Foundations of Physical Law from a Binary Distinguishability Relation

We develop a reconstruction of physical law from a single primitive: a binary distinguishability relation δ : X × X → {0, 1}, where δ(x,y)=0 iff x=y. All structure—geometric, dynamical, informational, and physical—is derived from the automorphism group Autδ(X), the set of all bijections on X that preserve δ. This framework yields a minimal, symmetry-based foundation from which metric form, dynamical evolution, conservation principles, and field-theoretic constructs emerge. The resulting formulation demonstrates that the fundamental content of physics can be reconstructed entirely from distinguishability and its symmetry-preserving transformations.