GROUND-STATE NEGATIVITY OF TRUNCATED CONNES-VAN SUIJLEKOM FORMS FOR DIRICHLET L-FUNCTIONS.

Abstract

The Connes-van Suijlekom (CvS) framework associates to a real distribution Don [0, L] a finite Galerkin matrix whose ground state encodes spectral data of an associated L-function. We construct the CvS kernel for real primitive Dirichlet characters $\chi_{D}$ D a fundamental discriminant, and validate it externally: the associated Weil distribution reproduces the explicit formula for $L(s,\chi_{5})$ against zeros computed independently from Hurwitz zeta functions, the two sides agreeing to the precision of the root-finding tolerance. We further establish an exact operator identity for even characters, $Q_{arcb}^{\chi_{D}}=Q_{arcb}^{\zeta}+log\vert{}D\vert{}\cdot I$ With the kernel so validated, we report certified numerical evidence that the least eigenvalue $\lambda_{min}$ of the truncated CvS form is negative and decreasing in the cutoff c for X5 and $X-3$ at fixed Galerkin dimension $N=20$, over $7\le c\le37$, while the corresponding quantity for ( remains below the certified archimedean truncation bound throughout. All eigenvalue enclosures are computed in ball arithmetic with explicit tail bounds and are gated on a resolution criterion We emphasise what this does and does not show. It does not bear on the Generalized Riemann Hypothesis: negativity of the finite-c form is consistent with GRH, and positivity of the finite-c form would not establish it. The conclusion we draw is that the truncated CvS quadratic form is not a positivity certificate for Dirichlet L-functions at the cutoffs examined, which is relevant to the open question of the behaviour of these truncations as $c\rightarrow\infty$