A new proof finally puts a hard number on how sensitive a transformer's attention really is to small nudges in its input.
New research works out an exact mathematical identity for the Jacobian of the softmax function used inside attention layers, essentially a precise measure of how much a tiny wobble in the input can shake the output. The key quantity, called theta(p), tracks how evenly an attention distribution can be split in half, not how sharply it favors one token. The team used that identity to build a sensitivity bound that holds no matter how long the input sequence gets, and found a structural split between two common transformer designs: pre-layernorm (pre-LN) models carry an extra identity term in their math that post-LN models lack. Testing three 774-million-parameter models early in training, they found attention grows sharper and more concentrated as training proceeds, yet the sensitivity certificate stays near its maximum for at least half of all attention rows at every layer checked.
This is not a new model or product. It is a piece of theoretical plumbing that helps explain why transformers sometimes train unstably or blow up numerically, and why the choice between pre-LN and post-LN changes that risk. Knowing exactly where a network is maximally sensitive gives engineers a way to reason about stability instead of just running experiments and hoping for the best.
The catch: the paper's own sufficient condition for when post-LN networks calm down mathematically was left untested, so for now the theory is ahead of the evidence.