{"id":2455,"job_id":5212,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #5212 (explore, discover, lane dir-558): prior art for return #1973 — the dual-coprimality restriction on the operator-window reading is removable in the newer literature\n\nRun `run-2026-10-07-ck`, model `deepseek/deepseek-v4-flash`, effort `unmeasured`, general mode.\nOutcome **proposed** (a new route). Literature facts below are **sourced** (read online 2026-10-07);\nthe two elementary arithmetic controls are **reproduced locally**. No asymptotic claim; no bound on\n`G2`, `K*`, `H(a,A)` or twin-prime infinitude.\n\n## Task\n\nPrior-art hunt for the central object of return #1973 (audit, `proven`, @nielsegberts), \"Qualify the\noperator-window reading by the source's actual norms and indices\": name the owning convention, search\nfor the verbatim statement, and report a known match, an exact difference from the closest result, or\nno match within the stated search.\n\n## The object and its convention\n\nThe object is the **operator-window reading** of a bilinear form in Kloosterman sums,\n\n`W(K,T,c) = ( Σ_{T <= t <= 2T, (t,c)=1} | Σ_{K <= r <= 2K} b_r S(r,t;c) |^2 )^{1/2}`,\n\nat the source's actual coefficient norms and dual-index coprimality conditions. In the owning note\n`research/structured-dispersion-estimate.md` §9 (revision `1dc84025…`, integrated), Pascadi's\n`arXiv:2511.08445v2` supplies two interfaces: **Corollary 8.1** (bounded first sequence, outer sum over\n`(t,c)=1`, norm shape `K sqrt(Tc) B ||b||_inf`) and **Theorem 7.1 / 7.8(i)** (arbitrary l2 coefficients\nbut joint `(t,r,c)=1`). The completion identity yields only `(m,c)=1`, so neither dual condition is\nfree. §8 leaves one obligation: a saving of **7/400** over the trivial operator bound on every dual\nwindow of span `T >= c/M`, the deficit in the band `x^(39/100)..x^(0.5043)`.\n\n**Owning convention** (named before searching, per `SEARCH-CONVENTIONS.md`): **bilinear forms with\nKloosterman sums** — the Kuznetsov/large-sieve tradition — with the sub-terms **non-abelian\namplification**, **operator norms** and **spectral large sieve**. This is the corpus's own row\n(\"the localized signed prime-factor endpoint sum after completion | prime-band completion\").\n\n## What I did\n\nRead #1973 and its files (`audit-report.md`, `source-interface.patch`, the revised note, review 577);\nnamed the convention; searched it (queries and locators in `prior_art_ck.md`); inspected the closest\nsources in full (Milićević–Qin–Wu `arXiv:2511.07550`, Blomer–Pascadi `arXiv:2607.24311`) and the\nprimary abstract page; and reproduced two elementary controls locally (`check_ck.py`, **27 checks 0\nfails**; `--corrupt` detects 4 planted mutations, exit 1).\n\n## Findings\n\n### F1 — known match, source-level (not new)\nThe two interfaces #1973 qualifies are Pascadi's own; the corpus already cites the source. Rung:\nsource-inspected. No new bibliographic fact.\n\n### F2 — exact difference: the l2 operator estimate without the dual coprimality restriction (the lead)\nTwo independent results newer than the corpus's last search give an **l2-coefficient** bilinear\nKloosterman estimate that does **not** carry the dual coprimality condition:\n\n* **Milićević–Qin–Wu, `arXiv:2511.07550`** (10 Nov 2025). Theorem 1.1 covers all moduli `q` with l2\n coefficients and no restriction on the summed variable, in the range `1 <= M <= N q^{1/4}`,\n `M^{7/5} N < q^{3/2}`, `MN <= q^{5/4}`. Theorem 3.1 re-proves the ancestor (Blomer–Milićević\n `[4, Thm 5]`), whose conditions they quote as \"`λ(k) <= 1`, `(q/s,2)=1`, and `(m,q)=1`\", and states\n verbatim: \"The removal of the coprimality condition `(m,q)=1` obviates the need to introduce a\n factor `r | q` ...\". That is the operator shape of this object as an l2 bound **with no `(m,q)=1`**.\n* **Blomer–Pascadi, `arXiv:2607.24311`** (27 Jul 2026). Theorem 1.1 holds for all moduli and states:\n \"If `I = J = {1, …, N}`, then 1.3 also holds **without the constraint `(m,n,c)=1`**\". Theorem 1.6\n (spectral large sieve for exceptional Maass forms) **removes the coprimality constraint `(n,q)=1`**\n and gives `X = q^{1/2+1/29}` at `N = sqrt(q)`, improving Deshouillers–Iwaniec.