Research proof index

Fix a cube $J$ in the forest’s remaining-cube family. Consider grid cubes $I$ at relative scale $s(J)-s'$, disjoint from the top cube $\mathcal I(u_1)$, whose enlarged balls meet $J$. The normalized average over $J$ of the square of the number of those enlarged balls containing each point is bounded by the scale-dependent constant $C(a,s')$.