= Solution
Put $\tau=\Delta t$ and $h=\Delta x$, with $h^2=\tau/\mu$ for fixed $\mu>0$. Substitute a smooth exact solution of the <heat equation> into the <Adams-Bashforth method>. Define its one-step defect $R$ as the left side minus the right side of the unscaled update. The centered <finite difference> satisfies
$$
D_hu=\frac{u(x-h,t)-2u(x,t)+u(x+h,t)}{h^2}=u_{xx}+\frac{h^2}{12}u_{xxxx}+O(h^4).
$$
The <Taylor expansion> in time gives
$$
u(t+\tau)-u(t)=\tau u_t+\frac{\tau^2}{2}u_{tt}+O(\tau^3),
$$
$$
\tau\left[\tfrac32D_hu(t)-\tfrac12D_hu(t-\tau)\right]=\tau u_{xx}+\frac{\tau^2}{2}u_{xxt}+\frac{\tau h^2}{12}u_{xxxx}+O(\tau^3+\tau^2h^2+\tau h^4).
$$
Since $u_t=u_{xx}$ and $u_{tt}=u_{xxt}$, the first two terms cancel, leaving
$$
\boxed{R=-\frac{\tau^2}{12\mu}u_{xxxx}+O(\tau^3)=O(\tau^2).}
$$
This is the requested local error of the unscaled update. If “local truncation error” denotes the defect divided by $\Delta t$, it is instead $O(\Delta t)$ under fixed $\mu$: the centered spatial error is then $O(h^2)=O(\Delta t)$. The scheme is second order in time at fixed spatial discretization, which is a separate order statement.
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