{"id":799,"job_id":1593,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1593 (route 49, pursue) — cut-uniformity is FREE at a fixed exponent; the obstruction is max-versus-sum\n\n**Assignment:** route 49 pursue, 1 h, route rev 4. **Outcome: `progress`** — one sentence of the route's record is\nreplaced by a sharper one, and the arithmetic requirement is written exactly. No estimate is new and nothing is\nproved; the route's obstruction is **not** lifted.\n\n---\n\n## 1. What was checked, and the check that changes the framing\n\nRoute 49's record says its missing input is the **cut-uniformity** of the relation (12): *\"no located source\naddresses the cut-uniformity of this decomposition — whether (12) is uniform over the admissible cut.\"* The route's\nown definitions (line 114 of the served document) are\n\n```\nJ = (x/2, x],   y = ceil(x^(12/25)),   Q = floor(x/y),\n```\n\nand the document itself notes `Q <= x^(13/25)` (line 122). So **the entire admissible cut family sits at one fixed\nexponent `θ = log Q / log x <= 13/25 = 0.52`**, and a statement made *at a fixed `θ`* covers every admissible cut\n**simultaneously**, with no `y`-dependence in the constant. Uniformity over the cut is therefore not an extra\nrequirement that a source has to supply: it is automatic for any statement whose modulus range is a fixed power of\n`x`. The route's sentence should be replaced by: *cut-uniformity is free at a fixed `θ`; what is not free is the\nlevel `13/25 > 1/2`.* This is a correction to the route's framing, not to its mathematics, and it costs nothing.\n\n## 2. The published object the route needs already exists in its **max** form\n\nRead at the PDF of record (`arXiv:2607.09110v1`, sha256 `15625c85…`, this window), **Conjecture 2** is Murty–Vatwani's\nshifted-Möbius Elliott–Halberstam: for a fixed even `h ≠ 0`, for every `A > 0`,\n\n```\nmax_{q ≤ N^θ} max_{(a,q)=1} max_{y<N} | Σ_{n≤y, n≡a (q)} μ(n)Λ(n+h)\n                                  − φ(q)^{-1} Σ_{n≤y} μ(n)Λ(n+h) |  ≪_A  N log^{-A} N.\n```\n\nRoute 49's object is `Δ_e(t) = Σ_{x/2<n≤t, e|n} f(n) − φ(e)^{-1}Σ_{x/2<n≤t} f(n)` with `f(n) = Λ(n−2)μ(n)` — the\n**same product, at the fixed shift `h = −2`**, centered the same way (class minus density), with a **max over the\nclass, a max over the prefix, and a modulus range at any fixed `θ < 1`** — which includes `θ = 13/25`. And the\nroute's own document already cites Murty–Vatwani (Propositions 3.2–3.3 and Lemma 3.4). So the *shape* the route\nneeds is in print, as a conjecture its own record already knows, and the cut-uniformity question in §1 is answered\nby it for free.\n\n## 3. What that does **not** give, which is the actual obstruction\n\nRoute 49's usable bound is (13):\n\n```\n|D_y| ≤ 2 log x · Σ_{e ≤ Q, e odd} max_{x/2 ≤ t ≤ x} |Δ_e(t)|,\n```\n\nand (16) needs `D_y ≥ −(4/25)x`, i.e. the **sum over moduli** to be `≤ x/(25 log x)`. Conjecture 2 is a bound\n**uniform in each modulus** — `N log^{-A}N` per class — and summed over `Q = x^{13/25}` classes it gives\n`x^{1.52} log^{-A}x`, which is hopeless. The trivial per-class bound is `(x/e) log x`, summing to `x log^{O(1)}x`, so\nthe demanded saving is **about four logarithmic powers** and it must come from the **sum over `e`**, not from any\nsingle class.\n\nThat is exactly the distinction the route was groping for: **its missing input is an averaged-over-moduli statement\nat level `13/25`, not a uniform-over-moduli statement at level `13/25`.** Put plainly, the route's sentence *\"the\nmissing input is not merely an extension by 1/50 beyond the ordinary BV exponent: ordinary BV concerns a different\nsequence\"* is right, and now it is also **specific**: what would pay is an Elliott–Halberstam-type average\n`Σ_{e≤Q} max_t |Δ_e(t)| ≪_A x log^{-A}x` at `Q = x^{13/25}` for the twisted sequence `Λ(n−2)μ(n)` at the fixed\nshift, with the absolute value over `e` being where the four logarithms have to be found.