A finite Guinand-Weil dictionary and archimedean tail order for the truncated Weil quadratic form

Abstract

We prove an exact finite dictionary from each real even finite Connes-van Suijlekom Galerkin coefficient vector to a band-limited Guinand-Weil test function: every value of the truncated Weil quadratic form is an exact sum over the nontrivial zeros of the Riemann zeta function. At Galerkin level N, finite signed sources factor through an exact source quotient of dimension 2N+1, and a non-collapsing positive-dimensional pole-neutral subfamily switches off the pole term. We also prove a finite-cutoff archimedean tail-order theorem. Past the Galerkin band, the omitted archimedean tail is a strictly positive definite, strictly totally positive Cauchy-Stieltjes increment. This yields a two-sided finite certification rule with an explicit budget B_T = (2N+1) rho (log(T/2 pi)+1)/(pi^2 T)(1+o(1)): finite-cutoff positivity certifies cutoff-free positivity, negative eigenvalues inside the band [-B_T, 0) certify nothing, and each doubling of T buys one binary digit. Resolving a 1e-59 spectral scale at c = 100 by brute cutoff would require T of order 1e63; a cutoff-free interval LDL^T certificate resolves it directly (n_+ = 401, n_- = 0 at 9000 bits). The manuscript includes a worked example verified against the first 512 nontrivial zeros of zeta. The companion package includes exact symbolic audits for the dictionary, source quotient, and pole-neutral corollary; a three-route verification of the dictionary identity (closed-form assembly, source-side evaluation, zero-side sums); interval tail budgets at c = 100, N = 200, T = 800; the exact-versus-asymptotic tail comparison; the Arb interval LDL^T certificate with its generator and provenance record; the eigenvalue-flow demonstration; figure generators; raw JSON outputs; and release checksums. The result is finite and forward-directional. It does not claim the Riemann Hypothesis, Weil positivity, global positivity, a prime-location bound, a next-prime theorem, a factoring result, or an inverse map from arbitrary Guinand-Weil test functions to finite coefficient vectors. Versions. This is version 1.3. Relative to version 1.2 the manuscript updates two references that have since appeared in print: Connes and van Suijlekom is now cited as Communications in Mathematical Physics 406 (2025), article 312, and Connes, Consani and Moscovici as a chapter in Applications of Noncommutative Geometry to Gauge Theories, Field Theories, and Quantum Space-Time, EMS Series of Lectures in Mathematics, EMS Press (2026), pages 39-76; no mathematical statement changes. The reproducibility archive corrects the exact-versus-asymptotic tail comparison, whose tabulated values of h_+'(t) at t = 50, 100 and 1000 were computed with an unconverged series acceleration, the largest discrepancy being a factor of 4.19 at t = 1000. The corrected values agree with the asymptotic expansion 1/t + 1/(12 t^3) to better than one part in 10^8, every tabulated row still satisfies the Lemma 3.1 envelope, and no theorem, proof, or reported bound changes. The archive also replaces its separate release manifest and verification report with a single verification document. Earlier versions remain available in the version history.