Read the equation
Both displacement vectors point outward from the center checkpoint: one toward t−k and the other toward t+k. Their normalized inner product measures the angle between those directions. Straight passage through the center gives −1, a right-angle turn gives 0, and return along the same side can give +1; this orientation is the reverse of an incoming-versus-outgoing velocity cosine.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| C_Z | Centered displacement cosine; c_Z in another version |
| Z | Factor or effective-operator path |
| t | Center checkpoint |
| k | Lag on each side |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
Assumptions & limits
The expression is undefined if either displacement has zero norm.
Original context
Read the surrounding source passage
4.2. Local bend and the role of the filter The released analysis code compares displacements from a center checkpoint to earlier and later neighbors [Model Organisms for Emergent Misalignment Project, 2025]. For either the output vector Zt=bt or effective matrix Zt=Ut, define CZ(t;k)=⟨Zt−k−Zt,Zt+k−Zt⟩∥Zt−k−Zt∥ ∥Zt+k−Zt∥. (4.5) Matrix comparisons use Frobenius quantities. A straight continuing path gives CZ=−1, because its backward and forward displacements point in opposite directions. A larger value indicates a larger local bend. This is a cosine between displacements, not between the states themselves and not a measure of movement amplitude. A zero displacement makes the corresponding angle undefined. I evaluate lags of five, ten, and 15 training steps on the regular grid. Each lag uses every center with both required neighbors. This preserves the training step offsets and does not impose the longest lag’s edge trimming on shorter lags.
Exact original notation
{C}_{Z}(t;k)=\frac{⟨{Z}_{t-k}-{Z}_{t},{Z}_{t+k}-{Z}_{t}⟩}{\Vert {Z}_{t-k}-{Z}_{t}\Vert \Vert {Z}_{t+k}-{Z}_{t}\Vert }.Related builds
Sources & provenance
{C}_{Z}(t;k)=\frac{⟨{Z}_{t-k}-{Z}_{t},{Z}_{t+k}-{Z}_{t}⟩}{\Vert {Z}_{t-k}-{Z}_{t}\Vert \Vert {Z}_{t+k}-{Z}_{t}\Vert }.
{C}_{Z}(t;k)=\frac{⟨{Z}_{t-k}-{Z}_{t},{Z}_{t+k}-{Z}_{t}⟩}{\Vert {Z}_{t-k}-{Z}_{t}\Vert \Vert {Z}_{t+k}-{Z}_{t}\Vert }.
{C}_{Z}(t;k)=\frac{⟨{Z}_{t-k}-{Z}_{t},{Z}_{t+k}-{Z}_{t}⟩}{\Vert {Z}_{t-k}-{Z}_{t}\Vert \Vert {Z}_{t+k}-{Z}_{t}\Vert }.
{C}_{Z}(t;k)=\frac{⟨{Z}_{t-k}-{Z}_{t},{Z}_{t+k}-{Z}_{t}⟩}{\Vert {Z}_{t-k}-{Z}_{t}\Vert \Vert {Z}_{t+k}-{Z}_{t}\Vert }.