Hamiltonian Active Inference: Escaping Local Traps via Cognitive Momentum -- Reconciling Macro-Scale Neurodynamics with Variational Inference

The Free Energy Principle (FEP) and Active Inference provide a normative theoretical framework for biological self-organization via variational free energy minimization. However, standard computational implementations ubiquitously employ a first-order overdamped approximation (dμ/dt = -∇_μ F(μ) + √(2T)ξ(t)), enforcing instantaneous velocity dissipation (τ_relax → 0). This creates a qualitative discrepancy with biological neurodynamics, such as prefrontal cortex (PFC) persistent activity, which dynamically sustains contextual representations across time. To resolve this bottleneck, we advance the framework of Hamiltonian Active Inference (HAI) and its search engine Momentum Exploration (ME) by formalizing a second-order canonical dynamical structure on phase space (μ, p). Introducing canonical Hamilton's equations parameterized by cognitive mass M (representing informational inertia sustained by NMDA-mediated recurrent circuits) and viscous damping Γ, we analytically derive exact upper and lower critical momentum thresholds (p_crit_upper, p_crit_lower) (Theorem 2.1). These bounds ensure inertial penetration in the noiseless limit (T → 0) and a momentum-driven transition where penetration probability saturates at P_cross ≈ 1.0 under finite noise T > 0 despite adverse thermal fluctuations and dissipative work loss. High-precision numerical simulations utilizing a phase-space-preserving BAOAB integrator confirm complete alignment with Theorem 2.1, demonstrating a sharp phase transition in penetration probability P_cross and a multi-order-of-magnitude collapse in mean escape time τ. Furthermore, as a hypothetical dynamical formulation, HAI qualitatively maps persistent neural activity (Default Mode Network vs. Task-Positive Networks) to phase-space momentum p_0, reframing neuroatypical hyperfocus (ADHD/2E) as a momentum-driven phase transition. Finally, HAI adiabatically reduces to the classical first-order FEP as a degenerate limit (M → 0), laying the conceptual groundwork for physics-informed neuromorphic AGI optimization by establishing localized trajectory bounds along specific anisotropic pathways (assuming dynamic momentum replenishment at each local attractor).