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This analysis choice takes Zₜ to be the output-side LoRA vector bₜ. It provides a compact coordinate path within the saved parameterization. Reciprocal rescaling of the two LoRA factors can change that path without changing the effective weight operator, so it is secondary to an invariant operator comparison.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| Z_t | Path variable |
| b_t | Output-side factor |
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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}={b}_{t}