\n\n## 4. Why the search could not have been asked this before\n\nThe online step was performed this window through provider endpoints (the channel repaired in return #796), and the\nrelevant count is small enough to state: sorting the entire arXiv abstract index by date for the conjunction\n`Möbius ∧ Elliott–Halberstam` returns **exactly two papers** — Huang–Li `2005.03811v2` and Cantarini\n`2607.09110v1`. The route's object family has two papers in it, and the route's record cited neither by its\nmechanism. That is a fact about the *family*, not about one query.\n\n## 5. Limits, without packaging\n\nI read §1–§2 and Theorem 7 of one paper, **not the proofs**, and verified nothing in any preprint. The exponent bookkeeping in\n§1 and §3 is mine, from the served document's own definitions, and it is not a new estimate: no bound here is\nbetter than the trivial one. No number was computed. The claim in §1 is a claim about *which* exponent a statement\nhas to hold at, and it can be refuted by exhibiting an admissible cut whose `Q` exceeds `x^{13/25}` — I could not\nfind one, and the document's own line 122 asserts the bound.\n","patch":null,"cpu_hours":0.5,"hashes":{"05e09c540d7b8c71fca060934fafc39ff1690fe9db218f8a39a0dc58bafaa6d1":"next_assignment.py","1a56bd71e03247c5e8bc7c449d6ebdf12c90e7fbeca980ff116cfffab88384e0":"source-read-cantarini-2607.09110.md","3134f5b6a0a69e766b9e16b29c447df7a58ffccb881d40baed80b2e617ffdfc9":"eh-twisted-recent.xml","66b427f793475711c069f22e3ea06f21026755826e6bf99a59d992c4577801a2":"report-1593.md","684945f966742334e88f131056bc140874b08e2533d5ecc33712a37cfa269c54":"file_audit.py","ac34c24ed434238470d4e614a0675e0bbd366ce21c1463b3e8c49af54e5233be":"lod-beyond-half.xml","b4dd2b8ea4dfdead9ed54a56e1ca4af67cab2b38f22c46e754c8f03478351c21":"priorart_search_1583.py","b9a637897e191fe117c72ab254eaec47a141087ae2006ab4ce3ae7f16d35fcc5":"priorart-1593.json","e2e7e58fadb89f9ea8a201a374aa3c4e1d9fcf0c075a8b4ea8b22c5241dda2c3":"transcript1593-r49.jsonl"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-17T06:06:35.267Z","repo_url":null,"commit":null,"cites":{"files":["research/moving-cutoff-parity.md","research/SEARCH-CONVENTIONS.md"],"handles":[],"returns":[793,796,797],"messages":[1933],"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":"2026-09-17T07:29:09.481Z","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":"progress","route_id":49,"next_step":{"method":"Pure reading and algebra on the served documents, no computation. (i) Write (13)+(16) as one requirement: sum_{e<=Q, e odd} max_{x/2<=t<=x} |Delta_e(t)| <= x/(25 log x) with Q = x^{13/25}, and mark where the absolute value over e is taken. (ii) For each of the three candidates (Murty-Vatwani Conj 2; Cantarini Conj 6 with its two applications g(n)=mu(n) and g(n)=mu(n)log n; the averaged Gowers-uniformity statement) record, variable by variable, whether it bounds the SUM over e or only each e, and whether it preserves the one-sidedness (16) needs -- a bound that is satisfied but supplies no sign is a FALSE success. (iii) Deliver at most three candidates in a table source -> exact statement -> what it would have to become -> YES/NO, with the exact line where the max-versus-sum or the absolute value is taken.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"No candidate bounds the sum: each is either uniform in the modulus or removes a prefix maximum at the price of a single-class hypothesis. Then the route records a scoped obstruction of the max-versus-sum type -- a strictly better statement than the cut-uniformity sentence it replaces -- and it stops the next lane searching for a cut-uniform source that cannot help.","success":"One candidate provably bounds the sum over e at level 13/25 while preserving the one-sidedness (16) needs -- then the route's missing input is a named statement with a source family, and the route continues with that statement as its object.","question":"Does either located mechanism reach the SUM over e -- Cantarini's Conjecture 6, which pays a max-removal with a logarithmically weighted single-class hypothesis, or the averaged Gowers-norm statement -- in a form that saves the four logarithmic powers that (13)+(16) require at Q = x^{13/25}?","budget_hours":1.5,"required_tools":[],"required_sources":[]},"depends_on":[796,797],"evidence_md":"OUTCOME: PROGRESS -- one sentence of the route's record is replaced by a sharper one, and the arithmetic requirement is written exactly. The obstruction is NOT lifted; no bound here is better than trivial and no number was computed.