The Chain: Hurwitz Convergence from Finite Primes to Zeta Zeros via the Connes–van Suijlekom Construction

We implement the Connes–van Suijlekom Weil quadratic form from the explicit formula using only arithmetic data (poles, primes, and the archimedean Gamma factor) with no zeta zeros as input, and recover 30 zeta zeros from 8 prime powers to 7–9 correct digits. The exact archimedean kernel is identified as Φ∞(t) = (1/2π) Re ψ(1/4 + it/2), with a separate −log π diagonal shift from the explicit formula. Building on this non-circular computation, we lay out a two-condition Hurwitz convergence chain connecting finite-prime arithmetic data to the zeros of ζ(s): (a) Operator norm control: ‖QW_Λ^explicit − QW^true‖op ≤ C · Σ{p>Λ²} log(p)/p^{3/2} with C bounded uniformly in Λ (verified numerically: at t_max = 200, C ∈ [1.54, 2.35]; at t_max = 400, C tightens to [0.54, 0.77]); (b) Spectral gap stability: gap ≥ log 5, proved unconditionally via the functional equation and conductor growth. If eigenvector convergence ‖ξ_Λ − ξ‖ → 0 is established, Steps 2–5 of the chain are proved: the reproducing kernel of the Paley–Wiener space controls uniform convergence of ξ̂_Λ → ξ̂ on compact subsets of {|Im z| < 1}, and Hurwitz's theorem forces the zeros of ξ̂ onto the real line, which is equivalent to RH. v3 update: We identify a structural obstruction in Step 1. The spectral gap is proved for the φ(d) × φ(d) character-space kernel K_d, whose eigenvalues are −2 Re[L'/L(1/2, χ)]. The eigenvector ξ, however, is a null vector of the (2N+1) × (2N+1) Fourier-mode QW, which is positive semidefinite with a high-dimensional null space. We prove (Proposition 5.5) that the character subspace and the null space of QW are structurally orthogonal (overlap < 10⁻⁵). The character-space gap does not directly control perturbations of ξ. This orthogonality — between arithmetic content (characters, gap, CRT) and analytic content (Paley–Wiener functions, null space, Hurwitz convergence) — is, we believe, the sharpest formulation of the "Connes wall" in the literature: the precise structural reason that the finite-prime spectral construction works numerically but resists proof of convergence. Steps 2–5 of the chain, the gap theorem, the non-circular computation, and the operator norm evidence remain valid. The obstruction is localized to Step 1: establishing eigenvector convergence by any mechanism would close the chain. This paper is self-contained: all foundational theorems (the Weil explicit formula, Hurwitz's theorem, Kato–Temple perturbation theory, Mertens' estimates, Paley–Wiener theory) are stated with references, and the spectral gap proof is given in full. All computations are reproducible; scripts are provided.