Proposition: State underdetermination

The fixed geometry, free wave equation, and stationary scattering matrix do not uniquely determine the outgoing occupation density.

Proof. Keep the metric, wave equation, scattering matrix, and infinity-input vacuum fixed. On any compact positive-frequency band, one admissible incoming horizon state is the Killing-frequency vacuum. For that choice, both incoming number covariances vanish, and (eq:quantumscatter) gives zero outgoing number covariance.

A second choice is the affine horizon vacuum restricted to the exterior half-horizon. Lemmas lem:fourier and lem:horizonoccupation give its positive occupation $n_{\mathrm{H}}(\omega )$, and Proposition prop:packet gives ${\Gamma }_{\ell }(\omega )n_{\mathrm{H}}(\omega )>0$.

Thus the same classical scattering data admit different outgoing occupations. The first choice is not asserted to have the same future horizon regularity as the second. That difference is precisely why a state and its regularity requirements must be part of a derivation.