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\label{thm:torus-degeneration}
The pinch map
$p : T^2 \to \mathbb{L} = S^1 \vee S^1$
is the unique continuous surjection
from~$T^2$ to a one-dimensional compact connected target
satisfying constraints \textup{(U1)--(U5)}.

Each constraint eliminates alternatives.

\emph{Step~1: Target topology.}
By (U1), (U2), (U5), the target is a compact connected
one-dimensional Hausdorff space---hence
homeomorphic to $S^1$, $[0,1]$,
or a finite graph.

\emph{Step~2: Gauge survival eliminates simple targets.}
By (U3), both $U(1)_\gamma$ and $U(1)_\eta$
must survive independently.
A single $S^1$ admits only one independent
$U(1)$-action; $[0,1]$ admits none.
The target must contain at least two circle factors.

\emph{Step~3: Connectivity forces the wedge.}
By (U2), the target is connected.
Two disjoint circles fail connectivity.
The minimal connected space
containing two independent circles
is $S^1 \vee S^1$.
Gluing along arcs would destroy
a $U(1)$-action or reduce dimension further.

\emph{Step~4: The identification is the diagonal.}
The quotient must collapse a one-dimensional subspace
to the wedge point.
Collapsing a $U(1)$-orbit $\{\theta_\gamma = \mathrm{const}\}$
or $\{\theta_\eta = \mathrm{const}\}$
would destroy that gauge factor.
The diagonal $\Delta = \{(\theta, \theta)\}$
is the unique closed one-dimensional submanifold
transverse to both gauge orbits,
so collapsing it preserves both actions.

\emph{Step~5: Continuity.}
The quotient $T^2 \to T^2/\Delta$ is continuous
by the universal property.
Since $\Delta$ is closed in the compact Hausdorff space $T^2$,
the quotient is Hausdorff.