Supplemental Mathematical Derivations and Computational Details for the Graviton Condensate Quantum Gravity Framework

This supplementary document provides detailed mathematical proofs, numerical calculations, and extended derivations supporting the main conclusions of the companion paper titled "Defenses, Self-Consistency Reinforcement, and Experimental Compatibility Verification for Core Controversies in the Graviton Condensate Quantum Gravity Framework." It includes five appendices: a complete groupoid-based proof of Lorentz invariance in discrete spacetime, full-energy-scale coupling constant fitting with numerical integration, topological charge conservation derived from groupoid homology, residue theorem treatment of loop integrals for ultraviolet completeness, and renormalization group evolution of core parameters. These technical supplements are essential for verifying the mathematical rigor and computational consistency of the proposed framework.