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 Q_Wλ^N 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 λ0^EVEN < λ0^ODD 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.