The central intertwining lifting is used to establish a maximum principle for the commutant lifting theorem. This maximum principle is used to prove that the central intertwining lifting is also a maximal entropy solution for the commutant lifting theorem, when T is a unilateral shift of finite multiplicity. The maximum principle is based on the residual spaces for intertwining liftings, and is motivated by Robinson's minimum energy delay principle for outer functions. A permanence property for the central intertwining lifting is also given.