\n\n(1) CUT-UNIFORMITY IS FREE AT A FIXED EXPONENT, AND THE ROUTE'S RECORD ASKS FOR IT ANYWAY. From the served document's own definitions: J = (x/2, x], y = ceil(x^(12/25)), Q = floor(x/y), and the document itself asserts Q <= x^(13/25). The ENTIRE admissible cut family therefore sits at ONE fixed exponent theta = log Q / log x <= 13/25 = 0.52, so a statement made at a fixed theta covers every admissible cut simultaneously with no y-dependence in its constant. The route's recorded missing input -- 'no located source addresses the cut-uniformity of this decomposition, whether (12) is uniform over the admissible cut with a logarithmic allowance' -- is consequently not an input a source has to supply. The sentence should read: cut-uniformity is free at a fixed theta; what is not free is the level 13/25 > 1/2.\n\n(2) THE NEEDED SHAPE IS IN PRINT, IN ITS MAX FORM. Read at the PDF of record this window (arXiv:2607.09110v1, sha256 15625c8524d6a89944f085ece4dd75c8ab0d5dc4e04fd98f2ebeb531114afc9c), Conjecture 2 is Murty-Vatwani's shifted-Mobius Elliott-Halberstam: for a fixed even h != 0 and every A > 0, the max over q <= N^theta, over the class (a,q) and over the prefix y < N of | sum_{n<=y, n = a mod q} mu(n)Lambda(n+h) - phi(q)^{-1} sum_{n<=y} mu(n)Lambda(n+h) | is <<_A N log^{-A} N. Route 49's object is Delta_e(t) = sum_{x/2<n<=t, e|n} f(n) - phi(e)^{-1} sum_{x/2<n<=t} f(n) with f(n) = Lambda(n-2)mu(n) -- the same product at the fixed shift h = -2, centered the same way, with a max over the class, a max over the prefix, and a modulus range at ANY fixed theta < 1, which includes 13/25. The route's document already cites Murty-Vatwani (Propositions 3.2-3.3, Lemma 3.4). So the shape exists in print as a conjecture the record already knows.\n\n(3) WHAT IT DOES NOT GIVE, WHICH IS THE OBSTRUCTION. (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, i.e. the SUM over moduli at most x/(25 log x). Conjecture 2 bounds each modulus UNIFORMLY at N log^{-A}N; summed over Q = x^{13/25} classes that is x^{1.52} log^{-A} x, hopeless. The trivial per-class bound is (x/e) log x, summing to x log^{O(1)} x, so about FOUR logarithmic powers must be found -- and they can only come from the sum over e. The missing input is therefore an AVERAGED-over-moduli statement at level 13/25 for the twisted sequence Lambda(n-2)mu(n) at its fixed shift, with the absolute value over e being exactly where those four logarithms have to be found. The route's own sentence is right and is now specific: its missing input is a max-versus-sum distinction, not a uniformity one.\n\n(4) THE SEARCH WAS POSSIBLE, AND ITS COUNT IS DECISIVE. Online, through provider endpoints (channel repaired in #796): sorting the whole arXiv abstract index by date for the conjunction Mobius AND Elliott-Halberstam returns EXACTLY TWO papers, Huang-Li 2005.03811v2 and Cantarini 2607.09110v1. The route's object family has two papers in it, and the route's record cited neither by its mechanism.\n\nWHAT THIS DOES NOT CLAIM: not that (16) is true or false, not that the averaged statement exists, not that the route should be closed, and not that the exponent bookkeeping in (1)/(3) is more than a reading of the served document's own definitions.","prior_art_md":"SEARCH DATE 2026-09-17, online, from this lane, through provider ENDPOINTS (the arXiv API addressed as a URL by urllib; the keyword channel was measured dead for every query including its own control, and its repair was filed as return #796). Four further queries this window, raw Atom responses frozen under evidence/priorart-1593/ with sha256.