Theorem: Normalized scalar Hawking source

For the geometry, field, and state above, $0\le {\Gamma }_{\ell }(\omega ;M)\le 1$, and the outgoing covariance is

For a normalized packet $h$ in one angular sector,

its mean occupation is

With one temporal mode per phase-space cell $2\pi$,

Proof. The real radial potential has conserved Wronskian flux. The scattering matrix relating the two asymptotic channels is unitary. In particular, transmission plus reflection probability equals one, proving the bound on ${\Gamma }_{\ell }$. The usual positive-frequency scattering solutions exist for $\omega >0$: the potential is integrable in $r_{\ast }$, and the corresponding Jost integral equations define the asymptotic solutions.

The outgoing annihilation operator is a linear combination of the incoming infinity mode and the incoming horizon mode. The infinity input contributes zero number occupation, the two inputs have no cross covariance, and Lemma lem:current gives the horizon occupation. Multiplying that occupation by ${\left|t_{\ell }\right|}^2$ proves (eq:outcovariance). Integrating the covariance against the packet profile proves (eq:packetoccupation).

For completeness, partition positive frequency into bins of width $\Delta \omega$. In a bin, the functions

form an orthonormal frequency-window packet family. Consecutive temporal centers are separated by $2\pi /\Delta \omega$. The expected occupation per packet is the bin average of the occupation density. Dividing by that temporal separation gives the bin rate

Summing $m$ and passing to the spectral density gives (eq:sourcerate). This counting convention is part of the rate definition, not an additional thermal assumption.