  We analyze nonequilibrium fluctuations of the averaging process on $\mathbb
T_\varepsilon^d$, a continuous degenerate Gibbs sampler running over the edges
of the discrete $d$-dimensional torus. We show that, if we start from a smooth
deterministic non-flat interface, recenter, blow-up by a non-standard
CLT-scaling factor $\theta_\varepsilon=\varepsilon^{-(d/2+1)}$, and rescale
diffusively, Gaussian fluctuations emerge in the limit $\varepsilon\to 0$.
These fluctuations are purely dynamical, zero at times $t=0$ and $t=\infty$,
and non-trivial for $t\in (0,\infty)$. We fully determine the correlation
matrix of the limiting noise, non-diagonal as soon as $d\ge 2$. The main
technical challenge in this stochastic homogenization procedure lies in a LLN
for a weighted space-time average of squared discrete gradients. We accomplish
this through a Poincar\'e inequality with respect to the underlying randomness
of the edge updates, a tool from Malliavin calculus in Poisson space. This
inequality, combined with sharp gradients' second moment estimates, yields
quantitative variance bounds without prior knowledge of the limiting mean. Our
method avoids higher (e.g., fourth) moment bounds, which seem inaccessible with
the present techniques.
