Emergence of Mass from Scale-Dependent Geometry

Building on a A Scale-Relative Model of SpaceTime[8], we propose that mass arises not as a fundamental source term, but as a scale-localised extremum of a geometric flow. Within a Scale Flow Equation governing the evolution of geometry under coarse-graining, such extrema persist over a finite interval of scales and are stabilised by spectral isolation from global geometric modes. Using Sobolev embedding and spectral arguments, we show that the existence of this encapsulation window depends sensitively on spatial dimension, with three dimensions representing a critical case in which localisation, stability, and finite renormalisation can coexist. Linearised fluctuations about these extrema obey a Klein–Gordon equation, where the effective mass emerges from the integrated second variation of the scale functional rather than from a fundamental parameter. The associated Compton wavelength appears naturally as the limiting localisation scale at the boundary of spectral isolation. Within this regime, coupling to global curvature is suppressed, while gravitational interaction arises as a consequence of scale-dependent delocalisation beyond the encapsulation window. This framework recasts mass, renormalisation, and gravitation as scale-dependent phases of a unified geometric flow, providing a geometric interpretation of mass and a dynamical perspective on the role of spatial dimension