  Restriction is a natural quasi-order on $d$-way tensors. We establish a
remarkable aspect of this quasi-order in the case of tensors over a fixed
finite field -- namely, that it is a well-quasi-order: it admits no infinite
antichains and no infinite strictly decreasing sequences. This result,
reminiscent of the graph minor theorem, has important consequences for an
arbitrary restriction-closed tensor property $X$. For instance, $X$ admits a
characterisation by finitely many forbidden restrictions and can be tested by
looking at subtensors of a fixed size. Our proof involves an induction over
polynomial generic representations, establishes a generalisation of the tensor
restriction theorem to other such representations (e.g. homogeneous polynomials
of a fixed degree), and also describes the coarse structure of any
restriction-closed property.
