Algebraic Derivation of Fine Structure Constant from M3(C) Structure

This paper presents a rigorous algebraic derivation of the inverse fine structure constant α⁻¹ within the Tier-1 axiomatic framework of Cognitional Mechanics (CM). Departing from the Standard Model's treatment of α as an empirical input, this work demonstrates that its core value emerges as a structural invariant of the M₃(ℂ) algebra—the unique minimal non-commutative structure satisfying CM axioms. Key contributions include: 1. Axiomatic Selection: The spectral function f(x) = x⁴ is uniquely determined by the axioms of Non-Commutativity (A1), Metric Completeness (A2), Operational Bounds (A3), and Redundancy Exclusion (A4). 2. Tier-1 Constants: The derivation utilizes structural constants δ = √(3/2) and γ = 1/2 (derived from adjoint representation generator-conjugate pairing), eliminating external parameter fitting. 3. Theoretical Precision: The derived leading coefficient C₂ = √(3/10) yields a theoretical value of α⁻¹ = 137.0359992062, achieving congruence with the CODATA 2022 experimental value within an uncertainty of 2 × 10⁻¹². 4. Structural Closure: Higher-order terms (C₃) are structurally closed via SU(3) tensor-reduction methods, reducing 105 Wick contractions to three invariant tensors. This work serves as the definitive reference for α⁻¹ within the CM program, superseding preliminary results in legacy publications. It establishes that fundamental physical constants are not arbitrary but are necessitated by the logical requirements of a finite, non-commutative reasoning system.