Spectral Completeness of ξ^2 and the Projection Structure of the Standard Model: Invariant Algebra of M3(C) under Axiomatic Closure

This paper establishes two related structural results within the Cognitional Mechanics (CM) framework. Part I proves that the spectral coupling constant ξ² = (μα)² is a complete statistic for the observational content of M₃(ℂ). The state space S is defined generatively as the union of O-orbits of the one-parameter family H_λ = diag(λ, λ, −2λ), where O is the operation algebra derived from axioms A1–A4. Normality of all elements of S holds by construction, eliminating Jordan structure entirely. The residual degree of freedom is λ ∈ ℝ⁺, uniquely recoverable from any state s via λ(s) = √(Tr(s²)/6). The map f(s) = Tr(s²)·(αμ₀)² satisfies f ∝ λ² and induces a bijection S/∼_f ≅ Im(f) of dimension exactly 1: all observational distinctions within M₃(ℂ) collapse to a single scalar. Part II classifies the 19 Standard Model parameters as eigenvalue-type (12), orbit-invariant (4), and basis-mixing (3) projections of Spec(D), and bounds the independent degrees of freedom to at most 11. The connecting corollary establishes that every Standard Model projection map πᵢ is a rational function of ξ² alone, so that the Standard Model parameter structure is a representation layer of the invariant algebra, not an independent ontological structure. Anomaly cancellation and asymptotic freedom are shown to be algebraic necessities of M₃(ℂ) at n = 3. A falsifiable prediction is derived: the ratio αₛ/α_em at the Planck scale equals Φ₃/Φ₆ = 13/7 ≈ 1.857, deviating from the Standard Model perturbative result by approximately 0.24. No free parameters are introduced at any stage. All logical gaps are closed within the CM axiom system. Published on April 26, 2026 doi: 10.5281/zenodo.19784769