  We describe the minimal space of polylogarithmic functions that is required
to express the six-particle amplitude in planar ${\cal N}=4$ super-Yang-Mills
theory through six and seven loops, in the NMHV and MHV sectors respectively.
This space respects a set of extended Steinmann relations that restrict the
iterated discontinuity structure of the amplitude, as well as a cosmic Galois
coaction principle that constrains the functions and the transcendental numbers
that can appear in the amplitude at special kinematic points. To put the
amplitude into this space, we must divide it by the BDS-like ansatz and by an
additional zeta-valued constant $\rho$. For this normalization, we conjecture
that the extended Steinmann relations and the coaction principle hold to all
orders in the coupling. We describe an iterative algorithm for constructing the
space of hexagon functions that respects both constraints. We highlight further
simplifications that begin to occur in this space of functions at weight eight,
and distill the implications of imposing the coaction principle to all orders.
Finally, we explore the restricted spaces of transcendental functions and
constants that appear in special kinematic configurations, which include
polylogarithms involving square, cube, fourth and sixth roots of unity.
