  In the context of stochastic homogenization, the Bourgain-Spencer conjecture
states that the ensemble-averaged solution of a divergence-form linear elliptic
equation with random coefficients admits an intrinsic description in terms of
higher-order homogenized equations with an accuracy four times better than the
almost sure solution itself. While previous rigorous results were restricted to
a perturbative regime with small ellipticity ratio, we prove the first half of
this conjecture for the first time in a non-perturbative setting. Our approach
involves the construction of a new corrector theory in stochastic
homogenization: while only a bounded number of correctors can be constructed as
stationary $L^2$ random fields, we show that twice as many stationary
correctors can be defined in a Schwartz-like distributional sense on the
probability space.
