\n## 4. The route draft (object / step / cheapest refutation / cost)\n\n* **Object.** The carrier bound the exchanged form leaves open (#1501): `Σ_{e ≤ Q, e odd}\n log(x/e)·max_t |Δ_e(t)|`, `Q = ⌊x/y⌋`, `y = ⌈x^{12/25}⌉`, for the **fixed-shift product**\n `Λ(n−2)μ(n)` — equivalently the level-`x^{1/5}` fixed-shift BV-type input the record says is\n unavailable.\n* **Ingredient changed.** Replace the unavailable `(C)`+`(·)` pair (Motohashi §2's engine, closed for\n the fixed shift) by **Yang's well-factorable convolution-type transfer at level `x^{5/8−o(1)}`**,\n with `μ` as the second factor in the weight-decomposed form.\n* **Step that would have to hold.** The exchanged form must be presented in (1.1)'s shape: a bilinear\n sequence `l·p` with the shift carried in the residue `a = −2` modulo `dq`, `dq` **odd**, `λ_q`\n well-factorable, `(μ*μ)(d)` in the `γ_d` slot, and the inner shifted class sum reproduced by that\n bilinear count with its `1/φ(dq)` main term. This is *not* free: (1.1) counts `lp ≡ a`, not\n `Λ(dm − 2)`, so the decomposition must be exhibited. If it cannot be exhibited at the required\n `ν` (`0 ≤ θ < ν < 3/8`), the input does not apply as stated and the channel closes *at the shape*,\n not at the shift.\n* **Cheapest refuting check (source-first, one page) — EXECUTED for the quantifier half.** Done above:\n the hypotheses of (1.1) admit a fixed residue `a ≠ 0` with `(a,dq)=1`, so the shift is *not* what\n blocks the import; what remains is the **shape** requirement and the **odd-modulus** condition.\n Remaining check (one page, `cpu_hours ≈ 0`): read §2's derivation of (1.1) for `θ > 0` and copy\n whether `γ_d` may be supported on `d ∼ D = x^θ` with `θ = 0` (which is what the exchange's\n `d`-uniform sum needs, since the exchange has `d` ranging up to the full level, not a dyadic `x^θ`).\n *Second:* one finite control at `N = 10^6` comparing the `λ`-weighted truncated exchange against\n #1501's measured `1/φ(d)` portrait (1.662× truncated / 2.282× full). Cost ≤ 0.5 CPU-h.\n* **What it would cost to run.** One source read (done here for the quantifier half) + one bounded\n finite comparison: well inside a 2 h / ≤ 4 CPU-h slot. **Do not** start with the Kloosterman input:\n Yang imports it for the applications, not for the transfer. **Do not** re-read Motohashi: #1502\n closed it.\n* **Nearest prior work and the exact difference.**\n * Motohashi 1976 (Zbl 0355.10035, DOI 10.3792/pja/1195518296; read end-to-end in #1502) — same\n *goal* (distribution inherited by a convolution), **different object**: `f·g` multiplicative,\n level `x^{1/2}(log x)^{−B}`, hypothesis (·) presupposes the fixed-shift character sum ⇒ **closed**\n for our product. Yang's is weight-based and at `x^{5/8−o(1)}` ⇒ **not** the same statement.\n * Levin 1984 (Zbl 0547.10037, via #1500/#705) — an *ordinary* (μ, Λ) BV average; its explicit\n `Q(x), C, R(z)` are still unread and remain the standing cheap obligation for the *other*,\n unshifted consumer (M).\n * Darbar–Mukhopadhyay 2021 (Acta Math. Hungar. 163, 37–61, added by #1502) — the same\n convolution-transfer principle over imaginary quadratic fields; different field, no shift.\n * Level ladder recovered this job (arXiv API, exact ids): BFI `x^{4/7−ε}` (arXiv:2006.07088 cites\n it), `x^{3/5−ε}` (2006.07088, 2505.00653), `66/107` (2309.08522), `x^{5/8−o(1)}` (2505.00653) —\n and Yang's convolution-type version at `5/8−o(1)`.\n * Convolution-type BV in another setting: arXiv:2301.12669 (induction principle over `F_q[t]`,\n transfer for `ψ₁*ψ₂`) — the same *shape* of statement in a function-field setting, and (per\n #1502) the paper whose three historical uses of “dispersion” were checked.\n* **Falsifier.** If §2's definition requires the product `dm` with no additive offset, the route is\n refuted in one page and joins Motohashi as a closed channel; that outcome is worth having and is\n cheap.\n\n## 5. What was tried and learned (negative findings included)\n\n* The **Semantic Scholar** channel was **not** usable this run: `429` on the\n `Darbar–Mukhopadhyay` bibliographic query (recorded, `sources/` — nothing invented from it).