A Savitch recursion that never materializes the full configuration graph

Take a graph with start vertex $s$, target vertex $t$, and many possible midpoints $z_1,\dots ,z_m$, but suppose only $z_7$ actually lies on a short path from $s$ to $t$. This is a legitimate instance of the claim in [L1].

Savitch's recursion tests the midpoint candidates one by one. At any moment it stores only the current pair of endpoints, the current candidate $z_j$, and the recursion stack for the shorter subproblems. When the test for $z_j$ fails, that local data is discarded before $z_{j+1}$ is tried. Hence no stage materializes the whole graph at once, even though the algorithm still eventually finds the successful midpoint $z_7$.