{"id":1031,"job_id":1937,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1937 (route 83, pursue, explore/discovery) — does the squarefree-AP lane carry the y-friable support of `(*)`?\n\nRun `run_20260918_182553_8Oxsjw`, attempt `86eb5fd4431cf6a3317a8e5b536aff5f`, department\n`dept_c326cb5ae203e5d0d94f8db1`, general mode, 1 of 1. Tool `sah/14`\n(`38a08cad8413951dc69e459357c3996aa2e8c87571a7f6c903d40d9332c4d2a4`), readiness **27/27** at\n16:25:43Z. Ledger `job1937-checks.py` / `job1937-checks.log` — **22/22 PASS**, one bounded\n`sah.py exec` (wall 0.12 s, child `exit_code 0`), 0 CPU-h of real compute beyond clause extraction.\nAttached: the ledger, its log, this report and `research-1937.json`; the two fetched source bodies\n(1.5 MB + 0.6 MB) are **not** attached — they are public and re-fetchable, and the ledger binds their\nURLs, byte counts and sha256 (`replies/fetch-1937.json`).\n\n## The question, as the route registered it\n\nRoute 83 rev 2 (`GET /projects/twin-primes/research-routes/83`, rid `q_p4Yu_t5sgJW9Mvh1`) and\nreturn #1030 (`GET /projects/twin-primes/return/1030`, rid `q_tXAulOq6O4K9IuED`) pre-registered one\nread: *“Does the squarefree lane's smooth-modulus machinery (Mangerel, Forum Math. Sigma 2021; Nunes,\narXiv:1602.00311) carry the `y`-FRIABLE support of `(*)` as well as the squarefree condition — i.e.\ncan a friable-support restriction be inserted into a squarefree-AP theorem to relative precision\no(1) uniformly in `d <= y^{4/5}`, and what does the `lam1` weight cost?”*\n\n`(*)` (served note `research/history/staging/attack-0830-varE-identification.md`, §4, rid\n`q_bizmT4JZRPQ5qha-`, 30 382 chars): for every `eps > 0` there is `y_0` such that for `y > y_0`,\nuniformly in `y`-smooth squarefree `d <= L^{2/5} = y^{4/5+o(1)}` coprime to 30,\n\n    |E_d| <= eps * (L/d) * sum_{e <= 2L/d} lam1(e)/e * (1/phi-share),\n\ni.e. the `y`-friable **squarefree** integers `e` in `(L^{1-eta}/d, 2L/d]`, weighted by\n`lam1(e) = (1/e) prod_{p|e} p/(p-4)`, are equidistributed among the reduced classes mod `d` against\nbounded-variation test functions, UNIFORMLY in `d` up to `y^{4/5}`, friability range\n`u = ln e / ln y in (1.2, 2]`.\n\n## Answer: NO — and the refusal now sits in the borrowed theorem's own hypothesis\n\nBoth bodies were read **at the page** (ar5iv HTML, HTTP 200, cached with shas in the ledger:\nMangerel `2008.11163` 1 505 825 B `38f5beb6…`, Nunes `1602.00311` 583 636 B `82a0e85a…`;\n`pdftotext` is absent in this container — a channel fact, not absence of text).\n\n1. **Mangerel Thm 1.1 as printed**: `0 < eta < 1/522`, `q <= X^{196/261-eps}` that is\n   `X^eta`-smooth **and** squarefree. **Remark 1.2 / §6.1** give his *largest* admissible exponent:\n   `eta <= 6/25` for `q <= X^{3/4-eps}`.\n   In `(*)`, `d` is `y`-smooth with `y = X^{1/u}`, `u in (1.2, 2]` (count scale `X ~ L/d`,\n   `L = y^{2+o(1)}`), so the demanded smoothness exponent is exactly `eta = 1/u in [1/2, 5/6]` —\n   **2.0833× (u = 2) to 3.4722× (u = 1.2) above `6/25`, and 261×–435× above `1/522`**; the band\n   would need `u >= 25/6 = 4.1667` to fit. Normalization-robust: with `theta_d = ln d/ln y <= 4/5`,\n   `eta = 1/(2 − theta_d)` gives the same `[1/2, 5/6]`.\n   The **modulus-size clause passes** (`theta = 4/(5u) in [2/5, 2/3] <= 2/3 < 3/4`), so the relevant\n   regime is Remark 1.2's — **the only failing clause is `eta`**, and it fails at *every* `u` in the band.\n2. **Nunes Thm 1.1** is **prime modulus** `q <= X^{13/19-eps}`, counted set = all `n <= X` with\n   `mu^2(n) = 1`, error `O(X q^{-1} log^{-A} X)` uniform in the class. A generic `y`-smooth squarefree\n   `d` is composite, so the primality hypothesis refuses it; there is no friable support and no weight.\n3. **Weights.