2.2 Fixed Point and Derivation of α⁻¹

This article derives the fine-structure constant α⁻¹ from the coding framework of Conjugate Spectral Geometry. The observer's position in the Birkhoff simplex is directly parameterized by the coding weights σ* = (σ_M, σ_C, σ_I) — σ* is the fixed point of the Birkhoff mixing matrix M_5 = T·diag(χ). The algebraic structure of the coding framework yields the zeroth-order coding weights σ₀ = (0.777912, 0.210603, 0.011484), determined by the structure constants {Λ=3, k₀=2, ΔΘ=5} and χ = (8, 9, 1) via Cramer's rule. Bott period echoes introduce higher-order corrections, shifting the zeroth-order σ₀ to the physical coding weights σ* = (0.777428, 0.211091, 0.011481). The algebraic action S(σ*) at σ* takes the value S = 137.036, consistent with the experimentally measured α⁻¹ = 137.035999084 (deviation ≈ 4×10⁻⁶).