Proof: Proof of Theorem thm:main

Proof. Lemma lem:geometry proves the surface gravity and its geometric normalization. Lemmas lem:pullback and lem:fourier prove the horizon covariance, its Fourier transform, and the distributional regulator removal. Lemmas lem:radial and lem:scattering construct the exact radial scattering coefficients. Proposition prop:packet proves the outgoing number covariance and packet formula. Proposition prop:rate proves the counting normalization.

The declared external theorem identifies the boundary prescription with the Hadamard bulk Unruh state on the stated Schwarzschild domain. No earlier internal manuscript is used as a substitute for that theorem. The exponent in the emission law is ${\beta }_{\mathrm{H}}\omega =8\pi GM\omega$, calculated rather than calibrated.