  We present the powerful module-intersection integration-by-parts (IBP)
method, suitable for multi-loop and multi-scale Feynman integral reduction.
Utilizing modern computational algebraic geometry techniques, this new method
successfully trims traditional IBP systems dramatically to much simpler
integral-relation systems on unitarity cuts. We demonstrate the power of this
method by explicitly carrying out the complete analytic reduction of two-loop
five-point non-planar hexagon-box integrals, with degree-four numerators, to a
basis of $73$ master integrals.
