CAT'S Theory Addendum: Empirical Convergence of the PMNS Matrix, Zero-Parameter Likelihood Evaluation, and the Diophantine Packing Functional

This addendum develops and evaluates a zero-parameter formulation of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix within CAT'S Theory. The framework models stable matter states through torus-knot topology on the S^3 manifold and formalizes a Diophantine packing functional based on scale-indexed best rational approximation. The theoretical construction fixes the four PMNS coordinates a priori as sin^2(theta_12) = 7/22, sin^2(theta_23) = 35/62, sin^2(theta_13) = 1/45, and delta_CP = 9pi/8 = 202.5 degrees, with N_free = 0. An explicit equivalence postulate between topological stability and Diophantine isolation is introduced to formalize the relationship between the geometric configuration space and the resulting observable coordinates. The empirical component defines a no-refit evaluation protocol in which the theoretical coordinates remain fixed while independent experimental likelihoods are evaluated against them. The analysis considers updated global neutrino-oscillation fits and independent solar-neutrino measurements, using numerical residuals and, where applicable, Delta-chi-squared likelihood evaluation. Particular attention is given to the atmospheric mixing coordinate sin^2(theta_23) = 35/62 approximately 0.56452 and its proximity to the updated Normal Ordering global-fit region. The solar and reactor mixing predictions and the CP-violating phase are likewise evaluated as fixed a priori coordinates. The purpose of the addendum is to distinguish mathematical closure, parameter-free prediction, and empirical confirmation within a reproducible falsification framework. The theoretical coordinates are not refitted to the observational data; future measurements therefore remain capable of falsifying the framework.