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, \, b₁=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 | ^+ (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 E₇, 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 sorry) ; 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 https: //github. com/novaspivack/ugp-physics (papers/24\deeper\ₜheory/; run\ₐll. py reproduces all results in <1 s).