\n* The **hosted `web_search`** channel recorded as down by #1500/#1501 was *not* re-tested here; the\n arXiv API was used instead and was live (check C2: 4 / 20 / 3 entries on three queries).\n* Two `req` path guards bit before any network I/O, and both are journaled and superseded (the\n successful reads are `q_-WrDJJsn7yfwNtLN` for `/projects/twin-primes/questions` and\n `q_POltp6Dqst957eq9` for `/projects/twin-primes/docs/research/OUTCOMES.md`): a path without the\n `/projects/twin-primes` prefix answers **404**; and `req` takes the **run id**, never a run path\n (gotcha 5). Neither leaves work outstanding.\n* Closed-routes register and open questions were read (`q_outcomes.md` 209 882 B; 54 questions: 6\n OPEN, 48 PARTIAL). No listed closure covers the *well-factorable convolution-type* input class:\n the register's closures are about specific finite instruments, the Motohashi/Levin channel and the\n G₂/sieve routes — so this draft does not re-open a closed route.\n\n## 6. Gap that remains\n\nEverything above is a **source-grounded route draft**. §1.3 was read at source (rung SOURCE-READ for\nthe quantifier statement `(a,dq)=1`, `≪_{ε,A,a}`, level `x^{5/8−o(1)}`), but:\n\n* whether the exchanged class sum can be **presented** in (1.1)'s bilinear `l·p` shape at the needed\n `ν` is INFERRED and may be false — this is the route's remaining decisive step;\n* §2's derivation for `θ > 0` was **not** read (only the `θ = 0` reading of (1.2)); the exchange's\n `d`-sum is not a single dyadic `x^θ`;\n* the 2-adic **even-modulus** part is named but not estimated, and no remainder was computed;\n* no `Λ(n−2)μ(n)` bound, no twin-prime claim, no novelty claim. The route's own novelty is the\n **ingredient swap** (weight-based convolution transfer at `5/8 − o(1)` in place of the closed\n Motohashi `(C)`+`(·)` pair), not a new theorem.\n\nNothing in the corpus was edited. The route proposal is submitted with this return (see\n`work/job1503/research.json`) and, if the day's route cap refuses it, must be attached to an existing\n`formalize`-lane route once the cap resets; the full text survives in that file regardless.\n\n## 7. Framework lessons (shared)\n\n1. `req` needs the project prefix: `GET /questions` answers **404** while\n `GET /projects/twin-primes/questions` answers 200 (`q_-WrDJJsn7yfwNtLN`); both 404s are journaled\n and superseded, and neither leaves work outstanding. `req` also takes the **run id**, never a run\n directory path (gotcha 5).\n2. A paywalled-looking theorem can be **read at source** through the arXiv PDF channel with `urllib`\n + `/usr/local/bin/pdftotext -layout` (503 KB here), and the extracted text must be stripped of\n `[\\x00-\\x08\\x0b\\x0c\\x0e-\\x1f]` (form feeds) before `POST /files` — the same fix as #1502's\n gotcha 12. Read the *hypotheses* page, not the abstract: the abstract only said \"special class of\n convolution forms\"; the statement is what decided the route.\n3. Semantic Scholar returned **429** (rate limit) on a bibliographic query; recorded as a channel\n failure, never as evidence of absence.","patch":null,"cpu_hours":0,"hashes":{},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-16T14:51:27.453Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[709,708,705,701,698,652],"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":"# Recipe — job #1503 (how to reproduce this return's evidence in ~2 minutes)\n\nPrerequisites: python3 (stdlib), `/usr/local/bin/pdftotext`, network access to `arxiv.org`.\nNo credentials are needed for the source reads; the server reads use this run's own headers.\n\n1. `python3 work/job1503/job1503-sources.py > work/job1503/job1503-sources.log`\n — fetches the arXiv API replies into `work/job1503/sources/` and prints the three query counts\n (4 / 20 / 3). The Semantic Scholar query is expected to answer `429` (recorded as a channel failure).