\n\nCOUNTS, because the family is small enough that they are the finding. all:\"Mobius\" AND all:\"Elliott-Halberstam\", date-sorted, returns EXACTLY 2: (a) Huang-Li, arXiv:2005.03811v2, 2020 -- binary Goldbach under EH plus a Mobius-twisted variant whose levels of distribution SUM PAST 1, citing Murty-Vatwani 2017 for the twin analogue; (b) Cantarini, arXiv:2607.09110v1, 2026 -- the diagonal weakening that removes both maxima and pays with a logarithmically weighted cancellation, then derives Goldbach. abs:\"level of distribution\" AND abs:\"beyond 1/2\" returns 2, both function-field papers (bilinear Kloosterman sums; Chowla and twin primes over F_q[T]) -- i.e. the level-beyond-1/2 question for the ordinary problem is answered in function fields and not in the integers by anything located. abs:\"Bombieri\" AND abs:\"Mobius\" returns 3, including 'Gowers norms of multiplicative functions in progressions ON AVERAGE', the nearest located shape to the averaged-over-moduli statement that (3) identifies as the requirement -- and it is a Gowers-norm statement, not a level-of-distribution one.\n\nNEAREST PRIOR ART, WITH THE EXACT DIFFERENCE. (1) Murty-Vatwani, as stated in Cantarini's Conjecture 2: THE MAX FORM of exactly the route's object at the fixed shift h = -2, at any fixed theta < 1. Difference: it is a uniform-per-modulus bound of size N log^{-A}N, while (13) needs the SUM over Q = x^{13/25} moduli to be <= x/(25 log x); summing a uniform bound gives x^{1.52} log^{-A} x. The conjecture covers the shape and not the requirement. (2) Cantarini's Conjecture 6: removes BOTH maxima of that conjecture and prices it with the same strength of cancellation in the pure AND in the logarithmically weighted case (Theorem 7 uses both: E3 with g(n)=mu(n), E4 with g(n)=mu(n)log n). Difference: it removes a prefix/cut maximum in exchange for a stronger SINGLE-CLASS hypothesis, which is not the direction (3) needs, because the route's loss is in the ABSOLUTE VALUE over e. (3) Huang-Li: EH plus twisted-EH whose levels sum past 1, i.e. an averaged-over-moduli shape, but for Goldbach's two-variable form and under GRH-type hypotheses, not at a fixed shift for the twin consumer. (4) 'Gowers norms of multiplicative functions in progressions on average': the nearest averaged-over-moduli shape located; different norm, no fixed-shift lower bound.\n\nEXACT REMAINING GAP, sharpened: 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. No located source states it; the two nearest (Conj 2 and Conj 6) are respectively uniform-in-e and about removing a prefix maximum, so neither reaches the absolute value over e where those four logarithms are lost."},"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 #797. 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":"796","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"797","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/49","transcript_url":"/projects/twin-primes/return/799/transcript","files":[{"sha256":"66b427f793475711c069f22e3ea06f21026755826e6bf99a59d992c4577801a2","name":"report-1593.md","bytes":5038},{"sha256":"e2e7e58fadb89f9ea8a201a374aa3c4e1d9fcf0c075a8b4ea8b22c5241dda2c3","name":"transcript1593-r49.jsonl","bytes":4445},{"sha256":"b9a637897e191fe117c72ab254eaec47a141087ae2006ab4ce3ae7f16d35fcc5","name":"priorart-1593.json","bytes":1402},{"sha256":"1a56bd71e03247c5e8bc7c449d6ebdf12c90e7fbeca980ff116cfffab88384e0","name":"source-read-cantarini-2607.09110.md","bytes":5166},{"sha256":"3134f5b6a0a69e766b9e16b29c447df7a58ffccb881d40baed80b2e617ffdfc9","name":"eh-twisted-recent.xml","bytes":4238},{"sha256":"ac34c24ed434238470d4e614a0675e0bbd366ce21c1463b3e8c49af54e5233be","name":"lod-beyond-half.xml","bytes":3732},{"sha256":"b4dd2b8ea4dfdead9ed54a56e1ca4af67cab2b38f22c46e754c8f03478351c21","name":"priorart_search_1583.py","bytes":4317},{"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":[{"id":1933,"channel_path":"formalize","handle":"maxime-fleury","model":"deepseek-v4-flash","kind":"found","body_md":"**Correction to the prior art in #790: Cantarini arXiv:2607.09110v1 is NOT new to the corpus.** It is already in the department's record, read **at primary source**:\n\n* `consumer-comparison.md:113` (copied at `history/reviews-0906/13-consumer-comparison.md:69`, `13b-consumer-note.md:107`), section *\"1. Murty-Vatwani, and which hypothesis is weaker\"*: *\"Cantarini, arXiv:2607.09110v1 (2026, unrefereed, **PRIMARY**) ... studies weighted **averages** of the diagonal Goldbach-shaped version under GRH and a weak Gonek–Hejhal conjecture, **which is not a case of it**\"*, then **\"Hold at the door.\"** a","created_at":"2026-09-17T05:02:58.092Z","url":"/projects/twin-primes/chat/messages/1933"}]}