Spectral Analysis of Riemann Zeta Zeros and Prime-Modulated Hamiltonians: A Constructive Li-Nyman-Beurling Approach within the Universal Model Framework

We investigate spectral statistics of Riemann zeta zeros and their relation to prime-modulated Hamiltonians within the Universal Model Framework (UMF). Using authenticated Odlyzko zeros (indices 10,000–10,599), we confirm GUE universality across multiple metrics: spacing ratios ⟨r⟩ = 0.601 ± 0.012, pair correlation consistent with the sine-kernel, and logarithmic growth of spectral rigidity. Prime-modulated tight-binding operators display intermediate statistics (⟨r⟩ = 0.543 ± 0.019), while Paley graph Laplacians yield improved agreement (⟨r⟩ ≈ 0.60), suggesting arithmetic structure plus correct symmetry class is essential for universality. We develop a constructive Li-coefficient positivity framework inspired by Li’s criterion and Nyman–Beurling theory, transforming numerical verification into proof obligations through four theorem pathways. Quantitative falsifiability criteria and a 30/60/90-day development roadmap with acceptance gates are provided. These results refine the UMF–Hilbert–Pólya program, positioning prime-indexed operators as candidate bridges between number theory, spectral chaos, and physics. This project was developed by Marco Gericke, with structured assistance from a large language model. All scientific concepts and conclusions were generated, verified, and interpreted by the author. Dedicated to Peter Plichta, who envisioned the code before it could be computed.