Non-Tonelli Finsler Geometry of Exotic Superconductivity: Metastable Vortex Phases and Geometric Phase Transitions

We develop a thermally coupled Ginzburg-Landau theory on Weakly Non-Tonelli (WNT) Finsler manifolds, extending classical vortex analysis beyond the Tonelli convexity paradigm. The WNT framework weakens global $1$-homogeneity and strict convexity while preserving superlinearity and local ellipticity, enabling a geometric treatment of superconductors whose anisotropic energy landscapes are nonconvex and temperature-dependent. Within this setting, we construct the generalized Legendre correspondence, Hamiltonian metric, and WNT Laplacian, proving existence and sharp Coulomb asymptotics of the three-dimensional Green kernel. We then establish the $Γ$--convergence of the WNT-GL energy and identify metastable vortex filaments minimizing a renormalized geometric functional. Finally, a dynamic $Γ$-limit yields an effective filament flow governed by the WNT Finsler curvature and thermally induced geometric forces, predicting curvature focusing and phase bifurcation at a critical transition temperature Tc. This theory unifies convex-analytic, geometric, and physical perspectives, showing that Non-Tonelli Finsler structures form a natural analytic bridge between classical Finsler geometry, anisotropic variational models, and the nonlinear thermodynamics of exotic superconductivity.