The Fine Structure Constant as an Algebraic Necessity of M3(C) Structure: A Structural Theorem from Operatiology and Noology

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: from mpmath import mp, mpf, pi, sqrtmp.dps = 50h = [mpf(1), mpf(1), mpf(-2)] # Cartan generator spectrume1 = sum(h) # 0e2 = h[0]*h[1] + h[0]*h[2] + h[1]*h[2] # -3e3 = h[0]*h[1]*h[2] # -2delta = sqrt(mpf(3)/2) # sqrt(|e2/e3|)gamma = mpf(1)/2 # 1/Phi_1(dim) = 1/(dim-1)D = (delta - 1)*delta*gammach2 = (e1**2 - 2*e2)/2 # 3ch3 = (e1**3 - 3*e1*e2 + 3*e3)/6 # -1N2 = mpf(10)/4 # Phi_4(3)/Phi_2(3) = 5/2N3 = mpf(7)/2 # Phi_6(3)/Phi_1(3) = 7/2C2 = sqrt(ch2/N2)*gamma # sqrt(6/5)/2f3 = abs(ch3)/N3 # 2/7 = Phi_1/Phi_6term0 = 6*pi/Dterm1 = C2*Dterm2 = f3*D**6alpha_inv = term0 + term1 + term2print("6pi/D =", term0)print("C2*D =", term1)print("(Phi1/Phi6)*D^6 =", term2)print("alpha^-1 =", alpha_inv)