\label{def:computational-bi-square}
Let $\Pi$ be a $\tau$-admissible NP problem
(Definition~\ref{def:tau-admissibility}) with TTM verifier $V$
(Definition~\ref{def:tau-tower-machine}) of interface width $k_0$.
The \textbf{computational bi-square} at depth $k \geq k_0$ is
\begin{equation}\label{eq:ch58-computational-bisquare}
    \begin{tikzcd}[column sep=3.5em, row sep=2.5em]
        V\bigl(\operatorname{Prim}(k{+}1)\bigr)
          \arrow[r, "\mathrm{res}_k"]
          \arrow[d, "\chi_\pm^{(k+1)}"']
        & V\bigl(\operatorname{Prim}(k)\bigr)
          \arrow[r, "\mathrm{wit}_k"]
          \arrow[d, "\chi_\pm^{(k)}"]
        & W(x, k)
          \arrow[d, "\pi_{\mathrm{CRT}}"]
        \\
        \chi_\pm \circ V\bigl(\operatorname{Prim}(k{+}1)\bigr)
          \arrow[r, "\mathrm{res}_k"']
        & \chi_\pm \circ V\bigl(\operatorname{Prim}(k)\bigr)
          \arrow[r, "\mathrm{wit}_k"']
        & {\displaystyle\prod_{i=1}^{k}} W(x, p_i)
    \end{tikzcd}
\end{equation}
where $V(\operatorname{Prim}(k))$ is the verifier's computation at
depth $k$ (a presheaf value), $\mathrm{res}_k$ is the tower restriction,
$\chi_\pm^{(k)}$ are the spectral characters,
$\mathrm{wit}_k$ extracts the witness
(Definition~\ref{def:np-witness-canonical-address}),
and $\pi_{\mathrm{CRT}}$ is the CRT decomposition
(Proposition~\ref{prop:crt-witness-decomposition}).
The left square is the \textbf{execution square};
the right square is the \textbf{witness square}.