A Structural Constraint Theorem for Stationary Circular Motion: Noether–Metric Cubic Closure, Scale-Free Universality, and Stability Geometry.

This deposit contains the manuscript “A Structural Constraint Theorem for Stationary Circular Motion: Noether–Metric Cubic Closure, Scale-Free Universality, and Stability Geometry.” Under minimal operational hypotheses (stationarity, axial symmetry, existence of circular families, and a local inertial definition of orbital speed $v$), the ratio $f^3/v^3$ is shown to be a purely metric identity in the ZAMO frame, independent of any gravitational field equation. Dimensional closure yields a uniqueness statement: any scale-free dimensionless scalar built solely from $(V, f, v)$ must be a function of the cubic combination $V f^3 / v^3$, with the cubic monomial as the unique power-law representative. In the first-order scale-free regime (no additional independent dimensional parameter beyond $(G, M, r)$ and spherical indistinguishability), isotropic volume scaling enforces $V(r) \propto r^d$ in $d$ spatial dimensions and implies $\Omega^2 \propto M/r^d$. In $d=3$, one recovers Keplerian scaling and obtains a density-controlled tempo $\Omega^2 \propto G \rho$ once $V(r)$ is specified. A no-go theorem proves that any leading-order deviation from $\Omega^2 \propto GM/r^3$ in such stationary circular regimes necessarily introduces at least one new independent scale, providing a structural classifier (“scale tag”) for theory-space deviations. Finally, the dimensionless stability invariant $I_{\rm stab} = \kappa^2/\Omega^2$ isolates a mass-independent stability geometry and vanishes exactly at the Kerr ISCO, providing a geometric marker of stability loss as the vanishing of a normalized radial Hessian.