Lemma: Stencil existence

Such stencils exist, with at most $C_D{\delta }^{-(D-1)}+2D$ generators.

Proof. Choose a symmetric $\delta /2$-net $\{u_i\}$ of the sphere. For $D\ge 2$, a maximal separated set gives the stated count by packing disjoint spherical caps; for $D=1$, the sphere has two points.

Round $(L-\sqrt{D})u_i$ to an integer vector $s_i$. The error is at most $\sqrt{D}/2$, so

These vectors are long and nonzero. For $\rho =L-\sqrt{D}$, normalization changes direction by at most $\sqrt{D}/\rho \le \delta /2$. Together with the net error this gives a $\delta$-net. Adjoin negatives and the unit generators.