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Let $h=\Delta x$, $k=\mu h$ and $M_r={\sum }_{j\in \{-2,-1,1,2\}}{a}_j{j}^r$. Along a smooth solution, ${\partial }_t^{r}u={\partial }_x^{r}u$. The exact difference $u(x,t+k)-u(x,t-k)$ contains only odd derivatives. Matching terms through degree four therefore requiresThe first two conditions give $a_{-1}+a_1=a_{-2}+a_2=0$, which also implies $M_4=0$. With $a_{-j}=-a_j$, the remaining equations are $a_1+2a_2=\mu$ and $a_1+8a_2={\mu }^3$. HenceThe fifth spatial moment is $M_5=10{\mu }^3-8\mu$. After dividing the exact-minus-scheme residual by $2k$, its first possible nonzero term isThis is the fourth-order two-step advection stencil. Thus the normalized local truncation error is $O(h^4)$ at fixed positive Courant number, as required. At $\mu =1$ or $2$ the formula translates the physical advection branch exactly, but consistency of a numerical method alone does not decide stability of all two-level perturbations.