Independent Reproduction and Convergence Analysis of the Connes-Consani-Moscovici Zeta Spectral Triple

Abstract

Version 2.4. We independently implement the Connes–Consani–Moscovici (CCM) operator construction described in arXiv:2511.22755 in Rust using arbitrary-precision MPFR/GMP arithmetic. This manuscript is an independent reproduction and empirical analysis of the CCM construction; it does not claim to prove the limiting convergence required for a proof of the Riemann Hypothesis. Changes in Version 2.4. This revision adopts Xcelerator Toolkit v0.14.2, adds the ultra research-capture interface, and improves remote artifact reuse and publication staging without changing the paper’s mathematical configurations or headline results. All eight numbered claim groups were rerun at their stated precisions, and the resulting numerical outputs and data artifacts were validated. Table 7 now reports the measured roundoff-scale relative differences between the natural and even-sector eigenvalues rather than describing three of them as exactly zero. This reporting correction does not change the conclusion that the two routes agree to hundreds of decimal digits. The reproduction extends the original headline result from approximately 55 matching decimal digits to 1019.0 measured matching decimal digits on the first nontrivial Riemann zeta zero. This result uses an HP-1000 requested-precision target with a 64-bit internal guard reserve and saturates the resulting guarded working precision. The intrinsic accuracy of that finite configuration is therefore not resolved by this run. Throughout the manuscript, matching digits are measured using the relative metric Dr = −log10(|νr − γr| / |γr|), where νr is the finite CCM spectral ordinate and γr is the corresponding Riemann zeta ordinate. The direct comparison with the original CCM headline also reports absolute error. The empirical analysis documents several numerical properties: As the basis size N increases, accuracy follows a smoothly saturating ramp. It is approximately linear only over the observed early range; the local gain then decreases as the first-zero accuracy approaches its finite-configuration plateau. The first-zero plateau closely tracks the Weil-eigenvalue depth Dε = −log10|εN|, but the two quantities are not identical. Their difference is a slowly varying finite eigenvalue-to-root conversion gap, approximately 2.7–3.2 digits over the sampled fixed-N cutoff sweep. The Weil-eigenvalue depth grows rapidly along the sampled cutoff-and-basis paths. Along the path N/√λ² ≈ 28, hundreds of eigenvalue-depth digits are gained over the listed prime-count doublings. Because both λ and N vary, these measurements do not define a prime-count-only law and do not establish whether the asymptotic decay is polynomial, exponential, or super-exponential. At fixed N = 120, first-zero accuracy increases monotonically over the sampled range 13 ≤ λ² ≤ 100 and closely follows −log10|εN|. No fixed-N monotonicity claim is made outside this finite sweep. The smallest eigenvector’s even symmetry, corresponding to the CCM Step 1 hypothesis, is observed at every tested configuration above the working-precision floor, extending through λ² = 1200 at HP-2000. At the tested configurations, the unrestricted natural full-space solve produces an essentially even eigenvector. Its smallest eigenvalue agrees with the reduced even-sector result to hundreds of decimal digits. The measured relative differences are at the arithmetic floor, reaching approximately 10−559 to 10−758 in the newly clarified entries. The even-sector restriction remains the faster reproduction default but is not required at the tested configurations to obtain the reported spectral values. At N = 10, accurate finite matches are obtained for several low zeta ordinates above the nominal lattice edge, with the corresponding secular roots lying above the largest pole N2. This shows that the lattice edge is not a hard root cutoff; no general root-localization claim is inferred from this finite example. Prime-power thresholds introduce localized slope kinks in the finite matrix path. In sampled accuracy and eigenvalue curves, these threshold effects appear as fine-scale perturbations on an otherwise smooth global envelope rather than producing the pronounced plateaus and jumps expected from a separate staircase law. Remarkably, a 21 × 21 matrix constructed from only six primes already reproduces 21.585 matching decimal digits of the first Riemann zero. This work has been shared with the CCM authors and reviewed by the lead author, Professor Alain Connes. Akiva Groskin also reviewed the work, independently reproduced key aspects of the truncated Weil-form computation, and cited earlier versions in his own research. This record archives the Version 2.4 manuscript. It is supplemented by a separate Zenodo software deposit containing the source code, claim scripts, documentation, and reproducibility infrastructure. The software concept DOI resolves to the latest available source release. The associated software pins Xcelerator Toolkit v0.14.2 and supports arbitrary precision, managed artifact reuse, seeded or independent finite-root acquisition, multiple parity policies, configurable research capture, and optional numerical certification. Reference ordinates used by the paper-claim scripts select which finite secular root is refined; they are not substituted into the computed result. Tagged paper source: https://github.com/TeamXcelerator/ccm-reproduction-and-convergence/tree/v2.4Paper source revision: b6df89beea06633b2af5746750c41bb9467fcf67Toolkit release: v0.14.2, revision f2f4539bd1e51b257ebf57ded0ad2147c61411b3