A Probe of the Odd Parity Sector of the CvS Galerkin Matrix

Abstract. A probe of the odd parity sector of the Connes-van Suijlekom Galerkin construction (A. Groskin, arXiv:2605.20224), which that paper lists as unprobed. The odd sector of the periodic construction is shown to coincide exactly, at finite size, with a specific finite sine space. On that sector, at c = 13, 17, 19, 23, the eigenvector belonging to the smallest eigenvalue yields a function whose unseeded sign-change scan returns the first ten zeta ordinates in order, with none missing and no additional sign-changing root detected; at c = 13 the scan extends to height 100 and returns exactly the twenty-nine ordinates below it. The scan builds its function from the eigenvector coefficients alone and calls for the ordinates only after its root list is frozen; a random odd vector in the same basis comes no closer than order 1e-1. Two further measurements are reported: the first-zero error tracks the smallest eigenvalue through a constant that grows with c in both parity sectors while the ratio between them moves only a few percent, and the Galerkin exponent is nearly insensitive to the archimedean cutoff. Everything is numerical, at four cutoffs. Nothing is claimed about the limit as the cutoff grows, or about the Riemann Hypothesis.