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. That Delta_n is positive semidefinite - so that any certificate S_kappa = A - kappa Delta_n >= 0 forces A >= 0 - is classical and needs only the isometric compression. What the reflection form adds is two consequences: Delta_n does not depend on r_0 or r_1; and the n actions T s_m cost O(nM) without cancellation, so that A, G and Delta_n are assembled in Theta(n^2 M) against Theta(n M^2) for a frontal assembly. Second, the window constant Gamma_{n,M} = ||(I - P_n) 1||^2 has a closed form and, at fixed n, a jet certified uniformly in M, 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 with non-zero endpoint value a = (Ku)(1), and provided the residual term is o(Gamma_{n,M}), the defect satisfies ||(I - P_n) K u||^2 = a^2 Gamma_{n,M} (1 + o(1)), hence, when n = o(M), Theta(1/n); and an exact rank-one identity shows that adjoining the single vector (I - P_n) 1 removes that term exactly, reducing the measured 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 measured 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 a factor of about 2394, the expansion-order analogue of Groskin's frequency-cutoff band. Every statement of a definite sign is decided in interval arithmetic; the exploratory tables are float64 and labelled as such; a checksummed replay package reproduces all of it. This note certifies no window, and makes no claim about the Riemann Hypothesis or about Weil positivity. This preprint record contains the article PDF, the complete LaTeX source archive, and SHA-256 checksums. This is a preprint: it has not been peer-reviewed, and its mathematics has not been independently verified. Anthropic Claude assisted with preparation; Julien Lange is the sole author and assumes responsibility.