Burnside Orbit Histogram Network (BOHN) and Symmetry-Breaking BOHN (SBOHN): From Group Orbits to Symmetry-Breaking Observables in Machine Learning Complete Documentation with Source Codes and Experimental Results

This work presents the development of the Burnside Orbit Histogram Network (BOHN) and Symmetry-Breaking BOHN (SBOHN), experimental data-representation methods based on group-action orbits, symmetry-breaking observables, and learned permutation transformations. BOHN constructs invariant representations from orbit statistics, whereas SBOHN extends them with information describing deviations from symmetry. The document covers automatic symmetry detection, evolutionary and differentiable permutation learning, Sinkhorn operators, fractal hierarchies, task adaptation, continual learning, and Mixture-of-Experts architectures. It also introduces FBOHN embedded in the RMIG-18 graph and reports experiments on the discovery of symbolic relational operators. The study is exploratory in character. It reports both positive and negative results, stability tests, comparisons with baseline models, identified limitations, and directions for further research. Source code and raw experimental results are included to facilitate verification and reproduction of the reported experiments.