$\text{Tr}(F^3)$ supersymmetric form factors and maximal transcendentality Part I: $\mathcal{N}=4$ super Yang-Mills

In the large top-mass limit, Higgs plus multi-gluon amplitudes in QCD can be computed using an effective field theory. This approach turns the computation of such amplitudes into that of form factors of operators of increasing classical dimension. In this paper we focus on the first finite top-mass correction, arising from the operator ${\rm Tr}(F^3)$, up to two loops and three gluons. Setting up the calculation in the maximally supersymmetric theory requires identification of an appropriate supersymmetric completion of ${\rm Tr}(F^3)$, which we recognise as a descendant of the Konishi operator. We provide detailed computations for both this operator and the component operator ${\rm Tr}(F^3)$, preparing the ground for the calculation in $\mathcal{N}<4$, to be detailed in a companion paper. Our results for both operators are expressed in terms of a few universal functions of transcendental degree four and below, some of which have appeared in other contexts, hinting at universality of such quantities. An important feature of the result is a delicate cancellation of unphysical poles appearing in soft/collinear limits of the remainders which links terms of different transcendentality. Our calculation provides another example of the principle of maximal transcendentality for observables with non-trivial kinematic dependence.