Quantum Prime--Zeta The Ising Universality of the Riemann Zeta Zeros: Hamiltonian Construction and Empirical Verification
Year: 2025
Type: article
Abstract: This paper presents the first explicit Hamiltonian construction for the Riemann zeta zeros and demonstrates its Ising universality class. Building upon our Quantum Prime-Zeta framework, we construct a quantum mechanical system whose eigenvalues exactly correspond to the non-trivial zeros of the Riemann zeta function. Through high-precision numerical analysis of 1858 eigenvalues generated by the spectral formula $t_n = \mu + n\Delta + \sin\theta_n$ with $\Delta = 2\pi/\log(\mu/2\pi)$, we report: (1) Brody parameter $\beta = 0.5052$ (95% CI: [0.4624, 0.5624]), definitively rejecting Poisson ($p\sim10^{-35}$) and GUE ($p\sim10^{-32}$) distributions; (2) 28 full sweep prime range slices produce central charge $c = 0.5$ with true error between ($0.0056 , 0.0091$) and correlation $>0.99$ from entanglement entropy scaling; (3) Exact quantum degeneracy obeying $(n-m)\Delta = \sin\theta_m - \sin\theta_n$; and (4) Universal scaling laws matching 2D Ising criticality. Furthermore, we establish a prime--zero duality: the critical Hamiltonian exhibits identical universal invariants whether constructed from zeta zeros or directly from the primes themselves. The Riemann Hypothesis emerges as a natural consequence of this underlying quantum critical system, completing the physical realization of the Hilbert-Pólya conjecture and establishing a profound connection between number theory and statistical physics. Code and Data are publicly available on Kaggle.
Author: SalahEldin Elsayed Elgamal, Waleed
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7082160804
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