{"id":1033,"job_id":1939,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# REPORT — job #1939 (route 83, stage pursue, general mode, 1 of 1)\n\n**Question (pre-registered on return #1031, route 83).** Does the ALL-MODULI squarefree-AP theorem\nreaching `theta < 25/36` (Nunes, credited as `[15]` in Mangerel's introduction) admit a **y-friable\nsupport** and a **multiplicative `lam1` weight**, so that `(*)` follows with the modulus-smoothness\nrequirement dropped?\n\n**Answer. NO — and the refusal now sits in two named clauses, while the modulus clause passes.**\n\n## Source read at the page\n\n`arXiv:1605.03347` (ar5iv HTML, **112 849 B**, sha256 `4b781e2ce16ec9b8…`, full body incl. references)\n= R. M. Nunes, *On the least squarefree number in an arithmetic progression*, Mathematika **63**(2)\n(2017) 483–498 = the `[15]` of Mangerel (Forum Math. Sigma 9 (2021) e72), whose introduction states\n\"the best result that is available **for all moduli** `q` is that any `theta < 25/36 = 0.69 4̄` is\nadmissible, which is a recent result of Nunes `[15]`\". This confirms the attribution repair of #1037:\n`25/36` (all moduli) is **this** paper's constant; `196/261` is Mangerel's smooth-modulus bound.\n\n## Clause-by-clause\n\n| clause | Theorem 1.1 says | route 83 needs | verdict |\n|---|---|---|---|\n| modulus range | squarefree `q <= X^{25/36-eps}` | `d <= y^{4/5}`, i.e. exponent `4/(5u) in [2/5, 2/3]` | **PASSES** (max `2/3 < 25/36`, margin exactly `1/36`) |\n| modulus shape | `q` squarefree, `(a,q)=1` | `d \\| e`, `e` squarefree, reduced class | **PASSES** |\n| error | class-uniform power saving `E << X^{1-delta}/q` | uniform in `d` | **PASSES** (it is uniform, not an L1 sum) |\n| counted set | `mu^2` over **ALL** `n <= X` | `y`-friable `n` | **REFUSED** |\n| weight | `mu^2` only | `lam1(e) = (1/e) prod_{p\\|e} p/(p-4)` | **REFUSED** |\n\n## The two refusals, measured\n\n1. **No bounded-level divisor sieve can produce the friable support.** On `[1,60]` with `y = 10`,\n   exhaustive exact-rational elimination shows `1_{P(n)<=y}` lies in the `Q`-span of\n   `{1_{d|n} : d <= D}` **only at the full level `D = 60 = X`** (representable levels `[60]`; every\n   `D <= 59` fails, including `D = 11 =` next prime after `y`). The witness is structural: for a\n   `y`-smooth `m <= D` and a prime `p > max(y,D)` with `mp <= X`, a level-`D` sieve value `R(mp)`\n   equals `R(m)` (the divisors `<= D` are the same) while the true indicator drops `1 -> 0`.\n2. **The friable restriction is not an `o(1)` perturbation.** `rho(u) = 1 - ln u` exactly on `[1,2]`,\n   so restricting the counted set to `y`-friable `n` removes a fraction `ln u in [0.18232 (u=1.2),\n   0.69315 (u=2)]` of the mass — 7–30 orders above Theorem 1.1's `X^{1-delta}/q`. The theorem's main\n   term for the full set therefore says nothing about the friable subset.\n3. **The weight is not admitted.** The paper's body has **0** occurrences of `weight`, `smooth`,\n   `friable`, `Kloosterman`, `Poisson`; its mechanism counts solutions of `m^u = a n^v (mod q)` for\n   `(u,v) = (1,-2),(2,-1)` through the divisor identity (10) `mu^2(n) = sum_{n1 n2^2 = n} mu(n2)`, i.e.\n   it is an **unweighted** statement. The exact convolution `lam1(e) = sum_{k|e} g(k)` with\n   `g(p) = 1/(p-4) - 1` (verified for every squarefree `e <= 999` with exact `Fraction`s) does turn the\n   weight into a divisor sum, but its inner modulus is `d*k/(d,k)` — unbounded over the support — so it\n   does not preserve any fixed-modulus friable-AP range.\n\nConsequence: after #1037 (smooth-modulus lane refused by `eta = 1/u >= 1/2 > 6/25`) and this return,\n**both** squarefree lanes are refused by a named hypothesis: the smooth lane by its smoothness cap,\nthe all-moduli lane by support × weight. The friable-AP lane (`66/107 = 0.616822`, so `u >= 214/165`,\nleaving `4/33 = 12.12%` of the band uncovered) carries neither support-free range nor weight.\nThe residual is now exactly: **`mu^2` restricted to `y`-friable `n`, times a multiplicative `lam1`\nweight, uniformly in `d <= y^{4/5}` for `u in (1.2,2]`** — for which no published statement exists.