\label{prop:poincare-gluing-guarantee}
Let $M$ be a closed $\tau$-manifold of dimension 3 arising from the gluing of local Hartogs bulks at the $E_0 \to E_1$ interface. If every loop of transition functions is contractible (i.e., $M$ is simply connected in the sense of Definition~\ref{def:simply-connected-tau}), then $M$ is homeomorphic to the 3-sphere $S^3$.

[Proof Strategy]
The proof proceeds in three stages, following the Mutual Determination template:

\textbf{Stage 1: Boundary-to-Interior.}
Given that all loops of transitions are contractible, the universal covering presheaf is trivial. By the Global Hartogs Theorem (I.T31), this forces the global topology to have no non-trivial local obstructions.

\textbf{Stage 2: Spectral Algebra Input.}
The spectral algebra $A_{\mathrm{spec}}(\mathbb{L})$ decomposes all transition functions into $\chi_+$ and $\chi_-$ components. Simple connectivity implies that both components factor through the trivial representation of $\pi_1$. This is the ``spectral purity'' condition.

\textbf{Stage 3: Interior Reconstruction.}
From spectral purity, the gluing data simplifies to a unique configuration. By the classification of closed 3-manifolds in the earned topology (a consequence of the Geometrization Theorem, which Book~III treats as \emph{established} classical input), the only simply connected closed 3-manifold is $S^3$.