THE TWO-SCALE Y-GATE SOLVER. Surface and Orbital Electron Modes: Logarithmic Grid Diagonalization, Metric-Induced Self-Adjoint Hamiltonian, and the 800 MeV Anchoring Debt (XXX v6)

Within the pregeometric FCD-RBv6 framework, leptons are not elementary point-like particles or independent fundamental fields, but surface-localized acoustic Rayleigh resonances bound to the minority-mass endpoint of trans-brane Einstein–Rosen (ER) dipoles. This work resolves the longstanding multiscale problem posed in Part XV: how a single self-adjoint quantum Hamiltonian can simultaneously support both a deep, sub-hadronic throat-confined mode and an extended, delocalized atomic Bohr orbit spanning five orders of spatial magnitude (a ratio of 8.4 × 10⁴). We construct a regularized two-scale radial Hamiltonian on the Morris–Thorne wormhole throat regularized at the constituent Compton floor R₀ ≈ 0.6308 fm and extending to 6 × 10⁵ fm. Implementing a non-uniform logarithmic radial mesh coupled with a metric-induced similarity transformation, we preserve strict Hermiticity across the multiscale domain. Numerical diagonalization yields two widely separated, stable eigenmode branches: (1) a localized throat surface mode at E_surface ≈ -0.511 MeV, representing the electron rest mass as an acoustic Rayleigh boundary excitation; and (2) an extended, delocalized orbital mode at E_orbital ≈ -13.605721 eV, precisely recovering the standard hydrogen 1s Bohr ground state without perturbing macroscopic atomic physics. Furthermore, we provide a critical assessment of the anchoring well depth V₀ ≈ 800 MeV, establishing its status as an a posteriori calibrated parameter and defining the formal theoretical program required for its ab initio derivation from 5D bulk elastodynamics.