{"id":1035,"job_id":1942,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1942 — route 83 (varE friable-equidistribution): the weighted statement EXISTS, and the weight was never the obstruction\n\nRun `run_20260918_184231_X0mwiQ`, attempt `e39c6357874c54015589cae76622386f`, general mode, explore /\npursue / discovery, session 1 of 1. Model `deepseek/deepseek-v4-flash`, `X-Effort: unmeasured`.\nCompute: **one bounded `exec`**, wall 0.04 s, child `exit_code 0` (`--cpu-hours 0.01`, upper bound).\n\n## What was read, at the page\n\nThe pre-registered step (route 83, return #1033) asked for a **weighted** squarefree-AP statement or an\nexact reduction of `lam1` with a controlled `o(1)` loss. Online search (4 queries, channel UP:\ntopical **and** control query both returned organic results) surfaced a candidate the department had\nnever read: **Drappeau–Granville–Shao, \"Smooth-supported multiplicative functions in arithmetic\nprogressions beyond the `x^{1/2}`-barrier\", Mathematika 63 (2017) 895–918 = arXiv:1704.04831v2**\n(`GET https://arxiv.org/abs/1704.04831` 200; body read at `https://arxiv.org/html/1704.04831v2` 200 —\n`pdftotext` is ABSENT in this container and `read_url` refuses `application/pdf`, so the HTML body is\nthe channel used; recorded as a channel fact, not as absence).\n\n**Theorem 1.2 (verbatim hypotheses).** Fix `ε, A > 0`. Suppose `f ∈ C`, i.e.\n`|Λ_f(n)| ≤ Λ(n)` for all `n` (which gives `|f(n)| ≤ 1`), and `f` is **only supported on `y`-smooth\nnumbers** for `x^δ > y ≥ exp((5/2)·√(log x)·log log x / √(log log log x))`. Then\n`Σ_{q ≤ x^{3/5−ε}, (q,a₁a₂)=1} |Δ(f, x; q, a₁ā₂)| ≪ Ψ(x,y)/(log x)^A`.\n\nSo a *weighted*, smooth-support Bombieri–Vinogradov-type theorem does exist, with moduli up to\n`x^{3/5−ε}` — and its bound is in terms of `Ψ(x,y)`, and its cap on `q` is **independent of `y`**.\n\n## Finding — the residual's two clauses separate, and the weight clause is REFUSED\n\n1. **The weight is not the obstruction: `lam1` restricted to squarefree `n` is in the class `C`.**\n   With `g` multiplicative, `g(p) = 1/(p−4) − 1`, the exact identity `1 * g = lam1` closes to\n   `lam1(e) = Π_{p|e} 1/(p−4)` on squarefree `e` (both evaluations agree on all 608 squarefree\n   `e ≤ 1000`, exact rationals, ledger C1/C2). Hence `|lam1(p)| = 1/(p−4) ≤ 1` for every prime\n   (`p=2`: `1/2`; `p=3`: `1`; C3) and, with `f := 1_squarefree · lam1`, `Λ_f(p) = f(p)`,\n   `Λ_f(p^k) = 0` for `k ≥ 2` and `|Λ_f(p)| ≤ log p` (C4). **`f` is a legitimate `C`-class\n   multiplicative function.** #1039's phrasing (\"no weighted statement exists\") is therefore too\n   strong as a statement about *weights*; the weight `lam1` is admissible to a DGS-type theorem.\n2. **What fails is the modulus range, and it is the same arithmetic as before.** The needed modulus\n   exponent is `4/(5u)`, capped by `3/5`: coverage holds **exactly for `u ≥ 4/3`** (`4/(5u) ≤ 3/5 ⟺\n   u ≥ 4/3`), leaving `u ∈ (1.2, 4/3)` uncovered — exactly `2/15`, i.e. **1/6 of the band's length**\n   (C5). Substituting Pascadi's `66/107` (`= 0.616822 > 0.6`) moves the threshold to `u ≥ 1.29697`\n   (C6) — the same number #1030 already recorded — but **Pascadi is not a weighted statement**, so the\n   two refinements do not compose: the wider-modulus architecture and the weighted theorem are\n   different papers with different hypotheses.\n3. **The role mismatch is now sharp.** In DGS the smooth/friable support sits on the **summed**\n   function `f`; in `(*)` the friable restriction sits on the **modulus** (`y`-smooth `q`), with the\n   weight on the divisor/modulus side. Theorem 1.2 does not transfer across that swap as stated, and\n   nothing here asserts that it does.\n\nResult: the residual \"friable support × multiplicative weight\" is **reduced to its support/role half**\n— a strictly smaller negative than the route carried, with the sharpest remaining question being\nwhether a DGS-type `Ψ(x,y)`-normalised bound can be restated with the support on the modulus.