variance_eq_half_resample_difference_pi

Global resampling (symmetrization) identity on a product cube. For a square-integrable functional $Z \in L^2(\mu)$ of independent coordinates ($\mu = \bigotimes_i \mu_i$ a product of probability measures over a finite index set), the variance equals one half of the expected squared difference of $Z$ evaluated at two independent draws $\omega, \omega' \sim \mu$:

This is the resampling identity $\operatorname{Var}(W) = \tfrac12 \mathbb{E}[(W-W')^2]$ for an independent copy, instantiated on the product space $\mu \otimes \mu$ with $W(\omega,\omega') = Z(\omega)$ and $W'(\omega,\omega') = Z(\omega')$: these are independent (disjoint factors) and identically distributed (both are $Z$ pushed by the measure-preserving coordinate projections). It is the all-coordinates-at-once form of the Efron–Stein resampling principle; the per-coordinate refinement replaces the full independent copy $\omega'$ by a single-coordinate resample, yielding $\operatorname{Var}(Z) \le \tfrac12 \sum_i \mathbb{E}[(Z - Z'_i)^2]$.