Invariant Spherical Mechanics: A Geometric Closure of Orbital Structure with Radial Projection

This work presents an extended formulation of Invariant Spherical Mechanics (ISM), a cycle-based geometric description of orbital motion. The framework characterizes orbital structure using a system-level circumferential scaling constant and a body-level cycle capacity, enabling mean orbital properties to be expressed without additional dynamical assumptions. A simple empirical relation is identified for radial motion: v_r = (2/π) ε v linking radial rate to tangential velocity through a dimensionless deformation parameter. The factor 2/π admits a geometric interpretation as a half-cycle averaged projection, suggesting that radial variation emerges as a cycle-integrated effect rather than an independent dynamical component. Together, these results provide a compact geometric–kinematic closure of orbital structure and observable rates within a cycle-based formulation.