Testable Scaled Mathematics: Structured Entropy, Hinges, and Emergent Forces

This preprint presents a minimal, testable mathematical framework for modeling physical systems without invoking point-particle ontology. Instead, the fundamental constituents are treated as non-point microstructures (“hinges”) whose interactions generate stability through a configuration-dependent form of structured (geometric) entropy.The work introduces an explicit entropy functional defined on configuration space, distinct from thermodynamic entropy, which penalizes both collapse and excessive dispersion while anchoring a reference scale. When incorporated into a standard Lagrangian formulation alongside elastic couplings, this structured entropy produces deterministic equations of motion, non-trivial equilibria, and intrinsic stability without fine-tuning.A minimal system of three hinges in three-dimensional space is analyzed as the smallest closed network capable of supporting curvature, internal tension, and non-colinear equilibria. Under coarse-graining and symmetry assumptions, the framework exhibits emergent inverse-square force scaling, demonstrating how long-range behavior can arise from local structured interactions rather than being imposed a priori.The mathematics is explicit, scalable, and directly testable through numerical simulation, allowing for falsifiable predictions regarding stability basins, attractors, and emergent force laws. While not derived from Zero Inversion Theory (ZIT), the framework is compatible with its core philosophy, emphasizing stability through balanced structural tendencies rather than externally imposed constraints.This work is intended as a foundational, exploratory contribution, offering a quantitative alternative to point-based mechanics and opening pathways toward modeling elastic networks, halo-like structures, and vacuum-analog fields using non-point microstructures.