Batch size is not the reduction size

Count entries per group, including any spatial axes being reduced.

For $m=1$, the mean is $x$ and the variance is zero. With $\varepsilon>0$, $\hat x=0$ and $y=\beta$ for every $x$. Therefore $\partial y/\partial x=0$ and $\partial y/\partial\gamma=0$. Only beta receives an upstream gradient from this scalar output. Substituting $m=1$ and $\hat x=0$ into (I.8.3) also gives zero.

For dense batch normalisation on a matrix, batch size one means one value per feature group. In spatial batch normalisation, the group instead contains $NHW$ values per channel. With $N=1$ and multiple spatial positions, it need not be a singleton. Highly correlated positions may still give poor statistics, but that is a different failure.

Layer normalisation over a feature width of one has the same algebraic degeneracy regardless of how many examples are in the batch: examples do not share its statistics. A framework may reject a degenerate training configuration before performing this calculation.