The Sklyanin Bracket and Cluster Adjacency at All Multiplicity

We argue that the Sklyanin Poisson bracket on Gr(4,n) can be used to efficiently test whether an amplitude in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory satisfies cluster adjacency. We use this test to show that cluster adjacency is satisfied by all one- and two-loop MHV amplitudes in this theory, once suitably regulated. Using this technique we also demonstrate that cluster adjacency implies the extended Steinmann relations at all particle multiplicities.