Quadrature sensitivity of deep spectra of truncated Weil forms, with corrections to two recent computational notes

The matrix elements of the truncated Weil quadratic form QWλN of Connes–van Suijlekom and Connes–Consani–Moscovici are computed, in all existing implementations, with a cutoff T on the archimedean integral. Our main point is methodological: deep-spectrum values computed at a single T carry no internal evidence of their own validity. Controlled T-sweeps under pre-registered gates show that the cutoff induces a noise band at the bottom of the computed spectrum: eigenvalues below a (c, T) -dependent error floor are undetermined, and spurious negative eigenvalues appear at the floor scale, in both parity sectors, reproducing exactly under increased working precision while failing to survive quadrature refinement — so arithmetic stability alone is not evidence of correctness. The resulting rule is a quadrature analogue of the familiar precision rule: trustworthy digits are those that agree between two values of T. We measure the floor at several cutoffs (e. g. c=100: ~1e-64 at T=800 and ~1e-105 at T=1200; c=53: ~4e-160 at T=1200) and document a validity envelope of the standard quadrature routine (at c=100, T=1600 the integrator saturates). Two applications follow. First, the negative even-sector eigenvalues reported by Groskin (arXiv: 2605. 20224) at c=100, computed at T=800 and left there as an unresolved puzzle, are quadrature artifacts in the audited configurations: they vanish, or shift by orders of magnitude, at T=1200; in particular the computed spectra show no genuine failure of positivity at c=100. Second, applying the same audit to our own note on the truncation dependence of sectorial ground-state ratios (Zenodo DOI 10. 5281/zenodo. 20614290), we find: the c=41 column is confirmed in full (ratio stable to 0. 2–1. 2% under T→1600; the reported rise with N is genuine) ; the c=29 values shift by 4–8%; the c=53, N=100 values are corrected by a factor ~5 (ratio 12922→2738) ; and the c=53, N≥120 entries were below the quadrature floor and must be regarded as undetermined. The sectorial ordering λ0EVEN < λ0ODD survives at every resolved point. No claims are made about Groskin's zero-recovery results, which concern different and better-conditioned quantities, nor about the continuum operator.