Fixed-regularization periodic PIC pair kernel

Fix $\epsilon >0$, $d\in \{1,2,3\}$, $L>0$, and a regularized one-species model with total charge $Q$ and total mass $M>0$. Let $c=Q\mathrm{/}M$. For each position $y\in {\mathbb{T}}_L^d$, define the continuous unit-source field by taking the zero-mean solution ${\varphi }_{\epsilon }^y$ from Periodic PIC regularization gives a smooth compatible Poisson solution for $\mu ={\delta }_y$, and setting ${\mathcal{E}}_{\epsilon }(x,y)={\int }_{{\mathbb{T}}_L^d}{\eta }_{\epsilon }(z-x)(-\mathrm{\nabla }{\varphi }_{\epsilon }^y(z))dz=\frac{1}{L^d}{\sum }_{k\in {\mathbb{Z}}^d\setminus \{0\}}\frac{-i{\kappa }_k}{\mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2}\left({\prod }_{a=1}^d{\alpha }_{\epsilon }^{2\mathrm{\mid }{k}_a\mathrm{\mid }}\right)e^{i{\kappa }_k\cdot (x-y)}\mathrm{.}$ The series converges absolutely with all derivatives by geometric summability. For the grid branch, let ${\varphi }_h^y$ be the mean-zero solution of $A_h{\varphi }_h^y=h^{-d}{w}^{\epsilon ,h}(y)-L^{-d}\mathbf{1}$ whose compatibility follows from Periodic PIC deposition conserves charge and satisfies Poisson compatibility, and put ${\mathcal{E}}_{\epsilon ,h}(x,y)={\sum }_{j\in J_N}{w}_j^{\epsilon ,h}(x)(-D^0{\varphi }_h^y{)}_j\mathrm{.}$ If $z=(x,v)$ and $z^{\mathrm{\prime }}=(y,w)$ are lifted phase-space states in ${\mathbb{R}}^{2d}$, the continuum and grid mean-field kernels are ${\mathbb{K}}_{\epsilon }(z,z^{\mathrm{\prime }})=(v,cQ{\mathcal{E}}_{\epsilon }(x,y)),{\mathbb{K}}_{\epsilon ,h}(z,z^{\mathrm{\prime }})=(v,cQ{\mathcal{E}}_{\epsilon ,h}(x,y))\mathrm{.}$ The spatial components are periodic in their position variables. A finite equal-weight system ${\dot{z}}_p=P^{-1}{\sum }_{r=1}^P{\mathbb{K}}_{\epsilon ,h}(z_p,z_r)$ uses particle charge $Q\mathrm{/}P$ and mass $M\mathrm{/}P$. No claim about the limit $\epsilon \to 0$ is part of this definition.