A finite thermal comparison state

For a prescribed finite Hamiltonian $H_R$ and temperature $T>0$, one may define

Proposition prop:featureless applies if a model proves $d_{\diamond }(\mathcal{N},{\mathcal{T}}_{{\omega }_T})\le \eta$. Defining (eq:finite-gibbs), or matching a selected occupation law to it, does not prove this hypothesis.

In particular, (eq:hawking-occupation) is an occupation statement for specified semiclassical preparation and mode data. It is not a uniform half-diamond estimate on an independently declared microscopic code. Absorption factors, packet selection, and correlations also prevent replacing the entire outgoing continuum by an unqualified finite Gibbs density.

A finite selection of bosonic modes still has an infinite occupation space. Applying the finite theorems requires a specified finite encoding or cutoff and a reference-preserving allowance for the discarded sector. Such an allowance must not be silently omitted from the complete residual. Finite-packet and regulated comparison interfaces are documented in CheungEn02E,CheungHawk02E; they are not imported as gravitational constructions.