Theorem: Em03-A mathematical release

For the declared fixed-background free-scalar model:

1. surface gravity is computed from the normalized Schwarzschild metric;

2. the affine horizon vacuum is specified before the thermal calculation;

3. its Killing-time covariance and distributional Fourier limit are proved;

4. the exact positive-frequency transmission probability is constructed from the Schwarzschild radial equation;

5. the outgoing compact-band number expectation and its mode-window rate are proved;

6. approximation errors are controlled by Corollary cor:error;

7. under the additional assumptions of Section sec:peeling, a compact-test late-time covariance limit has the proved remainder (eq:smearederror).

The bulk existence and Hadamard identification are the declared external input. No collapse packet limit, all-mode luminosity, backreaction, mass-loss equation, quantum-gravity correction, island, Page curve, microscopic evaporation channel, or inherited characteristic-coefficient admission is established by this release.

Proof. Assertions (i)–(v) are Theorem thm:main. Assertion (vi) is Corollary cor:error. Assertion (vii) is Theorem thm:peeling, with its hypotheses retained.

The proof contains no construction of a dynamical metric, graviton insertion, island extremization, entropy reconstruction, or inherited characteristic graph. The compact-test transfer theorem explicitly does not identify a noncompact collapse number packet. None of the unasserted conclusions follows as an additional part of the proved statements.