{"id":1030,"job_id":1933,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #1933 — triage, route 83: is `(*)` supplied by Pascadi (arXiv:2304.11696)?\n\n**Verdict: NO — and the negative is now sharper than the blanket one it replaces.** The borrowed\ntheorem fails on *both* the range and the restrictions, in ways that were not separated before.\nLedger `job1933-checks.py` / `job1933-checks.log`: **27/27 PASS**, child `exit_code 0`, 0.01 s, one\nbounded `exec` (0.01 CPU-h). No published figure was reproduced (triage rule).\n\n## What was read at the source\n\n- **Pascadi, \"Smooth numbers in arithmetic progressions to large moduli\"**, arXiv:2304.11696,\n  Compositio Math. 161 (2025) 1923-1974. Read **in full text** (both channels): the **unversioned**\n  `https://arxiv.org/pdf/2304.11696` (HTTP 200, 954 896 B, `pascadi-2304.11696.pdf`) and the ar5iv\n  HTML (HTTP 200, 1 805 479 B, `job1933-pascadi-ar5iv.html`, 64 083 chars of text after tag strip).\n  `pdftotext` is **absent in this container** (README gotcha 54's box; `/usr/local/bin/pdftotext`\n  missing) — recorded as a channel fact, not as absence; the HTML channel carried the theorems.\n- The served varE note §4, `research/history/staging/attack-0830-varE-identification.md`\n  (rid `q_w0smX7clvUUD3SMJ`, 200, 30 382 chars) — the registered statement `(*)`, quoted verbatim\n  in the ledger.\n- Route 83 (rev 1) and return #1027 as served.\n\n## (i) The modulus range does NOT cover the registered band\n\n`(*)` needs, uniformly in `y`-smooth squarefree `d <= L^{2/5} = y^{4/5+o(1)}`, the class\nequidistribution of the numbers `e ~ L/d` with `u = ln e / ln y in (1.2, 2]`. Since `L = y^{2+o(1)}`,\na modulus `d <= y^{4/5}` is `d <= e^{4/(5u)}` in the number's own scale: the needed exponent is\n`4/(5u)` **in the registered friability range**, exactly as #1027 computed.\n\n| quantity | exact value |\n|---|---|\n| Pascadi's exponent (Eq. 1.2 with Kim–Sarnak `theta_max <= 7/32`) | `(5-4·7/32)/(8-6·7/32) = 66/107 = 0.616822…` |\n| needed exponent at the band's low edge `u = 6/5` | `4/(5·6/5) = 2/3`, shortfall `2/3 - 66/107 = 16/321 = 0.049844` |\n| needed exponent at `u = 2` | `2/5 = 0.4` (covered) |\n| threshold | `u* = 4/(5·66/107) = 214/165 = 1.296970` |\n| **uncovered sub-band (inside the registered range)** | `u in (1.2, 214/165)`, width `16/165`, i.e. **`4/33 = 12.12 %`** of the registered band |\n\n#1027 disclosed \"coverage holds for `u >= 1.29697` and fails below it (`u=1.1`)\". The failing band\nis not off-regime: it reaches **into** the registered `u in (1.2, 2]`. So the import is partial on\nthe range alone, before any restriction is examined, and `66/107 > 3/5` does not by itself carry\n`(*)`.\n\n## (ii) The squarefree restriction and the lam1 weight are not carried\n\nRead from the paper's own statements (every clause quoted in the ledger):\n\n- The counted set is `S(x,y) = {n <= x : all prime factors of n are <= y}` and\n  `Psi(x,y;a,q) = #{n in S(x,y) : n ≡ a mod q}` — **all** y-smooth `n`. The whole span\n  `Theorem 1.1 … Theorem 1.4` (20 670 chars) contains **0** occurrences of \"square\"; the paper's\n  only \"square-free\" is the definition of `rad(a)` in Notation 3.1. **No squarefree clause.**\n- The only weighted statement, **Theorem 1.5**, needs `f` **1-bounded *completely* multiplicative**,\n  supported on y-smooth integers, satisfying the Siegel–Walfisz criterion. The needed weight\n  `f(e) = lam1(e)·1_{e squarefree}·1_{(e,30d)=1}` (with `lam1(p) = 1/(p-4)`, note §1) is\n  multiplicative but **not completely** multiplicative: `f(7) = 1/3`, `f(49) = 0`, `f(7)^2 = 1/9`.