Computational Verification of the Connes-Consani Adele Class Space Framework for the Riemann Hypothesis: Weil Positivity, Toeplitz Factorization, and Gap Mapping

This paper presents a computationally rigorous and independently cross-verified map of the internal consistency of the Connes-Consani adele class space framework for the Riemann Hypothesis. This paper does not claim a proof of the Riemann Hypothesis. Four primary results are documented and independently verified across five computational engines: (1) Frobenius eigenvalue consistency — |pʳho| = p^ (1/2) confirmed to machine precision for primes p in 2, 3, 5, 7, 11 across the first five known zeta zeros, with maximum deviation of 2. 22e-16 (one unit of double-precision machine epsilon). (2) PSWF Weil positivity — W (h) > 0 confirmed for prolate spheroidal wave function test functions across bandwidth parameters c = 1 through 5, using a corrected Slepian-Pollak ground state extraction. An implementation error (extraction of most oscillatory rather than ground state eigenvector) was identified through cross-engine comparison and corrected; analytical benchmarks at c = 0 verified to machine precision. (3) Toeplitz factorization (primary novel result) — the p-adic local Weil matrix admits the factorization Mₚ = AₚT Dₚ Aₚ, where Dₚ is positive definite by construction (all weights wₙ = log (p) * p^ (-n/2) > 0) and Aₚ has full rank (PSWFs are linearly independent, frequencies distinct). Positive semidefiniteness verified via Cholesky factorization for all tested primes (2 through 29) and bandwidths (c = 1 through 20). This result is the discrete p-adic analog of the archimedean Toeplitz structure proved by Connes (2020). (4) Five-lens zero ordinate screening — base conversion clustering, pairwise harmonic ratio analysis, GUE spacing comparison, Fibonacci/phi encoding, and prime lattice projection applied to the first 100 non-trivial zeros. All five lenses subtract cleanly under density-corrected Monte Carlo null models. GUE spacing is confirmed: KS statistic 0. 119 against critical value 0. 194 at alpha = 0. 05. The formal gap is precisely located: essential self-adjointness of the scaling operator H on L² (AQ/Q*, omega) with spectrum equal to the imaginary parts of the non-trivial zeta zeros remains unproven. The Connes-Consani-Moscovici (2024) prolate wave operator paper is the current frontier. A five-step proof outline for p-adic local Weil positivity is provided; Steps 1-5 are computationally established and theoretically supported; Step 6 (functional analysis limit as c tends to infinity) is the open problem.