Λ₂ᴀᴋ from G₂ Sump Geometry: Pion and Kaon Masses, the Gell-Mann–Oakes–Renner Relation, and Closure of the Lepton Mass Chain

Paper CXXIV established the G₂ glueball mass spectrum: 7Λ₂ᴀᴋ = 1519 MeV and 11Λ₂ᴀᴋ = 2387 MeV, giving Λ₂ᴀᴋ = 217 MeV. Here we show this agrees with PDG Λ₂ᴀᴋ(MS̅, nᵩ = 3) = 210⁺¹⁴/₋¹⁵ MeV at 0.5σ. We then connect Λ₂ᴀᴋ to the chiral condensate parameter B₀ via the G₂ root count: B₀ = nᵣₒₒₜₛ(G₂) ×Λ₂ᴀᴋ = 12 × 217 = 2604 MeV. From the Gell-Mann–Oakes–Renner (GOR) relation and framework-derived quark masses (Paper CXXV), we predict mπ⁰ = 133.2 MeV (−1.3 % from PDG 134.976 MeV) and mᴺ⁺ = 493.7 MeV (PDG). These pion masses then close the lepton mass chain from Paper CIV: mₑ = (αmπ⁰/2)(1 + 2α√7) = 0.5050 MeV (−1.1 % from PDG) and mμ = mπ⁺√(3/5) = 106.4 MeV (+0.7 % from PDG), both now fully derived with zero free parameters beyond Λ₂ᴀᴋ. The K⁺ mass is predicted at 504 MeV (PDG 493.7 MeV, +2.1 %) at leading-order GOR; next-order chiral perturbation theory corrections are identified. 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.