Remark: Why this is a genuine membership argument

No Green operator was used to define $L$. The proof starts from the explicit Wick square, establishes its joint products, and constructs local affiliated realizations in the already declared net. It does not claim the same result for an interacting theory or a reduced derivative-only net.

Essential self-adjointness of Wick squares is a substantive operator question in more general representations; see Sanders. Our all-real-smearing conclusion uses the special bounded one-particle kernel of the flat free scalar and is not presented as that more general theorem.