  We establish higher regularity properties of solutions to fully nonlinear
elliptic equations at interior critical points. The key novelty of our
estimates lies in the fact that they yield smoothness properties that go beyond
the inherent regularity limitations dictated by the heterogeneity of the
problem. We explore various scenarios, revealing a plethora of improved
regularity estimates. Notably, depending on the model's parameters, we
establish estimates that transcend the natural regularity regime of the model,
from $C^{0,\alpha_0}$ to $C^{1,\alpha_1}$ and further to $C^{2,\alpha_2}$, with
the potential for even higher estimates.
