Four-point correlator of planar supersymmetric Yang-Mills theory at twelve loops

We determine the four-point correlation function and amplitude in planar, maximally supersymmetric Yang-Mills theory to 12 loops. We find that the recently introduced “double-triangle” rule in fact implies the previously described square and pentagon rules; and when applied to 12 loops, it fully determines the 11-loop correlator and fixes all but 3 of the (619,981,403) 12-loop coefficients; these remaining coefficients can be subsequently fixed using the “(single-)triangle” rule. Not only do we confirm the Catalan conjecture for antiprism graphs, but we discover evidence for a greatly generalized Catalan conjecture for the coefficients of all polygon-framed fishnet graphs. We provide all contributions through 12 loops as Supplemental Material to this work.