Structural Properties of the Connes–van Suijlekom Truncated Weil Minimizer: Sobolev Scaling, Multi-Zero Universality, and L-Function Extension

The Connes-van Suijlekom truncated Weil quadratic form, indexed by a prime cutoff c, produces a ground state whose Fourier-Mellin zeros provably lie on the critical line. Whether these zeros converge to the Riemann zeros as c → ∞ is an open question identified by Connes (2026) and Connes-Consani-Moscovici (2025). We present the first independent implementation of the Connes-van Suijlekom Galerkin matrix and a systematic study across fifteen cutoffs c = 13 to 67, spanning 113 orders of magnitude in first-zero error. Beyond extending the data, we identify structural properties of the operator that constrain theoretical work on the convergence question: The Sobolev regularity of the ground-state eigenvector scales as s(c) ≈ 55 · log c − 128 (6 cutoffs), indicating that the eigenvector becomes dramatically smoother at higher cutoffs. All ten detectable zeros (γ₁ through γ₁₀) converge at rates within 3.8% of each other, reducing the convergence question to a single-zero problem. The eigenvector is approximately c-invariant (|⟨η_c1|η_c2⟩| > 0.950 across all 105 cutoff pairs) despite eigenvalues differing by 113 orders of magnitude. A partial extension to L(s, χ₃) shows convergence at three cutoffs (c = 13, 17, 37) but an oscillating positivity breakdown at c = 23, 29 — a structural finding about the character dependence of the CvS framework. No parametric convergence model captures the rate (nine tested, all fail), yet interval extension ΔL, not prime identity, is the dominant per-step driver (r = −0.96). The normalized Fourier-Mellin transform near γ₁ has essentially the same shape at all cutoffs (normalized slope varies 3.3%), consistent with convergence toward ξ(s). The bulk eigenvalue spectrum (λ₂ through λ₁₀₁) exhibits uniformly Poisson statistics (Brody β < 0.05), indicating no level repulsion. The spectral trace Tr(Q) decays as c⁻⁰·¹¹⁶ (R² = 0.994) while the Frobenius norm remains constant to within 2.3%. We make no claim about convergence rates, about Connes' conjecture, or about RH. All code, data, and ancillary files are publicly available.