Standard Model Gauge Dynamics from M3(C) Structure: Semigroup Flow, β-Function, and Coupling Ratios

This work establishes that the gauge dynamics of the Standard Model arise as structural necessities of the algebra M₃(ℂ) under Axioms 1–4 of the Cognitional Mechanics (CM) framework. Rather than assuming gauge groups, couplings, or renormalisation behaviour, the paper derives them from the minimal non‑commutative algebra capable of supporting internal distinction. The spectral space of the Dirac operator D_F is shown to possess a smooth structure without invoking Morse genericity, using only the isolated eigenvalue property and the geometry of the pure‑state space P(A_F) ≅ CP² ⊔ {pt}. The dynamical weight κ is obtained non‑circularly as the unique eigenfunction of the dilation operator D = λ d/dλ, giving κ = c₀ λ² with λ² ≡ ξ² at Tier‑1. A five‑functor chain F₄ ∘ H ∘ G ∘ F₂ ∘ F₁ maps the algebraic structure of M₃(ℂ) to the full gauge/sheaf structure of the Standard Model, with naturality verified at each stage. The Faddeev–Popov operator is proven elliptic independently of D_F, enabling BRST quantisation on the induced spectral manifold. The one‑loop β‑function coefficient b₀ = 11/3 is derived independently in three layers—algebraic (C_A = 3), geometric (scale invariance of a₄), and BRST (ghost pairing)—with n_f = 0 following from the Tier‑1 dissolution of particle ontology. The Standard Model gauge sector is identified as the colimit of a sheaf over spectral patches, and three structural predictions follow without free parameters, including α_s/α_em = 13/7 from the cyclotomic structure Φ₃(3) and Φ₆(3). All results arise internally from the CM axioms, with no Tier‑2 assumptions. This establishes that the Standard Model is not an externally assembled structure but a forced projection of M₃(ℂ) under axiomatic closure. Published on April 28, 2026 doi: 10.5281/zenodo.19833881