Accuracy of S¹ × ℝ² periodic summation methods. For all tests, the correction lookup table D(ρ, z) was computed on a 144 × 128 grid, varying the truncation parameter N_cut or the Ewald parameters α, n_max, and m_max. Using these tables, we reconstructed the full periodic force field F_{S¹ × ℝ²}(ρ, z) = F_ℝ³(ρ, z)+D(ρ, z) and compared it against a reference field. This reference field was generated via direct summation using N_cut = 10⁷ periodic images, requiring approximately 10⁴ s of computation time. Left: Mean force errors of direct periodic real-space summation as a function of N_cut. Although summation occurs only along a single direction (the z-axis), convergence remains exceptionally slow. Center: S¹ × ℝ² Ewald summation force accuracy as a function of the Ewald splitting parameter α with fixed n_max = 5 and m_max = 12 values. Using only 5 periodic images in real space and 12 modes in reciprocal space is sufficient to achieve single-precision (32-bit) floating-point accuracy when the optimal splitting parameter is applied. Right: Same as the center panel, but with fixed n_max = 7 and m_max = 14. Increasing both n_max and m_max by two improves the force accuracy by nearly three orders of magnitude (a factor of ∼600).