The compression defect of a discrete-sine Galerkin section: an exact reflection identity, a window-truncation law, and two certification traps

Abstract

Finite Galerkin certificates for the Weil quadratic form lose information in two independent ways: through what is discarded outside the band, and through the failure of the compression to be multiplicative inside it. The first loss was given an exact order theory by Groskin. This note treats the second. For a real symmetric Toeplitz section T compressed onto a window C_n of n low odd discrete-sine (DST-I) modes, write A and G for the compressions of T and T^2 and Delta_n = G - A^2 for the compression defect. We prove three exact finite statements. First, a reflection identity: in the DST-I basis T = D + K with D diagonal and K the Hankel reflection of the row, whence Delta_n = (2/M) S K (I - P_n) K S^T is a Gram matrix. Three consequences follow at once: Delta_n is positive semidefinite, so any certificate S = A - kappa Delta_n >= 0 forces A >= 0; Delta_n does not depend on r_0 or r_1; and Delta_n is computable in O(nM) operations without cancellation, against O(nM^2) for a frontal assembly. Second, the window constant Gamma_{n,M} = ||(I - P_n) 1||^2 has a closed form and a certified uniform jet, with Gamma_{n,M} tending to gamma_0(n), the tail (16/pi^2) times the sum of m^{-2} over odd m > 2n-1, which is (4/(pi^2 n))(1 + O(1/n)). Third, a window-truncation law: at fixed data u the defect satisfies ||(I - P_n) K u||^2 = a^2 Gamma_{n,M} (1 + o(1)) with a = (Ku)(1) the endpoint value, hence Theta(1/n); and an exact rank-one identity shows that adjoining the single vector (I - P_n) 1 removes precisely that term, reducing the defect by six orders of magnitude at n = 2560. We then isolate two traps that certified computations of this kind can fall into, both illustrated on an explicit archimedean-plus-prime minorant: a transfer constant sitting within 0.2% of its critical value, which makes the certified margin decay like 1/N and no uniform statement survive; and an archimedean minorant truncated at an order whose last retained term exceeds the signed eigenvalue by four orders of magnitude, the expansion-order analogue of Groskin's frequency-cutoff band. All numerical statements are backed by interval-arithmetic certificates and a checksummed replay package. No claim is made about the Riemann Hypothesis or about Weil positivity, and no window is certified beyond existing unconditional results. This preprint record contains the article PDF, the complete LaTeX source archive, and SHA-256 checksums. The companion reproducibility software and audit archive is available at https://doi.org/10.5281/zenodo.22239538. Anthropic Claude assisted with preparation; Julien Lange is the sole author and assumes responsibility.