Projection weak homotopy equivalence conjecture for twisted master operators

Let $\Gamma$ be the finite graph under consideration, let $\mathcal M_{\Gamma}$ be its parameter space, and let $\tilde{\mathcal M}_{\Gamma}\subset \mathbb R_+\times\mathcal M_{\Gamma}$ consist of those $(\beta_0,E,W)$ for which, for every $\lambda\in C^1(\Gamma;U(1))$ and every $\beta\geq\beta_0$, the twisted master operator has a non-degenerate ground state. Let

$$\pi:\tilde{\mathcal M}_{\Gamma}\to\mathcal M_{\Gamma},\qquad (\beta_0,E,W)\mapsto(E,W),$$

and let $\check{\mathcal M}_{\Gamma}$ be the image of $\pi$. Projection weak homotopy equivalence conjecture. The map

$$\pi:\tilde{\mathcal M}_{\Gamma}\to\check{\mathcal M}_{\Gamma}$$

is a weak homotopy equivalence. This identifies the analytically defined parameter space with its image up to weak homotopy, enabling the subsequent construction of a weak complex line bundle and its associated topological invariants.