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Let $S_{1}u$ be the sum of the four axial neighbors and $S_{2}u$ the sum of the four diagonal neighbors. The two PDF stencils defineThe reaction stencil has four axial weights $1/12$ and center weight $2/3$, so its weights sum to one. Taylor expansion givesFor example, the diagonal neighbors supply the $u_{xxyy}$ term needed to turn the fourth derivatives into the full squared Laplacian. Adding the reaction contribution producesOn a smooth solution of the constant-coefficient equation, both displayed leading terms vanish. Thus the compact fourth-order Helmholtz stencil hasThe cancellation uses the differential equation and constant $\lambda$; the nine-point finite-difference stencil alone still has an $O(h^2)$ operator error on an arbitrary smooth function. It also does not establish a uniform global fourth-order error without an appropriate inverse bound and compatible treatment of boundaries and resonances.