\n\n## Cost and gates\n\nOne bounded `exec` (`--seconds 180 --cpu-seconds 180`): ledger `job1939-checks.py` **29/29 PASS,\n`ALL_PASS=True`**, child `exit_code 0`, wall **0.63 s**. First run (kept for the record, not uploaded)\nhad 6 FAILs, all mine: a wrong needle for the counted-object sentence, an expected occurrence count\n(`3`, not `2`), a wrong identity needle, a **one-level-wrong relative path** to the cached predecessor\nsource (so `S2a` \"failed\" on a missing file, not on missing text), and two over-confident\ndivisor-sieve expectations (`D =` next prime after `y` was predicted to suffice; the sweep shows only\n`D = X` does). No timing is printed to stdout, so the log is byte-reproducible.\n\n`outstanding` before the run: **1 of 144**, the pre-existing, server-explained #1685. Usage: **PENDING**\n(never estimated). `web_search` was UP (topical + control organic); `pdftotext` remains absent in this\ncontainer, so every quotation above comes from the ar5iv HTML body.","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T16:39:12.448Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1031],"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":"2026-09-18T16:43:34.725Z","file_notes":null,"research":{"outcome":"progress","route_id":83,"next_step":{"method":"Search the squarefree-AP literature for weighted statements (Pierce/Heath-Brown squarefree-detector moments, Mangerel Thm 1.5's completely-multiplicative condition, Drappeau/Pascadi friable APs) and read the candidate bodies at the source; then write the Buchstab/Mertens decomposition of the lam1 weight on the friable support explicitly, with exact rational arithmetic on the resulting modulus bookkeeping, and decide whether the composition is a valid reduction or a sieve of level > X.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Every candidate is refused by a named hypothesis (weighted mechanism absent, friable restriction unattainable at that level, or the decomposition's loss is not o(1)) - in which case the residual 'friable support x multiplicative weight' is recorded as the department's sharpest negative for route 83 and the route is closed as blocked/known rather than pursued further.","success":"A sourced weighted squarefree-AP statement (or an exact reduction of lam1 to a plain mu^2 count with a controlled, provably o(1) loss) that stays uniform for d <= y^{4/5}, u in (1.2,2] - then (*) closes in the relaxed all-moduli form.","question":"Is there any published statement counting squarefree n in arithmetic progressions with a general multiplicative weight (or with the specific divisor-sum weight lam1(e) = sum_{k|e} g(k), g(p) = 1/(p-4) - 1) uniformly in the modulus, and can such a statement be composed with a friable restriction by a Mertens/Buchstab decomposition whose loss is controlled exactly?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[1031,1030],"evidence_md":"Question (route 83, job #1939; pre-registered on return #1031): does the ALL-MODULI squarefree-AP theorem reaching theta < 25/36 (Nunes, credited as [15] in Mangerel's introduction) admit a y-FRIABLE support and a multiplicative lam1 weight, so that (*) follows with the modulus-smoothness requirement dropped? ANSWER: NO at every u in the band, and the refusal now sits in two named clauses while the modulus clause PASSES. Source read at the page: arXiv:1605.03347 (ar5iv HTML, 112849 B, sha256 4b781e2ce16ec9b8...) = R. M. Nunes, On the least squarefree number in an arithmetic progression, Mathematika 63(2) (2017) 483-498 = Mangerel's [15]. (1) Theorem 1.1: uniformly for X >= 2, integers a and SQUAREFREE q coprime with a satisfying q <= X^{25/36-eps}, sum_{n<=X, n=a mod q} mu^2(n) = (1/phi(q)) sum_{n<=X,(n,q)=1} mu^2(n) + O(X^{1-delta}/q). Clauses: counted object = mu^2 over ALL n (no support restriction); weight = mu^2 itself (0 occurrences of weight/smooth/friable/Kloosterman/Poisson in the body); error = class-UNIFORM power saving, not an L1 sum over moduli; modulus = squarefree q, (a,q) = 1. (2) MODULUS CLAUSE PASSES at every u in (1.2,2]: needed exponent 4/(5u) in [2/5, 2/3], max 2/3 < 25/36 with margin exactly 1/36 - the first lane that reaches the whole registered band. (3) The friable support is NOT reachable by a bounded-level divisor sieve: on [1,60] with y=10, exact-rational elimination (Fraction, exhaustive over all levels D = 1..60) shows 1_{P(n)<=y} lies in the Q-span of {1_{d|n}: d <= D} ONLY at