The Spectral Coupling Invariant ξ^2 ≡ (µα)^2 as an Algebraic Necessity of M3(C) Structure: A Structural Theorem from Operatiology and Noology

This paper derives the spectral coupling invariant ξ² ≡ (μα)² in closed form from the axiom system {A1, A2, A4} of Operatiology, completing the Level 2 Class I certification mandated by the companion paper (DOI: 10.5281/zenodo.20577184), which established ξ² as the minimal generator of Inv(𝒪) ≃ ℝ[λ²] via A1 ⇒ su(3)-type ⇒ p₁=0 ⇒ Inv(𝒪)≃ℝ[λ²]. The derivation proceeds through the field ℚ(D₀), where D₀=(δ−1)δγ satisfies 16D₀²−24D₀+3=0, making ℚ(D₀) two-dimensional over ℚ. Since μ and α⁻¹ are elements of ℒ(M₃(ℂ))⊗ℚ(D₀), their ratio ξ=μ/α⁻¹ belongs to this field, and ξ²=(ξₐ+ξᵦD₀)² reduces exactly to A+B·D₀ — the unique mandatory representation forced by the two-dimensionality of ℚ(D₀), itself the algebraic expression of Axiom A4. No other value of ξ² is algebraically possible. The result is ξ² = A + B·D₀ = 179.534632529506…, with A = −1202.60331893592…, B = 10042.5954528296…, zero free parameters, and no reference to individually derived values of μ or α⁻¹. Agreement with benchmark (μ_CODATA2022/α⁻¹_Morel2020)² at +1.085σ, consistent with residuals of the μ (+1.44σ) and α⁻¹ (−0.81σ) derivations. The Morel–Parker 5.5σ discrepancy is identified as structural evidence that α is a Tier-3 projection of ξ², not an independent fundamental constant. The following code verifies the result at 50-digit precision. No value of μ or α⁻¹ is entered; the spectrum {1,1,−2}, dim=3, and π are the only inputs: from mpmath import mp, mpf, pi, sqrt, exp, eulerimport mathmp.dps = 50 h = [mpf(1), mpf(1), mpf(-2)]dim = 3e1 = 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)/(dim - 1)D0 = (delta - 1)*delta*gammagamma_E = eulerPhi1 = mpf(2); Phi3 = mpf(13); Phi6 = mpf(7)ch2 = (e1**2 - 2*e2)/2ch3 = (e1**3 - 3*e1*e2 + 3*e3)/6N2 = mpf(10)/4N3 = mpf(7)/2C2 = sqrt(ch2/N2)*gammaL6 = abs(ch3)/N3L0 = 6*pi**5L2 = Phi3/Phi6L4 = -delta**2/gamma * exp(-gamma_E)L5 = -Phi1/Phi3mu_a = L0 + L2*(-3)/16 + L4*(-99)/256 + L5*(-135)/256mu_b = L2*3/2 + L4*45/16 + L5*981/256ainv_a = 48*pi + L6*(-2943)/4096ainv_b = -32*pi + C2 + L6*2673/512N = ainv_a**2 + ainv_a*ainv_b*mpf(3)/2 + ainv_b**2*mpf(3)/16inv_a = (ainv_a + ainv_b*mpf(3)/2) / Ninv_b = -ainv_b / Nxi_a = mu_a*inv_a - 3*mu_b*inv_b/16xi_b = mu_a*inv_b + mu_b*inv_a + 3*mu_b*inv_b/2A = xi_a**2 - 3*xi_b**2/16B = 2*xi_a*xi_b + 3*xi_b**2/2print("A =", A)print("B =", B)print("xi^2 =", A + B*D0)