Proper-Time Extremisation on Discrete Complexes and the Geodesic Equation

We show that the discrete tick-cell construction of Paper I [1] enforces extremal proper time for worldlines. A worldline is a chain of tick cells. The condition that the total proper-time cost be stationary under endpoint-preserving deformations yields discrete Euler-Lagrange equations. The continuum limit recovers the geodesic equation. Symmetries of the background yield conserved charges via Noether's theorem: time-translation invariance gives energy; axisymmetry gives angular momentum. We identify the weak-field, slow-motion limit as a Newtonian readout map, where the gravitational potential and emerge as projections of tick extremisation onto the non-relativistic regime. No field equations are invoked. This is the kinematics of free motion in a curved background whose geometry was fixed by closure in Paper I.