What question did this study set out to answer?

This note aims to verify missing steps in the spectral realization methodology by Connes–Consani–Moscovici.

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

This note aims to verify missing steps in the spectral realization methodology by Connes–Consani–Moscovici.
Implemented semilocal Weil quadratic form using arbitrary-precision arithmetic (40-60 digits).
Evaluated the simple-even condition and the proximity of prolate ansatz at specific B6 windows.
Quantified eigenvalue ladder in computed spectra using precise numerical techniques.
Verified simple-even condition at B6 = 1.3, 1.5, 2.0, confirming evenness through numerical computation.
Measured Rayleigh excess r/B5B6 as approximately 0.14; spectral gap shows significant growth across windows.
Confirmed eigenvalue ladder ratio BC1/B5B6 increases significantly, indicating a large spectral gap.

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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