Forward path law is admissible for the finite masked model

The forward survival ratios telescope to ${\prod }_{j=1}^N{r}_j={\alpha }_N=0$, so for each coordinate the event of remaining clean at $N$ has probability zero; hence every $q_x$-positive path has $z_N=m^L$, while the defining path mass forces $z_0=x$. Fix such a path. A clean-to-clean forward transition copies its token, and the learned reverse row assigns that transition probability one. A clean-to-mask transition has learned reverse probability $c_j{h}_{i,j}(z_j)(x_i)>0$. A mask-to-mask transition can occur only for $j\ge 2$, because $z_0=x$ is clean; then ${\alpha }_{j-1}<1$ and its learned reverse probability $d_j=(1-{\alpha }_{j-1})\mathrm{/}(1-{\alpha }_j)>0$. These are all positive-probability forward transitions by [F1]. Thus every factor in the learned reverse path mass is positive, and the terminal prior and clean-state decoder are also positive; hence ${\gamma }_x(z_0,\dots ,z_N)>0$.

By [F3], at least one path has positive $q_x$ mass. Step 1.1 makes its joint model mass positive, so $Z_x>0$; [F4] gives $Z_x\le 1$. On every $q_x$-positive path, step 1.1 gives ${\gamma }_x>0$, hence ${\pi }_x>0$ and $q_x\ll {\pi }_x$.

The path space is finite. On each of its finitely many $q_x$-positive states, both $q_x(z)$ and ${\gamma }_x(z)$ are positive finite real numbers, so both logarithms are finite. Their finite weighted sum is finite; on $q_x=0$ states the variational convention contributes zero. This verifies [F5] and proves admissibility.