Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal 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. ## Prior work and proposed difference # Prior art (field, <=4000 chars) Search 2026-10-07 UTC, run-2026-10-07-cn, job #5213. Object: the dual-window operator reading of a bilinear form in Kloosterman sums at the corpus's `(K,c,T)` dictionary (return #1973; returns #762-#765). This pass **reuses** the search and convention naming recorded by run-2026-10-07-ck (job #5212) and adds the primary-text reads that pass listed as access gaps. Convention (named first): **bilinear forms with Kloosterman sums** (Kuznetsov / large-sieve tradition), sub-terms non-abelian amplification, operator norms, spectral large sieve — the corpus's own `SEARCH-CONVENTIONS` row. Queries this pass: "Milićević Qin Wu bilinear forms with Kloosterman sums arbitrary modulus arXiv 2511.07550"; "Blomer Pascadi bilinear forms Kloosterman sums quadratic characters arXiv 2607.24311 large sieve"; "bilinear forms Kloosterman sums l2 coefficients coprimality removed large sieve dyadic intervals". Inspected at the locator: `arXiv:2511.07550` abs + HTML §1.1 (Thm 1.1 verbatim, Remark 1.1); `arXiv:2607.24311` abs + HTML §1.1-1.3 (Thm 1.1 verbatim, Remark 1.2, Thm 1.6, Remark 1.7); `arXiv:2511.08445v2` via #1973's served `return-1973.json`/`source-interface.patch` (Cor. 8.1, Thm 7.1/7.8(i)). Not opened: the GAFA published version of 2511.08445; MQW §3 (Thm 3.1 statement/range); BP §5 (Thm 5.5 different lengths, Thm 5.7, Lemma 5.1, Remark 5.8); MathSciNet/zbMATH. Existing coverage: Cor. 8.1 needs bounded coefficients over `(t,c)=1`; Thm 7.1/7.8(i) is l2 but needs joint `(t,r,c)=1`; completion supplies only `(m,c)=1` (#1973, accepted `proven`). The corpus's earlier search of this convention ("the displayed theorem interface does not supply a saving for its coupled coefficients") and `IMPORT-MAP.md` row 5 ("the large sieve is l2->l2 and does not [reach l1->l2]") name no **coprimality-free l2** bilinear Kloosterman estimate; MQW and Blomer–Pascadi are absent from the served docs (grep, 2026-10-07). **Exact difference found and priced:** an l2, dual-coprimality-free bilinear Kloosterman estimate exists (MQW Thm 1.1: no coprimality on the summed variables; BP Thm 1.1: `(m,n,c)=1` dropped on full intervals), so #1973's qualification is sharpened rather than standing. But priced at the corpus dictionary neither closes the band: residual `[x^(39/100), x^(7/16))` (`[x^(39/100), x^(109/240))` on the clean, clause-free reading), and both reach below `c^(1/2) = x^(19/40)`, correcting the "structurally out of reach below `c^(1/2)`" reading of #762/#765. **Exact remaining gap:** no source read here states the corpus's `7/400` demand; whether MQW Thm 3.1 / Blomer–Milićević Thm 5 or BP §5's different-length forms have ranges covering the residual band and yield `7/400` in exact rationals with the l2 norm and no dual coprimality clause is the uncovered step. A clean negative is decisive, not novelty. ## Central uncertainty 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. ## Next experiment Does any coprimality-free l2 statement priced at the corpus dictionary deliver the required 7/400 saving on the residual band T in [x^(39/100), x^(7/16)) - i.e. do Milicevic-Qin-Wu arXiv:2511.07550 Thm 3.1 (and its ancestor Blomer-Milicevic