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\label{thm:boundary-minimality}
The boundary lemniscate $\mathbb{L} = S^1 \vee S^1$
is the \textbf{minimal} topological quotient
of the fiber torus~$T^2$
satisfying three constraints:
\begin{enumerate}
    \item[\textup{(M1)}] \textbf{Gauge preservation.}
          Both $U(1)_\gamma$ and $U(1)_\eta$
          survive as closed subgroups.
    \item[\textup{(M2)}] \textbf{Codimension increase.}
          $\dim Q < \dim T^2 = 2$.
    \item[\textup{(M3)}] \textbf{Crossing point.}
          The quotient admits a unique singular point
          where the two gauge factors meet.
\end{enumerate}
Any quotient~$Q$ of~$T^2$
satisfying \textup{(M1)--(M3)}
contains a homeomorphic copy of~$\mathbb{L}$.

\emph{Step~1.}
Constraint~(M2) forces $\dim Q \leq 1$,
so the quotient must collapse at least one direction.

\emph{Step~2.}
If $S^1_\gamma$ collapses to a point,
$U(1)_\gamma$ acts trivially on~$Q$,
violating~(M1).
Symmetrically for $S^1_\eta$.
Hence both circles survive.

\emph{Step~3.}
Both circles survive with $\dim Q = 1$.
Without a shared point,
$Q \supseteq S^1 \sqcup S^1$ is disconnected
with no crossing point, violating~(M3).
A single shared point yields
$Q \supseteq S^1 \vee S^1 = \mathbb{L}$.

\emph{Step~4.}
If $Q$ had fewer points than $\mathbb{L}$,
some identification would collapse
a portion of one $S^1$ factor, violating~(M1).
Hence $\mathbb{L}$ is minimal.