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 Πd‑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 (₃₈₌+₁) −₊ (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 = h0*h1 + h0*h2 + h1*h2 # -3e3 = h0*h1*h2 # -2delta = sqrt (mpf (3) /2) # sqrt (|e2/e3|) gamma = mpf (1) /2 # 1/Phi₁ (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 # Phi₄ (3) /Phi₂ (3) = 5/2N3 = mpf (7) /2 # Phi₆ (3) /Phi₁ (3) = 7/2C2 = sqrt (ch2/N2) *gamma # sqrt (6/5) /2f3 = abs (ch3) /N3 # 2/7 = Phi₁/Phi₆term0 = 6*pi/Dterm1 = C2*Dterm2 = f3*D**6alphaᵢnv = term0 + term1 + term2print ("6pi/D =", term0) print ("C2*D =", term1) print (" (Phi1/Phi6) *D⁶ =", term2) print ("alpha^-1 =", alphaᵢnv)