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 Q₄ at seed initialization predicts basin assignment across the four canonical seeds (one-way Analysis of Variance (ANOVA) p ≈ 0), establishing Q₄ 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 F₇, machine-certified in Lean~4 (110\ᵦ7\ₚoly\ᵣep, zero) ; when the center cell equals 1, this reduces to the NAND gate, providing a Cook-independent universality certificate~.