 Solution

For a chosen bead, the uniform displacement cube is inversion symmetric, so the proposal distribution satisfies $q(X\to Y)=q(Y\to X)$. The target canonical density is $\pi (X)\propto \exp [-\beta U(X)]$ for allowed configurations. With $a(X,Y)=\min [1,e^{-\beta (U(Y)-U(X))}]$,This proves detailed balance. Overlapping trials have zero target weight and are rejected. The proposal must retain rejected trials as self-transitions; repeatedly proposing until a nonoverlapping move appears would generally introduce a configuration-dependent proposal normalization and need a Hastings correction. Detailed balance establishes stationarity, while sufficient sampling also requires practical exploration of the accessible state space.