Numerical Framework and Code Architecture

This paper presents the computational and analytical realization of the Sc-Rubs persistence field and examines how threshold resonance governs field coherence and pattern stabilization. Using both symbolic and numerical evaluation, it reconstructs the recursive Laplacian field φ = PF τ(r) and verifies the Law of Persistence through iterative solution of ∇·(|∇φ|^{p−2} ∇φ) − β Δ²φ + ∂Vα/∂φ = 0.

Results demonstrate that stable geometric persistence emerges when the activation threshold α ≈ 0.3 is crossed, producing resonance lock between curvature and diffusion modes. The simulations confirm that roughly 70 % of the system’s residual energy forms the root-mean-square equilibrium value, matching analytical expectations from earlier papers. This state defines the coherence plateau — a regime where form, energy, and recursion achieve dynamic stability.

Computational modules written in Python reproduce the octahedral–spherical–cubic transitions under controlled parameter sweeps (β ≈ 24, 10 ≤ λ ≤ 80), demonstrating convergence toward scalar-field persistence. These results validate the complete Sc-Rubs framework as a unified scalar-field model of self-organization and geometric emergence.

This paper concludes the Sc-Rubs Modelling Series prior to the unified mathematical statement (Paper 8: Unified Scalar-Field Description).
For figures, code fragments, and supporting data, visit https://sc-rubs.cloud.
Related DOI: 10.5281/zenodo.17443937.