Symmetry-Induced Complexification of Distinguishability Representation

This work proves that, given a minimal real Hilbert representation of a distinguishability kernel and a strongly continuous one-parameter distinguishability symmetry, the representation admits a canonical compatible complex Hilbert structure on the invariant subspace where the symmetry generator acts nontrivially. The complex structure operator is constructed canonically via polar decomposition and functional calculus of the generator and is shown to be representation-intrinsic, transporting covariantly under canonical isometric isomorphisms between minimal realizations. The spectral theorem provides a classification of the invariant subspace into symmetry-invariant sectors, each inheriting a compatible complex Hilbert structure determined by the symmetry. The analysis also establishes a structural limitation: a single symmetry generator does not canonically determine quaternionic Hilbert structure, which requires additional independent symmetry generators.