  It is widely expected that NMHV amplitudes in planar, maximally
supersymmetric Yang-Mills theory require symbol letters that are not rationally
expressible in terms of momentum-twistor (or cluster) variables starting at two
loops for eight particles. Recent advances in loop integration technology have
made this an `experimentally testable' hypothesis: compute the amplitude at
some kinematic point, and see if algebraic symbol letters arise. We demonstrate
the feasibility of such a test by directly integrating the most difficult of
the two-loop topologies required. This integral, together with its rotated
image, suffices to determine the simplest NMHV component amplitude: the unique
component finite at this order. Although each of these integrals involve
algebraic symbol alphabets, the combination contributing to this amplitude
is---surprisingly---rational. We describe the steps involved in this analysis,
which requires several novel tricks of loop integration and also a considerable
degree of algebraic number theory. We find dramatic and unusual
simplifications, in which the two symbols initially expressed as almost ten
million terms in over two thousand letters combine in a form that can be
written in five thousand terms and twenty-five letters.
