{
  "scope": "Nonzero properly infinite separable-predual M; all modular actions on actual supports, including zero weights.",
  "regular_representation": "rho_t f(r)=f(r+t)",
  "carrier_scaling": "Theta_s p(psi)=p(exp(-s) psi)",
  "diagram": "Logical implications, not a metric or geometric model.",
  "proofs": [
    "CI10",
    "CI17–18",
    "CI21–27",
    "CI28–39",
    "OA-FLOW-DS Sections 2–6"
  ],
  "source_context": "M. Takesaki, Theory of Operator Algebras II, XII.4, printed403–405,417–419"
}