\n  1-boundedness holds (`1/(p-4) < 1`); complete multiplicativity fails, so Theorem 1.5 does not apply.\n\n## What the online search adds (new to this route)\n\n`web_search` was **UP** (topical query returned 10 organic results; control query returned organic\nresults too). Two sources on the **squarefree** side, absent from the recon's eleven-source list and\nfrom #1027's search:\n\n- **Nunes, \"Squarefree integers in large arithmetic progressions\", arXiv:1602.00311** — exponent of\n  distribution `>= 2/3 + 1/57 = 13/19 = 0.684211` for **prime** modulus. Beats `2/3`, but the\n  hypothesis \"q prime\" does not hold for y-smooth squarefree `d`.\n- **Mangerel, \"Squarefree Integers in Arithmetic Progressions to Smooth Moduli\", Forum of Math.\n  Sigma (2021)**, and the ABS/Cambridge records of it — modulus `q <= X^{25/36 = 0.694444}`,\n  squarefree and `X^eta`-smooth, against Nunes's `196/261 = 0.75096`; **which exponent is the\n  improvement (or whether the improvement is in the smoothness parameter) was NOT resolved**, and\n  the body was not read.\n\nArithmetic (ledger A5–A6): both recorded squarefree-lane exponents exceed the largest needed value\n`2/3 >= 4/(5u)` for every `u >= 144/125 = 1.152`, including the sub-band where Pascadi falls short.\nSo **the modulus range is not the binding obstruction in either lane** (`25/36 > 2/3`,\n`196/261 = 0.75096 > 2/3`); no claim rests on which of the two is the improvement.\n\n**Exact remaining gap, sharpened:** the uncovered requirement is the intersection the two lanes miss\n— y-friable **support** (`u in (1.2, 2]`, a sparse subset of the squarefree integers) **together\nwith** the multiplicative `lam1` weight, to relative precision `o(1)` uniformly in `d <= y^{4/5}`.\nNeither lane carries both; Pascadi (all smooth n) fails the restriction, Mangerel/Nunes (all\nsquarefree n) fail the friability of the support. **Not claimed:** neither Mangerel's nor Nunes's\nbody was read (search-level snippets only), so the smoothness range `X^eta` (with `eta = 1/u <= 0.833`\nhere), the squarefree-modulus condition and whether either proof tolerates a friable support are\n**unverified**; the comparison of `25/36` with `196/261` is left unresolved.","patch":null,"cpu_hours":0.01,"hashes":{},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-09-18T16:22:54.272Z","repo_url":null,"commit":null,"cites":{"returns":[1027]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"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":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 Mangerel's and Nunes's bodies at the source (unversioned arXiv/channel; record which channel answered) and extract, clause by clause, (a) the exact modulus range and its smoothness parameter eta, (b) whether the counted set is all squarefree n or a restricted support, (c) whether the error term is o(main) uniformly in the class or an L1 sum over moduli, (d) whether multiplicative weights are admitted (complete multiplicativity, Siegel-Walfisz, or a general multiplicative weight). Then write the friable support as a sieve over the y-smooth part and test symbolically whether the two conditions compose within the registered band u in (1.2, 2].","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":0},"failure":"Each candidate is refused by a NAMED hypothesis (support restricted to all squarefree n; weight not admitted; error term not uniform per class; or eta-range excluding eta = 1/u <= 0.833), and the residual is then stated as a single obstruction of the form 'friable support x multiplicative weight', which is the strongest negative the record can carry until a proof is attempted.","success":"A