  This work improves the existing central limit theorems (CLTs) for geometric
functionals of Gibbs processes in three aspects. First, we derive a CLT for
weakly stabilizing functionals, thereby improving on the previously used
assumption of exponential stabilization. Second, we show that this CLT holds
for interaction ranges up to the percolation threshold of the dominating
Poisson process. This avoids imprecise branching bounds from graphical
construction. Third, by constructing simultaneous couplings of several Palm
processes for Gibbs functionals, we provide a quantitative CLT in terms of
Kolmogorov bounds for normal approximation. An important conceptual ingredient
in these advances is the extension of disagreement coupling adapted to
unbounded windows and to the comparison at multiple spatial locations.
