Proposed Theoretical Framework for Boundary‑Defined Field Persistence and Emergent Geometry

This Analytical Edition develops the Laplace–biharmonic formulation of the Sc-Rubs persistence field, completing the proof structure introduced in Sc-Rubs: Unified Description of How Form Holds Together (Zenodo DOI 10.5281/zenodo.17443937).
The paper presents a self-contained theoretical framework in which boundary-defined persistence drives geometric emergence from scalar-field dynamics.

Building on variational and Laplace-continuation principles, the model introduces a resource-extended operator form

L0[ψ]+R[ψ]=0,\mathcal{L}_0[\psi] + \mathcal{R}[\psi] = 0,L0[ψ]+R[ψ]=0,

linking arithmetic, energetic, and geometric domains through a unified persistence law. Section 11 compares this formulation with Kyungu’s discrete-to-continuous Laplace inversion and Lutz’s quantum-thermodynamic correlation balance, showing that all three can be viewed as specific instances of the same operator–resource structure.

The manuscript defines canonical parameters ( β, λ, α, p ) governing field stiffness, truncation, and rectification; derives the equilibrium condition for emergent polyhedral morphology; and formalises the transition between smooth and discrete equilibrium states.
Results demonstrate how boundary persistence and rectifier coupling reproduce stable polyhedral geometries (octahedral to cubic) as natural equilibria of the scalar field.

All comparative equations are cited for theoretical context only; no external collaboration or co-authorship is implied.