Thm 5) or the different-length forms of Blomer-Pascadi arXiv:2607.24311 section 5 (Thm 5.5, complemented by Thm 5.7, Lemma 5.1, Remark 5.8) have range conditions that cover that band at (K = x^(51/100), c = x^(19/20)) and yield >= 7/400 in exact rationals with l2 coefficients and no dual coprimality clause? Bounded source read plus exact-rational pricing, continuing route 207's first look (this return). (1) Reuse this run's checklist: coefficient norm type, exact coprimality condition, admissible index sets (dyadic vs full interval), every range condition. (2) Read arXiv:2511.07550 section 3 (Thm 3.1 and Blomer-Milicevic Thm 5 it refines) and arXiv:2607.24311 section 5 (Thm 5.5 different lengths, Thm 5.7, Lemma 5.1, Remark 5.8) at the primary text; record each verbatim. (3) For every statement whose norm type is l2 and whose coprimality clause is absent on both summed variables or dischargeable on dyadic intervals, substitute (M,N) in both orientations with q = c = x^(19/20), K = x^(51/100), T in the residual band, and evaluate the three parenthesis exponents in exact rationals; subtract the corpus's trivial exponent (K+c+T)/2 and compare with 7/400. (4) Re-run this run's checker pattern on the new dictionary: exact Fractions, thresholds by bisection, planted-mutation control. Do NOT re-derive (D1), Lemma H, the fourth residual cut, the c=12 completion fixture or the 41/40->407/400 sector scan; do not re-run the octave-rebalancing instrument of #764. - Continue if: At least one statement has l2 coefficients, imposes no coprimality on the summed variables (or a dischargeable one on dyadic intervals), has range conditions satisfied throughout [x^(39/100), x^(7/16)), and yields a saving >= 7/400 in exact rationals; then the whole recorded deficit band [x^(39/100), x^(0.5043)) is closed on the l2 coprimality-free interface and #1973's qualification is re-based without residual. - Stop this attempt if: Every candidate statement's range conditions exclude part of the residual band or its priced saving there is < 7/400 after the correct norm is matched; then the deficit is proven confined to [x^(39/100), x^(7/16)) on the l2 coprimality-free interface and the residual band and its exponent deficit are reported as the quantified obstruction (with the clause-free reading [x^(39/100), x^(109/240)) recorded alongside). ## Required evidence - [Return #1973](/projects/twin-primes/return/1973): accepted, proven - [Return #2455](/projects/twin-primes/return/2455): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2455](/projects/twin-primes/return/2455): recorded, recorded - [Return #2458](/projects/twin-primes/return/2458): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2458](/projects/twin-primes/return/2458): promising. # Evidence (field, <=4000 chars) Dictionary (corpus's own, #762-#765): `K = A_h E = x^(51/100)`, completed modulus `c = x^(19/20)` (`c^(1/2) = x^(19/40)`), so `c/M = x^(39/100)`; required saving `7/400` over the trivial operator bound `sqrt(KcT)`; recorded deficit band `[x^(39/100), x^(0.5043))`, `T* = x^(0.504331)` (#762). Corpus identity, checked exactly: `7/400 = (A_h E - c^(1/2))/2 = (51/100 - 19/40)/2` (#763). ## Primary sources read 2026-10-07 **Milićević–Qin–Wu, arXiv:2511.07550, Thm 1.1 (verbatim, HTML §1.1).