Spectral Completeness of ξ² 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/αₑm 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.