  The construction of the conformal scalar propagator which has been obtained
in the preceding two projects as an analytic function of the Schwarzschild
black-hole space-time is completed with a boundary condition imposed by the
physical context through contour integration in the exterior vicinity of the
event horizon. It is shown that, as a consequence of the semi-classical
character which the emitted quanta have in that exterior vicinity, the particle
production by the Schwarzschild black hole which was formally established in
the preceding project is identical to thermal Hawking radiation. By extension,
it is established that such a particle production corresponds to a spectrum
which detracts from thermality by the amount predicted by Parikh and Wilczek if
energy conservation is properly imposed as a constraint on scalar propagation.
The results obtained herein support the case made by S. Hawking on the relation
between quantum propagation and observation of particles produced by a black
hole.
