The hierarchy problem can be systematically truncated after identifying correlated clusters. The simplest definitions follow after one identifies the clusters recursively. At the lowest level, one finds the class of single-particle expectation values (singlets) that are symbolized by ⟨ ⟨ 1 ⟩ ⟩ . Any two-particle expectation value ⟨ ⟨ 2 ⟩ ⟩ can be approximated by factorization ⟨ ⟨ 2 ⟩ ⟩ S = ⟨ ⟨ 1 ⟩ ⟩ ⟨ ⟨ 1 ⟩ ⟩ that contains a formal sum over all possible products of single-particle expectation values. More generally, ⟨ ⟨ 1 ⟩ ⟩ defines the singlets and ⟨ ⟨ N ⟩ ⟩ S is the singlet factorization of an N -particle expectation value. Physically, the singlet factorization among Fermions produces the Hartree–Fock approximation while for Bosons it yields the classical approximation where Boson operators are formally replaced by a coherent amplitude, i.e., B ^ ^ → → ⟨ ⟨ B ^ ^ ⟩ ⟩ . The singlet factorization constitutes the first level of the cluster-expansion representation.