Probabilty Space: Coordinated Structure in Galaxy Rotation Curve Extensions: An Empirical Validation

This work presents an empirical validation of coordinated structure in two-parameter radial extensions to baryonic galaxy rotation curves. Building on Krager (2025) (https: //doi. org/10. 5281/zenodo. 17794870), which introduced a structural reformulation of gravitational behavior at galaxy scale, the present study tests whether the fitted amplitude parameter (Λₑff) and transition scale (r₀) behave as independent halo-like degrees of freedom or exhibit systematic coordination across galaxies. Using the SPARC database and tightened quality-selection criteria, we find: • Λₑff scales systematically with central baryonic structure proxies. • r₀ tracks structural transition radii in the residual velocity-squared profile. • When expressed in dimensionless form, the combined parameters exhibit reduced effective dimensionality, clustering along a constrained manifold in projection space. Robustness tests across multiple selection regimes confirm that the observed dimensional reduction persists under varied thresholds. These results support the interpretation that the extension behaves as a coordinated deformation of baryonic dynamics rather than an arbitrary additive halo component. The study establishes the empirical foundation for further development of a broader structural gravitational program.