Uniform discrete derivative bound for the function $p_3$

Let $\text{\hminfamily{T}}_0^r$ be an isoradial Delaunay triangulation of the plane with isoradius $r$. For points $\mathtt{u},\mathtt{v}$ in this triangulation, define $p_3(\mathtt{u},\mathtt{v})$ by

$$p_3(\mathtt{u},\mathtt{v})=\sum_{i=1}^{2n} \mathrm{e}^{\mathrm{i}3\theta_i},\qquad \theta_i=\arg(z_i-z_{i-1}),$$

where $(z_0,z_1,\ldots,z_{2n})$ is a path on the rhombic lattice obtained from $\text{\hminfamily{T}}_0^r$, from $z_0=z(\mathtt{u})$ to $z_{2n}=z(\mathtt{v})$. For any nondegenerate triangle $\mathtt{t}$ in $\text{\hminfamily{T}}_0^r$, let $\nabla p_3(\mathtt{t})$ and $\overline\nabla p_3(\mathtt{t})$ denote the discrete derivatives with respect to $\mathtt{u}$ at the face $\mathtt{t}$.

Uniform discrete derivative bound. There is a uniform bound

$$|\nabla p_3(\mathtt{t})|\quad\text{and}\quad|\overline\nabla p_3(\mathtt{t})|\leq \mathtt{cst.}\,\frac{R(\mathtt{t})}{r^2},$$

where $\mathtt{cst.}=\mathrm{O}(1)$ is independent of the critical triangulation and of $\mathtt{t}$. The authors report $\mathtt{cst.}=6$ in the examples studied.

The bound is intended to control potentially nonuniform contributions from triangles with large circumradius or small area in perturbative estimates. Its uniform validity is supported by experimental studies and analytical estimates, but no proof or resolution is supplied here.