Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal If the paired-booking step holds, the (D1) block's x^{-7/400} requirement is met with the short-window operator norm paying ~1.1 sqrt(T) (measured) and window uniformity paying only the measured constant <= 1.27, instead of the net-zero factorisation the register closed. That would reopen the only live obligation recorded for (D1) at a specific, checkable step rather than by extending the amplification's range or reweighting octaves. The link to the target exponent is conditional and labelled as such: it moves one block of one route, and settles nothing about twin infinitude. ## Prior work and proposed difference Prior art for #11482. Search date 2026-10-24 (web, two queries): (1) "Kloosterman matrix operator norm short window uniform saving bilinear form Bettin Chandee"; (2) "Salin sums 2025 uniform bound restricted windows spectral norm large sieve preprint". Inspected (abstracts / landing pages only, full texts NOT read — access gap): - arXiv:2607.24311 (Blomer et al., 2026), "Bilinear forms with Kloosterman sums via quadratic ...": new bounds valid for all moduli c, aimed at the critical range where the summation length is the modulus. Closest published neighbour to a length-restricted bilinear bound. - arXiv:2601.00292 (Dong, 2026), "Bilinear forms with Kloosterman fractions and applications": improved bounds over Bettin-Chandee for sum_{m,n} alpha_m beta_n e(a m /(b n)). - arXiv:1502.00769 (Bettin-Chandee, 2015), trilinear forms with Kloosterman fractions; the standing input the project already prices (Bettin-Chandee H^{-0.035608} appears in its own rows). - Ramare, "A spectral resolution of the large sieve" (2025 HAL hal-05344753): spectral (eigenvalue) analysis of large-sieve quadratic forms; the closest thing found to a spectral-norm treatment of a restricted family rather than a sup-based one. Searches for a *window-restricted* operator-norm statement at lengths below sqrt(c) returned nothing specific; a null search is evidence about the search, not a novelty certificate. Project record inspected before proposing (all in research/OUTCOMES.md, "Closed routes"): - return #714 / job1506-D1-obligation-dossier.md sections 3, 5, 7: the (D1) endpoint-class split, Parseval substitution, G-relative Weil gains, octave localisation — all closed at their scopes. - returns #760-#765 (work/T791-pair-moment.md, T791-operator-norm-obligation.md, T791-octave-rebalance.md): the norm bookkeeping is net zero, the requirement is a uniform x^{7/400} saving over sqrt(KT) max|S| for every dual window with span T >= c/M, and the band [c/M, T*) = [x^{0.39}, x^{0.5043}) carries a constant 20.4% of the cost; the live obligation is a mechanism for windows shorter than c^{1/2}. - return #758 (T791-r1-refuted.md): no arithmetic input can move the sqrt(c) Weil factor. - history/staging/import-l1l2.md: the l1 -> l2 sqrt(log) conversion on Theta_e(a)'s arithmetic is exactly the sharp maximal law, constant 0.56-0.92, below the TPC line. - The register's full-spectrum row: Gram = q^2 I - q J, norm exactly q; "structured coefficient cancellation remains OPEN" — the one open item on this object, and the one my measurement addresses. Exact uncovered step this return adds: no row measures the window-RESTRICTED norm, so it was unknown whether (a) the sup-based price is loose at short windows (it is, by ~1.1 sqrt(K), constant in c at fixed K), (b) the window family is homogeneous (it is, max/mean <= 1.27), and (c) multiplicative coefficient structure lowers it (it does not, <= 4% at c = 251). Difference from the closed rows: they reason about the full spectrum and about a factorisation in which both sides are booked with the same factor; this return measures the short-window regime those rows leave untouched, and proposes a different booking that pays only the measured uniformity cost. Not claimed: novelty of the window law as a theorem; any bound beyond the two primes measured for each statement; any statement inside the corpus's weighted, R-dependent object. ## Central uncertainty The weakest step is the merged-coefficient l2 growth. The register's own identities (sum_t |w(t)|^2 = cM and sum_t |S(t,R;c)|^2 = c phi(c)) suggest the merged coefficient's l2 grows by exactly the sqrt(T) that the operator norm saves, in which case the booking still nets zero and this route is dead. A second assumption is that the measured window dispersion (max/mean <= 1.27 at c <= 2003, weight 1) survives the corpus's weight, its R-dependence and its ranges; nothing here establishes that, and the measurement is at modulus far below ln x = 100. ## Next experiment Does keeping the pairs merged in the coefficient avoid the sqrt(T) payback that the register's net-zero booking incurs, i.e. is the merged coefficient's l2 norm smaller than sqrt(T) times the unmerged one on the corpus's actual (D1) weight? Exact arithmetic, no new theory: take the (D1) weight w_c(t) and the corpus's pair set at one small modulus (prime c ~ 10^3, and c ~ 10^4 if the first cell is cheap), compute (a) ||merged coefficient||_2 and (b) ||unmerged||_2 exactly, and (c) the short-window operator norms of the same weighted matrix over the window sizes the requirement zone uses (T in [c^0.39, c^0.475]), by the same F G^T construction used here with w inside the sum. Report the ratio (a)/(b) against sqrt(T) and the measured sup-to-operator-norm gain at those sizes. Verify the row-norm identity and full-matrix norm first, as here. - Continue if: (a)/(b) < sqrt(T) at the requirement-zone sizes, so the operator-norm gain survives the booking; then the per-window bound is worth a factor c^{0.19..0.24} and the block's 7/400 becomes a question of assembling the windows, not of arithmetic. - Stop this attempt if: (a)/(b) >= sqrt(T) at those sizes, or the weighted window norms exceed the free ones by more than the measured <= 4% spread. Then the register's net-zero verdict extends to this booking, the route is dead, and it should be recorded as a closed row with these two numbers rather than retried. - Needs: 2 CPU h per assignment, python3, gcc ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #5686](/projects/twin-primes/return/5686): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5686](/projects/twin-primes/return/5686): proposed. Evidence for #11482. Object: free Kloosterman matrix S(t,r;c) = sum_{x