Paper12Dark matter under B framework

The aim of this paper is to formulate the dark-matter sector of the α-β-φ framework in aform suitable for formal scientific presentation while making its present scope explicit. In thisformulation, dark matter is not reduced either to a point-particle ansatz or to a purely geometriccorrection; it is modeled as an effective gravitating component that inherits its structure fromthe same filtered-vacuum architecture governing the visible sector. The manuscript organizes theradial mass, rotation-curve, and projected-mass observables, summarizes the current confrontationwith SPARC benchmarks including the point-by-point NGC 3198 test and the broader 40-galaxycampaign, and places the halo block in the observational landscape defined by rotation support,gravitational lensing, and large-scale matter mapping. The present version strengthens theformal status of the halo block by identifying an admissible branch, deriving the asymptoticconditions under which the radial family remains physically meaningful, clarifying the preciseobservational claim supported at the current stage, and linking the halo sector coherently to thebroader α-β-φ program.