Theorem: Fixed-background scalar emission law

For the geometry, field, state, and normalization of Section sec:scope, the following statements hold.

1. The normalized Schwarzschild Killing horizon has

   $$\kappa =\frac{1}{4GM},{\beta }_{\mathrm{H}}=\frac{2\pi }{\kappa }=8\pi GM.$$

2. The incoming horizon current, expressed in Killing coordinate $u$, has two-point distribution

   $$C_{\kappa }(s)=-\frac{{\kappa }^2}{16\pi }{\mathrm{csch}}^2\left(\frac{\kappa (s-\mathrm{i}0)}{2}\right),s=u-u^{\prime },$$

   whose Fourier transform is

   $${\hat{C}}_{\kappa }(\nu )=\frac{\nu }{2(1-e^{-{\beta }_{\mathrm{H}}\nu })},{\hat{C}}_{\kappa }(0)=\frac{1}{2{\beta }_{\mathrm{H}}}.$$

3. The radial scalar equation constructs a unitary two-channel scattering matrix. Its horizon-to-infinity transmission coefficient ${\tau }_{\ell }(\omega )$ defines

   $${\Gamma }_{\ell }(\omega )=|{\tau }_{\ell }(\omega ){|}^2,0<{\Gamma }_{\ell }(\omega )\le 1(\omega >0).$$

4. The outgoing number covariance is the distributional identity

   $${\varpi }_{\mathrm{U}}(a_{\mathrm{out}}(\omega {)}^{\ast }{a}_{\mathrm{out}}({\omega }^{\prime }))=\frac{{\Gamma }_{\ell }(\omega )}{e^{{\beta }_{\mathrm{H}}\omega }-1}\delta (\omega -{\omega }^{\prime }),$$

   and consequently

   $$\boxed{{\varpi }_{\mathrm{U}}(N_h)={{\int }_0}^{\infty }|h(\omega ){|}^2\frac{{\Gamma }_{\ell }(\omega )}{e^{8\pi GM\omega }-1}\mathrm{d}\omega .}$$

5. With the time–frequency counting convention of Proposition prop:rate, the number rate per angular mode is

   $$\boxed{f_{\ell }(\omega ;M)=\frac{1}{2\pi }\frac{{\Gamma }_{\ell }(\omega )}{e^{8\pi GM\omega }-1}.}$$

   This formula is derived from the boundary state and the radial scattering equation, not assumed as calibration data.