Path-Dependent Gravitation: Geodesic Levitation and Weight Anisotropy Induced by Spin-Curvature Coupling

The Weak Equivalence Principle (WEP) postulates that the trajectory of a free-falling test body is independent of its internal structure. However, this universality formally applies only to structureless point masses. In this work, we re-examine the dynamics of extended spinning bodies within the Schwarzschild spacetime framework. Starting from the premise that gravity manifests as spacetime curvature---geometrically analogous to a slope---we derive that tangential motion modifies the effective geodesic path of an object.We demonstrate analytically that for a rotating body with angular momentum parallel to the local gravitational field, the geometric factors in the gravitational and inertial sectors of the radial geodesic equation exhibit an exact cancellation at the critical orbital velocity $v_c = \sqrt{GM/R}$. This implies that ``weight''---defined as the force required to deviate an object from its natural geodesic---is not an intrinsic invariant but a dynamic quantity dependent on the geometric alignment between the velocity vector and the spacetime curvature.We extend this finding to the microscopic regime, proposing that polarized atomic nuclei should exhibit weight anisotropy dependent on spin orientation. The thermodynamic consequence---a ``Geometric Buoyancy'' effect causing spontaneous stratification of spin-polarized gases---is derived and formally verified using the Lean~4 theorem prover.