lin-0167
13.2 Why the bracket controls order
For finite strengths \(\varepsilon _i,\varepsilon _j\), the Baker–Campbell–Hausdorff expansion gives
\[ \exp (\varepsilon _iX_i)\exp (\varepsilon _jX_j) = \exp \! \left( \varepsilon _iX_i+\varepsilon _jX_j +\frac{\varepsilon _i\varepsilon _j}{2}[X_i,X_j] +O(\varepsilon ^3) \right) \]
[ Van-Brunt and Visser , 2016 ] . Reversing the order flips the leading bracket term. Hence pairwise order discrepancy is second order in adapter strength when the linear field approximation is valid.
Proposition
13.1
Linear order bound
Let \(a_i=I+\varepsilon \Delta _i\) and \(a_j=I+\varepsilon \Delta _j\). Then
\[ a_ia_j-a_ja_i = \varepsilon ^2[\Delta _i,\Delta _j], \]
and therefore
\[ \lVert (a_ia_j-a_ja_i)h\rVert \le \varepsilon ^2 \lVert [\Delta _i,\Delta _j]\rVert \lVert h\rVert . \]
Proof
Expand both products and cancel their equal zeroth- and first-order terms.
Across nonlinear layers, the commutator remains a local proxy rather than an exact expression for output difference. The strength-sweep experiments test where the second-order approximation remains informative.