Statistical Derivation of the Horizon–Instanton Relation in the Geometric Entropy Principle Framework

This paper establishes the relation between the instanton action and the number of de Sitter horizon microstates, S_inst ~ ln N, which is the central result underlying the energy scale hierarchy in the Geometric Entropy Principle framework. The derivation proceeds through three independent arguments: an entropic equilibrium condition on the finite Hilbert space of the horizon, a holographic cell volume argument derived from the Bekenstein-Hawking entropy, and an evaluation of the instanton moduli space prefactor showing it is subleading. The convergence of all three arguments supports the relation as a structural property of the framework and yields an order-of-magnitude estimate of the sigma-model coupling g² ~ 0.28.