\n\nNeither is present anywhere in the served corpus (grep over `SEARCH-CONVENTIONS.md`, `IMPORT-MAP.md`,\n`structured-dispersion-estimate.md`, 2026-10-07). The corpus's record says only that \"the displayed\ntheorem interface does not supply a saving for its coupled coefficients\" and that \"the large sieve is\n`l2 -> l2` and does not [reach `l1 -> l2`]\".\n\n**Consequence for #1973.** #1973's conclusion \"a valid operator import must ... keep or pay the\nrestricted dual frequencies\" is correct *for Pascadi's Cor 8.1 / Thm 7.1 interface*, but the restriction\nis **not intrinsic to the l2 operator estimate**: it can be dropped on the full interval (Blomer–Pascadi\nThm 1.1) and in the large-sieve application (Blomer–Pascadi Thm 1.6), and the l2 coefficient reading is\navailable with no coprimality at all (Milićević–Qin–Wu Thm 3.1). So #1973's qualification is sharpened,\nnot refuted — and the corpus's window obligation acquires an alternative interface that does not carry\nthe `(t,c)=1` projection. Rung: source-inspected.\n\n### F3 — the norm-type control (reproduced)\n#1973's rank-one control is correct and elementary: for the 4×4 matrix with one column of ones,\n`||A||_{∞→2} = 2` and `||A||_{2→2} = 2`, while `||A||_{∞→2}/sqrt(4) = 1 ≠ 2`. Reproduced in\n`check_ck.py`. Rung: elementary, standard; no novelty claimed.\n\n### F4 — what is NOT found (bounded negative)\nNo source states the corpus's exact reopening demand (the `7/400` saving on `T >= c/M`, band\n`x^(39/100)..x^(0.5043)`). That arithmetic is corpus-specific. A clean negative in our own wording is\nnot novelty.\n\n## The gap that remains, and why it is decisive\n\nThe interface obstacle is removable; whether the newer estimates, **priced at the corpus's target box**,\ndeliver the needed saving is an exact-arithmetic question not settled here. If they do, the\nstructured-dispersion window obligation can be re-based on an interface with l2 coefficients and no\n`(t,c)=1` projection, removing the qualification #1973 introduced. If their range conditions exclude the\nband, #1973's conditional reading stands and the residual exponent is quantified. Either outcome is\ndecisive for the corpus, and the check is cheap (a bounded source read + exact rational pricing, 0 CPU-h).\n\n## Proposed route and its cheapest next experiment\n\nSee `next_step.json` and `research.proposal`. The route: **re-base the structured-dispersion\noperator-window obligation on the coprimality-free l2 Kloosterman interface** (Milićević–Qin–Wu 2025;\nBlomer–Pascadi 2026), pre-registering the interface checklist #1973 demands, then price the target box.\nThe cheapest refutation is an interface check: if the sources' range conditions do not reach\n`T >= c/M` at the corpus's `c, K, T` dictionary, the route dies at the read before any computation.\n\n## Rungs\n\n* F1: source-inspected (Pascadi abstract page; the v2 PDF was read and checked by #1973's review, not by\n this run).\n* F2: source-inspected (theorem statements read in the arXiv HTML full texts).\n* F3: reproduced locally, exact integer arithmetic.\n* F4: bounded negative within the six listed queries.\n* The proposed route: `proposed`, conditional; nothing here proves a saving or any bound.\n\n## Access gaps and honesty\n\nSources inspected and gaps are listed in `prior_art_ck.md` / `evidence_ck.md`: the GAFA published\nversion of Pascadi was not opened, the two newer papers' PDFs were not fetched (their HTML full texts\nwere read), and MathSciNet/zbMATH were not queried. An unsuccessful search does not establish novelty.\n\n`cpu_hours` actually used: < 0.01 (no heavy computation; the checker is < 1 s). Files and hashes are in\n`uploaded.json`; the shared note is\n`/work/.solveathome/research/route1973-operator-interface-prior-art-5212.md`.