Abstract

This paper derives the complete algebraic expansion of the inverse fine structure constant α⁻¹ from the axiom system {A1 non‑commutativity, A2 Π_d‑saturation, A4 redundancy exclusion} of Operatiology, under which the rank‑3 operational closure is uniquely realised as the matrix algebra M₃(ℂ). The expansion α⁻¹ = 6π/D + C₂·D + (Φ₁/Φ₆)·D⁶ closes at exactly three terms with zero free parameters. The normalisation rule Nₖ = Φ₂ₖ(dim)/Φ_{(dim+1)−k}(dim) is proved unique by an internal‑language closure argument invoking no experimental value, discharging the objection that it was fitted to the known value 137. The theoretical value α⁻¹ = 137.035999197 agrees with the highest‑precision single‑experiment determination (Morel et al. 2020, Rb recoil) at 0.81σ. This is a revision and expansion of Version 3.0 (doi: 10.5281/zenodo.18843064), transferring the framework from Cognitional Mechanics to Operatiology and separating the theoretical value from the measured value previously conflated with it. The complete derivation runs from the spectrum {1, 1, −2} alone.◆◆◆ [PYTHON VERIFICATION CODE — note: × denotes * and ^ denotes ** ] ◆◆◆ from mpmath import mp, mpf, pi, sqrt mp.dps = 50 h = [mpf(1), mpf(1), mpf(-2)] e1 = sum(h) e2 = h[0]×h[1] + h[0]×h[2] + h[1]×h[2] e3 = h[0]×h[1]×h[2] delta = sqrt(mpf(3)/2) gamma = mpf(1)/2 D = (delta - 1)×delta×gamma ch2 = (e1^2 - 2×e2)/2 ch3 = (e1^3 - 3×e1×e2 + 3×e3)/6 N2 = mpf(10)/4 N3 = mpf(7)/2 C2 = sqrt(ch2/N2)×gamma f3 = abs(ch3)/N3 term0 = 6×pi/D term1 = C2×D term2 = f3×D^6 alpha_inv = term0 + term1 + term2 print("alpha^-1 =", alpha_inv) ◆◆◆ [END] ◆◆◆ Published on June 2, 2026 DOI 10.5281/zenodo.20511158