Threshold and Diffusion in Laplace Continuation Fields

c-Rubs: Recursive Bifurcation and Persistence Dynamics

This paper develops the recursive bifurcation basis of the Sc-Rubs model, expanding upon the persistence principle introduced in the archive edition. It demonstrates how form stability and transition arise through self-consistent scalar-field recursion, governed by a Laplace-like constraint and diffusion-driven curvature balance.

The model is expressed through the law of persistence, which links geometric continuity to dissipative field restoration. Numerical and symbolic treatments show that bifurcations follow a predictable route toward symmetry compression — octahedral, spherical, and cubic equilibria appear as stable attractors of the recursion field.

The analysis formalizes the role of threshold damping (α ≈ 0.3) as a rectifier controlling phase inversion, while λ (10 ≤ λ ≤ 80) and β ≈ 24 define truncation and stiffness transitions. Together these parameters illustrate how discrete polyhedral states emerge naturally from continuous field deformation.

For associated images, parameter maps, and supporting materials, visit sc-rubs.cloud.
This paper supplements Sc-Rubs Modelling Archive Edition (ISBN 978-1-919204-09-3) and relates to Sc-Rubs: Unified Description of How Form Holds Together (Zenodo DOI 10.5281/zenodo.17443937).