Spectral Analysis of Riemann Zeta Zeros and Prime-Modulated Hamiltonians: Towards a Refined Universal Model Framework

We compare spacing statistics of high-order Riemann zeros with spectra of several prime-structured Hamiltonians within the Universal Model Framework (UMF). Using Odlyzko’s dataset (zeros γn, indices 10,000–10,599), we test universality against Gaussian Unitary Ensemble (GUE) benchmarks. Riemann zeros yield ⟨r⟩ = 0.601 ± 0.012, consistent with GUE (0.60266), with pair correlations matching the sine kernel and spectral rigidity Δ₃(L) showing sub-Poisson logarithmic growth. Prime-modulated tight-binding Hamiltonians deviate significantly (⟨r⟩ = 0.543 ± 0.019, KS p < 0.001), while Paley graph Laplacians show closer agreement (⟨r⟩ ∈ [0.584, 0.612]). We define falsifiability criteria for UMF spectral predictions, including spacing-ratio thresholds, long-range Δ₃(L) bounds, and controlled FDR-corrected tests for prime-coded signatures. These results refine UMF operator design, suggesting that correct universality requires not only arithmetic modulation but also symmetry-class enforcement. The work strengthens the link between prime fractal geometry and the spectral universality of Riemann zeros, contributing to both mathematical physics and quantum chaos research. 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.