Unified Quantum Gravity Architectural Framework & Appendix A: Geometric Derivation of Alpha

UPDATE (Version 2.0 - February 10, 2026): This upload includes Appendix A, which details the explicit computational derivation of the Fine Structure Constant (α⁻¹) using the proprietary Evrmind HCA-Engine. Key Results in this Version: Identification of the electron's Fundamental Ground State at the (44, 345) Torus Knot configuration. Identification of the Hyper-Fine Harmonic State at the (3601, 3121) Torus Knot configuration. Derivation of the Geometric Impedance Formula: α⁻¹ ≈ Q² / (P · 2π²) Proof of convergence with CODATA 2018 experimental values to a precision of 10⁻⁶. Abstract: The Unified Quantum-Geometric Analytic Framework (UQGAF) proposes that fundamental physical constants are emergent properties of integer knot topology. This paper demonstrates that the electron is physically realizable as a standing wave on a Toroidal Manifold (T³). The Fine Structure Constant is derived as a geometric invariant required to stabilize a toroidal knot with exactly 3,121 longitudinal twists and 3,601 meridional loops.