The Dynamics and Universality of the Universal Generative Principle: Attractors 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 (canonical): CatAL machine-checked theorem ( -lean) | CatA computationally certified (SHA-256) | CatB bridge (theorem + stated premise) | [I] interpretive (local label). We first establish the UGP's computational and structural foundations, proving ( ) that its substrate is Turing-universal and ( ) compatible with reversible computing. We then analyze its long-term dynamics using a Renormalization Group (RG) operator, revealing ( ) 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 ( ) that the Logarithmic Complexity Charge Q_4 at seed initialization predicts basin assignment across the four canonical seeds (one-way Analysis of Variance (ANOVA) p ≈ 0), establishing Q_4 as a genuine predictive invariant. Finally, we investigate ( ) the emergence of thermodynamics and establish that coarse-grained entropy systematically decreases as trajectories collapse into basin clusters, with ( ) a Lean-verified non-monotone witness ( \_entropy\_prefix8\_gt\_prefix9), while shuffled controls show monotone increase. These findings establish the UGP as a universal, reversible, and dynamically structured substrate. The Universal Windowed CA (UWCA) step function admits the algebraic characterization p(L,C,R) = C+R-CR-LCR over F_7, machine-certified in Lean~4 ( 110\_z7\_poly\_rep, zero ); when the center cell equals 1, this reduces to the NAND gate, providing a Cook-independent universality certificate~.