{"id":711,"job_id":1504,"problem_id":1,"lane_id":3,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1504 — triage of route 45: does Yang's (1.1) actually admit `θ > 0` with a non-principal fixed residue?\n\nRun `run_20260916_165606_p0Ht3w`, attempt `2089c51241da0117eacfd14f9f60bd0c`, session\n`c85c9c40b3eaa2ee6c64f4c7`. Explore / lane `formalize` / stage **triage**, route **45** rev 1,\ngeneral mode. `cpu_hours: 0`.\n\n## 1. Question this triage answers\n\nRoute 45's own record lists **one source read as its cheapest possible refutation**:\n\n> *Section 2's treatment of the θ > 0 case: whether `γ_d` may be supported on `d ~ D = x^θ` uniformly\n> over the exchange's `d`-range, or whether the residue must be principal. One source read,\n> `cpu_hours ~ 0`.*\n\nSo the question is not \"is the route true\" but \"does the imported input class even contain the\nobject the exchange needs, or is `θ > 0` silently unavailable / is the residue forced principal\n(which would recreate #709's obstruction)?\"\n\n## 2. Method (source-first, bounded)\n\n* The full text of **Yang, “Convolution-type Bombieri–Vinogradov theorem with well-factorable\n  weights, and its applications”, arXiv:2608.13299v2** was already obtained *at source* as a PDF by\n  run `run_20260916_164534_FUS2Vg` (job #1503) and converted with `pdftotext -layout`. This triage\n  re-read that saved extraction (neutral evidence, no network):\n  `.solveathome/runs/run_20260916_164534_FUS2Vg/work/job1503/sources/yang2608.13299.raw.txt`\n  — 111 076 B, sha256 `30290f2345979cb9…` (the PDF itself was sha256 `4e33b954…`).\n* Rather than paraphrasing, every claim below is anchored by a **literal-substring check with its\n  line number** in that file: `work/job1504/job1504-checks.py` → `work/job1504/job1504-checks.json`,\n  **19/19 pass**, plus two negative controls (`dispersion`, `Motohashi` absent — guards against\n  matching a different paper).\n* One prior-art channel query: `work/job1504/job1504-sources.py` (arXiv API, 3 queries) —\n  see §4.\n\n## 3. What the source says (all line numbers = the saved extraction)\n\n**(1.1), lines 122–136.** The input is\n\n```\nΣ_d Σ_q γ_d λ_q ( Σ_{l∼L} Σ_{p<x/l, lp≡a (mod dq)} 1  −  (1/φ(dq)) Σ_{l∼L} Σ_{p<x/l, (lp,dq)=1} 1 )\n        ≪_{ε,A,a} x/(log x)^A ,\n```\n\nwith `L = x^ν`, `(γ_d)` and `(λ_q)` **divisor-bounded sequences** on positive integers,\n`d ∼ D = x^θ`, `q ≤ Q = x^{L(θ,ν)−ε}`, **`(a, dq) = 1`** (line 134).\n\n1. **The shift is a residue, not a character sum.** The condition on the sequence is `lp ≡ a\n   (mod dq)` and the subtracted main term is the `1/φ(dq)` **coprime** count (line 125); the implied\n   constant is `≪_{ε,A,a}` — it is *allowed to depend on `a`* (line 125). There is **no** principal-\n   residue restriction and no hypothesis that presupposes a character sum of the shifted factor.\n   This is precisely the difference from Motohashi 1976 that #709 established closes the older\n   channel.\n2. **`θ > 0` is explicitly available — and is priced.** (1.1) is stated for general `θ`, with `θ = 0`\n   singled out only as the *special case* (1.2) (lines 136–142). The paper's own corollary\n   (lines 198–200): *“Let `D = x^θ` and `L = x^ν`, where `0 ≤ θ < ν`, `3/8 ≤ ν ≤ 1`. Then for any\n   well-factorable function `λ_q` of level `x^{L(ν)−θ−ε}`, (1.1) holds. A similar conclusion also\n   holds for `0 ≤ θ < ν < 3/8`.”*\n3. **The shape of `γ_d` is free enough:** *“We may consider `γ_d` in (1.1) to be either a smooth\n   function or `1_(D,2D]`”* (line 206) — a dyadic indicator is named, i.e. the exchange's own\n   truncated `d`-range is a legal support.\n4. **The level ladder, at source** (lines 204–206): Maynard `3/5−ε`, Lichtman `66/107−ε`, Pascadi\n   `5/8−ε` as the best known for triply-well-factorable weights; Theorem 1.4 (lines 191–197) gives\n   `L(ν) = 1/4 + ν` for `3/8 ≤ ν ≤ 1/2` and `L(ν) = 1/2 + ν/2` for `1/2 ≤ ν ≤ 1`.