What question did this study set out to answer?

The aim is to verify missing steps in the spectral realization framework using the semilocal Weil quadratic form.

June 15, 2026Open Access

High-Precision Verification of the Spectral Realization by Zeta-Cycles: the Simple-Even Condition, the Prolate Proximity, and the Eigenvalue Ladder of QW_

Key Points

The aim is to verify missing steps in the spectral realization framework using the semilocal Weil quadratic form.
Implemented the semilocal Weil quadratic form in high-precision arithmetic (40-60 digits).
Reduced matrix computations of the quadratic form to one-dimensional integrals for efficiency.
Measured the proximity of the prolate ansatz to the true ground vector and quantified spectral properties.
Verified the simple-even condition across multiple windows (λ = 1.3, 1.5, 2.0) with substantial precision.
Quantified the first excited even eigenvalue's growth ratio, demonstrating a spectral gap much larger than the ground eigenvalue.
Reproduced critical identities and spectral realizations with exceptional accuracy over numerous orders of magnitude.

Abstract

This note reports an independent implementation of the semilocal Weil quadratic form \ (QW_\) of Connes–Consani–Moscovici (arXiv: 2511. 22755, hereafter CCM), together with numerical evidence — computed in arbitrary-precision floating-point arithmetic (mpmath, 40–60 digits) — bearing directly on the two missing steps identified in CCM, Section 8. The contributions are: A reduction of the matrix of \ (QW_N\) in the basis \ (Vₙ\) to \ (O (N) \) one-dimensional integrals of elementary functions, permitting fast evaluation at 40-digit precision. Verification of the simple-even condition (CCM, first missing step) at the windows \ (= 1. 3, \ 1. 5, \ 2. 0\), with both parity sectors computed so that evenness is verified rather than imposed. A direct quantitative measurement of the proximity of the prolate ansatz \ (k_\) to the true ground vector \ (_\) (CCM, second missing step): the normalized Rayleigh excess is \ (r/_ 0. 14\), and the resulting min–max bound \ (\| k_ - _\| 2r/ (₁-) \) is verified on the data. A quantification, in the computed spectra, of the sharply graded eigenvalue ladder whose existence is indicated in CCM, Section 8 (indication (3) ): the ratio \ (₁/_\) of the first excited even eigenvalue to the ground eigenvalue grows from \ (37\) to \ (1. 810³\) to \ (210⁵\) across the three windows, with the first excited even eigenvector lying in the span of the \ (E\) -images of the prolate cells \ (n = 0, 4, 8\) with squared-projection deficit \ (310^-6\) — so that the spectral gap relevant to the convergence strategy of CCM, Section 7, is exponentially larger than the ground eigenvalue. An independent reproduction of the identity \ (_ 1- () \) (CCM, Figure 4) over thirteen orders of magnitude, and of the spectral realization itself: with the primes \ (2, 3\) (and the prime power \ (4\) ) only, the zeros of \ (\) computed from our matrix are real to the working-precision floor (\ (10^-44\) at 40 digits, falling to \ (10^-66\) at 60 digits) and reproduce \ (₁\) to \ (2. 310^-9\). All results are packaged as five deterministic Python certificates (linked below), including a stability audit (truncation refinement, quadrature doubling, precision doubling, root-finder basin), which run in minutes on a laptop.

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