A Quantitative Convergence Law for the Connes-Consani-Moscovici Zeta Spectral Triple

The Connes–Consani–Moscovici (CCM) zeta spectral triple (arXiv:2511.22755) and its Connes–van Suijlekom refinement (arXiv:2511.23257) produce finite real symmetric matrices whose ground state yields an approximation ν₁ to the first nontrivial Riemann zero γ₁ that sharpens as three parameters grow: a prime cutoff λ², a basis size N, and the working precision P. The rate of this convergence in the basis size is named the central open problem in arXiv:2511.22755.

We characterize the convergence quantitatively, in two equivalent forms. Intrinsically, the relative error |ν₁−γ₁|/|γ₁| is controlled by a single two-channel decomposition,

E(N, λ²) ≤ ε_modes(N, λ²) + ε_primes(λ²),

a mode channel that vanishes as N→∞ and a prime channel that vanishes as λ²→∞, so that ν₁→γ₁ is exactly E→0 and requires both limits. The two channels are the finite-N transition and the extreme tail of a single prolate-type spectral edge. This form makes no reference to precision. Observed at finite working precision P, the same law is the matching-digit count D(λ², N, P) = −log₁₀(|ν₁−γ₁|/|γ₁|), which obeys

D = min(D_Wprec(P), D_nModes(N, λ²), D_Primes(λ²)),

the minimum of three budgets (precision, modes, prime content). Precision is the instrument resolution, absent from the intrinsic form. In digit space the minimum is forced, not fitted: D is a negative logarithm of a sum of independent errors, so the largest binds. This is why earlier single-expression digit fits fail.

We prove, under an explicit finite-cutoff positivity hypothesis (unconditional in the Yoshida–Bombieri small-support range), that convergence to γ₁ requires all three budgets to diverge. At a fixed cutoff the accuracy is capped at the prime ceiling D_Primes(λ²), so no basis size or precision reaches the zero and the cutoff must grow. This reduces convergence to the growth of the prime ceiling, whose leading coefficient is derived, parameter-free, from the prolate spheroidal eigenvalue asymptotic:

D_Primes ∼ (4π/ln10)λ².

The coefficient is rigorously anchored from one side by a variational floor bound; the reverse inequality, that the ceiling cannot be beaten at this rate, remains open.

The mode budget, the basis-size convergence rate left open by CCM, is the logarithm of a stretched-exponential convergence exp(−aN^q) whose exponent q(λ²) is the analyticity class of the eigenvector, derived from the archimedean Gamma factor with a prime correction from Mertens’ theorem. A by-product is that the eigenvalue-space Galerkin exponent measured by Groskin (2026) is a finite-window slice of the same curve.

Each component is stated with an explicit rigor level, separating what is proved from what is derived and empirically validated up to λ²=200. The results have been communicated to co-authors of arXiv:2511.22755.

GitHub: https://github.com/TeamXcelerator/ccm-convergence-rate