Theorem 4.7 (Buss 1986) A function \(f(\vec{x})\) belongs to \(\Box^P_i\) if and only if \(f(\vec{x})\) is provably total in \(\mathsf{S}^i_2\) relative to a \(\Sigma^b_i\)-definition – i.e. if and only if there is a \(\Sigma^b_i\)-formula \(\phi_f(\vec{x},y)\) such that \(\mathsf{S}^i_2 \vdash \forall \vec{x} \exists ! y \phi_f(\vec{x},y)\).