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

This note reports an independent implementation of the semilocal Weil quadratic form \(QW_\lambda\) 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 \(QW_\lambda^N\) in the basis \(V_n\) 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 \(\lambda = 1.3,\ 1.5,\ 2.0\), with both parity sectors computed so that evenness is verified rather than imposed. A direct quantitative measurement of the proximity of the prolate ansatz \(k_\lambda\) to the true ground vector \(\xi_\lambda\) ([CCM], second missing step): the normalized Rayleigh excess is \(r/\epsilon_\lambda \approx 0.14\), and the resulting min–max bound \(\|\tilde k_\lambda - \xi_\lambda\| \le \sqrt{2r/(\mu_1-\epsilon)}\) 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 \(\mu_1/\epsilon_\lambda\) of the first excited even eigenvalue to the ground eigenvalue grows from \(\approx 37\) to \(\approx 1.8\times10^3\) to \(\approx 2\times10^5\) across the three windows, with the first excited even eigenvector lying in the span of the \(\mathcal E\)-images of the prolate cells \(n = 0,4,8\) with squared-projection deficit \(\approx 3\times10^{-6}\) — 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 \(\epsilon_\lambda \approx 1-\chi(\lambda)\) ([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 \(\widehat\xi\) 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 \(\gamma_1\) to \(2.3\times10^{-9}\). 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.