Supplied static finite-mode Fermi occupation identity

This is a finite-mode conditional construction. Assume the Axiom of Countable Choice and the exact current interface for the batch-45 draft Explicit diagonal Fock domains and occupation trace, whose Statement matches its planned manifest entry on qsm-bose-fermi-gases. Its current carrier has a complete authored proof and precheck, plus a Step-1 ready authoring record; those records do not assert independent review approval. Let $J=\{1,\dots ,M\}$ be a nonempty finite set of supplied modes, let $m,c>0$, let ${\mathcal{N}}_j>0$ be supplied dimensionless lapse values, and let ${\widehat{p}}_j\in {\mathbb{R}}^3$ be supplied mode momenta. Put ${ϵ}_j=\sqrt{m^2{c}^4+c^2\mathrm{\mid }{\widehat{p}}_j{\mathrm{\mid }}^2},E_j={\mathcal{N}}_j{ϵ}_j,{\mathcal{H}}_F={\mathbb{C}}^{2^M},$ with occupation basis $\mathrm{\mid }{n}_1,\dots ,n_M⟩$, $n_j\in \{0,1\}$, and diagonal operators $H\mathrm{\mid }n⟩=({\sum }_j{E}_j{n}_j)\mathrm{\mid }n⟩$, $N\mathrm{\mid }n⟩=({\sum }_j{n}_j)\mathrm{\mid }n⟩$. For ${\beta }_K>0$ and finite ${\mu }_K\in \mathbb{R}$, conditional on that exact supplier result, ${\mathrm{\Xi }}_F=\mathrm{Tr}{e}^{-{\beta }_K(H-{\mu }_{K}N)}={\prod }_{j=1}^M(1+e^{-{\beta }_K(E_j-{\mu }_K)})<\mathrm{\infty },$ and the normalized diagonal state has occupation mean $\mathbb{E}[n_j]=\frac{1}{e^{{\beta }_K(E_j-{\mu }_K)}+1}\mathrm{.}$ With ${\beta }_j={\beta }_K{\mathcal{N}}_j$ and ${\mu }_j={\mu }_K\mathrm{/}{\mathcal{N}}_j$, this is exactly $\mathbb{E}[n_j]=\frac{1}{e^{{\beta }_j({ϵ}_j-{\mu }_j)}+1}\mathrm{.}$ No arbitrary curved-space Dirac spectrum or self-consistent quantum-gravity state is asserted.