Explicit Derivations and Numerical Verifications of Core Components in the S²-11DM²ET-X Model

This document presents the complete, explicit derivations, rigorous proofs, and numerical verifications of the core components of the S²-11DM²ET-X unification framework. All results are derived parameter-free from the axiom of exactly three fermion generations via the Hampton Qutrit Collatz Convergence (HQCC) theorem. The components detailed herein include the HQCC theorem (with applications in quantum computing), qutrit flux quantization, Π_brane projection operator, charged fermion and neutrino mass matrices (full 3×3 forms, eigenvalue calculations, PMNS eigenvalues/mixing, and CKM eigenvalue/mixing derivation), expanded G₂-manifold topology (Joyce constructions, Calabi–Yau flux threading proofs, moduli stabilization, and G₂ holonomy applications), G₂ 3-cycle windings, sub-harmonic assignments, and tower partition.