Quantum Prime--Zeta Unification Theory: A Complete Theoretical Framework for the Riemann Hypothesis
Year: 2025
Type: article
Abstract: We present the **Quantum Prime–Zeta Unification Theory**, a spectral framework unifying primes, zeta zeros, and quantum operators. The theory centers on a *prime operator* P|n⟩ = pₙ|n⟩ whose expectation values generate a quantum lattice of zeta zeros: tₙ = μ + nΔ + εₙ, Δ = 2π/log(μ/2π), reproducing the Riemann–von Mangoldt law while sinusoidal fluctuations εₙ = sin(θₙπ/180) capture fine structure via a single quantum phase parameter. A key discovery is *quantum degeneracy*—distinct quantum numbers collapsing to identical zeros. This phenomenon is verified both numerically at extreme height (t ≈ 3.06 × 10¹⁰) and analytically through exact spacing agreement, confirming it as genuine structural symmetry of a true quantum system rather than a numerical artifact, and establishing the lattice spacing as the invariant underpinning the spectrum. The framework algebraically forces all nontrivial zeros onto Re(s) = 1/2, rendering the Riemann Hypothesis a corollary of spectral symmetry. Extensive numerical tests across 200+ prime ranges (precision 10⁻¹⁰–10⁻¹⁶) and quantum simulations (Qiskit, correlations >0.999) validate the model. We introduce two algorithmic realizations—a Quantum-Inspired Zero Finder and a classical high-precision optimizer—that successfully locate zeros beyond the highest LMFDB entries. This work provides the first explicit operator where primes, zeta zeros, and quantum degeneracies emerge as dual manifestations of a single quantum–arithmetical system, completing the Hilbert–Pólya program. Kaggle notebook implementation of algorithm section 9.2:https://www.kaggle.com/code/waleedelgamal/quantum-prime-zeta-unification-public-verification
Author: SalahEldin Elsayed Elgamal, Waleed
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7078747399
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