Equipartition interpretation of the Koide relation: representation-theoretic formulation, covariance test, and neutrino mass predictions

The charged-lepton masses satisfy the Koide relation Q ≡ (Σm)/(Σ√m)² = 2/3 to a relative accuracy of 3×10⁻⁶, without an accepted explanation for four decades. We show that in the variables χᵢ = √mᵢ the relation is exactly equipartition of the norm between the singlet and doublet irreducible representations of the family group S₃, and prove that the unique S₃-invariant potential of degree ≤ 4 enforcing this as a minimum at every scale is V_eq = (n₁ − n₂)². The ratio Q is a renormalization-group quasi-invariant, exact in the on-shell scheme with a closed-form O(α) drift in MS-bar. Interpreting interactions as covariances between modes predicts that deviations from 2/3 grow with the coupling: across fermion triplets |Q − 2/3| rises monotonically over five orders of magnitude, with signs reproduced by a level-repulsion formula and fitted correlations matching C_F·α_s/π in the perturbative regime. Extended to neutrinos through the hyperoctahedral group B₃, the equipartition cone intersects exactly one sign orbit — a single reflection on the lightest mode — yielding, for normal ordering, Σm_ν = 59.16 ± 0.24 meV, m_β = 8.85 meV and m_ββ = 1.3–4.0 meV: a sharply falsifiable set of predictions for JUNO, cosmology and 0νββ searches. The local statistical significance of the relation is ~4.3σ, robust under look-elsewhere accounting, and independently supported by the trials-free 1981 out-of-sample prediction of the τ mass.