Classification of subsystems for graded-local nets with trivial superselection structure

We classify Haag-dual Poincar'e covariant subsystems $\text{B}\subset \text{F}$ of a graded-local net $\text{F}$ on 4D Minkowski spacetime which satisfies standard assumptions and has trivial superselection structure. The result applies to the canonical field net ${\text{F}}_{\text{A}}$ of a net $\text{A}$ of local observables satisfying natural assumptions. As a consequence, provided that it has no nontrivial internal symmetries, such an observable net $\text{A}$ is generated by (the abstract versions of) the local energy-momentum tensor density and the observable local gauge currents which appear in the algebraic formulation of the quantum Noether theorem. Moreover, for a net $\text{A}$ of local observables as above, we also classify the Poincar'e covariant local extensions $\text{B}\supset \text{A}$ which preserve the dynamics.