Finite Euclidean Mapping Framework (FEMF) - A Finite Euclidean Representation of General Relativity

We propose an alternative geometric perspective on the dark matter problem. In the mathematical formulation of General Relativity, spacetime is described as a Riemannian manifold, while observational measurements of galactic dynamics are interpreted within an effectively Euclidean framework. From a mathematical standpoint, a mapping from a Riemannian manifold to a Euclidean representation cannot, in general, be consistently defined when the boundary is taken to be infinite. Instead, such a mapping requires the introduction of a finite boundary scale. We argue that the implicit assumption of an unbounded Euclidean representation in observational analysis may reflect a long-standing inconsistency in the interpretation of velocities within General Relativity. In the observational framework, we implement a mathematical modulation in which the galactic center serves as a common anchor between the Riemannian description and the Euclidean representation. A finite boundary is then introduced, which deforms the Euclidean representation through a radial mapping. We test this framework empirically using the SPARC database. We present basic statistical diagnostics together with sixteen representative rotation-curve reconstructions. In our view, these results show a consistent and visually strong agreement between the model and the observed data across a wide range of systems. Furthermore, a robust scaling relation emerges between the boundary scale and the galactic radius R(B) ≈ 1.5 R(gal), with a correlation coefficient of r = 0.94). These findings suggest that part of the observed discrepancy in galactic dynamics may originate from the geometric mapping between the underlying Riemannian description and its Euclidean observational representation.