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

Summary

This preprint claims the first complete disproof of Rockafellar's sum conjecture — the 1970 question of whether the sum of two maximally monotone operators on a general Banach space must be maximally monotone whenever dom A ∩ int(dom B) ≠ ∅. The author establishes a general construction theorem (Theorem 1): for graphs G = {(La, a)} defined by a bounded linear map M: X* → H and a Lipschitz curve constraint Ma = F(r(a)), the entire monotone polar is computed explicitly (Eq. 8), maximality reduces to unsolvability of r(p) = τ, Mp = (τ, η) on [−1,1] (Eq. 9), and adding the everywhere-defined positive rank-one operator P = ⟨·,g⟩g produces a nonmaximal sum, witnessed by (0,0) lying in the polar but off the graph. On c0(I), block-sum and triangular maps (Definitions 6–8) verify every hypothesis; the explicit curve ωs(n) = max{1 − sn^{1/4}, 0}/4n yields S = π²/96 < 1/8 and bounds V(gra A) ≤ 2σ < 1/4 and ⟨x,b⟩ > 7/8 on gra(A+P). An explicitly constructed bounded surjection Q: ℓ1 → c0 with ‖Q‖ = 1 and lifting bound 2‖x‖∞ (Lemma 5) transfers the counterexample to standard ℓ1 (Theorem 3) via a pullback lemma (Lemma 1). A Lean formalization of the c0 instance and pullback lemma is linked. Key limitations: the detailed proofs are disclosed as AI-generated (GPT-5.6), the Lean artifact's coverage of Theorem 1's generality is unstated, and a claim of this magnitude requires independent expert verification.