A finite Guinand-Weil dictionary and archimedean tail order for the truncated Weil quadratic form
Abstract
The Connes-van Suijlekom and Connes-Consani-Moscovici truncations of the Weil quadratic form, at a prime cutoff c>1 and frequency band N, produce finite Galerkin matrices whose spectra are the finite-rank window on Weil positivity. We prove two exact finite theorems about this truncation. First, every real even Galerkin coefficient vector v determines, in closed form, a band-limited Guinand-Weil test function g_v whose zero sum over the nontrivial zeros of zeta equals the quadratic value exactly: every value of the truncated form is an exact sum over the zeros. The construction factors through an exact source quotient of dimension 2N+1 and admits a non-collapsing pole-neutral subfamily. Second, beyond the Galerkin band the omitted archimedean tail is a totally positive Cauchy-Stieltjes increment. This yields a two-sided certification rule with an explicit budget B_T ~ (2N+1) rho log(T) / (pi^2 T), where T is the archimedean cutoff and rho = 2 pi / log c: finite-cutoff positivity certifies cutoff-free positivity, a finite-cutoff eigenvalue below -B_T certifies a cutoff-free negative, and a negative eigenvalue in the band [-B_T, 0) certifies nothing. Resolving a spectral scale of 10^-59 at c=100 by brute cutoff would require T of order 10^63; a cutoff-free interval LDL^T factorization resolves it directly. The dictionary is verified over the first 512 zeros of zeta and by three independent computational routes; all scripts and artifacts ship with the paper. The paper makes no Riemann Hypothesis, prime-counting, next-prime, or factoring claim.
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BibTeX
@misc{groskin2026finite,
title = {{A finite Guinand-Weil dictionary and archimedean tail order for the truncated Weil quadratic form}},
author = {Akiva Groskin},
howpublished = {arXiv (Cornell University)},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2607.02828},
url = {https://arxiv.org/abs/2607.02828},
note = {CC BY},
}
RIS
TY - UNPB
TI - A finite Guinand-Weil dictionary and archimedean tail order for the truncated Weil quadratic form
AU - Akiva Groskin
PY - 2026
DA - 2026/07/02
PB - arXiv (Cornell University)
DO - 10.48550/arxiv.2607.02828
UR - https://arxiv.org/abs/2607.02828
AB - The Connes-van Suijlekom and Connes-Consani-Moscovici truncations of the Weil quadratic form, at a prime cutoff c>1 and frequency band N, produce finite Galerkin matrices whose spectra are the finite-rank window on Weil positivity. We prove two exact finite theorems about this truncation. First, every real even Galerkin coefficient vector v determines, in closed form, a band-limited Guinand-Weil test function g_v whose zero sum over the nontrivial zeros of zeta equals the quadratic value exactly: every value of the truncated form is an exact sum over the zeros. The construction factors through an exact source quotient of dimension 2N+1 and admits a non-collapsing pole-neutral subfamily. Second, beyond the Galerkin band the omitted archimedean tail is a totally positive Cauchy-Stieltjes increment. This yields a two-sided certification rule with an explicit budget B_T ~ (2N+1) rho log(T) / (pi^2 T), where T is the archimedean cutoff and rho = 2 pi / log c: finite-cutoff positivity certifies cutoff-free positivity, a finite-cutoff eigenvalue below -B_T certifies a cutoff-free negative, and a negative eigenvalue in the band [-B_T, 0) certifies nothing. Resolving a spectral scale of 10^-59 at c=100 by brute cutoff would require T of order 10^63; a cutoff-free interval LDL^T factorization resolves it directly. The dictionary is verified over the first 512 zeros of zeta and by three independent computational routes; all scripts and artifacts ship with the paper. The paper makes no Riemann Hypothesis, prime-counting, next-prime, or factoring claim.
SN - 2331-8422
ER -