Ground-State Negativity of Truncated Connes-Van Suijlekom Forms for Dirichlet L-Functions.

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 χD D a fundamental discriminant, and validate it externally: the associated Weil distribution reproduces the explicit formula for L(s,χ₅) 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, Qarcb^χD=Qarcbζ+logD· I With the kernel so validated, we report certified numerical evidence that the least eigenvalue λₘᵢₙ 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≤ c≤37, 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→∞