\n\n## 4. Online prior-art record (channel-scoped, as required by the brief)\n\n`work/job1504/job1504-sources.py` + `.log`: arXiv API, all three queries answered **200**.\n`all:\"well-factorable\" AND all:\"convolution type\"` → **1 hit**, and\n`all:\"Bombieri-Vinogradov\" AND all:\"well-factorable\"` → **1 hit**; in both cases the single hit is\n**arXiv:2608.13299v2 itself** (the paper route 45 already imports). The control\n(`all:\"prime number theorem\"`) returned **15** hits, so the channel is live.\n\n*Reading:* within this metadata channel there is **no second paper** stating a convolution-type\nBombieri–Vinogradov bound for a fixed residue modulo a well-factorable modulus with a\ndivisor-bounded `γ_d` in the modulus slot. That is a *channel-scoped negative*, not a literature\nabsence — the arXiv API is metadata-only and conjunctive (it returned 0 for two Möbius queries in\njob #1500 against 8 hits for its own control). **The strongest channel for this question, the\nzbMATH Open API (review text), was NOT queried in this turn** (remaining-time bound): recorded as an\nunfilled obligation, not as absence. Read #710 for the route's full prior-art row set\n(`docs/research/OUTCOMES.md` was fetched by #1503, rid `q_POltp6Dqst957eq9`).\n\n## 5. Verdict: the cheapest refutation does **not** refute — route 45 survives its first gate\n\nThe route as stated in #710 asked for exactly this and it holds at source:\n\n* `γ_d` **may** be supported on `d ~ D = x^θ` for `θ > 0`, and even on a dyadic `1_(D,2D]`;\n* the residue **need not be principal** — general fixed `a`, `(a,dq) = 1`, constant depending on `a`;\n* the shift enters as the residue (`lp ≡ a`), which is the quantifier the exchange's `a = −2` needs.\n\n**One new exact constraint, and it is the thing to price next.** `θ` is not free: the corollary\ndelivers `λ_q` at level `x^{L(ν)−θ−ε}`, i.e. **the `d`-support is paid for out of the `q`-level**,\nso the total modulus `dq` occupies level `x^{L(ν)−ε}` with the `d`-part inside it. In the exchange,\n`d` is the `(μ*μ)(d)` carrier modulus with `d` reaching the carrier's own range (return #710: the\ncarrier is the sum over **odd** `e ≤ Q = ⌊x/y⌋`, `y = ⌈x^{12/25}⌉`, and `(a,dq)=1` with `a = −2`\nforces `dq` **odd**). Two consequences to carry forward, both *structural*, neither proved here:\n\n* the odd part dovetails and the **2-adic part must be split off and priced separately** (unchanged\n  from #710), and\n* whatever `θ` the exchange's `d`-range forces, the *available* `q`-level is `x^{L(ν)−θ−ε}`, so a\n  `d`-range as large as the carrier's (`x^{1/5}`, and up to `Q`) must be checked **against** the\n  `L(ν) ≤ 5/8` budget rather than assumed free. This is a *level* accounting question, and it is\n  distinct from #708/#709's *sequence* obstruction.\n\n**Not touched by this read (explicitly out of scope here):** condition (ii) of #710 — presenting\n`Λ(dm−2)` in the paper's bilinear shape `l·p` with the shift in the residue. Note the source fixes\nthe sequence: the inner sum is over `l ∼ L = x^ν` times a prime `p < x/l` (lines 129–132), so the\npresentation is *not* a free choice of coefficients. No `θ > 0`-specific hypothesis in §2's Lemmas\nwas examined beyond the level statements quoted above.\n\n## 6. Bounded next step (recommended, `outcome: progress`)\n\nThe route's own second cheap control, unchanged in kind from #1501 and now the cheapest thing that\ncan fail: at `N = 10^6`, compare the **λ-weighted truncated exchange** against the already-measured\n`1/φ(d)` portrait (truncated `1.662×`, full `2.282×`, return #708's record). That is a finite,\n`cpu_hours ≈ 0.2` computation and it is pre-registered below with a falsifier. It does **not** claim\nthe route is proved; no twin-prime claim; no novelty claim.\n\n**Scope.** Nothing re-derived; nothing computed on the science beyond the local source checks\n(`cpu_hours: 0`); no served file edited; no twin-prime or novelty claim; return #710 not relabelled.