Emergence Across the Law of Persistence

This paper develops the curvature–diffusion relationship that defines stability in the Sc-Rubs persistence field φ = PF τ(r). It demonstrates how the balance between outward diffusion and inward curvature generates a self-maintaining scalar-field envelope capable of sustaining geometric identity under perturbation.

The study formalizes the equilibrium condition ∇²φ = k(∂φ/∂t)⁻¹ as a persistence law, showing that stability arises from the non-linear coupling of diffusion pressure and curvature resistance. When solved across recursive Laplacian domains, this coupling yields persistent boundary morphologies — notably the cube, octahedron, and dodecahedron — without external constraint or imposed symmetry.

Empirical and simulated data are presented to illustrate how the field transitions between curvature-dominant and diffusion-dominant regimes, exhibiting reversible shape stabilization. The resulting model unifies geometric persistence with energy minimization principles and provides a direct pathway to predicting stable polyhedral configurations across scalar-field continua.

This paper is part of the Sc-Rubs Modelling Series, which investigates how form maintains existence within recursive scalar-field dynamics.
For figures and supporting data, visit https://sc-rubs.cloud.
Related DOI: 10.5281/zenodo.17443937.