= Solution
There is a genuine indexing defect in the printed PDF: its new value is at $m+1$, while every old-time coefficient is centered at $m$. On a fixed grid the literal residual contains
$$
\frac{u(x+\delta,t+k)-u(x,t)}k-L_\delta u(x,t)
=\frac\delta k u_x+O(1)+O(k)+O(\delta^2),\qquad k=\mu\delta^2.
$$
For a general solution the leading term is $u_x/(\mu\delta)$, so this is not a consistent diffusion discretization. It also leaves one new interior value unspecified while imposing a generally incompatible update on a prescribed boundary value. The calculation that follows is for the intended same-node update $U_m^{n+1}=U_m^n+kL_\delta U_m^n$; that correction is necessary, not an unnoticed transcription change.
For the <midpoint flux stencil for one-dimensional diffusion>, define
$$
L_\delta u(x)=\frac{a(x-\delta/2)[u(x-\delta)-u(x)]+a(x+\delta/2)[u(x+\delta)-u(x)]}{\delta^2}.
$$
Assume sufficient smoothness of $a$ and the solution for the displayed remainders. Multiplying the centered <Taylor expansions> of $a$ and $u$ gives
$$
L_\delta u=(au_x)_x+\delta^2\left(\frac{a u_{xxxx}}{12}+\frac{a'u_{xxx}}6+\frac{a''u_{xx}}8+\frac{a^{(3)}u_x}{24}\right)+O(\delta^4).
$$
The first time difference is $(u(t+k)-u(t))/k=u_t+ku_{tt}/2+O(k^2)$. After using the <diffusion equation>, the <normalized local truncation error> is
$$
\tau_m^n=\frac{k}{2}u_{tt}-\delta^2\left(\frac{a u_{xxxx}}{12}+\frac{a'u_{xxx}}6+\frac{a''u_{xx}}8+\frac{a^{(3)}u_x}{24}\right)+O(k^2+\delta^4).
$$
Therefore, for the corrected update,
$$
\boxed{\tau=O(k+\delta^2),\qquad d=k\tau=O(k^2+k\delta^2).}
$$
The first is the residual divided by the time step; the second is the unnormalized one-step defect. At fixed $\mu=k/\delta^2$ they are respectively $O(\delta^2)$ and $O(\delta^4)$. A rate based on derivatives is not justified for a merely bounded nonsmooth $a$.
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