Tiling without a subspace complement: a sharp counterexample in dimension nine over the field with two elements

We construct an explicit 64-point subset of $\mathbb F_2^9$ which tiles by translation but has no linear-subspace tiling complement. Together with our earlier dimension-eight theorem and a product argument, this shows that every nonempty tile in $\mathbb F_2^d$ has a linear-subspace complement preci