Equivalence between Sole--$\chi$ TS\texorpdfstring{$=1$}{=1} Dynamics and the Schwarzschild Geodesic Equation

This technical note establishes, in a fully explicit manner, the equivalence between the radial dynamics obtained from the Sole--$\chi$ gravity sector with a TS$=1$ metric and the standard radial geodesic equation in the Schwarzschild spacetime.

The aim is twofold.

First, using only the notation and assumptions already fixed in Chapters G0, G1, and G2, we derive the Sole--$\chi$ radial acceleration law in a form that keeps the Lorentz factor $\gamma(V_r)$ explicit as long as possible.

Second, we independently derive the corresponding radial geodesic equation in general relativity for the same metric class and show that, under static and spherically symmetric vacuum conditions, the two results match term by term.

The analysis is independent of whether one adopts the Einstein field equations themselves as axioms; instead, it focuses on how the choice of metric, together with the definition of inertial and energy constants, determines the equations of motion.

The range of validity and the limitations of the comparison are stated explicitly so as to avoid misinterpretations by referees or critical readers.