A Numerical Realization of Suzuki's Weil-Quadratic-Form Operator: The Archimedean Spectral Law, its Universality, and an Operator Form of Weil's Positivity Criterion
Abstract
This paper presents the first numerical realization of Suzuki's Weil-Quadratic-Form operator, a candidate for the Hilbert--Pólya program linking spectral positivity to the Riemann Hypothesis (RH). Suzuki's 2026 construction was purely theoretical; here, the operator is instantiated via P1 finite-element discretization and Richardson extrapolation. Key results include: (R1) In the prime-free regime, the spectrum follows a closed Archimedean law Ak(a) = log(1/a) + log(k−2) + B0 + O(a), with B0 = log q − 2log 2, confirmed to 30-digit precision. (R2) A Mellin double-pole argument proves the head coefficient B(ν) and shows B0 depends only on the conductor q, independent of the Archimedean parameter. (R2b) The degree d of an L-function appears directly as the logarithmic slope of the spectrum. (R3) Total spectral intensity follows the prime number theorem, S(a) ∼ (2a)3/6. (R4) Nontrivial zeros are not eigenvalues but occur in the explicit-formula error term of the prime symbol. (R5) The best-match line σ*(a) descends toward the critical line. (R6) Weil's positivity criterion is realized in operator form: bounded residual growth corresponds to all zeros on the line, while an injected off-line zero causes exponential blow-up. (R7) The lowest eigenvalue λ1(a) is strictly positive, decays superexponentially, and passes smoothly through the first prime threshold. (R8) The characteristic function W(a,0;z) is computed for the first time, with all zeros confirmed real. (R9) Indirect traces of GUE statistics appear in the moment structure, even where direct detection is blocked. The authors emphasize that this work does not prove RH. All results are Archimedean and universal, with significance lying in the faithful numerical realization of classical identities rather than new arithmetic.
Related
- A trace Paley-Wiener theorem for GLn(F)backslash GLn(E), with E/F a quadratic extension of non-Archimedean local fields
- A finite Guinand-Weil dictionary and archimedean tail order for the truncated Weil quadratic form
- Thickening realization and positivity properties of canonical bases
- An energy-decreasing algorithm for the finite element approximation of ferronematic equilibrium states and its numerical realization
- Universality Criterion Sets for Definite Quadratic Lattices over Polynomial Rings
BibTeX
@misc{kim2026numerical,
title = {{A Numerical Realization of Suzuki's Weil-Quadratic-Form Operator: The Archimedean Spectral Law, its Universality, and an Operator Form of Weil's Positivity Criterion}},
author = {Taebong Kim and Youngsik Hong and Minsik Kim and Sunyoung Choi and Jaewon Jang and Minseo Kim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.24830},
eprint = {2607.24830},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.24830},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Numerical Realization of Suzuki's Weil-Quadratic-Form Operator: The Archimedean Spectral Law, its Universality, and an Operator Form of Weil's Positivity Criterion
AU - Taebong Kim
AU - Youngsik Hong
AU - Minsik Kim
AU - Sunyoung Choi
AU - Jaewon Jang
AU - Minseo Kim
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2607.24830
UR - https://arxiv.org/abs/2607.24830
AB - This paper presents the first numerical realization of Suzuki's Weil-Quadratic-Form operator, a candidate for the Hilbert--Pólya program linking spectral positivity to the Riemann Hypothesis (RH). Suzuki's 2026 construction was purely theoretical; here, the operator is instantiated via P1 finite-element discretization and Richardson extrapolation. Key results include: (R1) In the prime-free regime, the spectrum follows a closed Archimedean law A_k(a) = log(1/a) + log(k−2) + B_0 + O(a), with B_0 = log q − 2log 2, confirmed to 30-digit precision. (R2) A Mellin double-pole argument proves the head coefficient B(ν) and shows B_0 depends only on the conductor q, independent of the Archimedean parameter. (R2b) The degree d of an L-function appears directly as the logarithmic slope of the spectrum. (R3) Total spectral intensity follows the prime number theorem, S(a) ∼ (2a)^3/6. (R4) Nontrivial zeros are not eigenvalues but occur in the explicit-formula error term of the prime symbol. (R5) The best-match line σ^*(a) descends toward the critical line. (R6) Weil's positivity criterion is realized in operator form: bounded residual growth corresponds to all zeros on the line, while an injected off-line zero causes exponential blow-up. (R7) The lowest eigenvalue λ_1(a) is strictly positive, decays superexponentially, and passes smoothly through the first prime threshold. (R8) The characteristic function W(a,0;z) is computed for the first time, with all zeros confirmed real. (R9) Indirect traces of GUE statistics appear in the moment structure, even where direct detection is blocked. The authors emphasize that this work does not prove RH. All results are Archimedean and universal, with significance lying in the faithful numerical realization of classical identities rather than new arithmetic.
ER -