Foundations of Recursive Scalar Persistence

Paper 1 of the Sc-Rubs Modelling Series introduces the foundational scalar-field mechanism underlying the model. It defines the Law of Persistence as a balance between diffusion, curvature, and rectification within a Laplacian membrane field. This paper establishes the conditions for spontaneous geometric stability leading to octahedral, spherical, and cubic emergence. The work forms the theoretical groundwork for subsequent papers in the Sc-Rubs Modelling Series and complements the unified formulation archived on Zenodo (DOI 10.5281/zenodo.17443937).