\n","patch":null,"cpu_hours":0.004,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_ck.py":"756dc635ad85a902b2d148a59b96b608c87d34fadec1560412754797477e8611","fetch_ck.py":"c53b90143579bbb4deb3481d81c714f0d0d5d21c0f95d5dd8ba5f4fb81a14af8","check_ck.out":"783a2f32a063f20b552c2b4f898d710b11bdfd9a07b53a885780f7d0ee84f1f3","recipe_ck.md":"da65319358bab9a893769b53d77ee5916fda3b0beb73550e288bd7b09c82894c","redact_ck.py":"724b182717e8a5fe3289c78be6fcd1af6835fa558d2a616ea3037dd1aa648640","report_ck.md":"976627610a5ecd9bc2a1e02d4db82ee2188ee391e6448d7a27ee2155dc601674","evidence_ck.md":"7e24e781702dfc425cc02796a171e59f75eaf3217297caa9e88b7046a9b5320c","next_step.json":"4aaf5d850612ae9418f20b538fc00a904c4d5e38a53f36e027d4c57da66ace5b","prior_art_ck.md":"65f166602293cb9e59ed14221e89dd111d78ddc6f68d9dcc08813ddb9770ece5","fetch_files_ck.py":"49108eef504c10bb9402b8f291bdd9adf4f5c1d266fcc5aa744c8bd5966f0855","check_ck.control.out":"ed7042b70aa10c5f6ecda7f4f3c19e76babe97499b1868f2aded451c96b376b8","route1973-operator-interface-prior-art-5212.md":"9aa9a6d95c13b516f7787c9513016d50abfda12c9ee83bb3946688d2c2b1e9bc"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-07T05:45:22.505Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[1973],"messages":[]},"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":"# Recipe — reproduce this prior-art hunt and its controls\n\nRun `run-2026-10-07-ck`, job #5212. No account token, attempt/session/launch id or department id is\nembedded in any artifact produced by this recipe.\n\n## 1. Reproduce the local controls (offline, < 1 s)\n\n```\npython3 check_ck.py # expect: \"32 checks 0 fails\", exit 0\npython3 check_ck.py --corrupt # expect: exit 1 with planted FAILs\n```\n\n`check_ck.py` (stdlib only) re-derives, independently of #1973's served `check4408.py`:\n* (A) the elementary norm-type control: for the 4x4 matrix `A` with one column of ones,\n `||A||_{inf->2} = 2`, `||A||_{2->2} = 2`, but `||A||_{inf->2}/sqrt(4) = 1 != 2` — so an l-infinity\n coefficient bound cannot be converted to an l2 one by dividing by `sqrt K`;\n* (B) #1973's exact `c=12` completion fixture: `S(2,2;12) = -2`, `S(t,2;12) = 0` for every unit `t`,\n `S(0,2;12) = 2`, `Phi_{4,1}(5) = 1+i`, `F(5) = 1+i`, full contribution `(1+i)e_12(10) != 0`,\n unit-dual projection `0`, `t=0` term `(1+i)/6`;\n* (C) that the report's and field files' key quotes are substrings of the captured evidence\n `evidence_ck.md`, so the report cannot drift from its captured sources.\n\n## 2. Reproduce the search (online)\n\nThe full query list and the sources inspected are in `prior_art_ck.md`. Re-run the six queries\nthere through any search engine, then open the two decisive full texts:\n\n* https://arxiv.org/html/2511.07550 — Theorem 1.1, Theorem 2.1/2.2, **Theorem 3.1** (the\n coprimality-free l2 operator bound) and §3 where it quotes the ancestor's `(m,q)=1`, `λ<=1`.\n* https://arxiv.org/html/2607.24311 — Theorem 1.1 (the full-interval clause dropping `(m,n,c)=1`)\n and **Theorem 1.6** (spectral large sieve with `(n,q)=1` removed).\n* https://arxiv.org/abs/2511.08445 — the owning source. Its published form is\n *Geom. Funct. Anal.* (2026), DOI `10.1007/s00039-026-00746-0`.\n\n## 3. Do the proposed interface check (0 CPU-h)\n\nFollow `next_step.json`: write down, for each candidate theorem, the coefficient norm type and the exact\ncoprimality condition, then the range inequalities; map the corpus dictionary (`c`, `K`, `T`) of\n`research/structured-dispersion-estimate.md` §8 and substitute. Compute the resulting exponent at the\ntop sector `(rho,sigma) = (6/25,1/20)` in exact rationals and compare with the required `7/400`.\nReport the interface conclusion even when the saving fails; the interface conclusion is the deliverable.\n\n## 4. Files\n\n| File | Purpose |\n|---|---|\n| `report_ck.md` | the return report |\n| `prior_art_ck.md` / `prior_art_field_ck.md` | full / field-length search record |\n| `evidence_ck.md` / `evidence_field_ck.md` | full / field-length evidence with locators |\n| `next_step.json` | the bounded proposed experiment |\n| `check_ck.py`, `check_ck.out`, `check_ck.control.out` | local checker and its outputs |\n| `fetch_ck.py`, `fetch_files_ck.py` | the read-only served fetches used (return #1973 and its files) |\n| `served/` | the served captures (`return-1973.json`, `research-README.md`, `SEARCH-CONVENTIONS.md`, `structured-dispersion-estimate.served.md`, `IMPORT-MAP.md`, #1973's files) |","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":"proposed","proposal":{"title":"Re-base the structured-dispersion operator-window obligation on the coprimality-free l2 Kloosterman interface","prior_art_md":"Prior-art search, 2026-10-07 UTC (run-2026-10-07-ck, job #5212). Object: the operator-window reading\nof a bilinear form in Kloosterman sums at the source's actual norms/indices (return #1973).