Resilient Cognitive Architectures via Anosov Flows on Lorentz Manifolds: A Topological Approach to Functional Identity Preservation

Hierarchical data structures fundamental to language and reasoning exhibit exponential volume growth incompatible with the polynomial capacity of Euclidean embedding spaces, causing systematic representation collapse (over-squashing) in deep architectures. While static hyperbolic models—including Lorentzian Residual Networks (LResNet) and Intrinsic Lorentzian Neural Networks (ILNN)—partially address this geometric capacity bottleneck, they lack mechanisms for maintaining structural integrity under perturbation. We introduce a unified framework integrating Lorentz geometry, Anosov dynamical systems, and persistent homology within active inference. The Minimal Mathematical Model of Functional Dynamics (MMFD) operates as a second-order system on Ln where internal states evolve as structurally stable Anosov flows, regulated by a topological loss Ltopo . Our central theoretical contribution (Theorem 4.4) proves that Ltopo acts as a Lyapunov function with convergence rate µ > 0 bounded below by the Forman-Ricci curvature κF of the embedding graph—transforming topological data analy sis into a control-theoretic guarantee. Empirical validation demonstrates convergence rates of µ = 0.84 ± 0.02 (R2 = 0.98) for the hybrid model versus µ = 0.11 ± 0.03 for static hyperbolic baselines and µ < 0.05 for Euclidean architectures. Ablation studies confirm Lorentzian geometry contributes 87% of performance gains, while sensitivity analysis identifies optimal topological regularization at λtopo ∈ 0.05, 0.20. The framework resolves the fundamental tension between representational flexibility and structural stability through the synergy of precision control, self-model preservation, and homeostatic regulation.