Pairing Is a Coupling: Design Sensitivity in Activation Patching

Activation patching replaces an internal state on a corrupt prompt with the corresponding state from a clean prompt and averages the resulting score changes over paired examples. Although the two prompt pools may be fixed, the average also depends on which clean row is matched to which corrupt row, a choice left implicit in the activation-patching studies we reviewed. We formalize this matching as a coupling of the empirical marginals and prove an exact finite-pool criterion: the mean patch effect is pairing-invariant if and only if the component's unit-effect kernel is additively separable, or equivalently, every $2\times2$ mixed difference vanishes. With an admissibility mask, the analogous condition is a vanishing alternating sum around every cycle of the usable-support graph. Otherwise, the attainable means form an interval whose endpoints are permutations computable by linear assignment. Thus, once the kernel is available, evaluating the pairing-design sensitivity $\Gamma_j$ requires two assignment problems per component. A worked kernel gives mean effects of $6$ and $1$ under two legitimate pairings of the same rows, motivating the reporting of $(L_j,\theta_j,U_j)$ alongside any patching average.