Theorem: Exact induced sine effect

For a centered quasifree independent probe state ${\sigma }_{\psi }$,

Its induced sine effect is

Proof. For a future-supported test vector $H$, advanced comparison gives the scattering-test transformation

Green's identity, evaluated in a past Cauchy region where the theories agree, verifies the transformation. The conserved coupled symplectic current shows that it is symplectic and therefore induces the Weyl scattering morphism.

For $H=(0,z/ϵ)$, its components are exactly $B_{ϵ},z_{ϵ}$. The two fields are separate tensor factors, so no mixed Weyl phase occurs. The centered probe characteristic function is real and nonnegative. Taking the imaginary part after applying $\mathrm{id}\otimes {\sigma }_{\psi }$ gives (eq:inducedsine). Thus the sine identity is not inferred by using a cosine probability for the opposite source signs.