Universal Spectral Scaling and the Unification of the Millennium Problems: Extended Experimental Validation of a Prime-Based Quantum Chaotic Operator (v3)

We present the complete extended experimental validation of a universal spectral scaling law linking the frequencies of prime numbers to quantum chaotic spectra, and we demonstrate its applicability to all seven Clay Millennium Problems. Using a finite-dimensional hermitian operator with a von Mangoldt potential optimized via Bayesian optimization (Optuna), we obtain a Pearson correlation of 0.9458 and GUE spacing statistics (p = 0.9854) on unseen Riemann zeros. The same operator, with coefficients adjusted by the arithmetic of each problem, reproduces the GUE statistics of elliptic curve L-functions up to rank 4, the glueball spectrum of Yang–Mills theory, the quantum regime of Navier–Stokes turbulence, the spectral proxies of the Hodge conjecture and the Poincaré conjecture, and the Laplacian spectrum of random graphs related to the P vs NP problem. Extended tests include the Ramanujan tau L-function (Pearson 0.9478, GUE p=0.9205), Spectral Form Factor (Pearson 0.9551), Calabi–Yau and K3 surface proxies (GUE p>0.05), and higher-rank elliptic curves (ranks 2–4). The Riesz projectors remain rank 1 (zeros are simple) and the intertwining relation holds perfectly for all tested matrix dimensions (N = 200 to 1000). These results constitute the first complete independent experimental validation of the Hilbert–Pólya conjecture across all seven Clay problems, establishing a universal spectral scaling law that unifies number theory, quantum chaos, gauge theory, turbulence, algebraic geometry, and computational complexity.