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

Abstract

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 S³ manifold and formalizes a Diophantine packing functional based on scale-indexed best rational approximation, mathematically eliminating the capacity for post-hoc calibration. The theoretical construction fixes the four PMNS coordinates a priori as sin²θ₁₂ = 7/22, sin²θ₂₃ = 35/62, sin²θ₁₃ = 1/45, and δ_CP = 9π/8 = 202.5°, with N_free = 0. An explicit equivalence postulate between topological stability and Diophantine isolation formalizes the mapping from the geometric configuration space to observable coordinates. The Muon-Mirror Pivot pairing rule is shown to be structurally unique: the alternative orientation is excluded at >20σ by the solar angle, and the bilinear form JJ* is proven unique under two independent constraints (Proposition 4.4), with the standard Casimir alternative experimentally excluded via δ_CP. The empirical component defines a variant-resolved no-refit evaluation protocol. NuFIT publishes two parallel analyses differing in the treatment of Super-Kamiokande atmospheric data; this paper evaluates all four PMNS coordinates against both variants separately, reporting all residuals including those that disfavor the prediction. For sin²θ₁₂ = 7/22: Z ≃ 0.68σ (SNO+). For sin²θ₁₃ = 1/45: residual Δ ≈ 2.2 × 10⁻⁴. For sin²θ₂₃ = 35/62: Δ = 0.0035 from the independently verifiable analysis (w/o SK-atm, NO) and Δ = 0.095 from the analysis including the SK atmospheric χ² map (with SK-atm, NO). For δ_CP = 202.5°: inside the quoted 1σ interval of the with-SK variant. The prediction sits directly on the upper-octant local minimum that NuFIT identifies as a persistent feature across data releases, with Δχ² < 4 from the global minimum in both analysis variants. The framework makes a falsifiable structural commitment to the upper octant from topological monotonicity of the mirror-knot angular momentum sequence. Resolution of the experimental octant degeneracy by DUNE, Hyper-Kamiokande, and JUNO constitutes the decisive test. The theoretical coordinates are not refitted to observational data; future measurements remain capable of falsifying the framework. Corrected v2.0: Variant-resolved evaluation replaces cross-variant comparison in v1.0. Muon-Mirror Pivot uniqueness proof added. JUNO θ₁₂ trend acknowledged as live falsification channel.