We study decentralized minimization of an average of strongly convex, smooth local objectives over an intersection of agent-private closed convex sets, where each agent knows only its own objective and its own set and agents communicate over a gossip network. We show that the complexity is controlled by a single geometric scalar, which we call the constraint gap: the reciprocal square of the averaged linear regularity modulus of the collection of sets. For affine sets it coincides with a spectral gap of the averaged projection operator. We prove that a positive constraint gap forces the strong conical hull intersection property, bound the perturbation of the solution map, and show that for affine local sets the dual condition number is at most the product of the objective condition number, the gossip condition number, and the reciprocal constraint gap, with an explicit family attaining that bound exactly. We give a Chebyshev-accelerated dual method whose oracle and communication complexities are square roots of these quantities and validate the theory numerically. A companion paper shows that the same operator governs a known primal method and proves a matching lower bound in the constraint gap.