\label{thm:hartogs-flow-theorem}
Let $f_0$ be $\tau$-admissible initial data on a clopen cylinder domain $U \subset \tau^3$. Then the Hartogs flow operator $H_{\mathrm{flow}}$ satisfies:
\begin{enumerate}
\item[\emph{(i)}] \textbf{Unique continuation.} At each primorial level $\mathrm{Prim}(n)$, the extension $\mathrm{Ext}_n(f_0|_{\partial U_n})$ is the unique $\tau$-holomorphic function on $U_n$ restricting to $f_0|_{\partial U_n}$ on the boundary.

\item[\emph{(ii)}] \textbf{Tower coherence.} The continuations are compatible across primorial levels: for every $n \geq 1$, the diagram
\begin{equation}
\begin{tikzcd}[column sep=large]
\Gamma(\partial U_{n+1}, \mathcal{O}_{H_\tau}) \ar[r, "\mathrm{Ext}_{n{+}1}"] \ar[d, "\mathrm{res}"]
& \Gamma(U_{n+1}, \mathcal{O}_{H_\tau}) \ar[d, "\mathrm{res}"] \\
\Gamma(\partial U_n, \mathcal{O}_{H_\tau}) \ar[r, "\mathrm{Ext}_n"]
& \Gamma(U_n, \mathcal{O}_{H_\tau})
\end{tikzcd}
\label{eq:ch36-tower-coherence}
\end{equation}
commutes, where $\mathrm{res}$ denotes restriction along the tower map $\mathrm{Prim}(n{+}1) \to \mathrm{Prim}(n)$.

\item[\emph{(iii)}] \textbf{Sector preservation.} The continuation preserves the spectral trichotomy (Theorem~III.T14): if $f_0$ belongs to sector $S \in \{B, C, X\}$, then $H_{\mathrm{flow}}(f_0)$ belongs to sector $S$.
\end{enumerate}