CKM and PMNS Predictions with CKM Matrix from a Foundation–Projection Two-Angle Rotation

This appendix derives leading quark- and lepton-mixing observables in MFET from Fibonacci F-symbol matrix elements and the anyonic phase θ*, and supplies a physical derivation of the substrate effective dimensionality that controls the CKM-versus-PMNS scale separation. The leading CKM elements are obtained as Vᵤs = 1/φ³ = 0. 236, Vcb = 0. 04097 (0. 08% from PDG), and the Wolfenstein parameter A = 0. 808 (0. 4%), with Vᵤb partially closed. For lepton mixing, the leading PMNS angle gives sin θ₁₂ ≈ 0. 578 (≈5%). The key addition over earlier versions is the derivation of dₑff^∂ = 1 for the boundary substrate from the information density flowing from bulk to boundary — one boundary dimension is consumed by that flux, leaving one free for braid dynamics — so the substrate asymmetry driving the mixing-scale separation acquires a physical origin rather than being assumed. It also constructs the CKM matrix as a composition of two unitary rotations, VCKM = F (θF) †·P (θ_∂): a "foundation" rotation by an internal angle θF that sets the dominant mixing, followed by a "projection" rotation by a boundary angle θ_∂ that selectively suppresses third-generation amplitudes. The resulting closed forms — Vᵤs = cos θ_∂·sin θF, Vcb = cos θF·sin θ_∂, and Vᵤb = sin θF·sin θ_∂ — reproduce the observed |Vᵤs| ≫ |Vcb| ≫ |Vᵤb| hierarchy from two angles, with CP violation entering through a phase carried by the foundation rotation. The two angles are identified with MFET's bulk (Hausdorff dimension 3) and boundary (spectral dimension 2) substrates, tying the construction to the same substrate distinction used for the gauge and Yukawa sectors. The appendix presents the explicit element-by-element matrix multiplication and the weak-versus-mass basis definitions.