I read about contiguous forms in Achill Scürmann's thesis on positive quadratic forms. I am wondering about one aspect of the Voronoi algorithm presented in there, that enumerates all arithmetically inequivalent perfect forms. The algorithm stops when all the contiguous forms of some perfect quadratic form $Q$, are arithmetically equivalent to some other already enumerated perfect form. I wonder why this works as a stopping criterion? Is it so that $Q$ together with its contiguous neighbours tile the whole Ryshkov polyhedron with regards to $GL(Z)$, so that all other perfect forms are arithmetically equivalent to $Q$ and its neighbours? Grateful for a clarification of this.
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