Conditional asymptotic occupation in a supplied collapse channel

Fix $\mathrm{\ell }$ and a normalized $f$ supported in $K$. The postulate identifies the near-horizon occupation with $2S_k(\omega )\mathrm{/}\omega$; by [F1] and $\theta =2\pi \mathrm{/}k$, this is $n_{\theta }(\omega )$. Work with the joint incoming number quadratic form supplied by the direct-integral Fock channel in [A2]. Write $q_K$ for its conjugate-linear-first, linear-second polarization on ${\mathcal{H}}_K\oplus {\mathcal{H}}_K$, with the first summand the signal and the second the incoming vacuum; its diagonal is the adopted input occupation. The frequency-diagonal signal form is $q_K((g,0),(h,0))={\int }_K\overline{g(\omega )}h(\omega )n_{\theta }(\omega )d\omega$. It is bounded, since $n_{\theta }$ is continuous and bounded on the compact positive-frequency band $K$. The vacuum condition gives $q_K((0,h),(0,h))=0$. These are the quadratic-moment data of the two input channels already adopted in [A2]; they require neither a pure signal vector nor a normal density-matrix representation, and no assertion that a monochromatic annihilator belongs to the compactly smeared field algebra is used. To see that there is no additional cross-moment assumption, put $F=(g,0)$ and $E=(0,h)$. Positivity and sesquilinearity give $0\le q_K(F+zE,F+zE)=q_K(F,F)+2\mathrm{Re}(z{q}_K(F,E))$ for every $z\in \mathbb{C}$. Unless $q_K(F,E)=0$, a suitable phase and arbitrarily large modulus of $z$ make this negative. Hence both cross pairings vanish. In quadratic-moment Hilbert-space language, quotienting the input form by its null vectors and completing sends every vacuum annihilation label $E$ to the zero vector; its pairing with every finite signal annihilation image is therefore zero. Thus all cross pairings are fixed by positivity, with no operator product-domain or extra finite-cross-moment axiom. The passive map in [A2], with the same real coefficients as [F2], sends the output packet label to $J_{\mathrm{\ell }}f=(\sqrt{{\mathrm{\Gamma }}_{\mathrm{\ell }}}f,\sqrt{1-{\mathrm{\Gamma }}_{\mathrm{\ell }}}f)$. The coefficients are bounded measurable functions on $K$, so $J_{\mathrm{\ell }}$ is defined on ${\mathcal{H}}_K$ and $\mathrm{\parallel }{J}_{\mathrm{\ell }}f{\mathrm{\parallel }}^2=\mathrm{\parallel }f{\mathrm{\parallel }}^2$. Pulling back the adopted input number form gives $B_{\mathrm{\ell },\mathrm{\infty }}(f,f)=q_K(J_{\mathrm{\ell }}f,J_{\mathrm{\ell }}f)={\int }_K\mathrm{\mid }f(\omega ){\mathrm{\mid }}^2{\mathrm{\Gamma }}_{\mathrm{\ell }}(\omega )n_{\theta }(\omega )d\omega$. This calculates the diagonal of the adopted packet form directly, rather than treating $b^{\ast }b$ as an unbounded operator on an unspecified domain or extending [F2]'s vector theorem by assertion.

Operator-norm convergence in [A2] implies $\mathrm{\mid }{B}_{\mathrm{\ell },s}(f,f)-B_{\mathrm{\ell },\mathrm{\infty }}(f,f)\mathrm{\mid }\le \mathrm{\parallel }{B}_{\mathrm{\ell },s}-B_{\mathrm{\ell },\mathrm{\infty }}\mathrm{\parallel }\mathrm{\parallel }f{\mathrm{\parallel }}_2^2\to 0$; using $\mathrm{\parallel }f{\mathrm{\parallel }}_2=1$ and the value computed in step 1.1 proves the finite-packet limit.

Set $g(\omega )={\mathrm{\Gamma }}_{\mathrm{\ell }}(\omega )n_{\theta }(\omega )$ and $d{\mu }_{ϵ}=\mathrm{\mid }{f}_{ϵ}(\omega ){\mathrm{\mid }}^{2}d\omega$. The common band $K$ is compact and bounded away from zero, and [F3] gives $n_{\theta }(\omega )>0$ there; with the postulated continuous ${\mathrm{\Gamma }}_{\mathrm{\ell }}$, $g$ is bounded and continuous on $K$. For any $\eta >0$, continuity gives a neighborhood $V$ of ${\omega }_0$ in $K$ on which $\mathrm{\mid }g(\omega )-g({\omega }_0)\mathrm{\mid }<\eta$. Choose a continuous $q:K\to [0,1]$ with $q({\omega }_0)=0$ and $q=1$ outside $V$. Weak convergence ${\mu }_{ϵ}\rightharpoonup {\delta }_{{\omega }_0}$ gives $\int qd{\mu }_{ϵ}\to 0$, hence ${\mu }_{ϵ}(K\setminus V)\to 0$. Therefore $\mid {\int }_{K}gd{\mu }_{ϵ}-g({\omega }_0)\mid \le \eta +2\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}{\mu }_{ϵ}(K\setminus V)\to \eta$; as $\eta$ is arbitrary, the integral tends to $g({\omega }_0)$.

Apply step 2.1 at each fixed $ϵ$, then step 2.2 to the resulting spectral integral. This order yields ${\mathrm{\Gamma }}_{\mathrm{\ell }}({\omega }_0)\mathrm{/}(e^{2\pi {\omega }_0\mathrm{/}k}-1)$ and does not replace a finite-width integral by its central-frequency value.

By the definition of $T_{\mathrm{\infty }}$ in the Statement, $\theta \omega =\mathrm{\hslash }\omega \mathrm{/}(k_B{T}_{\mathrm{\infty }})$ for every $\omega >0$; solving $\theta =2\pi \mathrm{/}k$ gives $T_{\mathrm{\infty }}=\mathrm{\hslash }k\mathrm{/}(2\pi k_B)$. Substituting the model value $k=c^3\mathrm{/}(4GM)$ from [F4] gives $T_{\mathrm{\infty }}=\mathrm{\hslash }{c}^3\mathrm{/}(8\pi GM{k}_B)$. Its units are $(\mathrm{J}\mathrm{s})({\mathrm{s}}^{-1})\mathrm{/}(\mathrm{J}{\mathrm{K}}^{-1})=\mathrm{K}$. The conclusion is a model-derived spectral parameter and contains no observed temperature.