For any three-component Bogoliubov-de Gennes (BdG) system whose quasiparticle gaps form a real Z3-equivariant triplet ∆ k = A+B cos(θ+2πk/3) with all ∆ k ≥ 0, the Koide ratio Q ≡ ∑ k m k /(∑ k √ m k) 2 satisfies the exact identity Q = 1 3 + B 2 /(6A 2), so that within this positive-gap region the empirical charged-lepton value Q = 2/3 is equivalent to |B/A| = √ 2. Outside the positive-gap region the identity fails; a counterexample at (B/A, θ) = (√ 2, π/3) gives Q ≈ 0.4094. The algebraic identity is not new mathematics; it is implicit in Brannen's parametrisation and the broader Koide literature. The contribution of this note is twofold. First, we separate existence from selection: the Koide puzzle reduces to a closed-form theorem on the existence side, leaving two narrower selection questions (Why |B/A| = √ 2? and Why θ = 2/9?). Second, we argue that the BdG framing supplies the physical reason for why a Z 3 symmetry acts on √ m rather than on m: the quasiparticle mass is |∆ k | 2 for an order-parameter gap on which the symmetry naturally acts. Numerical verification against Particle Data Group charged-lepton masses and a fully runnable reproduction script are provided.