Paper CCII: The Koide Quantity K = 2/3 from G2 Root Geometry: A Sharper Z3-Symmetry Proof

We present the sharpest version of the framework's derivation of the Koide formula K = 2/3 for the charged leptons. Paper CLXXXVI established this result as Tier 1 of the broader G2 Degeneracy Theorem. The present paper isolates the Z3 Weyl subgroup argument and gives an explicit construction. The three charged leptons share Klein ladder rung k = 1 (λ1 = 2 − √ 3), occupying weight-space vertices separated by the canonical G2 short-root angle 120◦= 2π/3. Two trigonometric identities valid for any parametrization angle force the Koide quantity exactly to 2/3, independent of lepton mass values. The framework therefore predicts QKoide = 2/3 exactly, robust against future refinements of me, mµ, mτ. Observed value Qobs = 0.666609 matches 2/3 to 0.009%. The next-to-leading correction (from the fbulk- coupled rung-k = 2 mixing) is at the 0.5% level and matches the framework's Tier 2 prediction K = 2/3 + (2fbulk/9 · sin2(δ/2)). Five testable predictions. Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.