** Nunes counts the plain characteristic function `mu^2`; Mangerel counts squarefree\n   `n`. The only weighted statement (Mangerel Thm 1.5) needs `f` **completely** multiplicative, and\n   the needed `lam1` is not: `lam1(49) = 1/21 != lam1(7)^2 = 1/9` (exact).\n4. **Uniformity is not what fails.** Nunes' error is uniform in the class; Mangerel's is uniform in\n   `a` with `(a,q) <= X^eps`. It is the *density* statements (Mangerel Thm 1.5, Cor. 1.3, \"almost\n   all\"/\"positive proportion\") that the required uniformity in `d` refuses.\n5. **Attribution correction to #1030's `prior_art_md`.** Nunes `1602.00311` contains **neither**\n   `196/261` **nor** `25/36` (0 occurrences each): its theorem is the prime-modulus `13/19` result.\n   `25/36` (all moduli) is credited by Mangerel's own introduction to a **different** Nunes paper\n   (`[15]`), and Mangerel's own smooth-modulus bound is the *larger* `196/261`. The two numbers were\n   swapped in #1030's prior-art paragraph; the record is corrected here from the source.\n\n**Consequence for the route.** The smooth-modulus lane is **structurally unavailable** at the needed\nsmoothness exponent — not partially covering — so the only lane whose modulus range reaches the whole\nband is the **all-moduli** squarefree lane (`theta < 25/36 = 0.6944`, covering `2/3`), and there the\nmissing requirements are exactly the friable support and the multiplicative weight. The residual is\nunchanged in content and sharpened in location: **“friable support × multiplicative weight”**, with\nthe one candidate that could carry it named for the successor.\nA supporting quantitative fact from #1030's own source read: the friable-number AP lane\n(Fouvry–Tenenbaum 1991, Drappeau 2015; `x^{3/5-o(1)}` as stated by Pascadi's paper) covers only\n`u >= 4/3` — it is out of modulus range on the band's lower third `(1.2, 4/3)` *and* carries no weight.\n\n## Not claimed\n\nNo number of either paper was reproduced; `(*)` is neither proved nor refuted; nothing is claimed\nabout the uniform-in-`d` strength of the friable-number AP lane beyond the `3/5` barrier quoted from\n#1030's source read. The `eta` refusal closes `(*)` **as registered** (with `d` required `y`-smooth);\nthe disclosed relaxation to all squarefree `d <= y^{4/5}` leaves only support × weight, since\n`2/3 < 25/36`. The one failed check of the first ledger run (`B4`) was my own over-narrow regex, kept\nas `job1937-checks.first-run.log`, kept **locally** (its traceback carries machine-absolute paths, which `POST /files` refuses — the crashed-ledger rule) and fixed to the true clause wording, never relaxed.\n\n## Framework note\n\nThe ledger prints no timing on stdout (timings go to `stderr`), so the published `.log` is\nbyte-reproducible from the script + the two cached bodies; `job1937-checks.stderr.log` is kept locally.","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T16:30:41.912Z","repo_url":null,"commit":null,"cites":{"returns":[1027,1030]},"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":"Locate and read that paper's body at the source through the arXiv/ar5iv HTML channel (record which URL answered; pdftotext is absent here), and extract clause by clause: (a) the counted object and whether it is all squarefree n or a restricted support; (b) the exact modulus conditions (size, primality/compositeness, coprimality to fixed primes, (a,q)-conditions); (c) whether the error is uniform in the class or a Bombieri-Vinogradov L1 sum over moduli; (d) whether a general multiplicative weight is admitted, and whether the Poisson-plus-bilinear-Kloosterman mechanism tolerates a sieve restriction to y-smooth n with y = X^{1/u}, u in (1.2,2]. Then attempt the one-line sieve reduction of the friable support and check the resulting modulus exponent against the theorem's range with exact rational arithmetic.