All-loop singularities of scattering amplitudes in massless planar theories

In massless quantum field theories the Landau equations are invariant under graph operations familiar from the theory of electrical circuits. Using a theorem on the $Y$-$\Delta$ reducibility of planar circuits we prove that the set of first-type Landau singularities of an $n$-particle scattering amplitude in any massless planar theory, in any spacetime dimension $D$, at any finite loop order in perturbation theory, is a subset of those of a certain $n$-particle $\lfloor{(n{-}2)^2/4}\rfloor$-loop "ziggurat" graph. We determine this singularity locus explicitly for $D=4$ and $n=6$ and find that it corresponds precisely to the vanishing of the symbol letters familiar from the hexagon bootstrap in SYM theory. Further implications for SYM theory are discussed.