ugp-lean: A Machine-Checked Formalization of the Universal Generative Principle

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 formalization of this framework in \lean{} 4 against Mathlib, comprising 51 modules with a strict zero-sorry, zero-custom-axiom policy on the core proof path. \textbf{What makes \ugp{} special, and what this paper proves.} The \ugp{} exhibits three distinct, interlocking properties, each of which we formalize and machine-check: (1) Algebraic uniqueness. The ridge sieve, prime-lock criterion, and the \rsuc{} theorem together prove that (1,73,823) is the unique lexicographic minimum of the residual seed class at n=10---forced by the arithmetic, not chosen. The \gte{} update map T is proved to derive the canonical three-step orbit (1,73,823)\to(9,42,1023)\to(5,275,65535) rather than merely stating it. The Quarter-Lock identity k_M = k_{G_2} + \tfrac{1}{4}k_L^2 is an unconditional algebraic invariant. (2) Structural richne