A tableau is legal exactly when the start row is correct and every local window is legal

If $T$ is an actual computation history, then its first row is the start configuration. Each pair of adjacent rows is either one legal transition or a repeated halting row, so every padded $2\times 3$ window is legal by [L2].

Conversely, assume the two bullet conditions in the statement hold. In particular, every local window is legal in the sense of [L2].

Fix adjacent rows $r$ and $r+1$. If row $r$ is already halting, [L2] forces row $r+1$ to repeat it exactly. Otherwise row $r$ has a unique head neighborhood, and the overlapping legal windows force every cell outside that neighborhood to stay unchanged while the local neighborhood realizes exactly one allowed transition of $N$. So row $r+1$ is either the same halting configuration or the one-step successor of row $r$.

Because the first row is correct and every later row follows the previous one in one of the two ways allowed by step 2.1, the whole tableau is a valid branch computation history in the sense of [L1]. Hence [L1] gives the converse implication.