Metastable Transitions and $\Gamma$-Convergent Eyring-Kramers Asymptotics in Landau-QCD Gradient Systems

We develop a rigorous analytical framework for metastable stochastic transitions in Landau-type gradient systems inspired by QCD phenomenology. The functional $F(\sigma;u)=\int_\Omega [\frac{\kappa}{2}|\nabla\sigma|^2+V(\sigma;u)]\,dx$, depending smoothly on a control parameter $u\in\mathcal U$, is analyzed through the Euler-Lagrange map $\mathcal{E}(\sigma;u)=-\kappa\Delta\sigma+V'(\sigma;u)$ and its Hessian $\mathcal{L}_{\sigma,u}=-\kappa\Delta+V''(\sigma;u)$. By combining variational methods, $\Gamma$- and Mosco convergence, and spectral perturbation theory, we establish the persistence and stability of local minima and index-one saddles under parameter deformations and variational discretizations. The associated mountain-pass solutions form Cerf-continuous branches away from the discriminant set $\mathcal D=\{u:\det\mathcal L_{\sigma,u}=0\}$, whose crossings produce only fold or cusp catastrophes in generic one- and two-parameter slices. The $\Gamma$-limit is taken with respect to the $L^2(\Omega)$ topology, ensuring compactness, convergence of gradient flows, and spectral continuity of $\mathcal L_{\sigma,u}$. As a consequence, the Eyring-Kramers formula for the mean transition time between metastable wells retains quantitative validity under both parameter deformations and discretization refinement, with convergent free-energy barriers, unstable eigenvalues, and zeta-regularized determinant ratios. This construction unifies the classical intuition of Eyring, Kramers, and Langer with modern variational and spectral analysis, providing a mathematically consistent and physically transparent foundation for metastable decay and phase conversion in Landau-QCD-type systems.