The Arithmetic Uniqueness of the Standard Model: Asymptotic Sparsity, Galois Structure, and the Positive-Root Theorem

The Universal Generative Principle (UGP) addresses the Standard Model parameter spectrum, classifying each sector by epistemic status~, from a uniquely selected integer seed at ridge level n=10 via a deterministic arithmetic cascade. We investigate whether UGP is itself a shadow of a deeper mathematical structure. Four main results. (1)~The Asymptotic Sparsity Theorem proves the joint arithmetic-admissibility and physical-viability constraint has exactly one solution across all ridge levels: (n=10,b_1=73). The arithmetic component is Lean-certified (unconditional); the physical-viability component uses the CODATA-derived instantiation factor (A/D). (2)~The Positive Root Theorem identifies an exact arithmetic correspondence between the SM bare gauge-coupling numerators and the positive-root counts |Φ^+(G)| of the respective gauge groups; the Chirality Theorem shows the squareness/non-squareness of these numerators is the Lean-certified arithmetic signature of the vector-like/chiral distinction. (3)~Galois structure: the UGP algebraic constants lie in the unique minimal cyclotomic field Q( _120) whose conductor is the lcm of the Coxeter numbers of the SM gauge algebras; algebras with h 120 (notably E_7, h=18) are algebraically impossible to contain in Q( _120) — a structural Tower Law theorem, Lean-certified zero-sorry (Corollary~). (4)~The WZW route is falsified: the Wess-Zumino-Witten construction cannot reproduce the UGP bare coupling rationals, closing a natural alternative-parent hypothesis. Epistemic status: all main arithmetic theorems are A_ Lean (Lean-certified, zero ); physics-bridge interpretations and CODATA-conditioned results are A/D. A front-matter claim-status table lists every result's status explicitly. All computations are reproducible via the open-source scripts at ( /24\_deeper\_theory/; \_all.py reproduces all results in <1~s).