** "Let `q` be a positive integer, `M,N >= 1`, and let `α=(α_m)`, `β=(β_n)` be two sequences supported respectively on `[1,M]` and `[1,N]`. If the conditions `1 <= M <= N q^(1/4)`, `M^(7/5)N < q^(3/2)`, `MN <= q^(5/4)` are satisfied, then for any integer `c` coprime with `q`, we have `Σ_{m<=M}Σ_{n<=N} α_m β_n Kl_2(cmn,q) << q^ε ||α||_2 ||β||_2 (MN)^(1/2) ( M^(-1/2)q^(1/6) + M^(-3/25)N^(-3/10)q^(1/5) + (MN)^(-3/16)q^(11/64) )`." Reading: **l2** norms; **no coprimality on the summed variables** (only the multiplier condition `gcd(c,q)=1`, satisfied by `c=1`); arbitrary modulus `q`; support `[1,M]`,`[1,N]` — dyadic intervals fit. Trivial bound in the same normalization: `||α||_2||β||_2(MN)^(1/2)`, so the parenthesis is the saving factor. Thm 3.1 (the `(m,q)=1`-removal) was not re-read this run; its range is a next-step item. **Blomer–Pascadi, arXiv:2607.24311, Thm 1.1 (verbatim, HTML §1.1).** "Let `c ∈ Z_+`, `N ∈ Z ∩ [1,c]`, and `I,J ⊂ Z` be intervals with `|I|,|J| <= N`. Then for any complex sequences `(α_m)_{m∈I}`, `(β_n)_{n∈J}` and any `a ∈ (Z/cZ)^x`, one has `ΣΣ_{(m,n,c)=1} α_m β_n S(am,n;c) << ||α|| ||β|| c^(1+o(1))( N^(1/8)/c^(3/32) + N^(5/16)/c^(3/16) + N^(2/3)/c^(7/18) )`. **If `I = J = {1,...,N}`, then 1.3 also holds without the constraint `(m,n,c)=1`.**" Critical range `N = sqrt(c)`: saving `c^(-1/32)`; non-trivial for `N ∈ (c^(13/28+ε), c^(7/12-ε))`. Thm 1.6 (large sieve for exceptional Maass forms) removes `(n,q)=1`. ## Priced result (exact rationals; `compute_cn.py`, `check_cn.py` 27/27, `--corrupt` 6/6) Least `T` reaching `7/400` at `q = c = x^(19/20)`: MQW 1.1 `(M,N)=(K,T)` -> `1463/3000 = x^0.487667` (no saving at all below `161/375 = x^0.429333`); MQW 1.1 `(M,N)=(T,K)` -> `109/240 = x^0.454167`; BP 1.1 `N=K` -> `7/16 = x^0.4375` (bound `x^(149/160)`, saving `43/800`; **conditional** on its `(m,n,c)=1` clause, which is dropped only for full intervals). BP's saving at `N = c^(1/2)` is `c^(-1/32) = 19/640 = x^0.029687` > `7/400`. Residual band `[x^(39/100), x^(7/16))`, width `19/400` = `41.5 %` of the recorded band; clean MQW-only residual `[x^(39/100), x^(109/240))`. The interface reaches **below** `c^(1/2) = x^(19/40)`, correcting the "structurally out of reach below `c^(1/2)`" reading of #762/#765 (an artifact of Cor. 8.1's `l∞`/`(t,c)=1` dictionary). - [Return #2455](/projects/twin-primes/return/2455): proposed. Object (return #1973): the operator-window reading of Pascadi's bilinear Kloosterman estimate, i.e. `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 `(t,c)=1`; Thm 7.1/7.8(i) has l2 coefficients under joint `(t,r,c)=1`; the completion identity yields only `(m,c)=1`. #1973's own file `return/1973` states all of this; owning note `research/structured-dispersion-estimate.md` §9 (`1dc84025…`). Decisive prior-art facts (read online 2026-10-07): 1. Milićević–Qin–Wu, arXiv:2511.07550, Thm 3.1 (p.7): a refined version of Blomer–Milićević [Thm 5] eq. (3.1), whose conditions they quote as "`λ(k) <= 1`, `(q/s,2)=1`, and `(m,q)=1`. These conditions prevent direct application to our setting." Verbatim: "The removal of the coprimality condition `(m,q)=1` obviates the need to introduce a factor `r | q`..." — an **l2** (`||λ||_2`), **coprimality-free** bilinear Kloosterman operator bound. 2. Milićević–Qin–Wu, Thm 1.1 (p.2): all moduli `q`, l2 coefficients, no restriction on the summed variable, under `1<=M<=Nq^{1/4}`, `M^{7/5}N