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We give a general procedure for constructing an extended phase space forYang-Mills theory at null infinity, capable of handling the asymptoticsymmetries and construction of charges responsible for sub$^n$-leading softtheorems at all orders. The procedure is coordinate and gauge-choiceindependent, and can be fed into the calculation of both tree and loop-levelsoft limits. We find a hierarchy in the extended phase space controlled by theBernoulli numbers arising in Todd genus computations. We give an explicitexample of a calculation at tree level, in radial gauge, where we also uncoverrecursion relations at all orders for the equations of motion and charges.

A General Hierarchy of Charges at Null Infinity via the Todd Polynomials