Sandpiles prediction and crossover on \(\mathbb {Z}^2\) within Moore neighborhood

The computational complexity of predicting sandpiles on \(\mathbb {Z}^2\) ${\mathbb{Z}}^2$ is not settled yet, neither for von Neumann nor for Moore neighborhood (is it in \({\textsf{NC}}\) $\mathsf{NC}$ ? is it \({\textsf{P}}\) $\mathsf{P}$ -complete?). In this work we study the sandpile model considering all the 256 possible sub-neighborhoods within the Moore neighborhood. Surprisingly, we found that 12 of them have a \({\textsf{P}}\) $\mathsf{P}$ -complete prediction problem, while for the remaining 244 neighborhoods, we prove that they do not admit a crossover gate, i.e., for them, it is impossible to cross information, if the bit of information is the presence (or absence) of an avalanche.