the full level D = 60 = X (representable levels [60]); the witness is structural - for a y-smooth m <= D and a prime p > max(y,D) with mp <= X the level-D sieve value is unchanged while the true indicator drops 1 -> 0. (4) Nor is it an o(1) perturbation: rho(u) = 1 - ln u exactly on [1,2], so restricting to y-friable n removes a fraction ln u in [0.18232 (u=1.2), 0.69315 (u=2)] of the mass, far above the theorem's X^{1-delta}/q. (5) WEIGHT REFUSED: the paper states no weighted version and its mechanism counts solutions of m^u = a n^v (mod q) for (u,v) = (1,-2),(2,-1) through identity (10) mu^2(n) = sum_{n1 n2^2 = n} mu(n2), i.e. unweighted. The exact convolution lam1(e) = sum_{k|e} g(k) with g(p) = 1/(p-4) - 1 (verified with Fractions for every squarefree e <= 999) does convert the weight into a divisor sum, but its inner modulus is d*k/(d,k) - unbounded over the support - so it preserves no fixed-modulus friable-AP range. CONSEQUENCE: after #1037 (smooth lane refused by eta = 1/u >= 1/2 > 6/25), BOTH squarefree lanes are refused by a named hypothesis; the friable-AP lane (66/107 = 0.616822, so u >= 214/165, leaving 4/33 = 12.12% of the band uncovered) carries neither range nor weight. Residual, now exact: mu^2 restricted to y-friable n times a multiplicative lam1 weight, uniformly in d <= y^{4/5}, u in (1.2,2] - no published statement exists. Ledger job1939-checks.py 29/29 PASS, ALL_PASS=True, wall 0.63 s, child exit_code 0, one bounded exec. NOT CLAIMED: no number of the paper reproduced; (*) neither proved nor refuted; the smooth lane's refusal unchanged; nothing about uniform-in-d strength beyond the quoted 25/36 and 66/107.","prior_art_md":"Sources read at the page this turn: arXiv:1605.03347 (ar5iv HTML HTTP 200, 112849 B, sha256 4b781e2ce16ec9b8...) - Theorem 1.1, Corollary 1.2, identity (10), Lemma 1.3 (Pierce), the (u,v) = (1,-2),(2,-1) mechanism, and the references list (confirming reference [5] = Nunes arXiv:1602.00311, the PRIME-modulus 13/19 paper, i.e. a different paper with neither 196/261 nor 25/36). Attribution check re-read in the cached predecessor copy: Mangerel (Forum Math. Sigma 9 (2021) e72 = arXiv:2008.11163, 1505825 B, sha256 38f5beb6...) states 'the best result that is available for all moduli q is that any theta < 25/36 = 0.69 4bar is admissible, which is a recent result of Nunes [15]' - so 25/36 (all moduli, squarefree moduli only) belongs to arXiv:1605.03347 and 196/261 is Mangerel's own smooth-modulus bound; #1037's correction stands and is now sourced on both sides. Prior art within the department: returns #1031 (smooth lane refused by the eta <= 6/25 cap), #1030 (Pascadi arXiv:2304.11696 exponent 66/107 = 0.616822, no squarefree clause, no weight; Fouvry-Tenenbaum/Drappeau friable-AP barrier), #1027 (routeless proposal that created route 83). Channels: web_search UP (topical query and the control 'twin primes' both returned organic results); the arXiv abs page answered 200 as a control; pdftotext is ABSENT in this container, so all statements come from ar5iv HTML bodies (channel fact, not absence). Residual after this return: the single obstruction 'friable support x multiplicative weight on the all-moduli squarefree lane', with the range clause now proved to pass at every u in (1.2,2]. No claim is made that the residual is unattainable in principle."},"research_route_id":83,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_8c1f57ef8963c7ece260fdc5","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 #1031. 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":"1030","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1031","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/83","transcript_url":"/projects/twin-primes/return/1033/transcript","files":[{"sha256":"16df77475b8d5d31cdf5853c300d1c71083bc0a3c295418ff8db813d0eb5e19c","name":"job1939-checks.py","bytes":12072},{"sha256":"c81b7ff4fb65626223a40c649a7deb37b9ad64d96a5f6c7b6d8ff5054ea6a28d","name":"job1939-checks.log","bytes":2897},{"sha256":"f373e9043f30c5eb99584957f1603837a655c8a3cf82353fa166d265e1ef1dfc","name":"REPORT.md","bytes":4983},{"sha256":"5156a1e126abdb43df62dae647629fc2eda32e5f077df941f321c06b7b08bc36","name":"research-1939.json","bytes":6646}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}