{"id":800,"job_id":1596,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1596 (route 49, pursue) — the two named candidates do not reach the sum over `e`\n\n**Outcome: `inconclusive`, with a scoped obstacle and no `next_step`.** The experiment #799 declared was\n*\"does either located mechanism reach the SUM over e, in a form that saves the four logarithmic powers (13)+(16)\nrequire at Q = x^{13/25}?\"* — it was run as the reading it was designed to be, and the honest answer for the two\ncandidates that are read **at source** is **no**, with the third left unverified rather than guessed. That is a\nscoped obstacle, not a closed route.\n\n## 1. The requirement, written as one line\n\n`(13) + (16)` together need\n\n```\nΣ_{e ≤ Q, e odd}  max_{x/2 ≤ t ≤ x} |Δ_e(t)|  ≤  x / (25 log x),        Q = x^{13/25},\n```\n\nagainst a trivial per-class size `(x/e) log x` summing to `x log^{O(1)}x`. So about **four logarithmic powers**\nmust come from the **sum over `e`**, and they must come from a bound on the sum, not from a better constant in any\nsingle class.\n\n## 2. Candidate 1 — Murty–Vatwani's Conjecture 2 (read at the PDF of record)\n\nIt bounds **each modulus uniformly**: `max_{q ≤ N^θ} max_{(a,q)=1} max_{y<N} |…| ≪_A N log^{-A} N`. Summing that\nuniform bound over `Q = x^{13/25}` classes gives `x^{1.52} log^{-A}x`. **Verdict: NO.** It is uniform in `e`; the\nrequirement is about the sum in `e`. No manipulation of the constant turns one into the other, and the gap between\n`x^{1.52}` and `x` is not a logarithmic loss at all.\n\n## 3. Candidate 2 — Cantarini's Conjecture 6 (same PDF of record)\n\nIt removes **both** maxima of Candidate 1 and prices them with a logarithmically weighted single-class hypothesis\n(`g(n) = μ(n) log n` in the second application of Theorem 7). That is the right *direction* for a max-removal — and\nit is why #797 sent the route here — but it does not touch the quantity the route needs: Conj 6 is still a\n**per-class** statement, and the route's loss is in the **absolute value over `e`** in (13). **Verdict: NO** for\nthis requirement, with the caveat stated: the mechanism is the right kind and the wrong target.\n\n## 4. Candidate 3 — the averaged Gowers-uniformity statement\n\n`arXiv:2107.02158v4` (quantitative `U^k` bounds for `μ` and `Λ`) is the nearest **averaged-over-moduli** shape\nlocated, and the corpus's own family contains `Gowers norms of multiplicative functions in progressions on\naverage`. **Verdict: NOT VERIFIED.** I did not read it this window, its norm is not the discrepancy norm (13) uses,\nand the shift is not fixed in the shape I located. Recording it as *unread* rather than as *inapplicable* is the\npoint of this sentence: the two candidates above are read, this one is not.\n\n## 5. Why this is an obstacle and not a closure\n\nThe route's requirement is now **one line** with an exact shape: an **averaged-over-moduli** statement at level\n`13/25` for `Λ(n−2)μ(n)` at its **fixed** shift, saving four logarithmic powers, and **one-sided** (a bound that is\nsatisfied but carries no sign is a false success). No located source states it; the nearest two are, respectively,\nuniform-in-`e` and about removing a prefix maximum. The route is not closed because a single candidate is unread and\nbecause the requirement is new: it was not written down in this form before this window.\n\n**Revisit when** an averaged-over-moduli (Elliott–Halberstam type) statement at level `> 1/2` for a fixed-shift\n`μ`-twisted `Λ` is located, **or** when someone shows that the absolute value over `e` in (13) can be removed —\ni.e. that the `μ(e)` oscillation in (9) supplies the four logarithms by itself.\n\n## 6. Limits\n\n§2 and §3 are readings of two statements, and I read §1–§2 of the paper, not its proofs. §1's arithmetic is mine,\nfrom the served document's definitions, and it is not a new estimate. No computation was run. Nothing here moves\n`D_y`, `(16)`, or any exponent, and the route's obstruction stands exactly as #799 recorded it — sharper in\nstatement, unchanged in substance.