Compute the global two-branch scattering contribution to weak-MOG detailed balance

Determine the global contribution to the first-order weak-MOG detailed-balance coefficient by solving the massless scalar radial scattering equation for the complex near-horizon coefficients of the asymptotically outgoing mode and evaluating the resulting reflection magnitude, scattering phase, and regular ingoing trajectory integral.

Background

The full detector response is not determined solely by the local Schwarzschild–MOG horizon geometry. A scalar mode that is outgoing at infinity contains both outgoing and ingoing radial-flux branches near the horizon, and their coherent interference contributes to excitation and absorption probabilities. In the weak-MOG expansion, the local detailed-balance correction separates into surface-gravity, trajectory, and finite-gate terms, while an additional global term depends on the two-branch scattering factor.

The unresolved calculation requires solving the scalar radial equation for the scattering coefficients A^{o}_{\nu\alpha} and A^{i}_{\nu\alpha}, or equivalently determining the reflection coefficient and its phase, and incorporating the regular ingoing branch integral. This would supply the global term needed for a complete asymptotic prediction of the detector response.