Sharp Lower Bounds on the Haraux Function Beyond Reflexivity
We prove that the sharp $\frac{1}{2}$ lower bound for the Haraux function holds for every maximally monotone operator of type~(NI) on an arbitrary real Banach space. This resolves the nonreflexive extension raised by the recent reflexive result. We establish an exact decomposition at each graph point, where the local contribution to the Haraux function and a nonnegative residual together equal $\frac{1}{2}$ times the weighted squared displacement. Since the equivalence between type~(NI) and quasidensity provides graph points whose residuals tend to zero, this decomposition also yields the sharp bound without requiring a graph point at which the residual vanishes. Moreover, for every operator with a nonempty graph, this decomposition yields a lower bound involving the residual infimum. For maximally monotone operators, this decomposition also yields a new characterization of type~(NI) in terms of the Haraux function. Finally, on $c_0$, we give a maximally monotone operator of type~(NI) for which the residual infimum is zero at some target but is not attained. This shows that the existence of a graph point at which the residual vanishes is strictly stronger than the vanishing of the residual infimum required in our proof.