SDRIS: The Geometric Origin of Entropy

This paper presents a formal derivation of entropy and the arrow of time within the Static-Dynamic Recursive Information Space (SDRIS). We resolve the fundamental paradox of structural persistence: why localized information states (particles) remain invariant despite the inherent dissipative nature of the recursive vacuum. We demonstrate that entropy is a geometric necessity arising from the topological tortuosity and the accumulation of lattice noise over N=30 recursive layers, with the finite-depth saturation value given by ₃₀ = _ ₍=₁^30 ^-n 0. 001119. We introduce the Diophantine Shield—based on Fermat's theorem on sums of two squares—as a non-linear filter that safeguards integer-resonant states (N=5, 13) from phase decoherence. Furthermore, we model the vacuum's stability via hydrodynamic self-stabilization and the Ouroboros Cycle, identifying the N=6 hexagonal tiling as a zero-entropy sink for cumulative lattice noise. This synthesis provides a unified, generative mechanism for thermodynamics, the arrow of time, and the structural stability of the Standard Model against the dissipative flow of recursive information processing.