By Law, Every Zero-Mean Risk Is the Difference of Two Equally Distributed Risks

What the math gives to ML

The paper gives an explicit coupling construction showing that any mean-zero scalar law can be realized as the difference of two identically distributed random variables. The transferable asset is the circle-rotation coupling: it preserves the marginal distribution of both endpoints while producing an exactly prescribed signed residual distribution. This suggests distribution-preserving residual or noise-injection modules in which a learned latent distribution is maintained at both endpoints while their difference follows a calibrated target law. The most direct experiment is to replace ordinary scalar noise or residual perturbations with this coupling and test whether marginal drift, calibration, or training stability improves.