Corrected Mathematical Framework for Structured Entropy and Hinge-Based Microdynamics

This repository presents a corrected and fully specified mathematical framework for modeling non-point microstructures as interacting hinges governed by elastic constraints and configurational (geometric) entropy.The work resolves algebraic errors, derivational gaps, and interpretational ambiguities present in earlier formulations by providing explicit variational derivations of the governing force laws. In particular, the entropy-derived interaction is shown to scale inversely with separation distance, correcting a prior omission that led to unphysical constant-magnitude forces. Earlier claims of inverse-square emergent behavior under coarse-graining are formally withdrawn; the correct large-scale behavior is identified as a logarithmic effective potential producing a 1/r1/r1/r force.The framework is analytically tractable and demonstrated using a minimal three-hinge system, which exhibits stable, finite equilibria. The entropy employed is configurational rather than thermodynamic, encoding geometric degrees of freedom in configuration space and acting as a dynamical constraint analogous to entropic forces in soft-matter and polymer physics.This work provides a minimal, internally consistent, and testable foundation for structured, scale-dependent dynamics beyond point-particle idealizations. The model is suitable for numerical simulation and further theoretical extension.