The Koide Invariant as an Algebraic Necessity of M3(C) Structure: Scalar–Traceless Norm Balance from the Uniqueness of n=3

The Koide invariant K = (m_e + m_mu + m_tau) / (√m_e + √m_mu + √m_tau)² = 2/3 has remained an unexplained empirical relation for over forty years. Within the Standard Model, the three charged lepton masses enter as independent free parameters of the Yukawa sector, and no structural reason for K = 2/3 exists. This paper establishes K = 2/3 as an algebraic necessity of the unique minimal algebra M₃(ℂ) determined by the Operatiology axioms {A1, A2, A4}, reducing the three apparently independent charged lepton mass parameters of the Standard Model to a single-parameter family by algebraic constraint. The derivation proceeds as follows. Any Hermitian element T in M₃(ℂ) admits the canonical decomposition T = aI + X where X in sl(3,ℂ) is traceless. The Koide invariant reduces exactly to K = 1/3 + ||X||²_HS / (3||aI||²_HS), so K = 2/3 is equivalent to the equal-norm condition ||X||²_HS = ||aI||²_HS. Three structural results of the corpus -- Pid-separation, Spectral Completeness (Inv(O) = R[X²], dim(S/~) = 1), and Gauge Dynamics (gauge-invariant observables = Tr(H²), Tr(H³)) -- jointly force X prop. H, so X = cH for some scalar c. The A4 normalisation constant gamma = 1/Phi₁(n) = 1/(n−1) satisfies the algebraic identity gamma = ||I||²_HS / ||H||²_HS. The A4 canonical normalisation implements W_II = gamma, where W_II := ||aI||² / (||aI||² + ||cH||²) is the fractional scalar-sector weight. At n = 3, gamma = 1/2, which fixes |c|² = a²/2 and gives ||X||²_HS = |c|² · 6 = 3a² = ||aI||²_HS, hence K = 2/3. The phase parameter theta of the standard Koide parametrisation √m_i = M(1 + √2 cos(theta + 2pii/3)) labels points on the circle S¹ of admissible eigenvalue triples; K = 2/3 holds for every point independently of theta, and theta is fixed at the Tier-3 level by measurement. The following code verifies the equal-norm condition and the Koide invariant using CODATA 2022 experimental values. ◆◆◆ [PYTHON VERIFICATION CODE — note: × denotes *, ^ denotes **, and numpy_ denotes np.] ◆◆◆ import numpy as numpy_ import math m_e = 0.51099895 m_mu = 105.6583755 m_tau = 1776.86 lam = numpy_.array([math.sqrt(m_e), math.sqrt(m_mu), math.sqrt(m_tau)]) a = numpy_.mean(lam) eps = lam - a norm_eps_sq = numpy_.sum(eps**2) norm_aI_sq = 3 * a**2 print(f"a = {a:.6f} MeV^(1/2)") print(f"||eps||^2 = {norm_eps_sq:.4f}") print(f"3a^2 = {norm_aI_sq:.4f}") print(f"||eps||^2 / 3a^2 = {norm_eps_sq/norm_aI_sq:.8f}") K = numpy_.sum(lam**2) / numpy_.sum(lam)**2 print(f"K (experimental) = {K:.8f}") print(f"2/3 = {2/3:.8f}") print(f"|K - 2/3| = {abs(K - 2/3):.4e}") print("(Residual attributable to experimental scatter of m_tau, not to the derivation)") print("Tier-2 result: K = 2/3 [no free parameter]") ◆◆◆ [END] ◆◆◆ Published on September 8, 2026 DOI 10.5281/zenodo.22661799