SFVdSB/Option_A__Bounce_Lorentz_Violation_Scale_in_the_SFVdSB: Option A: Bounce Lorentz Violation Scale in the SFVdSB

We estimate the photon-sector Lorentz-violation (LV) scale in the SFV/dSB model using Option A: the trusted two-field O(4) bounce (Golden1078) sets the transverse geometry and a bounce-derived gauge-kinetic weight f(y). We enforce the two-sided (Z2) embedding used in the dark-energy (DE) work and use the bounce thick-wall ratio R/w ≃ 3.35 as the only sharpness control (no independent localization-length knob). Solving the Sturm–Liouville problem yields a localized gauge zero-mode and a first excitation m1. Converting to physical units with the brane mapping gives the baseline spectral scale E∗ ≡ min(Eloc, E1) ≈ 2.28 × 1013 GeV. We then "close the loop" at first pass by computing the quadratic dispersion coefficient ξ2 from overlap integrals, reusing the same 1D Gaussian localization machinery developed for quark/lepton masses. This converts the baseline spectral scale into an observable quadratic LV scale, whose value can shift upward or downward depending on the brane-current profile J(y) and tower couplings.