Comprehensive Mathematical Addendum to the Al-Ani Fabric Theory: Complete Step-by-Step Derivations from the Mother Equation to the Standard Model Spectrum

This comprehensive addendum resolves the structural, mathematical, and physical conceptual gaps in the main Al-Ani Fabric Theory paper. We provide explicit, gapless derivations for the Mother Equation (from maximum entropy), the S-Matrix, the emergence of quantum mechanics, and the fundamental constants. Furthermore, we resolve four deep physical conceptual challenges: the stabilization of the 7th zero mode via bending stiffness, the compatibility of the holistic quantum potential with Bell's inequalities, the consistency of our LIV predictions with LHAASO bounds, and the topological origin of fractional electric charge quantization. We also provide the resolution of the Cosmological Constant Problem, an empirical validation using ALICE heavy-flavor elliptic flow data, and a paradigm shift from continuum fields to rigid topological constraints.