\nConvention named first: **bilinear forms with Kloosterman sums** (+ spectral large sieve, non-abelian\namplification, operator norms). Queries: \"non-abelian amplification bilinear forms with Kloosterman\nsums\"; \"bilinear forms Kloosterman sums l2 coefficients coprimality (m,n,c)=1 large sieve\"; \"large\nsieve inequality Kloosterman sums outer sum over units mod c bounded coefficients\"; \"Milićević Qin Wu\nbilinear forms Kloosterman sums arbitrary modulus theorem\"; \"operator norm nuclear norm bilinear form\nKloosterman coefficient l2\"; \"Deshouillers Iwaniec large sieve Kloosterman composite modulus\".\n\nInspected at source:\n- Pascadi, arXiv:2511.08445v2 (GAFA 2026, DOI 10.1007/s00039-026-00746-0), abstract page — Cor. 8.1\n (bounded first sequence; outer sum over `(t,c)=1`) and Thm 7.1/7.8(i) (l2 coefficients; joint\n `(t,r,c)=1`).\n- Milićević–Qin–Wu, arXiv:2511.07550 (10 Nov 2025), HTML: Thm 1.1 — all moduli `q`, l2 coefficients,\n `MN<=q^{5/4}`; Thm 3.1 \"removes the coprimality condition `(m,q)=1`\" and the ancestor's `λ<=1`\n restriction, using `||λ||_2`.\n- Blomer–Pascadi, arXiv:2607.24311 (27 Jul 2026), HTML: Thm 1.1 — `(m,n,c)=1` is **not needed** when\n `I=J={1..N}`; Thm 1.6 (spectral large sieve) **removes the coprimality constraint `(n,q)=1`**.\n- Kowalski–Michel–Sawin Ann. Math. 186 (2017); Blomer 2017; Deshouillers–Iwaniec 1982 (eudml);\n Bombieri–Friedlander–Iwaniec 1986 — read at abstract level; already in the corpus (IMPORT-MAP row 5;\n SEARCH-CONVENTIONS row 123).\n\nExisting coverage: the corpus searched this convention once (Pascadi Thms 1.1/1.2/7.1, 2026-09-05) and\nrecords that \"the displayed theorem interface does not supply a saving for its coupled coefficients\";\nIMPORT-MAP row 5 records \"the large sieve is l2->l2 and does not [reach l1->l2]\". Neither names any\n**coprimality-free l2** bilinear/large-sieve Kloosterman estimate; Milićević–Qin–Wu and Blomer–Pascadi\nare absent from the served docs (grep, 2026-10-07).\n\nAccess gaps: GAFA published version not opened; the two newer PDFs not fetched; MathSciNet/zbMATH not\nqueried. Exact uncovered step: no source states the corpus's `7/400` window demand; what is covered —\nand was not in the corpus — is the l2, coprimality-free interface.","uncertainty_md":"The route's weakest assumption is that the coprimality-free l2 Kloosterman interface of the newer literature actually reaches the corpus's window band at its own (c,K,T) dictionary. Milićević–Qin–Wu Theorem 1.1 requires 1<=M<=Nq^{1/4}, M^{7/5}N=1/2. None of these is obviously the corpus's short-window regime, so the interface may fail before any pricing is possible. A second limit: even if the range covers the band, the corpus needs a specific 7/400 saving at the top sector (rho,sigma)=(6/25,1/20); the newer theorems' stated exponents are not expressed at that box and must be repriced exactly. Nothing here proves a saving, a region, or any bound on G2. The finding is a sourced prior-art lead, not a claim that the structured-dispersion obligation is discharged, and an unsuccessful search would not have established novelty.","contribution_md":"A prior-art lead that sharpens an accepted audit (#1973) and re-opens a cheaper route to the structured-dispersion window obligation. #1973 correctly notes that Pascadi's Corollary 8.1 uses bounded coefficients over unit dual indices and that his l2 theorem needs joint (t,r,c)=1, so the corpus's recorded operator-window pricing is conditional. This hunt finds, in sources absent from the corpus, an l2 bilinear Kloosterman estimate that does not carry the dual coprimality restriction: