The two-loop remainder function for eight and nine particles

Two-loop MHV amplitudes in planar ${\cal N} = 4$ supersymmetric Yang Mills theory are known to exhibit many intriguing forms of cluster-algebraic structure. We leverage this structure upgrade the symbols eight- and nine-particle complete analytic functions. This is done by systematically projecting onto components these that take different functional forms, matching each component an ansatz multiple polylogarithms with negative cluster-coordinate arguments. The remaining additive constant can be determined analytically comparing collinear limit amplitude lower-multiplicity results. also observe nonclassical part admits a unique decomposition terms specific $A_3$ cluster polylogarithm, explore numerical behavior remainder function along lines positive region.