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 $\tau$ and the accumulation of lattice noise $\eta$ over $N=30$ recursive layers, with the finite-depth saturation value given by $\eta_{30} = \delta_\tau \sum_{n=1}^{30} \Phi^{-n} \approx 0.001119$. 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.