Metric-Dependent Active Inference: Gauge-Theoretic Degeneracy and the Geometric Extension of the Free Energy Principle

Scalar objective functions such as variational free energy in the Free Energy Principle (FEP) model cognition as Bayesian active inference within a fixed background geometry. However, they do not explicitly parameterize deformable state-space metrics, potentially conflating dimensional freedom with geometric rigidity. We formalize this limitation as Gauge-Theoretic Degeneracy: distinct network architectures may project onto identical scalar energy coordinates under mean-field approximations. The Λ³ framework introduces a geometric extension in which precision-weighting (Dₚ) functions as an endogenous metric deformation operator. We derive the Spectral Rank Contraction Theorem, linking localized curvature constraints to global dimensionality collapse via Cheeger conductance bounds and exact Ornstein–Uhlenbeck Lyapunov solutions. Empirical evaluation in ABIDE resting-state fMRI data (N=172) reveals a statistically robust inverse association (r = −0. 220, p = 0. 0037) between Forman–Ricci curvature and effective rank. Gaussian Mixture Modeling identifies two geometric biotypes (Silhouette = 0. 4409): a resilient high-dimensional regime and a contracted low-dimensional regime. The Iso-energetic Equipartition Theorem illustrates scalar invariance under mean-field energy preservation, while the Pharmacological Invariance Theorem bounds homogeneous linear scaling. The framework further predicts Topological Hysteresis consistent with Kramers-type escape dynamics. This geometric extension provides a formal treatment of scalar degeneracy and generates falsifiable predictions for geometry-informed computational psychiatry. Keywords: Active Inference, Information Geometry, Spectral Graph Theory, Effective Rank, Curvature, Topological Hysteresis.