Relational Geometric Mechanics VII: Generation Structure — The Koide Formula from the SU(2) Adjoint Well in the Quantum Foam Manifold

This is the seventh paper in the Relational Geometric Mechanics (RGM) series. We derive the Koide formula for the charged lepton masses from the SU(2) adjoint structure of the fermionic wave signature in the quantum foam manifold QF. The charged lepton generations — electron, muon, and tau — are identified as the three stable bound states of a topological well in the generation coordinate of QF. The well potential is fixed by the SU(2) Killing form with no free parameters, producing a critically-flattened quartic well (V''(0) = 0 exactly) that supports exactly three bound states. Projection onto the SU(2) adjoint descriptor subspace gives a 3-site tight-binding Hamiltonian whose hopping amplitude t = √2 is forced by the SU(2) raising operator matrix element. The Brannen parametrization then requires κ = √2, giving the Koide ratio K = 2/3 exactly. Empirical confirmation: κ = 1.41419748 from the measured lepton masses, deviating from √2 by 0.0001%.