Reciprocal rescaling leaves the rank-one operator unchanged

Here st is the effective scaling coefficient from the adapter configuration, at is the input-side direction, and bt the output-side direction. Standard and rank-stabilized scaling differ in general; at rank one their rank denominators coincide. Configuration values, per-layer patterns, orientation conventions, and optional methods such as DoRA must be checked rather than assumed [1,4].

Comparing bt alone is useful within a fixed parameterization, but the factorization is not unique. For any nonzero scalar c:

cbtat/cT=btatT.

In particular, reversing the signs of both factors changes the apparent direction of each while leaving the effective matrix unchanged. A factor-level rotation therefore deserves a corresponding effective-update check. This does not assert that the published rotation was a gauge artifact; that question requires the actual tensors.

6.2 Low-memory exact formulas

The Frobenius inner product between two rank-one updates can be computed without constructing the dense matrices:

\left(c{b}_{t}\right){\left({a}_{t}/c\right)}^{T}={b}_{t}{a}_{t}^{T}.