Entropic Geometry and Symmetry Breaking in Lie-Group Free-Energy Minimization

We present a geometric formulation of entropic free-energy minimization as Riemannian gradient descent on Lie-group orbits endowed with the Fisher information metric. This approach reveals how symmetry structures constrain the dynamics of information and entropy reduction, linking variational inference to geometric thermodynamics. We establish well-posedness, Lyapunov monotonicity, and convergence theorems, and derive a second-variation criterion explaining entropic symmetry breaking and bifurcations. Examples on Gaussian families under translations and rotations illustrate the interplay between group invariance and adaptive stability. The results provide a unified view connecting information geometry, thermodynamics, and the Free Energy Principle through a group-theoretic lens.