  We show that approximate graph colouring is not solved by any level of the
affine integer programming (AIP) hierarchy. To establish the result, we
translate the problem of exhibiting a graph fooling a level of the AIP
hierarchy into the problem of constructing a highly symmetric crystal tensor.
In order to prove the existence of crystals in arbitrary dimension, we provide
a combinatorial characterisation for realisable systems of tensors; i.e., sets
of low-dimensional tensors that can be realised as the projections of a single
high-dimensional tensor.
