The Koide Relation as a Null Condition on the sl(2, R) Killing Cone: Derivation from Lee–Wick Vacuum Stability

We present a derivation of the Koide lepton mass relation Q = (m_e + m_mu + m_tau) / (sqrt(m_e) + sqrt(m_mu) + sqrt(m_tau))^2 = 2/3 from the requirement of vacuum stability in the higher-derivative Lee-Wick (LW) extension of the Standard Model. The derivation proceeds through the following logical chain: (1) The Box^2 operator, proven unique for universal vacuum-energy cancellation in a companion paper (arXiv:2512.16955), introduces a LW partner with opposite Krein-space norm, giving the lepton internal space V_ell signature (2,2). (2) The real 2x2 matrix representation of V_ell is isomorphic to M(2,R). The tracelessness condition tr(q) = 0, shown here to follow algebraically from ad(I_2) = 0 (a consequence of the Killing form structure rather than an additional assumption), restricts V_ell to a 3- dimensional subspace isomorphic to sl(2,R), naturally accommodating three lepton generations. (3) The Killing form on sl(2,R) has signature (2,1) and coincides, up to a positive overall factor, with the Koide metric etaab = 3*deltaab - 2. The value Q = 2/3 is shown to be a topological invariant of the (2,1) signature alone. (4) The Lee-Wick Coleman-Weinberg effective potential, evaluated at the natural UV/IR matching point mu = M_LW, takes the form V_eff = g * [kappa(v,v)]^2 with g > 0. Its global minimum is therefore the null cone kappa(v,v) = 0, which is equivalent to the Koide relation. (5) The coordinate identification v^a = sqrt(m_a) is established via the on-shell residue condition Z_a = 1 and a non-relativistic reduction argument, with the overall proportionality constant shown to cancel from Q. Each step is explicitly labeled as a Theorem [T], Proposition [P], or open problem [A]. The one remaining open problem is the two-loop RGE stability of the matching condition mu = M_LW. This is Part of a research series. Part I (arXiv:2512.16955) establishes the uniqueness of the Box^2 operator.