Paper 2 - Exact Critical Orbit Classification of a Truncated G2-Invariant Sextic Potential

We study a coercively regularized adjoint-invariant sextic potential on the compact real form of the exceptional Lie algebra of type \(G_2\), viewed as a \(14\)-dimensional real vector space. The unregularized model \[ V(\Phi)=\mu^2 I_2(\Phi)+\kappa I_2(\Phi)^2+\lambda I_6(\Phi) \] is not bounded below when \(\lambda\neq 0\). Adding the stabilizing term \(\nu I_2(\Phi)^3\) with \(\nu>|\lambda|\) restores coercivity and produces a well-defined Gibbs family for every inverse temperature \(\beta>0\). After reduction to the Cartan subalgebra and angular minimization, the ground-state problem becomes a one-dimensional cubic minimization in \(u=r^2\): \[ f(u)=\mu^2u+\kappa u^2+\Delta u^3,\qquad \Delta:=\nu-|\lambda|>0. \] This reduced model exhibits a first-order level crossing at \[ \kappa_c=-2\sqrt{\Delta\mu^2}, \] where the global minimizer moves from the origin to a singular nonzero \(G_2\)-orbit of dimension \(10\). The main results are as follows. Sections~\ref{sec:existence}--\ref{sec:groundstate} give rigorous proofs of the Gibbs measure existence and the ground-state level crossing. Sections~\ref{sec:rigorous_asymptotics}--\ref{sec:thermo_kl} then provide a detailed asymptotic analysis based on uniform Laplace methods, yielding exact scaling laws and universal information-geometric signatures.In particular, we derive a boundary‑Laplace low‑temperature expansion for the partition function, with the origin and the nonzero orbit contributing on different asymptotic scales. The Fisher information for the parameter \(\kappa\) displays two regimes: a broken-phase law \(g_{\kappa\kappa}=\beta\,\mathcal{A}(\kappa)+O(1)\) for \(\kappa\kappa_c\). In the crossover window, the metric is governed by a shifted logistic profile whose weak limit contains a Dirac mass of weight \(\mu^2/\Delta\) at \(\kappa_c\). Finally, we prove that the local thermodynamic length contributed by the crossover equals \(\pi\), a universal constant independent of the model parameters. The leading contribution to the three-parameter Fisher information matrix in coordinates \((\mu^2,\kappa,\Delta)\) is rank one, so any scalar-curvature calculation must begin from subleading terms.