Supported transfer does not transfer a state

For example a flat retained-past shadow may obey

A sufficient ball-laboratory condition is

together with time and regularity reserves. The two bounds protect different requirements CheungXBST,CheungXCST.

Equality of retarded and advanced operations in such a supported window fixes a commutator there, not a symmetric covariance. It does not imply $W_F=W_{\mathrm{flat}}$. The nonzero smooth Wick-square remainder is an explicit counterexample to that inference.

The direct Hadamard singularity is identified by Theorems thm:analytic–thm:hadamard, not by transferring a completion state. No supported-transfer theorem for a different extension is assumed without a corresponding proof.