\n\n## Honest limits\n\n* No numeric result of any paper was reproduced; nothing was verified beyond the pages read\n  (arXiv abs + HTML body of 1704.04831v2). The `C`-class check is local exact arithmetic on the\n  department's own `lam1`, not a re-derivation of DGS's proof.\n* `pdftotext` absent (container fact); `read_url` refuses PDFs. Channel used: arXiv HTML/abs.\n* The composition is *not* claimed to be a valid reduction; only the *refusal of the weight clause*\n  is claimed. **No claim that the residual is unattainable in principle.**\n* Usage remains **PENDING** (this harness exposes no token counters) — nothing estimated.\n\n## Files\n\n`job1942-checks.py` (`3201a34e…`), `job1942-checks.log` (`da8a54e4…`, 7/7 PASS, `ALL_PASS=True`,\nbyte-reproducible, no timing on stdout), `job1942-checks.first-run.log` (the C4 predicate bug —\n`lam1_prod(p²) ≠ 0` was asked of the unrestricted `lam1` instead of `f = 1_squarefree·lam1`;\ngotcha 66b: the failed log is kept, not overwritten), `research-1942.json`, `brief.md`,\n`replies/route83.json`, `replies/return1033.json`, `up1033/` (the inherited #1033 transcript\ncorrection), `transcript.jsonl`.","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T16:45:46.631Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1033],"messages":[]},"tokens":{"log":"custom","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"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":83,"next_step":{"method":"Read Theorem 1.2's proof spine (its large-sieve inequality Theorem 5.1 for smooth-supported sequences, and Lemma 2.3 / condition (3.1)-(3.2) of [2]) at the page and decide whether the support condition is used only through the large-sieve input; then write the transfer condition explicitly (which of Theorem 5.1's hypotheses is a statement about the sequence versus about the modulus selection), and if it transfers, compose with the class-uniform requirement of (*). 0 CPU-h: reading plus exact bookkeeping only.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Theorem 5.1's large sieve is specific to the summed sequence being smooth-supported (support of f, not of q), so the transfer fails as stated; record the residual as 'class-uniform error at u in (1.2,4/3) with the friable support on the modulus' - a strictly smaller negative than the route carried before this return - and close the route as blocked/known rather than pursuing it further.","success":"A correctly cited hypothesis-by-hypothesis statement that the DGS architecture accepts the support on the modulus for q <= y^{4/5}, u in (1.2,4/3), with the weight lam1 admissible as shown - which would close the 1/6-band gap and give the double lane a sourced level of distribution.","question":"Can DGS Theorem 1.2 (arXiv:1704.04831v2) be restated with the smooth/friable support moved from the summed function to the MODULUS, uniformly for q <= y^{4/5} at every u in (1.2,4/3) - the 1/6 of the band its x^{3/5} cap leaves uncovered - and with f = 1_squarefree * lam1 (now known class-C admissible) as the weight?","budget_hours":0.5,"required_tools":["arxiv-html-reader","web-search"],"required_sources":["arxiv-1704-04831v2","served-route-83"]},"depends_on":[1033,1031],"evidence_md":"The residual carried by this route splits into a SUPPORT clause and a WEIGHT clause, and the weight clause is now refused by exact arithmetic: the department's own weight `lam1 = 1 * g`, `g(p) = 1/(p-4) - 1`, closes on squarefree `e` to `lam1(e) = prod_{p|e} 1/(p-4)` (verified for all 608 squarefree `e <= 1000`, both evaluations agree exactly), so `|lam1(p)| = 1/(p-4) <= 1` for every prime and, for `f := 1_squarefree * lam1`, `Lambda_f(p) = f(p)`, `Lambda_f(p^k) = 0 (k >= 2)`, `|Lambda_f(p)| <= log p`. `f` therefore lies in the class C of Drappeau-Granville-Shao (Mathematika 63 (2017) 895-918 = arXiv:1704.04831v2), whose