ugp-lean: A Machine-Checked Formalization of the Universal Generative Principle
Nova Spivack · Zenodo (CERN European Organization for Nuclear Research) · 2026
The Universal Generative Principle (UGP) is a deterministic arithmetic framework defined over integer ridges R_n = 2^n - 16. It produces, from first principles and without free parameters, a unique canonical seed---the triple (1,73,823)---whose orbit under the Generative Triple Evolution (GTE) map is rigidly determined. We present ugp-lean, a machine-checked Lean 4 formalization of the UGP/GTE framework comprising 442 modules, all zero sorry. This version reflects the Turing-universality and vacuity remediation (Rounds 98-100 + Phase 1-3 remediation, 2026-07-06): a genuine Turing-universality route via register-machine (Minsky 1967 2-counter machine) simulation is established and machine-certified; a vacuous proof stub (True AND True) previously used for this purpose has been removed; the unsound Cook-independent algebraic universality route has been retracted (NAND functional completeness is a finite Boolean result, not a Turing-universality certificate, and the two routes are now clearly distinguished); 17 additional vacuous/tautological proofs across the library have been corrected or honestly rescoped; the axiom inventory has been audited to 98 disclosed named axioms with a reproducible counting methodology; and a stale Rule-110 theorem name was corrected throughout. This version also corrects a theorem-table attribution (P34 -> P44 reference fix) and updates the formalization count to reflect the intrinsic three-tape area-scaling construction. This version adds full theorem documentation for the three-tape area-scaling modules (zero sorry, zero new axioms), corrects the module count to 442, and fixes a bug in the architecture diagram that was silently hiding three layer nodes (Spacetime, Substrate, Algebra); all layers are now correctly displayed in the diagram. Canonical current state: 442 modules, 98 disclosed named axioms (all named, none hidden), zero sorry throughout, zero standard Lean/Mathlib logical axioms beyond propext/Classical.choice/Quot.sound. All core algebraic uniqueness, structural richness, and computational universality results remain fully machine-certified. Remaining open formalizations (GH convergence, entanglement area law, Page-Wootters Born bridge, cobordism/quark/vertex bridges) are explicitly labeled as open in the paper and repository.