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Life sciences · Preprint
arXiv · September 9, 2026
Raises a question worth testing. It does not answer one.
This is a theoretical mathematics paper that constructs explicit counterexamples to Rockafellar's sum conjecture, demonstrating that two maximally monotone operators satisfying the interior-domain condition can have a sum that is not maximally monotone. The work provides a general construction theorem and verifies it on specific Banach spaces (c₀ and ℓ¹), resolving a longstanding open problem in convex analysis.
Preprint.
Counterexample on c₀ where two maximally monotone operators satisfying interior-domain condition have nonmaximal sum Counterexample on ℓ¹ with usual norm constructed via bounded linear surjection from ℓ¹ onto c₀ General construction theorem computes entire monotone polar and establishes necessary and sufficient condition for maximal monotonicity
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This is a mathematical theory paper presenting counterexamples to a longstanding conjecture; it raises and resolves a theoretical question rather than testing a clinical or empirical hypothesis.
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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. We give one counterexample on $c_0$ and another on $\ell^1$ with its usual norm. 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 $c_0$, thereby obtaining a counterexample to the conjecture. Furthermore, we construct a bounded linear surjection from $\ell^1$ onto $c_0$ and use it to obtain the counterexample on $\ell^1$.
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