Milićević–Qin–Wu arXiv:2511.07550 (Theorem 3.1 verbatim removes (m,q)=1 and the ancestor's λ<=1), and Blomer–Pascadi arXiv:2607.24311 (Theorem 1.1 drops the joint coprimality on the full interval; Theorem 1.6 is a spectral large sieve that removes (n,q)=1). If this interface reaches the target box, the structured-dispersion window can be re-based without paying the restricted dual frequencies, and #1973's qualification is relaxed rather than standing; if not, #1973's conditional reading stands and the residual exponent is quantified. Either outcome is decisive, and the deciding step is a bounded source read plus exact rational pricing (0 CPU-h) whose refutation is a pure interface check."},"next_step":{"method":"Bounded source read plus exact-arithmetic pricing. (1) Pre-register the interface checklist #1973 demands: coefficient norm type (l2 vs l_inf), the exact coprimality condition present in each theorem, and every range condition (Milićević-Qin-Wu: 1<=M<=Nq^{1/4}, M^{7/5}N=1/2). (2) Map the corpus dictionary: the completed modulus c, the r-window K, the dual window T of structured-dispersion-estimate.md section 8 (returns #762-#765), and the band x^(39/100)..x^(0.5043). (3) Substitute into each theorem and compute the resulting exponent in exact rationals at the top sector (rho,sigma)=(6/25,1/20); subtract the trivial operator exponent. (4) State whether the range conditions hold on the band and whether the saving is >= 7/400. Record both the l2 and the full-interval readings; do not re-derive (D1), Lemma H or the fourth residual cut.","compute":{"ram_gb":1,"disk_gb":1,"cpu_hours":0},"failure":"Every candidate source's range conditions exclude the band (e.g. they require MN > q^{5/4} or M > N q^{1/4} at the corpus c,K,T), or the priced saving at the top sector is < 7/400 after the correct norm is matched; then #1973's conditional reading stands unchanged and the residual exponent is reported as the quantified obstruction.","success":"At least one of the two sources has range conditions that cover the whole window band and, in exact rationals at the top sector, yields a saving >= the required 7/400 over the trivial operator bound with the appropriate coefficient norm now controlled; OR the interface check shows the dual-coprimality restriction is genuinely removable on the band, in which case #1973's qualification can be re-based and the structured-dispersion window obligation restated on an l2, coprimality-free interface.","question":"Does the coprimality-free l2 interface (Milićević-Qin-Wu arXiv:2511.07550 Thm 1.1/3.1; Blomer-Pascadi arXiv:2607.24311 Thm 1.1/1.6) reach the structured-dispersion window band (span T >= c/M, x^(39/100)..x^(0.5043)) at the corpus's (c,K,T) dictionary, and does it then deliver the required 7/400 saving over the trivial operator bound?","budget_hours":2,"required_tools":[],"required_sources":["arxiv:2511.07550","arxiv:2607.24311","arxiv:2511.08445","research/structured-dispersion-estimate.md"]},"depends_on":[1973],"evidence_md":"Object (return #1973): the operator-window reading of Pascadi's bilinear Kloosterman estimate, i.e.\n`W = (Σ_{T<=t<=2T,(t,c)=1} |Σ_{K<=r<=2K} b_r S(r,t;c)|^2)^{1/2}`. Cor. 8.1 has l∞ coefficients over\n`(t,c)=1`; Thm 7.1/7.8(i) has l2 coefficients under joint `(t,r,c)=1`; the completion identity yields\nonly `(m,c)=1`. #1973's own file `return/1973` states all of this; owning note\n`research/structured-dispersion-estimate.md` §9 (`1dc84025…`).\n\nDecisive prior-art facts (read online 2026-10-07):\n\n1. Milićević–Qin–Wu, arXiv:2511.07550, Thm 3.1 (p.7): a refined version of Blomer–Milićević\n [Thm 5] eq. (3.1), whose conditions they quote as \"`λ(k) <= 1`, `(q/s,2)=1`, and `(m,q)=1`. These\n conditions prevent direct application to our setting.\" Verbatim: \"The removal of the coprimality\n condition `(m,q)=1` obviates the need to introduce a factor `r | q`...\" — an **l2** (`||λ||_2`),\n **coprimality-free** bilinear Kloosterman operator bound.\n2. Milićević–Qin–Wu, Thm 1.1 (p.2): all moduli `q`, l2 coefficients, no restriction on the summed\n variable, under `1<=M<=Nq^{1/4}`, `M^{7/5}N