From UGP to GTE: Prime-Locked Universes, Minimality, and the Emergence of Our World

We develop the Universal Generative Principle (UGP), a deterministic and parameter-free arithmetic framework proposed as a foundational model for generating complex, structured universes. The UGP is governed by a duality between a rigid algebraic law and a universal computational capacity whose interplay strongly constrains the physically relevant trajectories in the parameter space. The first pillar is algebraic rigidity. We prove the Kernel Symmetry Theorem: all lawful dynamics are constrained by the Quarter-Lock relation (k_ = k_ + 14k_), a codimension-1 constraint on all possible physical evolutions. The second pillar is computational universality. We construct a Universal Windowed Cellular Automaton (UWCA) on the UGP's arithmetic substrate and prove it is Turing-universal via a direct simulation of Rule 110, establishing the UGP as a universal computational medium. The resolution of this Necessity-Contingency duality is the paper's core result. At operational level n=10, the algebraic laws filter the universal substrate's computational possibilities: this process isolates a unique lexicographically minimal mirror-dual seed and canonical three-step orbit, (1, 73, 823) (9, 42, 1023) (5, 275, 65535), with zero free parameters at prediction time (see the DOF ledger in Appendix app: dof). The derivation includes a proof of Fibonacci lift rigidity, forcing the quotient gap |q₂-q₁|=13 = F₁3/F₁2 to equal the 13th Fibonacci number's ratio. The argument that identifies this as the unique law-compatible world-seed under the full PSC\, +\, mirror\, +\, MDL package is given in the companion paper SpivackUGPUniqueness. We further establish: the state space is conjectured to form a classifying topos for the geometric theory of survivors; FO-decidability holds on finite windows with ⁰₁-completeness for general reachability; and self-reference is established via Lawvere's fixed-point theorem and Kleene's recursion theorem. All core theorems are formalized and machine-checked in the ugp-lean artifact SpivackUGPFormalization, ugp-lean. These results present the UGP as a self-contained framework where a fundamental algebraic law selects a canonical minimal program from a universal computational medium, offering a potential path toward a parameter-free description of reality.