Asymptotic mean value Laplacian on equiregular sub-Riemannian manifolds

Let $(M,\mathcal{D},g)$ be a smooth equiregular sub-Riemannian manifold equipped with a smooth positive measure $\mu$. We study the small-scale limit of the metric-ball mean-value operator $$A_{h}f(x)=\frac{1}{h^2\mu (B(x,h))}{\int }_{B(x,h)}(f(q)-f(x))d\mu (q)\mathrm{.}$$ Exact homogeneity yields the pointwise limit on Carnot groups. On a general equiregular manifold, convergence for every smooth test function is equivalent to convergence of the rescaled horizontal first moments of metric balls in first-kind privileged coordinates. This criterion is independent of $\mu$; when it holds, the principal symbol is determined by the normalized second-moment tensor of the tangent unit ball, and the drift satisfies an explicit change-of-measure formula. In step at most two, we verify the criterion by combining a real-analytic finite-jet reduction with tame integration, which rules out oscillation of the normalized moments. We also compute the limit on Lie groups and prove unconditional distributional convergence of the volume-weighted operators on every equiregular manifold.