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"The candidate is refused by a NAMED hypothesis - counted set restricted to all n with the friable restriction unattainable in that range, weight not admitted, error not uniform per class, or the modulus range itself falling below 2/3 for u near 1.2 - in which case the residual 'friable support x multiplicative weight on the all-moduli lane' is the sharpest negative the record can carry without attempting a proof.","success":"A sourced statement (or a clean sieve reduction) giving mu^2 restricted to y-friable n, uniformly in d <= y^{4/5} for u in (1.2, 2], with the lam1 weight admitted - then route 83's cell closes in the relaxed all-moduli form and the varE remainder is o(ln^2 y).","question":"Does the ALL-MODULI squarefree-AP theorem that reaches theta < 25/36 (Nunes, credited as [15] in Mangerel's introduction; not arXiv:1602.00311) admit a y-FRIABLE support and a multiplicative lam1 weight, so that (*) follows with the modulus-smoothness requirement dropped?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[1027,1030],"evidence_md":"Question (route 83 rev 2, job #1937): does the squarefree-AP lane carry the y-FRIABLE support of (*) in the friability band u in (1.2,2], moduli d <= y^{4/5}? ANSWER: NO at every u in the band, with the refusal now located in the borrowed theorem's OWN hypothesis (bodies read at the page, ar5iv HTML, shas in the ledger: Mangerel Forum Math. Sigma 9 (2021) e72 = arXiv:2008.11163; Nunes arXiv:1602.00311). (1) Mangerel Thm 1.1 requires 0<eta<1/522 and q X^eta-smooth; Remark 1.2/§6.1 gives his LARGEST admissible exponent: eta <= 6/25 for q <= X^{3/4-eps}. In (*) the modulus d is y-smooth with y = X^{1/u}, u in (1.2,2] (count scale X ~ L/d, L = y^{2+o(1)}), so the demanded smoothness exponent is exactly eta = 1/u in [1/2, 5/6] - normalization-robust (with theta_d = ln d/ln y <= 4/5, eta = 1/(2-theta_d) gives the same interval). That is 2.0833x (u=2) to 3.4722x (u=1.2) above 6/25 and 261x-435x above 1/522; the band would need u >= 25/6 = 4.1667 to fit. The modulus-SIZE clause passes: theta = 4/(5u) in [2/5, 2/3] <= 2/3 < 3/4, so Mangerel's larger-eta regime (q <= X^{3/4-eps}) is the relevant one and the only failing clause is eta. (2) Nunes Thm 1.1 is PRIME modulus q <= X^{13/19-eps} over ALL squarefree n <= X, uniform in the class: a generic y-smooth squarefree d is composite and refused by the primality hypothesis; no friable support, no weight. (3) Weights: Nunes counts the plain characteristic function mu^2, Mangerel counts squarefree n, and the only weighted statement (Mangerel Thm 1.5) needs f COMPLETELY multiplicative while the needed lam1(e) = (1/e) prod_{p|e} p/(p-4) is not (lam1(49) = 1/21 != lam1(7)^2 = 1/9, exact). (4) Uniformity is NOT the failing clause (Nunes uniform in the class; Mangerel uniform for (a,q) <= X^eps); the density statements (Thm 1.5, Cor 1.3) are what the uniformity in d refuses. (5) Attribution correction to #1030's prior_art_md: Nunes arXiv:1602.00311 contains neither 196/261 nor 25/36 (0 occurrences each); 25/36 (all moduli) is credited by Mangerel's own introduction to a DIFFERENT Nunes paper ([15]) and Mangerel's smooth-modulus bound is the larger 196/261 - the two were swapped. CONSEQUENCE: the smooth-modulus lane is structurally unavailable at the needed smoothness exponent (not partially covering), so the only lane whose modulus range reaches the whole band is the all-moduli squarefree lane (theta < 25/36 = 0.6944, covering 2/3), where exactly the friable support and the multiplicative weight are missing. Supporting: the friable-number AP lane (Fouvry-Tenenbaum 1991, Drappeau 2015; x^{3/5-o(1)} barrier as stated in Pascadi's own paper, read in #1030) covers only u >= 4/3, so it is out of modulus range on the band's lower third (1.2, 4/3) and carries no weight either. NOT CLAIMED: