Abstract

We give an audited formulation of the standard free-scalar emission law on a fixed Schwarzschild background. The geometry, time normalization, field, state, observable, and scattering problem are specified separately. Surface gravity is calculated from the metric. A vacuum defined by an affine horizon coordinate is pulled back to Killing time, and its distributional Fourier transform is evaluated without inserting a temperature or an occupation table. Schwarzschild transmission probabilities are constructed from the radial wave equation. For every normalized positive-frequency wavepacket supported in a compact band away from zero, the resulting number expectation is the greybody-weighted Bose occupation integral. A trace-class time–frequency window gives the corresponding number-rate normalization.

The boundary calculation and its regulator removal are proved here. The bulk existence and Hadamard property of the specified Unruh state are a precisely declared external input, not a result attributed to an earlier manuscript in the series. We also prove a compact-test late-time estimate under explicit peeling and state-regularity hypotheses. This estimate is not silently promoted to a collapse wavepacket theorem. The supplied manuscript record is audited with its retrieval limitations retained. No quantum-gravity correction, backreaction, evaporation law, island construction, or reopening of an inherited admission gate is claimed.