Classical RK4 step for the fixed-grid regularized PIC particle ODE

Input an odd periodic grid with $N\ge 3$, a fixed $\epsilon >0$, particle count $P\ge 1$, total charge $Q$, total mass $M>0$, a state $Z=(z_1,\dots ,z_P)$ with $z_p=(X_p,V_p)\in {\mathbb{R}}^{2d}$, and step $\mathrm{\Delta }t>0$. The grid, mass, and shape must satisfy the domains in Periodic uniform PIC grid and inner product, Fixed-regularization periodic PIC pair kernel, and Periodic PIC deposition shape and gather operators; otherwise return status invalid-input. Use the exact-evaluation phase field of Fixed-regularization periodic PIC pair kernel: $F_p(Z)=\frac{1}{P}{\sum }_{r=1}^P{\mathbb{K}}_{\epsilon ,h}(z_p,z_r)\mathrm{.}$ At each of four classical RK stages, evaluate this complete field at that stage's particle state: deposit every charge $Q\mathrm{/}P$, subtract the neutralizing background, solve its zero-mean Poisson system with Fourier diagonal solver for the periodic centered-difference Poisson equation, compute $E_h=-D^0{\varphi }_h$, and gather $E_h$ at each stage position $X_p$. Multiply the gathered field by $c=Q\mathrm{/}M$ and return $F_p=(V_p,c(G_X{E}_h{)}_p)$, using that stage's velocity $V_p$. With $k_1=F(Z)$, $k_2=F(Z+\frac{\mathrm{\Delta }t}{2}{k}_1)$, $k_3=F(Z+\frac{\mathrm{\Delta }t}{2}{k}_2)$, and $k_4=F(Z+\mathrm{\Delta }t{k}_3)$, return $Z^{+}=Z+\frac{\mathrm{\Delta }t}{6}(k_1+2k_2+2k_3+k_4)\mathrm{.}$ Every stage source is compatible by Periodic PIC deposition conserves charge and satisfies Poisson compatibility, and The periodic PIC Poisson algorithm returns the unique gauged solution supplies each exact finite field solve. Thus no field-solve-failure branch occurs for admissible input. All operations use exact real or complex arithmetic; no floating-point stability or error bound is asserted.