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 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. Strict parameter accounting: zero continuous free parameters, one binary discrete branch eliminated by settled data at >20σ — a total hypothesis-space cost of one bit, fully discharged before any prediction is evaluated. 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 — notably, NuFIT adopts sin²θ₁₃ = 0.0222 as a fixed input in their solar/KamLAND analyses, coinciding with 1/45 = 0.02222 to the quoted precision. For sin²θ₂₃ = 35/62: Δ = 0.0035 from the independently verifiable analysis (w/o SK-atm, NO) and Δ = 0.095 from the with-SK analysis (NO). The prediction sits within ~0.004 of NuFIT's upper-octant local minimum in both analysis variants — not only the one that favors it globally. For δ_CP = 202.5°: inside the quoted 1σ interval of the with-SK variant. The mass-splitting ratio Δm²₃₁/Δm²₂₁ = 34.03 is evaluated under the same Boundary Lock protocol at 0.6–1.4% from NuFIT 6.0 values, the least precise but honestly reported coordinate in the sector. 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 v2.1: Variant-resolved evaluation replaces cross-variant comparison in v1.0. Muon-Mirror Pivot uniqueness proof and one-bit parameter accounting added. Mass-splitting ratio Boundary Lock added. JUNO θ₁₂ trend acknowledged as live falsification channel. θ₁₃ standard candle coincidence noted.