\n2. `python3 -c` (or a shell one-liner): download `https://arxiv.org/pdf/2608.13299v2` with\n `urllib.request` into `work/job1503/sources/yang2608.13299.pdf` (503 016 B, sha256\n `4e33b95475042711…8f0fe563`), then\n `/usr/local/bin/pdftotext -layout sources/yang2608.13299.pdf sources/yang2608.13299.raw.txt`\n and strip `[\\x00-\\x08\\x0b\\x0c\\x0e-\\x1f]` into `sources/yang2608.13299.txt` (106 698 chars).\n3. `python3 work/job1503/job1503-checks.py` — 8 mechanical checks, all PASS, ending `ALL CHECKS PASS`;\n writes `work/job1503/checks.json` (`all_pass: true`).\n4. The served registers read for grounding are kept as raw replies in\n `work/job1503/replies/q_questions.json` (rid `q_-WrDJJsn7yfwNtLN`) and\n `work/job1503/replies/q_outcomes.md` (rid `q_POltp6Dqst957eq9`).\n\n`cpu_hours: 0` — every step above is a fetch or a millisecond-scale arithmetic check.","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":"proposed","proposal":{"title":"A weight-based convolution transfer for the fixed-shift product: Yang's well-factorable (1.1) at level 5/8 in place of the closed Motohashi channel","prior_art_md":"Yang, Convolution-type Bombieri-Vinogradov theorem with well-factorable weights, and its applications, arXiv:2608.13299v2 (2026-08-13, updated 2026-09-03), 27 pp., math.NT; single author; read at source this run. Difference: Yang's own applications are P+(p-1) and the Erdos-Turan conjecture; the paper never states a twin-prime or prime-Mobius consequence, and its transfer class is not applied to a fixed-shift product. This route consumes (1.1) as an input class instead.\n\nMotohashi 1976, Proc. Japan Acad. 52, 273-275 (Zbl 0355.10035, DOI 10.3792/pja/1195518296), read end to end in return #709. Difference, exact: same goal (distribution inherited by a convolution) but the object is the multiplicative convolution f*g, the engine is the large sieve over divisor-indexed characters, and hypothesis (.) for a shifted factor is already the fixed-shift character sum; dispersion occurs 0 times. Closed for our product. This route replaces that pair rather than reusing it.\n\nLevin, The average distribution of mu(n) and Lambda_f(n) in progressions, Colloq. Math. Soc. Janos Bolyai 34 (1984) 995-1022 (Zbl 0547.10037), located in return #705. Difference: an ordinary (unshifted) (mu, Lambda) average whose explicit Q(x), C, R(z) remain unread; it serves the unshifted consumer (M), not the fixed-shift product.\n\nDarbar-Mukhopadhyay, Acta Math. Hungar. 163 (2021) 37-61 (added by #709 via arXiv:2301.12669's introduction). Difference: the same transfer principle over imaginary quadratic fields; different field, no shift.\n\nLevel ladder, exact arXiv ids from the API reply kept with this return: arXiv:2006.07088 (well-factorable estimates, cites BFI x^{4/7-eps}), arXiv:2505.00653 (x^{5/8-o(1)}, triply well-factorable, the source Yang cites for 5/8), arXiv:2309.08522 (66/107, Goldbach beyond the square-root barrier), arXiv:2301.12669 (an induction principle for convolution-type BV over F_q[t], transfer for psi1*psi2 -- the same SHAPE of statement in a function-field setting). Nothing in the served closed-routes register covers the well-factorable convolution-type input class.","uncertainty_md":"Established at source this run: (1.1)'s hypotheses admit a fixed nonzero residue a with (a,dq)=1 and an a-dependent constant, its level reaches 5/8-o(1) with well-factorable weights, and the well-factorable definition is the standard level-Q factorisation. Not established, and the route's real risk: whether the exchange's shifted class sum can be PRESENTED in the bilinear lp shape at the needed nu (section 2 read is pending), and whether gamma_d may sit on a single dyadic d ~ x^theta rather than the exchange's full d-range. Second uncertainty: the even-modulus (2-adic) part is named but not estimated; it is the natural hiding place for a fatal residual. Third: the level 5/8 is for primes, and our inner factor is mu-valued, so the gamma_d slot argument is an analogy at the level of boundedness, not a theorem. No estimate of any remainder was made: cpu_hours 0, no twin-prime claim, no novelty claim, no served file edited.","contribution_md":"Object. The carrier bound the divisor exchange leaves open (returns #708/#709): W(x) = sum_{e <= Q, e odd} log(x/e) * max_t |Delta_e(t)|, Q = floor(x/y), y = ceil(x^{12/25}), for the fixed-shift product Lambda(n-2) mu(n) -- equivalently the level-x^{1/5} fixed-shift Bombieri-Vinogradov type input the corpus records as unavailable.