Geometric decomposition of the $d$-dimensional hard-sphere partition function

We introduce a geometric decomposition of the hard-sphere partition function. Using a close-packing-inspired geometric bound on the available insertion volume, made rigorous when a corresponding local density certificate is available, we establish a reference upper bound $Q^{\ast }$ on the configurational integral. Factoring this upper bound out of the statistical geometric partition function of Speedy yields a new form for the $d$-dimensional partition function, $Q(N,V,T)=Q^{\ast }\exp (-N\mathcal{J})$, where $\mathcal{J}$ depends strictly on the boundary-to-volume ratio of the voids and the close-packing density. Overall, this work deepens our statistical geometric understanding of the hard-sphere system.