I measured softmax's precision cliff. In fp32, cross-entropy returns exactly zero at margin 16.

Jon Barron's paper claims standard softmax/cross-entropy has two numerical bugs with precision cliffs at 16.6 in fp32. I reproduced both in NumPy: the cliff is real, it moves with your reduction tree (15.8-16.7), and the distillation gradient has a ~5e-8 error floor that the log-space fix removes entirely.