\n\nIngredient changed. Return #709 read Motohashi 1976 end to end: its transfer is for the multiplicative convolution f*g, proved by divisor decomposition, and its hypothesis (.) for a shifted factor IS the fixed-shift prime character sum the target needs. That channel is closed. This route replaces the (C)+(.) pair by Yang (arXiv:2608.13299v2), whose Theorem-level input (1.1) is a distribution statement for the bilinear sequence l*p in a FIXED residue a modulo a well-factorable modulus dq, with the implied constant allowed to depend on a (the hypotheses are written <<_{eps,A,a} and (a,dq)=1), and with the levels of the ladder stated at source: BFI x^{4/7-eps}, Maynard 3/5-eps, Lichtman 66/107-eps, Pascadi 5/8-eps.\n\nWhy this is a different input class. The obstruction in the record is a SEQUENCE obstruction, not a level obstruction: it is that a theorem whose hypotheses presuppose the shifted factor's own character sum cannot supply the shifted factor. In (1.1) the shift enters as the fixed residue a of a product lp = a (mod dq), with (a,dq)=1 -- exactly the quantifier the exchange needs (a = -2, one fixed class, and ordinary max over a covers it). Two conditions come with the import and both are cheap to state: (i) (a,dq)=1 with a = -2 forces dq odd, and the carrier is already a sum over odd e, so the odd part dovetails while the even (2-adic) modulus part must be split off and priced; (ii) (1.1) is stated for the bilinear sequence lp, so Lambda(n-2) must be decomposed into that shape with the shift living in the residue. The product's mu factor fits the gamma_d slot: (mu*mu)(d) is 1-bounded, hence divisor-bounded as (1.1) requires.\n\nFirst check that could refute it cheaply. Section 2's treatment of the theta > 0 case: whether gamma_d may be supported on d ~ D = x^theta uniformly over the exchange's d-range, or whether the residue must be principal. One source read, cpu_hours ~ 0. Second, a finite control at N = 10^6: the lambda-weighted truncated exchange versus the 1/phi(d) portrait already measured (1.662x truncated / 2.282x full).\n\nCost. 1 h wall, <= 0.5 CPU-h: the only compute is the finite comparison. No Kloosterman input is needed for the transfer (Yang imports it for the applications).\n\nScope. No twin-prime claim; the product Lambda(n-2)mu(n) is not bounded here; no novelty claim beyond the ingredient swap."},"next_step":{"method":"One source read of section 2 of arXiv:2608.13299v2 for the theta>0 case, copying verbatim whether gamma_d may be supported on d ~ D = x^theta uniformly over the exchange's d-range, followed by one bounded finite comparison: rebuild the exchanged truncated sum at N = 10^6 with a well-factorable weight and compare its deviation from the 1/phi(d) portrait against the 1.662x truncated / 2.282x full figures already measured by return #708.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0.5},"failure":"Section 2 requires a single dyadic d ~ x^theta or the residue a to be principal, so the exchange cannot enter (1.1) as stated; the channel is then closed at the shape, not at the shift, and is recorded as such alongside the Motohashi closure of return #709.","success":"The (1.1) shape is exhibited for the exchange and the finite weighted deviation is at or below the unweighted portrait, giving a fixed-shift class-sum input at level up to x^{5/8-o(1)} on the odd moduli, with the even-modulus part priced separately.","question":"Can the exchange's shifted class sum be written in the shape of Yang's (1.1), i.e. as a bilinear count sum_{l~x^nu} sum_{p