Finite box fails to cure point gravitational collapse

Given: In SI units, let $B={\prod }_{j=1}^3(a_j,b_j)\subset {\mathbb{R}}^3$ with finite $a_j<b_j$ in metres. Let $q_i\in B$ be positions, $p_i\in {\mathbb{R}}^3$ momenta in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, $m>0$ a mass in $\mathrm{k}\mathrm{g}$, $G>0$ in ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, $\beta >0$ in ${\mathrm{J}}^{-1}$, and $h_{\ast }>0$ a fixed action reference in $\mathrm{J}\mathrm{s}$. On $\mathrm{\Gamma }=\{(q_1,q_2,p_1,p_2)\in B^2\times {\mathbb{R}}^6:q_1\mathrm{\ne }{q}_2\}$, define the point Hamiltonian in joules by $H=\frac{\mathrm{\mid }{p}_1{\mathrm{\mid }}^2+\mathrm{\mid }{p}_2{\mathrm{\mid }}^2}{2m}-\frac{G{m}^2}{\mathrm{\mid }{q}_1-q_2\mathrm{\mid }}$ and its dimensionless canonical partition integral by $Z_2(\beta )={\int }_{\mathrm{\Gamma }}{e}^{-\beta H}{d}^3{q}_1{d}^3{q}_2{d}^3{p}_1{d}^3{p}_2\mathrm{/}(2!h_{\ast }^6)$, using product Lebesgue measure. The collision diagonal is excluded, with no positive minimum separation imposed.

For two equal positive masses and the stated point Hamiltonian in a bounded three-dimensional box with nonempty interior, the canonical partition integral with the dimensionless phase reference is infinite for every $\beta >0$. (Point collapse and unscaled gravity nonextensivity, clause 1)