A Counterexample to Nivat's Conjecture for a Non-Convex Window of Full Affine Span

We construct an exact cluster $F\subseteq\mathbb{Z}^2$ of cardinality $8$ with full affine span, together with an $F$-tiling $T$, such that the orbit closure of $T$ in $\{0,1\}^{\mathbb{Z}^2}$ does not contain a $1$-periodic $F$-tiling. Since every $F$-tiling is a low-complexity configuration with respect to the window $\bar F := \{-a : a \in F\}$, this supplies a "non-degenerate" counterexample, in a strong sense, to Nivat's conjecture for non-convex windows. This answers, in the negative, a question of Kari and Moutot (2023) whether every such counterexample must be degenerate, in the sense that the probing window is contained in a coset of a proper finite-index sublattice. We complement this with a positive result: for every exact cluster $F$ of full affine span whose cardinality is the square of a prime, every $F$-tiling has a $1$-periodic $F$-tiling in its orbit closure. Together with Szegedy's theorem that every tiling by a cluster of prime cardinality is $1$-periodic, this shows that no cluster of fewer than $8$ cells can exhibit the phenomenon, with the possible exception of cardinality $6$, which we leave open.