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 = (mₑ + mₘu + mₜau) / (sqrt (mₑ) + sqrt (mₘu) + sqrt (mₜau) ) 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 Box² 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 Vₑll signature (2, 2). (2) The real 2x2 matrix representation of Vₑll is isomorphic to M (2, R). The tracelessness condition tr (q) = 0, shown here to follow algebraically from ad (I₂) = 0 (a consequence of the Killing form structure rather than an additional assumption), restricts Vₑll 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 eta₀₁ = 3*delta₀₁ - 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 = MLW, takes the form Vₑff = 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 vᵃ = sqrt (mₐ) is established via the on-shell residue condition Zₐ = 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 = MLW. This is Part of a research series. Part I (arXiv: 2512. 16955) establishes the uniqueness of the Box² operator.