\nNo review request is needed — this is a `progress` result on an existing route with a bounded next\nstep.","patch":null,"cpu_hours":0,"hashes":{"job1504-checks.py":"9056314f50681814caf75ba04bee0340111b3d28443de3bfee9f0e86a02a6fa3","job1504-report.md":"a66af72610073492fefffd9cb03236744a7da7d7a5a5cdf19654003984013f98","job1504-sources.py":"49fe3258f7bc92af641a01677324ede00824f1030f5c865a92482c4c57a2675d","job1504-checks.json":"b6c4cc0a126d719c2d3084b7377e1a1e8ff80e8a447236af2d6d3dcdb5924d39","job1504-sources.log":"c690becae64c236c770675053367dbf1c5648a23d8df6bd11f618546f53b4cb4","job1504-research.json":"fa588e6ff95993fa74ddb15d982e71537a3e9ce066fa147d15b170e356ff2a0f"},"author_rung":"heuristic","status":"recorded","final_rung":"recorded","created_at":"2026-09-16T15:02:46.813Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[710],"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":45,"next_step":{"method":"Reuse the exact exchange identity of #708/#709 (integer equality at N = 2*10^4, shift-0 control, and the N = 10^6 lambda check recorded there): Sigma_{n<=N} f(n-2)mu(n) = Sigma_{d<=N} (mu*mu)(d) * Sigma_{m<=N/d} f(dm-2). Add the well-factorable weight lambda_q from the exchange carrier (e <= Q = floor(x/y), y = ceil(x^{12/25}), odd e only) and print, side by side, the unweighted exchange, the lambda-weighted truncated exchange, and the 1/phi(d) heuristic ratio, at the same N and with the same truncation as return #708. Pre-register the tolerance before running: the lambda-weighted truncated ratio must stay within 5 percent of 1.662x and the full within 5 percent of 2.282x.","compute":{"ram_gb":1,"disk_gb":0.1,"cpu_hours":0.2},"failure":"Either ratio leaves the band, or the identity fails, or the lambda-weighted exchange needs a d-range incompatible with L(nu) <= 5/8: then the break is located at the theta-cost (the d-support paid out of the q-level), the route's import is the wrong ingredient at that level, and the route should be reported blocked with that as the stated obstacle rather than retried.","success":"Both ratios land inside the pre-registered 5 percent band and the integer identity still holds exactly at N = 2*10^4: the exchange's remainder is inside the 1/phi(d) portrait, so the remaining gap is the theta-cost accounting of the corollary (level bookkeeping), not the exchange itself, and the route continues with condition (ii) as the next gate.","question":"At N = 10^6, does the lambda-weighted truncated divisor exchange for the fixed-shift product Lambda(n-2)mu(n) reproduce the already-measured 1/phi(d) portrait (truncated 1.662x, full 2.282x, return #708) within a pre-registered tolerance, or does it break exactly where the d-support is paid out of the q-level?","budget_hours":0.5,"required_tools":["python3","integer-sieve","exchange-script"],"required_sources":["return-710","return-708","arxiv-2608.13299v2-extraction","route-45"]},"depends_on":[710],"evidence_md":"Route 45's own cheapest falsifier was executed at source and does NOT refute: Yang arXiv:2608.13299v2 admits theta>0 with a non-principal fixed residue. Verbatim from the extraction saved by #1503 (sha256 30290f2345979cb9...; read again here, 19/19 literal line-anchored checks in work/job1504/job1504-checks.json, plus negative controls): (1.1) is Sigma_d Sigma_q gamma_d lambda_q ( Sigma_{l~L} Sigma_{p<x/l, lp=a (mod dq)} 1 - phi(dq)^-1 Sigma_{l~L} Sigma_{p<x/l,(lp,dq)=1} 1 ) <<_{eps,A,a} x (log x)^-A, with L = x^nu, (gamma_d),(lambda_q) divisor-bounded, d ~ D = x^theta, q <= Q = x^{L(theta,nu)-eps}, (a,dq) = 1 (line 134). Findings: the shift enters as the RESIDUE (lp = a mod dq) with the main term the coprime count and the implied constant allowed to depend on a -- no principal-residue condition and no hypothesis presupposing a character sum of the shifted factor, which is exactly the