Mathematical Scaffolding and a Structural Toy Model for Equilibria Field Theory

This paper presents a stand-alone technical formalisation of Equilibria Field Theory (EFT), an ontological framework in which the physical universe is grounded in an infinite, atemporal, equilibrium-seeking substrate. Within EFT, spacetime, forces, matter, and dynamical evolution are not taken as fundamental, but are interpreted as emergent descriptions of structured deviation from equilibrium. The purpose of this work is not to propose a completed physical theory, but to provide mathematical scaffolding sufficient to render the EFT ontology precise, internally coherent, and empirically vulnerable. A scalar Purity Field is introduced as the primary order parameter, together with a complexity functional and a universal collapse threshold governing structural stability. A field-theoretic representation is developed to illustrate how mass, inertia, and gravitational behaviour can arise as impedance effects associated with equilibrium gradients. In addition, a simple structural toy model is presented to demonstrate how local capacity conservation, complexity-limited collapse, and repurification processes can give rise to steady-state cosmological behaviour without invoking fundamental time or primordial expansion. Within this framework, black holes are interpreted as functional repurification domains rather than singularities, and accelerated cosmic expansion is treated as a consequence of equilibrium restoration. The mathematical constructions presented here are explicitly representational and incomplete. Their role is to clarify structure, expose constraints, and identify points of empirical sensitivity rather than to close the theory. This document is intended as a technical companion to the broader conceptual development of Equilibria Field Theory and as a foundation for future formal and observational work. This document serves as a technical companion to Equilibria Field Theory II: Ontology and Structural Consequences (Zenodo, 2025). It provides extended mathematical scaffolding and illustrative toy models that support, but do not replace, the ontological development presented there.