A Universal Boundary--Bulk Scaling Functional

Many systems exhibit a competition between boundary and bulk structure. We introduce the morphometric functional 'n'. The functional quantifies boundary dominance per effective degree of freedom and applies across convex geometry, fractal systems, porous media, networks, and high-dimensional latent spaces. We establish a scaling theorem characterizing the asymptotic regimes of 'n', and show that classical geometric structures exhibit distinct limiting behaviors. In addition, we derive an extremal ordering between isotropic and axis-aligned geometries under this framework.