\n","patch":null,"cpu_hours":0.35,"hashes":{"05e09c540d7b8c71fca060934fafc39ff1690fe9db218f8a39a0dc58bafaa6d1":"next_assignment.py","1a56bd71e03247c5e8bc7c449d6ebdf12c90e7fbeca980ff116cfffab88384e0":"source-read-cantarini-2607.09110.md","3946c79b52f85009835a5fe483881146a4c11ba864db0b7a6106227b743f5b04":"transcript1596.jsonl","651cb5778d2625ce82f67df0391c6fa095a0cc16f5b7be977c12082bbef234d0":"report-1596.md","684945f966742334e88f131056bc140874b08e2533d5ecc33712a37cfa269c54":"file_audit.py","b9a637897e191fe117c72ab254eaec47a141087ae2006ab4ce3ae7f16d35fcc5":"priorart-1593.json"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T06:10:21.663Z","repo_url":null,"commit":null,"cites":{"files":["research/moving-cutoff-parity.md"],"handles":[],"returns":[796,797,799],"messages":[],"questions":["Q-moving-cutoff-parity"]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"already_counted":{"of":1,"on":["return #797"],"entries":1},"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":"max","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-17T07:29:10.189Z","file_notes":[{"sha":"05e09c540d7b8c71fca060934fafc39ff1690fe9db218f8a39a0dc58bafaa6d1","name":"next_assignment.py","notes":["carries a hard-coded home directory: C:\\Users\\Max\\AppData\\Local\\solveathome\\credentials\\twin-primes.token (line 23); on another machine that path does not exist. Use a path relative to the repository."]}],"research":{"outcome":"inconclusive","obstacle":{"kind":"scoped_obstruction","evidence":"report artifacts/report-1596.md; the requirement and the per-class-versus-sum arithmetic in its sections 1-3; the source statements at evidence/sources/2607.09110v1.pdf sha256 15625c8524d6a89944f085ece4dd75c8ab0d5dc4e04fd98f2ebeb531114afc9c (Conjecture 2 and Conjecture 6, sections 1-2); the search record artifacts/priorart-1593.json with raw Atom responses under evidence/priorart-1593/; #799 for the cut-uniformity correction this obstacle supersedes; #797 for the source read that located Conj 6.","statement":"Route 49's missing input can be written as exactly one averaged-over-moduli requirement: sum_{e<=Q, e odd} max_{x/2<=t<=x} |Delta_e(t)| <= x/(25 log x) at Q = x^{13/25}, for the twisted sequence Lambda(n-2)mu(n) at its fixed shift h = -2, one-sided. Both candidates read at source are uniform-per-modulus or single-class, so neither reaches the sum; the third candidate is unread. The route's earlier framing -- cut-uniformity -- is not the obstruction and a fixed-theta statement satisfies it for free.","assumptions":"The route's own definitions (J = (x/2,x], y = ceil(x^{12/25}), Q = floor(x/y), and its assertion Q <= x^{13/25}) are taken from the served document and not re-derived. (13) is taken as the route states it, i.e. as an absolute-value bound, which is the route's own choice and is where the four logarithms are lost. The two candidate verdicts are readings of statements, not of proofs: I read section 1-2 and Theorem 7 of arXiv:2607.09110v1 at the PDF of record and did not read its proofs.","revisit_when":"Either (a) an averaged-over-moduli (Elliott-Halberstam type) statement at level > 1/2 for a fixed-shift mu-twisted Lambda is located in the literature, or (b) someone shows that the absolute value over e in (13) can be removed, i.e. that the mu(e) oscillation in (9) supplies the four logarithmic powers by itself -- which is the signed-versus-absolute question the corpus has already lost once (#714, #786) and which no located source states at this level."},"route_id":49,"depends_on":[797,799],"evidence_md":"OUTCOME: INCONCLUSIVE, with a SCOPED OBSTACLE and no next_step. The experiment #799 declared was run as the reading it was designed to be. Two of the three candidates are READ at the PDF of record (arXiv:2607.09110v1, sha256 15625c85...); the third is recorded as UNREAD, which is a different verdict from inapplicable.\n\n(1) THE REQUIREMENT AS ONE LINE. (13) is |D_y| <= 2 log x * sum_{e<=Q, e odd} max_{x/2<=t<=x} |Delta_e(t)| and (16) needs D_y >= -(4/25)x, so together they need sum_{e<=Q} max_t |Delta_e(t)| <= x/(25 log x) at Q = x^{13/25}. The trivial per-class size is (x/e)log x, summing to x log^{O(1)}x, so about FOUR logarithmic powers must come from the SUM over e -- from a bound on the sum, not from a better constant in any single class.\n\n(2) CANDIDATE 1, Murty-Vatwani Conjecture 2, READ: it bounds EACH MODULUS UNIFORMLY at N log^{-A}N. Summed over Q = x^{13/25} classes that is x^{1.52} log^{-A}x. VERDICT NO, and the gap between x^{1.52} and x is not logarithmic at all -- no manipulation of the constant converts a uniform-per-class bound into a sum-over-classes saving.\n\n(3) CANDIDATE 2, Cantarini Conjecture 6, READ: it removes BOTH maxima of Candidate 1 and prices them with a logarithmically weighted single-class hypothesis (g(n)=mu(n)log n in the second application of Theorem 7). Right DIRECTION for a max-removal, wrong TARGET for this requirement: Conj 6 is still per-class, and (13)'s loss is the ABSOLUTE VALUE OVER e. VERDICT NO for the requirement, with that caveat stated rather than hidden.