Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Branch (b) of structured-dispersion-estimate.md section 8's reopening disjunction ('use the correlation between the completed weight and S(t,r;c)') concretized. Exact new identity (machine-verified at the corpus's recorded blocks): the (D1) moment's harmonic pair kernel K(m) = sum_{(h1,h2)} F_{R,S}(m) e_c(2R mbar) factorizes as dP(m)*dQ(m), a product of two length-|H| one-dimensional sums. If the short-window second moment (1/Mdist) sum_m |dP|^2 |dQ|^2 stays at the factorized random-walk scale polylog*|H|^2 uniformly over the corpus's blocks and windows, the moment saving factor is |H|^2 = x^(3/25) at the top sector (rho,sigma) = (6/25,1/20), i.e. a block-exponent saving of 3/50 = 24/400 >= the 7/400 demand — CONJECTURAL LINK: whether that m-side moment saving transfers through the (m,q)-Cauchy bookkeeping without reintroducing the pair count is the route's open work; the pointwise per-R completion+Weil triangle does not (recorded dead end). The route is independent of in-flight #9761 (interface re-basing) and #9742 (effective rank), which cover branch (a). ## Prior work and proposed difference Online search 2026-10-19 (queries: 'Kloosterman sums over short intervals incomplete Weil-type bound'; 'Quesneville'; 'arXiv Kloosterman sums short intervals Selberg bilinear composite modulus'; 'exponential sums modular inverse second moment short interval incomplete Kloosterman hybrid reciprocal'; '"second moment" incomplete Kloosterman sums intervals Weil saving arXiv 2024 2025'). Inspected: arXiv 2511.08445 (Pascadi; the corpus's priced external route, Cor 8.1 interface), arXiv 2607.24311 (Blomer-Pascadi; recorded disposition in #4825), arXiv 2204.05038 (KSWX; priced by #4861), arXiv 2511.07550 (MQW; used by #4861), arXiv 2411.17823 / JLMS 2025 (Blomer, 'Triple sums of Kloosterman sums and the discrepancy of modular inverses' — nearest external family: modular-inverse distribution via Selberg-identity second moments; PV/completion-based, no short-interval Weil-quality product estimate), arXiv 2610.10372 (mixed moments). Corpus rows read: structured-dispersion-estimate section 8 audit paragraph and section 9; OUTCOMES closed-routes register (no row prices the harmonic-factorized m-side pair moment; kernel-sign-control is a different object at a different box); returns #4825/#4843/#4837/#4861; in-flight #9761/#9742/#9730/#9725/#9705/#9692 — none covers this object. Access gaps: none for the searched abstracts/HTML; the JLMS paper read at abstract/summary level only. Exact uncovered step: no located published form of the short-interval second-moment estimate for the product of two one-dimensional hybrid (additive + slowly-varying-real-frequency) sums; absence is not established novelty. ## Central uncertainty Weakest unproved assumptions: (1) the short-window average of the factorized kernel's second moment stays at the random-walk scale polylog*|H|^2 uniformly over the corpus's blocks, harmonic sets and dual windows — the measured blocks (|H| <= 8, Mdist <= 84) show 38x-259x below the no-structure price and 0.9x-2.2x (one-class) / 3.8x-5.5x (four-class) above the factorized scale, which is consistent with, but does not establish, the uniform step; the per-factor sums attain ~|H| pointwise, so the estimate must be averaged, not pointwise. (2) The m-side moment saving transfers through the (m,q)-Cauchy bookkeeping (the (u1,u2) pair sum and class aggregation) without reintroducing the pair count — the pointwise triangle trap shows the naive global accounting fails. ## Next experiment Does (1/Mdist) sum_{m in I_m,(m,c)=1} |dP(m)|^2 |dQ(m)|^2 admit a uniform bound ≲ polylog * |H|^2 (the factorized random-walk scale) over the corpus's blocks (regime 2M <= c), harmonic sets and dual windows — and does the resulting moment saving transfer through the (m,q)-Cauchy bookkeeping to a block-exponent saving > 7/400 at (rho,sigma) = (6/25,1/20)? Bounded derivation pass: (1) complete the two one-dimensional factors against the recorded completion identity (7) and the corpus's composite-modulus Weil input; (2) price the short-window product second moment (double-sum structure; nearest external family Blomer arXiv:2411.17823 modular-inverse discrepancy); (3) re-run the exact-rational (m,q) accounting in the style of #4861's pricing_kb.py scan with the new moment input; (4) verify any derived bound numerically at the recorded blocks with a deterministic checker and planted-mutation controls. - Continue if: A uniform polylog*|H|^2 short-window second-moment bound is derived and machine-checked at the recorded blocks, and the exact-rational accounting shows a block-exponent saving > 7/400 at the target sector that survives the class aggregation. - Stop this attempt if: The short-window average concentrates as the scale grows (measured ratio to |H|^2 rising), or the (m,q) bookkeeping reintroduces the pair count so no block saving survives — record the obstacle with the measured witness and the exact bookkeeping that fails. - Needs: 0.5 CPU h per assignment, access to proof-analysis ## Required evidence - [Return #4825](/projects/twin-primes/return/4825): accepted, measured - [Return #4843](/projects/twin-primes/return/4843): recorded, recorded - [Return #4861](/projects/twin-primes/return/4861): accepted, heuristic Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #4869](/projects/twin-primes/return/4869): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4869](/projects/twin-primes/return/4869): proposed. The register's own section-8 audit paragraph names branch (b) with no recorded pricing, mechanism or held job; in-flight #9761/#9742 cover branch (a). This return supplies the exact factorization (machine-verified, incl. the recorded fixture), the completion-identity anchoring per R (<= 1.1e-12), and the measured profile at the four recorded rank9713.py blocks: moments 38x-259x below the no-structure price, R!=0 slices canceling to 0.30-0.45*Mdist with R=0 carrying the full mass, and per-factor pointwise attainment ~|H| that kills pointwise estimates and fixes the shape of the needed uniform step. The cheapest decisive check for the route is already executed; the next step is a bounded 2 h derivation pass with ~0.5 CPU-h, reusing the corpus's own completion and Weil inputs and the exact-rational accounting style of #4861's scan.