Brane Induced Curvature Mass: Geometric Halos, Galactic Rotation Curves and Relativistic Embedding

This work introduces the Brane Induced Curvature Mass (BICM) model, a geometric framework in which the apparent dark matter in galaxies arises from curvature energy on a tensionful two dimensional brane. A scalar height field h generates an effective surface mass density proportional to its gradient energy, reproducing galactic rotation curves and weak lensing signals without invoking particle dark matter. Baryons deform the brane through a Helmholtz type coupling, creating extended geometric halos with a single screening scale. The model is formulated through a local action, embedded in a relativistic scalar tensor framework and supported by numerical fits to 175 SPARC galaxies, where BICM performs competitively with NFW and MOND. The manuscript also outlines cosmological implications from brane tension and discusses quantum and nonlinear extensions of the theory.