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 non-commutativity, A2 Π_d-saturation, A4 redundancy exclusion} of Operatiology, completing the Level 2 Class I certification mandated by the companion paper (DOI: 10.5281/zenodo.20577184). 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 each elements of L(M₃(ℂ)) with coefficients in Q(D₀), their ratio ξ=μ/α⁻¹ belongs to this field, and ξ²=(ξ_a+ξ_b·D₀)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. 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 complete derivation runs from the spectrum {1,1,−2} alone. ◆◆◆ [PYTHON VERIFICATION CODE — note: × denotes * and ^ denotes **] ◆◆◆ from mpmath import mp, mpf, pi, sqrt, exp, euler import math mp.dps = 50 h = [mpf(1), mpf(1), mpf(-2)] dim = 3 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)/(dim-1) D0 = (delta-1)×delta×gamma gamma_E = euler Phi1 = mpf(2); Phi3 = mpf(13); Phi6 = mpf(7) 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 L6 = abs(ch3)/N3 L0 = 6×pi^5 L2 = Phi3/Phi6 L4 = -delta^2/gamma × exp(-gamma_E) L5 = -Phi1/Phi3 mu_a = L0 + L2×(-3)/16 + L4×(-99)/256 + L5×(-135)/256 mu_b = L2×3/2 + L4×45/16 + L5×981/256 ainv_a = 48×pi + L6×(-2943)/4096 ainv_b = -32×pi + C2 + L6×2673/512 N = ainv_a^2 + ainv_a×ainv_b×mpf(3)/2 + ainv_b^2×mpf(3)/16 inv_a = (ainv_a + ainv_b×mpf(3)/2) / N inv_b = -ainv_b / N xi_a = mu_a×inv_a - 3×mu_b×inv_b/16 xi_b = mu_a×inv_b + mu_b×inv_a + 3×mu_b×inv_b/2 A = xi_a^2 - 3×xi_b^2/16 B = 2×xi_a×xi_b + 3×xi_b^2/2 print("A =", A) print("B =", B) print("xi^2 =", A + B×D0) ◆◆◆ [END] ◆◆◆ Published on June 7, 2026 DOI 10.5281/zenodo.20581357