lin-0166
13.1 Adapters as local intervention fields
For a frozen weight \(W_\ell \), a rank-\(r\) LoRA adapter contributes
\[ \Delta _i^\ell =B_i^\ell A_i^\ell , \qquad W_\ell \longmapsto W_\ell +\Delta _i^\ell \]
[ Hu et al. , 2022 ] . At hidden state \(h\), the corresponding residual map is
\[ a_i^\ell (h)=h+\Delta _i^\ell h. \]
The adapter can be read as a small intervention field on representation space. For two declared-compatible adapters, the learning sketch contains
\[ a_i^\ell a_j^\ell \sim a_j^\ell a_i^\ell . \]
In the linear residual realization,
\[ a_i^\ell a_j^\ell -a_j^\ell a_i^\ell = \Delta _i^\ell \Delta _j^\ell - \Delta _j^\ell \Delta _i^\ell = [\Delta _i^\ell ,\Delta _j^\ell ]. \]
Thus the matrix commutator is an exact coordinate observer of the two-route obstruction at one insertion site.