Classical Common-Envelope 𝑈(1)_𝐺 Charge-Three Body Factor in the Axis Model: Continuum Existence, Hamiltonian Stability, and Branch-Resolved Fragmentation

This work establishes the classical common-envelope charge-three body factor connecting the localized Axis Parent cell to downstream matter-carrier and fermion interfaces. Computer-assisted continuum certificates establish exact localization, linear Hamiltonian stability, a classical Poincare orbit, and energetic separation from the certified localized one-plus-two and three-one-cell fragmentation branches. Combined with the conditional three-frame singlet, the result provides a stable classical common substrate compatible with downstream charged- and neutral-lepton constructions. Quantum particle existence, spin/statistics, electroweak component resolution, electromagnetic identity, and flavor remain open interfaces. A complete reproducibility archive accompanies the manuscript.