Tidal Deformations and Analytical Derivation of Surface Gravity for Deformed Horizons

We present an analytical derivation of the horizon-averaged surface gravity κ(τ ) for anaxisymmetrically deformed topological core under the horizon within an adopted metric de-formation model. By evaluating a quadrupolar perturbation governed by the Legendre poly-nomial Y20(θ), we compute the explicit distortion of the interior metric tensor in the vicinityof the transition surface. We demonstrate that the linear contributions of the deformationparameter τ to the surface potential cancel identically under global angular integration overthe compact boundary manifold due to spherical orthogonality. By incorporating relativis-tic causality, we show that the propagation of elastic space-time deformations is boundedby the speed of light c. Within the adopted deformation framework, we analyze how theretardation of potential lines across the stretched manifold fields a geometric modificationof the horizon-averaged potential gradient, demonstrating that the horizon-averaged surfacegravity exhibits a negative quadratic scaling, κ(τ ) = κ0(1 − βτ 2), where β > 0. This nega-tive sign is structurally consistent with the geometric stretching of the horizon, closing thethermodynamic consistency loop of entropy-driven cosmogenesis.