Translation Limits in Coarse-Grained Ising Models

This project explores the structural and translational limits in coarse-grained Ising models using a novel theoretical framework integrating structural deviation theory and symbolic translation layers. The findings offer a new perspective on emergent behaviors in discrete physical systems, challenging traditional interpretations of coarse-graining and reversibility. This version includes additional refinements following feedback from initial journal submission and is made publicly available for transparency and further development. This is Version 1.6 of the manuscript. Minor revisions were made after initial journal submissions, including clarification of Section 4.4, reference expansion, and formatting improvements. The core theoretical results and conclusions remain unchanged.