GPT-2 medium: a cut Lk is the residual stream entering block k, at the last token of the host prompt. L0 is the embedding; the stream after the last block is the final readout, not a cut. The host's other positions are held at their own values at the same depth.GPT-2 medium block: pre-layer-norm, attention then MLP, each with its own addition. The cut is the raw stream before ln_1.
The median radius is the sphere a rotation rides on and the median norm is what relative displacement is divided by, so both sit in the denominators of every panel below. Centre over spread says how far the activations sit from the origin relative to their own scatter: where it is large, rotating about the origin and about the bank mean are materially different measurements.
Arora, Ge, Neyshabur and Zhang’s Figure 1 construction: a perturbation of a fixed fraction of a state’s norm is added at one cut, and each line shows how much of it survives at every deeper readout, relative to what that readout carries. Each point is a mean over bases and directions, and the band is the tenth to ninetieth percentile across bases — an attenuation every base agrees on and an average over disagreement are different findings, and the mean alone cannot tell them apart.
A plane through three real activations at the perturbation cut, A, B and C, on the axes of Janiak et al. (arXiv:2409.17113): along A to B, and the component of C perpendicular to it. The two maps answer different questions. Left: a finite comparison against the anchors: the KL from each anchor’s own output distribution to the grid point’s, or the L2 distance at a chosen readout, taking the nearest anchor: dark is one activation’s territory, bright seams are where territories meet. Right: a local rate at the same point: how fast the output moves per unit step, by default the largest Fisher speed within the plane, which says where the function has a wall but not what lies on either side. A seam on the left without a ridge on the right is a midpoint between two activations rather than a wall.
Every grid point is evaluated inside one fixed document, the host, at its final position, with the other positions holding that document’s own states at the cut; this is how the released perturbation code injects a state. The host is named under the maps. An anchor taken from another document is therefore read inside a context it did not come from, which is part of what the plane measures.
The part carries every readout deeper than the cut from one pass, and the walls grow with depth: a plane flat one block after the cut can be walled twenty blocks later, which is why the readout is a control rather than a fixed choice.