Complete Analysis of the SMaC/UPG Parameter Catalog and Implications for Dark Matter Paradigm

10.5281/zenodo.18406855 We present a complete analysis of the SMaC/UPG Galaxy Parameter Catalog (doi:10.5281/zenodo.18395426), a derivative dataset based on SPARC observations. The analysis reveals a new symmetry in galaxy dynamics that challenges the standard dark matter paradigm. Based on the complete SMaC/UPG derivative parameter catalog for 175 SPARC disk galaxies, we establish the existence of a precise power law: the characteristic radius ℓ is proportional to ρ−0.501±0.011 eff . We demonstrate that this law is not merely an empirical correlation but represents a direct observational consequence of a fundamental scale symmetry in galaxy dynamics – invariance under the nontrivial similarity transformation (r, ρ, M ) → (λr, λ−2ρ, λM ). This symmetry, absent in the Newtonian limit of General Relativity, quantitatively explains rotationcurve anomalies without invoking dark matter and indicates that Einstein’s equation Gμν ∝ Tμν on galactic scales may not be a fundamental law but rather an effectivedescription of the dynamics of a deeper entity – the primary space-matter continuum (SMC/ПМК). The SMaC/UPG catalog establishes new empirical criteria for anycomplete theory of gravity. Keywords: modified gravity, dark matter, scale invariance, SMaC, SPARC, cosmology, space-matter continuum ∗ORCID: 0009-0006-5152-069X,