Finite-Chart Riemannian-to-Euclidean Mapping for Galactic Rotation Curves in SPARC Galaxies

We propose an alternative geometric perspective on the dark matter problem.Rather than introducing additional unseen mass, we consider the possibilitythat the observed discrepancies in galactic rotation curves arise from themapping between different geometric descriptions of space. Specifically, we examine the translation between a Riemannian spacetimeframework and an effective Euclidean representation used in the analysis ofgalactic dynamics. We argue that such a mapping requires, in general, theintroduction of a finite boundary scale in order to remain mathematicallywell-defined. Using the SPARC database, we implement this finite-boundary geometricframework and apply it to a large sample of galaxies. The results arepresented through rotation curve reconstructions, empirical scalingrelations, and basic statistical summaries. We find that the model reproduces the overall structure of rotation curvesacross a wide range of systems, with a consistent scaling relation emergingbetween the boundary scale and the galactic radius. While not a completestatistical treatment, the results suggest that geometric effects alone mayaccount for a significant portion of the observed dynamics.