Modular Phase Coherence in the Riemann Spectrum

Modular Phase Coherence in the Riemann Spectrum: Evidence for $\mathbb{Z}/6\mathbb{Z}$ Structure from the Saturation of the Spectral Signal-to-Noise Ratio We present robust empirical evidence demonstrating that the non-trivial zeros of the Riemann zeta function encode the deterministic structure of prime numbers through a rigid $\mathbb{Z}/6\mathbb{Z}$ modular phase coherence. While the local spacing of the zeros is famously governed by the Gaussian Unitary Ensemble (GUE) of random matrix theory, this analysis reveals that long-range phase correlations are governed by the sieve of Eratosthenes. By analysing the first $N = 10^5$ zeros obtained from the L-functions and Modular Forms Database (LMFDB), we evaluate the spectral form factor and the distribution of phases $\theta_n(x) = \gamma_n \ln x \pmod{2\pi}$ across the six congruence classes modulo 6. Key Discoveries: Extreme Modular Dichotomy: Phases evaluated at prime channels ($x \equiv 1,5 \pmod{6}$) exhibit massive non-uniformity and phase-locking (Kolmogorov-Smirnov $p$-value $\sim 10^{-75}$). Conversely, forbidden channels ($x \equiv 0,2,3,4 \pmod{6}$) are statistically indistinguishable from uniform white noise. SNR Saturation: The Signal-to-Noise Ratio (SNR) between prime and forbidden channels drastically violates the diffusive random-walk growth predicted by standard GUE theory. Instead, it undergoes an abrupt saturation to an asymptotic constant value of $\mathrm{SNR}_{\text{sat}} = 12.69 \pm 0.01$. The Modular Thouless Time: The transition to the saturated arithmetic regime is mathematically modelled and shown to occur at a critical scale of $N_{\text{sat}} \approx 132$ zeros, which exactly corresponds to the spectral resolution of the prime $p=7$. These findings quantitatively invalidate the purely diffusive GUE model for long-range correlations (yielding a mean absolute percentage error of 2.5% for our saturation model versus 37.8% for GUE). The results prove that the Riemann spectrum operates as a structured chaotic system: locally universal but globally organised by a $\mathbb{Z}/6\mathbb{Z}$ arithmetic topology. 💻 Reproducibility & Code: The complete Python source code, datasets, and Jupyter notebooks required to reproduce the statistical tests and the SNR saturation curve are openly available in the companion GitHub repository: https://github.com/NachoPeinador/RIEMANN_Z6 📄 Companion Theoretical Paper: The analytic foundations of this phenomenon, including the exact arithmetic identities and the formulation of a modified random matrix ensemble, are detailed in our companion paper: "The Riemann-GUE Ensemble: Reconciling Local Chaos with Global Arithmetic Order".