While your point about numerical stability is correct in general, there are no numerical stability issues here and I think this conception, which I've seen in more than one place now, stems from a fundamental misunderstanding of the paper's results. While they _did_ come up with a faster TPU/GPU algorithm too, the primary result is not a fast matmul approximation, it is an exact algorithm comprising of stepwise addition/multiplication operations, and hence is numerically stable and should work for any ring (https://ncatlab.org/nlab/show/ring). AlphaTensor itself does not do the matrix multiplication, it was used to perform an (efficiently pruned) tree search over the space of operations to find an efficient, stable algorithm.