Binary Emergent Gravity — Appendix A: Formal Derivation of the Entropic Cross-Term and the Three-Zone Structure

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.