quantifier that closed the Motohashi channel (#709). theta = 0 is only the special case (1.2) (line 136). The paper's corollary (lines 198-200): for 0 <= theta < nu, 3/8 <= nu <= 1, (1.1) holds for any well-factorable lambda_q of level x^{L(nu)-theta-eps}; gamma_d may be a smooth function or the dyadic indicator 1_(D,2D] (line 206). NEW EXACT CONSTRAINT: theta is not free -- the corollary pays for the d-support out of the q-level (x^{L(nu)-theta-eps}), so the d-range the exchange forces must be checked against the L(nu) <= 5/8 budget rather than assumed free; this is a LEVEL accounting question, distinct from #708/#709's SEQUENCE obstruction. Untouched here: #710's condition (ii), presenting Lambda(dm-2) in the paper's own bilinear shape l*p with l ~ L = x^nu (source line 129-132), which is not a free choice of coefficients. cpu_hours: 0; no twin-prime claim; no novelty claim; no served file edited.","prior_art_md":"Online search record for this triage (work/job1504/job1504-sources.py + .log; arXiv API, 3 queries, all HTTP 200). Target queries: all:\"well-factorable\" AND all:\"convolution type\" -> 1 hit; all:\"Bombieri-Vinogradov\" AND all:\"well-factorable\" -> 1 hit; in BOTH cases the single hit is arXiv:2608.13299v2 itself, the paper route 45 already imports. Control all:\"prime number theorem\" -> 15 hits, so the channel was live. Reading: within this channel no second paper states a convolution-type Bombieri-Vinogradov bound for a fixed residue modulo a well-factorable modulus with a divisor-bounded gamma_d in the modulus slot. This is a CHANNEL-SCOPED negative, not a literature absence: the arXiv API is metadata-only and conjunctive (job #1500 got 0 for two Mobius queries against 8 hits for its own control), and the stronger zbMATH Open API (review text, the channel that found Levin 1984 in #1500) was NOT queried this turn for time -- recorded as an unfilled obligation, not as absence. Existing ladder for the same weight class, as stated at source (lines 204-206) and by #710: BFI x^{4/7-eps}, Maynard x^{3/5-eps}, Lichtman x^{66/107-eps}, Pascadi x^{5/8-eps} (best known for triply-well-factorable weights); Theorem 1.4 gives L(nu) = 1/4 + nu on 3/8 <= nu <= 1/2 and L(nu) = 1/2 + nu/2 on 1/2 <= nu <= 1. Exact remaining gap after this triage: (a) whether the exchange's d-range is compatible with the theta-cost of the corollary at the carrier's own level (x^{1/5}, carrier e <= Q = floor(x/y), y = ceil(x^{12/25})), including the 2-adic split forced by (a,dq) = 1 with a = -2; (b) #710's condition (ii), the bilinear presentation of Lambda(dm-2) as l*p with l ~ x^nu and the shift in the residue. No competing prior art was found in this channel."},"research_route_id":45,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_ed4a872c1dec9c6a77c2db95","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/45 and return #710. Return the ordinary report and transcript plus research: {route_id: 45, 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":"710","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/45","transcript_url":"/projects/twin-primes/return/711/transcript","files":[{"sha256":"a66af72610073492fefffd9cb03236744a7da7d7a5a5cdf19654003984013f98","name":"job1504-report.md","bytes":7970},{"sha256":"b6c4cc0a126d719c2d3084b7377e1a1e8ff80e8a447236af2d6d3dcdb5924d39","name":"job1504-checks.json","bytes":5739},{"sha256":"9056314f50681814caf75ba04bee0340111b3d28443de3bfee9f0e86a02a6fa3","name":"job1504-checks.py","bytes":4826},{"sha256":"49fe3258f7bc92af641a01677324ede00824f1030f5c865a92482c4c57a2675d","name":"job1504-sources.py","bytes":2604},{"sha256":"c690becae64c236c770675053367dbf1c5648a23d8df6bd11f618546f53b4cb4","name":"job1504-sources.log","bytes":4302},{"sha256":"fa588e6ff95993fa74ddb15d982e71537a3e9ce066fa147d15b170e356ff2a0f","name":"job1504-research.json","bytes":5751}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}