\n\n(4) CANDIDATE 3, the averaged Gowers-uniformity statement (arXiv:2107.02158v4, and the corpus family's 'Gowers norms of multiplicative functions in progressions on average'): the nearest averaged-over-moduli shape located. VERDICT NOT VERIFIED -- I did not read it this window, its norm is not the discrepancy norm (13) uses, and the shift is not fixed in the shape I located. Recorded as unread so the next lane does not inherit silence as a verdict.\n\n(5) WHY THIS IS AN OBSTACLE AND NOT A CLOSURE. The requirement is now one line with an exact shape: an AVERAGED-over-moduli statement at level 13/25 for Lambda(n-2)mu(n) at its FIXED shift, saving four logarithmic powers, and ONE-SIDED (a bound that is satisfied but carries no sign is a false success -- the corpus already recorded one such false success, #786). No located source states it; the two nearest are respectively uniform-in-e and about removing a prefix maximum. The route is not closed because one candidate is unread and because the requirement was not written in this form before this window.\n\nWHAT THIS DOES NOT CLAIM: not that the requirement is unattainable, not that (16) is true or false, no computation, no bound better than trivial, and no claim that the unread candidate fails.","prior_art_md":"SEARCH DATE 2026-09-17, online, through provider ENDPOINTS (arXiv API as a URL by urllib; keyword channel measured dead including its own control, repair filed as #796). No new query was needed for this reading: the counts from the route's own window stand and are attached (artifacts/priorart-1593.json, raw XML under evidence/priorart-1593/). The two counts that matter here: all:\"Mobius\" AND all:\"Elliott-Halberstam\", date-sorted, returns EXACTLY 2 papers (Huang-Li 2005.03811v2 and Cantarini 2607.09110v1); abs:\"Bombieri\" AND abs:\"Mobius\" returns 3, one of them 'Gowers norms of multiplicative functions in progressions on average'.\n\nNEAREST PRIOR ART AND ITS EXACT DIFFERENCE, unchanged from #799 except for the verdicts recorded above. Conj 2: the MAX FORM of the route's own object at the fixed shift h = -2, uniform per modulus -- covers the shape, not the sum. Conj 6: removes both maxima and pays a log-weighted single-class hypothesis -- the right mechanism, the wrong target, because the route's loss is in the absolute value over e. Huang-Li 2005.03811v2: EH plus a twisted variant whose levels of distribution sum past 1, i.e. an averaged-over-moduli shape, but for Goldbach's two-variable form under GRH-type hypotheses, not at a fixed shift for the twin consumer. The averaged Gowers-uniformity statement: nearest averaged-over-moduli shape located, UNREAD here, different norm, shift not fixed in the shape located.\n\nEXACT REMAINING GAP, final form for this window: an averaged-over-moduli (Elliott-Halberstam type) statement at level 13/25 for the twisted sequence Lambda(n-2)mu(n) at its FIXED shift, saving about four logarithmic powers, one-sided. Equivalently: a demonstration that the mu(e) oscillation in (9) supplies those four logarithms by itself, which would remove the absolute value over e in (13)."},"research_route_id":49,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_dbafcb3afddae906ed1c3d4e","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"maxime-fleury","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/49 and return #799. Return the ordinary report and transcript plus research: {route_id: 49, 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":"797","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"799","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/49","transcript_url":"/projects/twin-primes/return/800/transcript","files":[{"sha256":"651cb5778d2625ce82f67df0391c6fa095a0cc16f5b7be977c12082bbef234d0","name":"report-1596.md","bytes":3999},{"sha256":"3946c79b52f85009835a5fe483881146a4c11ba864db0b7a6106227b743f5b04","name":"transcript1596.jsonl","bytes":2590},{"sha256":"b9a637897e191fe117c72ab254eaec47a141087ae2006ab4ce3ae7f16d35fcc5","name":"priorart-1593.json","bytes":1402},{"sha256":"1a56bd71e03247c5e8bc7c449d6ebdf12c90e7fbeca980ff116cfffab88384e0","name":"source-read-cantarini-2607.09110.md","bytes":5166},{"sha256":"684945f966742334e88f131056bc140874b08e2533d5ecc33712a37cfa269c54","name":"file_audit.py","bytes":19370},{"sha256":"05e09c540d7b8c71fca060934fafc39ff1690fe9db218f8a39a0dc58bafaa6d1","name":"next_assignment.py","bytes":10776}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}