  Multi-loop scattering amplitudes/null polygonal Wilson loops in ${\mathcal
N}=4$ super-Yang-Mills are known to simplify significantly in reduced
kinematics, where external legs/edges lie in an $1+1$ dimensional subspace of
Minkowski spacetime (or boundary of the $\rm AdS_3$ subspace). Since the edges
of a $2n$-gon with even and odd labels go along two different null directions,
the kinematics is reduced to two copies of $G(2,n)/T \sim A_{n{-}3}$. In the
simplest octagon case, we conjecture that all loop amplitudes and Feynman
integrals are given in terms of two overlapping $A_2$ functions (a special case
of two-dimensional harmonic polylogarithms): in addition to the letters $v,
1+v, w, 1+w$ of $A_1 \times A_1$, there are two letters $v-w, 1- v w$ mixing
the two sectors but they never appear together in the same term; these are the
reduced version of four-mass-box algebraic letters. Evidence supporting our
conjecture includes all known octagon amplitudes as well as new computations of
multi-loop integrals in reduced kinematics. By leveraging this alphabet and
conditions on first and last entries, we initiate a bootstrap program in
reduced kinematics: within the remarkably simple space of overlapping $A_2$
functions, we easily obtain octagon amplitudes up to two-loop NMHV and
three-loop MHV. We also briefly comment on the generalization to $2n$-gons in
terms of $A_2$ functions and beyond.
