Landau Singularities of the 7-Point Ziggurat I

The (first-type Landau) singularities of any $n$-point massless planar Feynman integral, at any loop order, are encoded in a certain ''ziggurat graph''. We compute the leading Landau singularities of the 7-point ziggurat graph and find perfect agreement with a subset of the ''heptagon symbol alphabet'' that has appeared in the context of planar $\mathcal{N}=4$ super-Yang-Mills theory. The remaining heptagon symbol letters are found in subleading Landau singularities, which we address in a companion paper.