  In a previous work, we proposed an integrability setup for computing
non-planar corrections to correlation functions in $\mathcal{N}=4$ super
Yang-Mills theory at any value of the coupling constant. The procedure consists
of drawing all possible tree-level graphs on a Riemann surface of given genus,
completing each graph to a triangulation, inserting a hexagon form factor into
each face, and summing over a complete set of states on each edge of the
triangulation. The summation over graphs can be interpreted as a quantization
of the string moduli space integration. The quantization requires a careful
treatment of the moduli space boundaries, which is realized by subtracting
degenerate Riemann surfaces; this procedure is called stratification. In this
work, we precisely formulate our proposal and perform several perturbative
checks. These checks require hitherto unknown multi-particle mirror
contributions at one loop, which we also compute.
