Gluing Ladder Feynman Diagrams into Fishnets

We use integrability at weak coupling to compute fishnet diagrams for four-point correlation functions in planar phi(4) theory. The results are always multilinear combinations of ladder integrals, which are in turn built out of classical polylogarithms. The Steinmann relations provide a powerful constraint on such linear combinations, leading to a natural conjecture for any fishnet diagram as the determinant of a matrix of ladder integrals.