= Solution
Take the endpoint values $u_0=u_M=0$ and evolve the interior indices $1,\ldots,M-1$. This matches $\Delta x=1/M$ on $[0,1]$. If the printed upper index $m=M$ is retained, the odd ghost extension $u_{M+1}=-u_{M-1}$ makes its boundary equation identically compatible; it does not add an interior unknown.
Writing $h=\Delta x$ and $\tau=\Delta t$, this is the <theta method> with parameter $a$:
$$
\frac{U^{n+1}-U^n}{\tau}=aD_{xx}U^{n+1}+(1-a)D_{xx}U^n.
$$
For a smooth exact heat solution, its normalized residual at $t_n$ is
$$
\mathcal R=(\tfrac12-a)\tau u_{xxxx}-\frac{h^2}{12}u_{xxxx}
+O(\tau^2+\tau h^2+h^4).
$$
Thus the generic consistency orders are \b[first in time and second in space for $a\ne1/2$], and \b[second in both time and space for $a=1/2$], the <Crank-Nicolson method>. These are independent-step orders. Along the special parabolic refinement $\tau=\mu h^2$, the leading combined residual cancels if $a=1/2-1/(12\mu)$, giving a higher combined smooth-data order; it does not change the generic temporal order at independently varied meshes.
The given <L2 initial condition> need not have pointwise values or four derivatives. Initialize by a bounded cell-average or projection operator, as in <L2-compatible initialization of grid data>. The Taylor estimate applies to smooth data or sufficiently positive times after heat smoothing, rather than an unqualified uniform high-order estimate at the initial instant for arbitrary rough data. Stability and convergence for rough data are distinct from this consistency order.
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