Lemma: Singularity data do not select a state

Suppose a bosonic physical two-point function $W$ has been constructed, and $v$ is a smooth real solution of its linear field equation. For every $c\ge 0$,

has the same antisymmetric part and singular wavefront data, and is positive whenever $W$ is positive. Distinct $c$'s can therefore give different states with the same stated local singularity condition.

Proof. The addition is a smooth symmetric bisolution. Its pairing with a conjugate test pair is $c|⟨v,f⟩{|}^2\ge 0$. It changes neither the antisymmetric part nor the wavefront set modulo smooth terms.