I derive the volume dependence of bound states from a cluster-cluster picture with nucleon degrees of freedom. A constant factor called the geometric factor appears in the generalization from pointlike particles to clusters. I show that this factor is obtained explicitly when the underlying overlap integral is evaluated directly in the original many-body space. I achieve this by constructing Jacobi coordinates on the lattice with periodic boundary conditions. The derivation requires only the interaction between the clusters to be short ranged, with the nonperturbative Coulomb force included. The factor emerges as a geometric property of the many-body configuration space rather than a multiplicity of physical channels. I validate my derivation using many-body calculations; in particular, I find this factor to be essential in extracting asymptotic normalization constants from lattice calculations of the ${}^{16}\mathrm{O}$ ground state.

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• Few-body systems
• Hydrostatic stellar nucleosynthesis
• Lattice field theory
• Nuclear many-body theory