# Concise takeaways

- Evaluate the entire input score at W+iV, with independent real and imaginary Gaussian errors sharing the known covariance. Rotational invariance plus holomorphic mean value yields exact cancellation under integrability; covariance may be singular.
- Conditional centering of correction error given the clean observation eliminates its cross term with clean sampling noise. Input growth, amplitude control and clean-score variance are separate assumptions.
- For n observations and m simulations per observation, variance is v_obs/n+v_sim/(nm). Simulation averaging alone leaves observation variability.
- For step exponent 1/2<alpha<1, uniform PR averaging has covariance H^{-1}S_*H^{-1}. At alpha=1 use the manuscript's separate kernel covariance and spectral condition c lambda_min(H)>1/2.
- The correction identity is classical. Neural-network experiments do not verify the global envelope assumptions; non-Gaussian extensions require the stated conjugate-law integrability.