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This choice takes Zₜ to be the effective update matrix Uₜ. It makes path geometry insensitive to reciprocal scaling of its two rank-one factors. It still describes parameter-operator change and does not by itself measure behavior or a declaration.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| Z_t | Path variable |
| U_t | Effective update matrix |
How it works
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Original context
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When both effective norms are nonzero, their normalized inner product gives the effective cosine. At the fixed positive scale used here, it equals the product of the two factor cosines. These matrix metrics are invariant under Eq. (4.2), whereas individual factor norms are not. 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.
Exact original notation
{Z}_{t}={U}_{t}