  We model the thermodynamics of local structures within the hard sphere liquid
at arbitrary volume fractions through the \textit{morphometric} calculation of
$n$-body correlations. We calculate absolute free energies of local geometric
motifs in excellent quantitative agreement with molecular dynamics simulations
across the liquid and supercooled liquid regimes. We find a bimodality in the
density library of states where five-fold symmetric structures appear lower in
free energy than four-fold symmetric structures, and from a single reaction
path predict a relaxation barrier which scales linearly in the compressibility
factor. The method provides a new route to assess changes in the free energy
landscape at volume fractions dynamically inaccessible to conventional
techniques.
