Quantum Theory as a Unitary Representation of Distinguishability Symmetries

This technical note shows that quantum theory arises from unitary representations of subgroups of Autδ(X), where δ encodes the primitive notion of distinguishability. Distinguishable states become orthogonal basis elements, superposition follows from the linear structure required to preserve symmetry, and unitary dynamics is generated by one-parameter subgroups of the symmetry group. Measurement corresponds to projection onto invariant subspaces, and the Born rule emerges from the geometry of normalized distinguishability. The resulting formulation reconstructs core quantum behavior without assuming quantum axioms, deriving it instead from symmetry and δ.