Metric Structure from Orbit Classes of Distinguishability-Preserving Symmetries

This short technical note derives metric structure directly from the distinguishability relation δ and the symmetry subgroups of Autδ(X). By decomposing X×X into orbit classes under a chosen subgroup G, we show that distances arise as invariant labels assigned to these orbits. This reconstruction demonstrates that metric geometry is not fundamental but a consequence of distinguishability patterns and symmetry preservation. The result connects δ to Euclidean, Lorentzian, graph-like, and ultrametric geometries through subgroup choice and orbit type assignment.