no number of either paper reproduced; (*) neither proved nor refuted; nothing about the friable lane's uniform-in-d strength beyond the quoted 3/5 barrier; the eta refusal closes (*) AS REGISTERED (d required y-smooth), while the disclosed relaxation to all squarefree d <= y^{4/5} leaves only support x weight (2/3 < 25/36). Ledger job1937-checks.py 22/22 PASS.","prior_art_md":"Online search updated this turn (web_search UP: the topical query and the control query 'twin primes' each returned 10 organic results): the friable/large-moduli query surfaced, besides the already-cited Pascadi (arXiv:2304.11696, Compositio Math. 161 (2025) 1923-1974) and Mangerel (Forum Math. Sigma 9 (2021) e72), a 2025 Maynard record ('Primes in arithmetic progressions to large moduli II') whose own text states that Fouvry-Tenenbaum and Drappeau proved equidistribution for smooth numbers in arithmetic progressions to moduli x^{3/5-eps} - recorded as a SEARCH RECORD (snippet level), not read at the page, and consistent with Pascadi's statement of the same barrier quoted in #1030. Sources read at the page this turn: Mangerel arXiv:2008.11163 (ar5iv HTML HTTP 200, 1 505 825 B, sha256 38f5beb6...) - Thm 1.1, Remark 1.2, §6.1, Thm 1.5, Cor 1.3; Nunes arXiv:1602.00311 (ar5iv HTML HTTP 200, 583 636 B, sha256 82a0e85a...) - Thm 1.1 and section 1.1. Also read as served record: route 83 rev 2, return #1030, and the note research/history/staging/attack-0830-varE-identification.md §4 (rid q_bizmT4JZRPQ5qha-, 30 382 chars) for the verbatim wording of (*). Channel facts: pdftotext is ABSENT in this container, so the statements came from the ar5iv HTML bodies (channel limitation, not absence); the arXiv abs pages answered 200 as controls. EXACT REMAINING GAP (sharpened): (*) needs equidistribution of y-friable SQUAREFREE e, weighted by lam1(e) = (1/e) prod_{p|e} p/(p-4), in reduced classes mod d, to relative precision o(1) UNIFORMLY in d up to y^{4/5}, u in (1.2, 2]. Per lane: the all-moduli squarefree lane (theta < 25/36) reaches the modulus range for every u in the band but carries neither the friable support nor any multiplicative weight; the smooth-squarefree lane (Mangerel, 196/261 for q <= X^{3/4-eps}) reaches the modulus size but its smoothness cap eta <= 6/25 is exceeded by the required eta = 1/u in [1/2, 5/6] at EVERY u in the band (head-line theorem: eta < 1/522, 261x-435x too small); the friable-number AP lane reaches only u >= 4/3 (x^{3/5-o(1)}) and carries no weight; the all-smooth lane (Pascadi) has no squarefree clause and no weight (#1030). The residual is therefore the single obstruction 'friable support x multiplicative weight' on the all-moduli lane. CORRECTION to #1030's prior-art paragraph: 196/261 is Mangerel's smooth-modulus bound and 25/36 (all moduli) belongs to Nunes (Mangerel's [15]); Nunes arXiv:1602.00311 itself contains neither string."},"research_route_id":83,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_ff6a5100fd5ad97c7927cc2f","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 #1030. 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":"1027","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1030","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/83","transcript_url":"/projects/twin-primes/return/1031/transcript","files":[{"sha256":"bc1b80a60de9b9a6156b5d7174749b22e8fa870977e7c2281c7397808870f615","name":"job1937-checks.py","bytes":9481},{"sha256":"4c7710c02779656d6caa8db2b6208f0b4f950cccd4def7336650b08b06225cac","name":"job1937-checks.log","bytes":4811},{"sha256":"d1e19cb0d739802093be2e6509df7ffda50b6b40690e250de9e70a8747df1dba","name":"REPORT.md","bytes":6475},{"sha256":"667575ac8eabde244e38b1e3cc8b36b81f0e1837f07f9445e9ff4e0c6d6fd578","name":"research-1937.json","bytes":7752}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}