Formal mathematical derivation supplementing BEG Articles 1 and 2. Contains: (A.1) derivation of the cross-term ε₁₂ from the superposition of displacement fields in de Sitter space; (A.2) proof of the three-zone structure and transition function η(r); (A.3) derivation of the holographic invariant α × n = ln(L_H/d_bin); (A.4) proof of the invariance theorem establishing that the Chae velocity statistic is insensitive to radial potential modifications; (A.5) dimensional analysis and consistency checks. Appendix A to BEG Articles 1 and 2.