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

Version 2.3.

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.

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 D_r = −log₁₀(|ν_r − γ_r| / |γ_r|), where ν_r is the finite CCM spectral ordinate and γ_r is the corresponding Riemann zeta ordinate. The direct comparison with the original CCM headline also reports absolute error.

The empirical analysis establishes 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 D_ε = −log₁₀|ε_N|, 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|ε_N|. 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 is numerically equivalent, at the reported precision, to the result from the reduced even-sector solve. The even-sector restriction remains the faster reproduction default but is not required in the tested configurations to obtain these 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 2.3 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 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 source repository: https://github.com/TeamXcelerator/ccm-reproduction-and-convergence/tree/v2.3