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February 6, 2026Open Access

Spectral-Arithmetic Duality: Modular Phase Coherence and the Riemann-GUE Ensemble

Key Points

The research aims to uncover modular phase coherence within the non-trivial zeros of the Riemann zeta function and its implications for number theory.
Integrated Digital Signal Processing with analytical number theory.
Validated findings using a dataset of $N=10^5$ Riemann zeta zeros.
Developed a computational lab with a JIT-compiled pipeline for replicating research phases.
Documented empirical breakdown of GUE universality with statistical models.
Identified the spectral Signal-to-Noise Ratio saturation at 12.69, implying modular structure influences on arithmetic.
Achieved a Kolmogorov-Smirnov $p$-value of 0.27, suggesting compatibility with GUE models.
Demonstrated a 33.33% reduction in factorization search space with a 'Modular Factorization Reactor'.
Established that Mersenne primes exhibit 100% polarization in the $1 mod 6$ channel.

Abstract

Spectral-Arithmetic Duality: Modular Phase Coherence and the Riemann-GUE Ensemble Abstract This dataset and software suite provide the experimental and analytical framework for the discovery of Modular Phase Coherence within the non-trivial zeros of the Riemann zeta function. By integrating Digital Signal Processing (DSP) with analytical number theory, this research documents a structured breakdown of GUE (Gaussian Unitary Ensemble) universality in the mesoscopic regime. The core finding is the saturation of the spectral Signal-to-Noise Ratio (SNR) at the constant 12. 69, a value derived from the quadratic identity L (2, ₀^ (6) ) = ²/9. This reveals that the Riemann spectrum operates as an "arithmetic crystal, " where the Z/6Z modular structure acts as a noise-free channel for arithmetic information. Key Scientific Contributions The Riemann-GUE Ensemble: A new statistical model that reconciles local GUE chaos with global modular rigidity. Validated with N=10⁵ zeros, achieving a Kolmogorov-Smirnov p-value of 0. 27 (failing to reject the model, unlike pure GUE). SNR Saturation Dynamics: Empirical evidence of universality breakdown where the SNR deviates from GUE diffusive growth to a hard saturation at 12. 69 0. 01. Computational Efficiency: A "Modular Factorization Reactor" that leverages Z/6Z resonance to reduce the factorization search space by 33. 33%, validated via JIT-compiled benchmarking. Mersenne Polarization: Analytical and empirical proof of the absolute modular rigidity of Mersenne primes (Mₚ), which exhibit 100% polarization in the 1 6 channel. Contents of this Record The Research Paper (PDF/LaTeX): Full theoretical derivation, including the "Modular Thouless Time" concept and the L (2) identity proof. Computational Lab (Jupyter/Python): A complete JIT-compiled (Numba) pipeline for replicating the 7 phases of the research: from statistical anomaly detection to factorization speedup. Dataset: High-precision non-trivial zeros used for the 10⁵ sample validation. Technical Specifications Framework: Python 3. 10+ / Numba (JIT) / NumPy / SciPy. Field: Number Theory, Quantum Chaos, Information Theory, Computational Complexity. License: PolyForm Noncommercial 1. 0. 0 (Research & Open Science).

Spectral-Arithmetic Duality: Modular Phase Coherence and the Riemann-GUE Ensemble

Key Points

Abstract

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