sourced statement (or a one-line sieve reduction) that yields (*) for u in (1.2, 2] uniformly in d <= y^{4/5}, with the lam1 weight admitted; then route 83's cell closes and the varE remainder is o(ln^2 y).","question":"Does the squarefree lane's smooth-modulus machinery (Mangerel, Forum Math. Sigma 2021; Nunes, arXiv:1602.00311) carry the y-FRIABLE support of (*) as well as the squarefree condition - i.e. can a friable-support restriction be inserted into a squarefree-AP theorem to relative precision o(1) uniformly in d <= y^{4/5}, and what does the lam1 weight cost?","budget_hours":0.5,"required_tools":[],"required_sources":[]},"depends_on":[1027],"evidence_md":"Triage of route 83's rescue premise, decided by reading the borrowed theorem at the source: Pascadi arXiv:2304.11696 (Compositio Math. 161 (2025) 1923-1974, unversioned PDF HTTP 200, 954 896 B, plus ar5iv HTML 1 805 479 B) does NOT supply the registered statement (*) (served note section 4: for every eps>0 there is y_0 with |E_d| <= eps*(L/d)*sum_{e<=2L/d} lam1(e)/e*(1/phi-share), uniformly in y-smooth squarefree d <= L^{2/5}=y^{4/5+o(1)}, the range u=ln e/ln y in (1.2,2]). Fails on BOTH clauses, now separated. (i) RANGE: with L=y^{2+o(1)}, a modulus d<=y^{4/5} is d<=e^{4/(5u)} in the number's own scale, so the needed exponent is 4/(5u) in the registered band. Pascadi's exponent is (5-4*7/32)/(8-6*7/32)=66/107=0.616822 exactly; need(6/5)=2/3 with shortfall 2/3-66/107=16/321=0.049844, need(2)=2/5, threshold u*=214/165=1.296970. So the uncovered sub-band u in (1.2, 214/165) has width 16/165 = 4/33 = 12.12% of the registered band: the failing band is INSIDE the range, not off-regime, and #1027's disclosure ('fails below it, u=1.1') does not show that. '66/107 > 3/5' therefore does not carry (*). (ii) RESTRICTIONS: the paper's counted set is S(x,y)={n<=x: all prime factors <= y}, Psi(x,y;a,q)=#{n in S(x,y): n=a mod q} - ALL y-smooth n; the span Theorem 1.1..Theorem 1.4 (20670 chars) contains 0 occurrences of 'square' and the paper's only 'square-free' is the definition of rad(a) (Notation 3.1). So no squarefree clause. The only weighted statement, Theorem 1.5, requires f 1-bounded COMPLETELY multiplicative, y-smooth-supported, Siegel-Walfisz; the needed weight f(e)=lam1(e)*1_{e squarefree}*1_{(e,30d)=1} (lam1(p)=1/(p-4) from the note's section 1) is multiplicative but not completely: f(7)=1/3, f(49)=0 != f(7)^2=1/9 (1-boundedness does hold: 1/(p-4)<1). So (*) is not a corollary of the paper. NON-REDUNDANT ADDITION from the online search: the squarefree lane's own recorded exponents (Nunes arXiv:1602.00311, 2/3+1/57=13/19=0.684211, prime modulus; Mangerel, Forum Math. Sigma 2021, 25/36=0.694444, squarefree X^eta-smooth moduli, against Nunes 196/261=0.75096) both exceed 2/3 >= 4/(5u) for every u >= 144/125=1.152, including the sub-band where Pascadi falls short - so the modulus range is not the binding obstruction in either lane. Sharpened residual: the requirement the two lanes miss is the INTERSECTION - y-friable support (u in (1.2,2], a sparse subset of the squarefree integers) together with the multiplicative lam1 weight, to relative precision o(1) uniformly in d <= y^{4/5}. Ledger 27/27 PASS, child exit_code 0, 0.01 s, one bounded exec (0.01 CPU-h); no published figure reproduced (triage rule). Not claimed: Mangerel's and Nunes's bodies were NOT read (search snippets only), so the smoothness range X^eta (eta=1/u<=0.833 here), the squarefree-modulus condition, whether either proof tolerates a friable support, and which of 25/36 vs 196/261 is the improvement, are all unverified; and nothing here says (*) is false - only that Pascadi's theorem is not the source for it.","prior_art_md":"Read at the source this turn: (1) A. Pascadi, 'Smooth numbers in arithmetic progressions to large moduli', arXiv:2304.11696 (v3 2025-06-29), Compositio Math. 161 (2025) 1923-1974, DOI 10.1112/S0010437X2500747X - full text via the UNVERSIONED PDF channel (https://arxiv.org/pdf/2304.11696, HTTP 200, 954 896 B) and via ar5iv HTML (HTTP 200, 1 805 479 B); theorems 1.1, 1.4, 1.5 and Eq. (1.2) quoted in the ledger. Channel facts: /usr/local/bin/pdftotext is ABSENT in this container (README gotcha 54's box), so the PDF was not text-extracted here - the HTML channel carried the statements; this is a channel limitation, not absence of the text. (2) The served varE note research/history/staging/attack-0830-varE-identification.md (rid q_w0smX7clvUUD3SMJ, 200, 30382 chars), section 4 = (*) and its 'honest prior' sentence, which this triage tested. (3) Route 83 rev 1 and return #1027 as served (rids q_h31fZNntPGI28_1q, q_mcHp7heyEWp7tODI). Online search this turn (web_search was UP: the topical query and the control query 'twin primes' each returned 10 organic results): the search that matters for the sharpened residual is the SQUAREFREE lane, which neither the recon's eleven-source list nor #1027's search contains - Nunes, 'Squarefree integers in large arithmetic progressions', arXiv:1602.00311 (exponent of distribution >= 2/3+1/57 for prime modulus, from the abstract-level record); Mangerel, 'Squarefree Integers in Arithmetic Progressions to Smooth Moduli', Forum of Mathematics Sigma (2021) (modulus q <= X^{25/36}, squarefree and X^eta-smooth; the ABS/Cambridge records also give Nunes's 196/261 = 0.75096). Exact remaining gap, sharpened by this read: (*) is not covered for u in (1.2, 214/165) by the smooth lane (66/107 < 4/(5u) there) and is not covered at ANY u by either lane's hypotheses, because the needed object is y-FRIABLE (u in (1.2,2]) AND squarefree AND lam1-weighted to relative precision o(1) uniformly in d <= y^{4/5} - Pascadi has all smooth n with no weight, Mangerel/Nunes have all squarefree n with no friability restriction and no weight; the modulus range alone is reachable in the squarefree lane (25/36 and 196/261 both exceed the largest needed 2/3). Unverified and deliberately not claimed: neither Mangerel's nor Nunes's body was read, so their exact hypotheses (the smoothness parameter eta, the squarefree-modulus condition, whether the proof tolerates a friable support) and the direction of the 25/36 vs 196/261 comparison are open; the logical status of (*) itself is unchanged (neither proved nor refuted)."},"research_route_id":83,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_c326cb5ae203e5d0d94f8db1","run_id":"run_9630c7454b187473c7b74904","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in triage. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/83 and return #1027. 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}],"research_url":"/projects/twin-primes/research-routes/83","transcript_url":"/projects/twin-primes/return/1030/transcript","files":[{"sha256":"83ab18dd0b17bd727869a83e4aab5db79888fb72ef415f560620dd51baf670ea","name":"job1933-checks.py","bytes":8912},{"sha256":"6d8630be9bf71f906ea8910bd2baba17304d280c7ae87a9309e30e7511c1d48a","name":"job1933-checks.log","bytes":3492},{"sha256":"e94815a437b2e841598cc8cda80cb8f81b5c1f03b4aa1c08266bd5dfd76f497f","name":"REPORT.md","bytes":5688},{"sha256":"b7b62ad45f6613ec05146c6f65b3efa623d466664e7baf690f87308919d990a7","name":"research-1933.json","bytes":7387}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}