Cost Geometry Determines Causal Allocation in Optimal Transport-Identified Counterfactuals

Backtracking counterfactuals seek alternative worlds that satisfy a desired outcome while keeping the causal mechanisms fixed. Existing approaches select among feasible worlds using distances, naturalness, or related cost criteria, but the causal role of this choice remains unclear. Recent optimal transport-based identification results provide conditions under which the underlying counterfactual mechanism can be fixed, allowing us to isolate the remaining world-selection geometry. We show that this geometry controls \emph{causal allocation}, the division of a required change between upstream causes and local disturbances. We characterize this allocation from scalar to multivariate settings. In scalar mechanisms, geometry determines the upstream allocation fraction; in multiple dimensions, this generalizes to a direction-dependent allocation operator with strengths $\lambda_i=\sigma_i^2/(1+\sigma_i^2)$. Controlled experiments validate these predictions with analytic and learned transport mechanisms. These results show that even when the counterfactual mechanism is identified, cost geometry remains a distinct determinant of backtracking behavior.