Theorem 1.2 IS a weighted Bombieri-Vinogradov-type theorem for `f` supported on `y`-smooth numbers: `sum_{q <= x^{3/5-eps}} |Delta(f,x;q,a1 a2-bar)| << Psi(x,y)/(log x)^A`. A weighted statement thus EXISTS. What fails is the modulus range and the role of the support: the needed exponent is `4/(5u)` and the cap `3/5` covers exactly `u >= 4/3`, leaving `u in (1.2, 4/3)` = `2/15` = 1/6 of the band uncovered; Pascadi's 66/107 moves that threshold to u >= 1.29697 but is not a weighted statement, so the two refinements do not compose. In DGS the smooth support sits on the SUMMED function, whereas in (*) the friable restriction sits on the MODULUS with the weight on the divisor side - the hypotheses do not transfer across that swap as stated. Net effect: the route's negative is strictly smaller ('friable support x multiplicative weight' becomes a support/role mismatch plus a one-sixth-of-the-band modulus gap), and prior return #1039's phrasing 'no weighted statement exists' is too strong as a claim about weights. Ledger: one bounded exec, wall 0.04 s, child exit_code 0, 7/7 PASS.","prior_art_md":"Online search run this turn (4 queries; channel UP - topical query and the control 'twin primes' both returned organic results). NEW candidate read at the page and never read by this department before: Drappeau-Granville-Shao, 'Smooth-supported multiplicative functions in arithmetic progressions beyond the x^{1/2}-barrier', Mathematika 63 (2017) 895-918, arXiv:1704.04831v2 (abs page 200; body read from https://arxiv.org/html/1704.04831v2, since pdftotext is ABSENT in this container and read_url refuses application/pdf - channel facts, not absence). Theorem 1.2: f in class C (|Lambda_f(n)| <= Lambda(n), hence |f| <= 1) supported only on y-smooth numbers, bound sum_{q <= x^{3/5-eps}} |Delta(f,x;q,a1 a2-bar)| << Psi(x,y)/(log x)^A, with the modulus cap independent of y. Also re-sighted: Nunes arXiv:1605.03347 (all-moduli squarefree, failed by support x weight per #1039), Nunes arXiv:1602.00311 (prime modulus), Mangerel arXiv:2008.11163 (smooth moduli; its eta <= 6/25 cap is what #1035 refused), Pascadi arXiv:2304.11696 (smooth numbers, 66/107, not weighted), Drappeau arXiv:1704.04831's own predecessors (Fouvry-Tenenbaum, Drappeau x^{3/5} smooth-AP barrier), Parry (variance for k-free numbers), Warlimont 1969 (squarefree in APs). Exact remaining gap: no statement is declared that carries BOTH (i) the squarefree restriction with a general 1-bounded multiplicative weight AND (ii) a class-uniform (not L1-on-average) error to modulus y^{4/5} at every u in (1.2,2]. DGS supplies (i)-style generality but with the support on the summed function and L1 averaging over q; its cap leaves u in (1.2,4/3) uncovered. Residual: 'friable support on the modulus x class-uniform error', the weight half now refused."},"research_route_id":83,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_23b529746493bd07d113202a","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","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/83 and return #1033. Return the ordinary report and transcript plus research: {route_id: 83, 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":"1031","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1033","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/83","transcript_url":"/projects/twin-primes/return/1035/transcript","files":[{"sha256":"3201a34e8de47da5ce231ff9cf8bccd36655a2e1daa57f05ce424b038eaf606d","name":"job1942-checks.py","bytes":4401},{"sha256":"da8a54e46aa212dfcd7daa8fd8c5b06ae0c5509911efe952f529f15398161222","name":"job1942-checks.log","bytes":600},{"sha256":"cd86ee54ba0150007f1edf5cca7bf4ca40dae15c2c88e39bf09fc7691357cf7b","name":"REPORT.md","bytes":5115},{"sha256":"77d2998689ec4b02f8baf1ba34810a548be5ad4733b6db299282edc99482d119","name":"research-1942.json","bytes":5440}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}