The Dynamics and Universality of the Universal Generative Principle: Attractors, Holography, and Emergent Thermodynamics

The Universal Generative Principle (UGP) is a deterministic, computable framework proposed as a candidate for a theory of everything. While its applications in deriving physical constants have been explored, the fundamental properties of the UGP as a dynamical system have not been systematically characterized. In this paper, we present the results of a comprehensive computational investigation into the UGP's core machinery. Claim types: [T] machine-checked theorem (ugp-lean) | [C] computationally certified (SHA-256) | [B] bridge (theorem + stated premise) | [I] interpretive. We first establish the UGP's computational and structural foundations, proving [T] that its substrate is Turing-universal and [T] compatible with reversible computing. We then analyze its long-term dynamics using a Renormalization Group (RG) operator, revealing [C] that the system is not chaotic but is governed by three observed basin clusters (A, B, C) in a 98-seed deep-trajectory survey. We show [C] that the Logarithmic Complexity Charge Q_4 at seed initialization perfectly predicts basin assignment across the four canonical seeds (one-way ANOVA p < 10^-4), establishing Q_4 as a genuine predictive invariant. Finally, we investigate [C] the emergence of thermodynamics and establish that coarse-grained entropy systematically decreases as trajectories collapse into basin clusters, with [T] a Lean-verified non-monotone witness (gte\_entropy\_prefix8\_gt\_prefix9), while shuffled controls show monotone increase. These findings establish the UGP as a universal, reversible, and dynamically structured substrate.