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When G₂ symmetry breaks to SU(3), one confined branch closes on an SU(3)₃ topological phase. The charged-lepton mass amplitude is represented by a Z₃-equivariant Bogoliubov–de Gennes operator on the three family labels; the physical masses are m_k = Δ_k². The cyclic closure condition [Δ, S] = 0 forces the mass operator to be circulant, with eigenvalues Δ_k = A + B cos(θ + 2πk/3). The chiral SU(3)₃ modular tensor category and the octonionic G₂/SU(3) Clebsch–Gordan data fix the amplitude ratio B/A = √2; the fundamental conformal weight gives the phase θ = h(3) = 2/9. A branch-normalized scale closure fixes the UV coupling α_{G₂}(M_Pl) = 1/(24π), and the trace-lift normalization gives c_eff = 1/2. One-loop dimensional transmutation then yields m_k = (1/2) M_Pl exp(−9π²/2) [1 + √2 cos(2/9 + 2πk/3)]², reproducing the electron, muon, and tau masses with a common-mode residual of 0.12%. The Koide relation Q = 2/3 follows as an algebraic consequence of |B/A| = √2 on the positive branch. A companion supporting math note with all calculation-level details is included.