Quantitative Tests of S-Field Topological Dark Matter: Rotation Curves and the Small-Scale Crises

We provide a definitive quantitative resolution to the "cusp-core crisis" of Cold Dark Matter (CDM) models using the Spatial Substrate S-Field (S-field) theory. In this framework, dark matter is reconstructed as topological distortions of the space substrate, eliminating the need for unknown particles. By systematically fitting high-resolution rotation curves from the SPARC database across three distinct galaxy types (LSB, HSB, and Giants), we demonstrate that the S-field model naturally yields a smooth density core purely through topological relaxation, without invoking ad-hoc baryonic feedback. Furthermore, in a strictly fair comparison (4-parameter vs. 4-parameter), the model achieves excellent fits (chi^2_nu ~ 1 for most galaxies; slightly higher residuals in extreme dwarfs due to non-circular motions, yet still successfully capturing the core profile) and shows highly competitive Bayesian Information Criterion (BIC) performance. Most importantly, through a rigorous field-theoretical derivation from the S-field action (utilizing Derrick's theorem and scale transformations), we analytically derive the Baryonic Tully-Fisher Relation (BTFR) for the first time in a purely geometric framework. The theoretical prediction is perfectly consistent with the observed fitted slope of 0.86. Our results provide a purely geometric and unified picture for the nature of dark matter.