Exact-particle-number positional energy shell

Use ${\mathrm{\Lambda }}_n=[-n-\frac{1}{2},n+\frac{1}{2}{)}^d$, its side length $L_n=2n+1$ and dimensionless volume $V_n=L_n^d$, the periodic positional energy $H_{L_n,\mathrm{p}\mathrm{e}\mathrm{r}}$ of Stable regularly tempered classical positional preparation, and the exact-$N$ measure ${\mu }_{{\mathrm{\Lambda }}_n,N}$ of Indistinguishable-particle positional measure. Given $N_n\in {\mathbb{N}}_0$, a reduced configuration-energy center $e\in \mathbb{R}$ joules per dimensionless reference volume, and a width ${\delta }_n>0$ in the same units, define $S_n(N_n,e,{\delta }_n)=\{\omega \subset {\mathrm{\Lambda }}_n:\mathrm{\mid }\omega \mathrm{\mid }=N_n,\mathrm{\mid }{H}_{L_n,\mathrm{p}\mathrm{e}\mathrm{r}}(\omega )\mathrm{/}{V}_n-e\mathrm{\mid }\le {\delta }_n\}\mathrm{.}$ When $0<{\mu }_{{\mathrm{\Lambda }}_n,N_n}(S_n)<\mathrm{\infty }$, the uniform positional shell law is $M_n(A)=\frac{{\mu }_{{\mathrm{\Lambda }}_n,N_n}(A\cap S_n)}{{\mu }_{{\mathrm{\Lambda }}_n,N_n}(S_n)}\mathrm{.}$ A shrinking energy-density shell uses some specified sequence ${\delta }_n\downarrow 0$. Its total energy width is $2{\delta }_n{V}_n$, which is $o(V_n)$ but need not be microscopic or bounded. Existence of a normalizing shell and the existence of a suitable shrinking sequence are separate theorem obligations. This is not a surface measure on an exact-energy hypersurface and not a phase-space shell for $K(p)+H_N^{\mathrm{p}\mathrm{e}\mathrm{r}}$.