arXiv:2611.13916v1 Announce Type: cross Abstract: This paper develops a dual-space preconditioning framework for variational inequalities and root-finding problems beyond classical cocoercivity and Lipschitz continuity assumptions. To this end, we introduce both a novel cocoercivity-type condition and a relaxed Lipschitz continuity condition, and derive convergence of deterministic and stochastic methods under these generalized conditions. For isotropic preconditioners, the resulting algorithms can be interpreted as variable-stepsize variants of classical methods, extending recent analyses based on $(L_0, L_1)$-type growth conditions while allowing simpler proofs and larger stepsize ranges. We also show that the framework naturally handles variational inequalities through weak Minty-type arguments. Finally, we study the associated continuous-time dynamics and derive a variational and optimal control interpretation of the preconditioned flow. Overall, the results show that dual-space preconditioning provides a flexible mechanism for designing and analyzing operator methods beyond the standard Euclidean Lipschitz setting.