Exact archimedean entries for truncated Weil forms: closed-form implementation, a precision-stable library defect, and corrected deep spectra

We implement the closed-form archimedean matrix elements of the truncated Weil quadratic form QW_lambda^N given by Connes-Consani-Moscovici (arXiv:2511.22755, Section 4.3), eliminating the archimedean quadrature cutoff T entirely — and with it the quadrature noise floor documented in our previous note (DOI 10.5281/zenodo.20650146). The implementation is validated against the defining x-space integrals to 1e-250..1e-300 and reproduces the zero-recovery table of CCM at lambda=sqrt(13), N=120 to the published accuracy; it is roughly two orders of magnitude faster than the quadrature route.

The main content of this note is an instrumental finding. Deep spectra computed with these closed forms at c=53 (N=140) and c=100 (N=100) exhibited tiny negative eigenvalues (1e-161..1e-131) in both parity sectors. These survived an audit that would have satisfied the standards of the existing computational literature: entry-level validation against defining integrals, bitwise agreement of the spectra at 300 and 400 decimal digits, eigenpair residuals of order 1e-400, solver-independent negative counts (LDL inertia versus Jacobi), and Cauchy interlacing consistency across N=100,120,140. All ten negatives were nevertheless artifacts: mpmath's lerchphi returns values wrong by up to 2.7e-127 in the parameter range needed (0<z<<1, a=1/4+it, t~10..50), and the error is locally stable under precision increase — it plateaus across wide precision ranges (identical at dps=300..500), which defeats the standard "recompute at higher precision and compare" gate by construction. Replacing lerchphi by the defining series removes every negative: the corrected spectra are positive, with ground states +1.1e-192 (even) and +2.4e-188 (odd) at c=53, N=140, descending in a clean geometric ladder, as the CCM framework predicts. We reported the defect upstream (mpmath issue #1099).

The methodological conclusion extends the two-cutoff rule of our previous note: precision-stability is not correctness. Special-function implementations used in critical computations must be validated against their defining series or integrals in the exact parameter range of use. We also record a 0.4544 discrepancy between the combined constant c(L)+w(L) printed in CCM Section 4.3 and the sum of the displayed expressions for c(L) and w(L); our computations decisively support the printed combined value.

No claim about the Riemann Hypothesis is made in either direction: the corrected computations show no failure of positivity at the audited configurations reported here.