Diffusive in plain sight: An inconspicuous law of market impact

Abstract

Decomposing impact as the difference between realized and counterfactual returns and requiring both to be diffusive yields an identity that restricts admissible impact scaling at the level of individual participants. This constraint implies the square-root law in the information-neutral regime and a crossover to linear impact under strong informational coupling, consistent with empirical observations. In the weak-coupling regime, cumulative market impact is itself diffusive -- a diagnostic that many propagator and latent liquidity models fail to satisfy.