Nonmaximal sums of maximally monotone operators under Rockafellar's constraint qualification

We construct counterexamples to Rockafellar's sum conjecture in which two maximally monotone operators satisfy the interior-domain condition but their sum is not maximally monotone, thereby providing the complete disproof of the conjecture. We establish a general construction theorem that computes the entire monotone polar of a class of graphs, gives a necessary and sufficient condition for their maximal monotonicity, and shows how a positive rank-one perturbation yields a nonmaximal sum under this condition. We verify the theorem's hypotheses and its maximality criterion on c0, thereby obtaining a counterexample to the conjecture. Furthermore, we construct a bounded linear surjection from ℓ1 onto c0 and use it to obtain the counterexample on ℓ1. Lean formalizations of the c0 counterexample and the pullback lemma are also provided.