Temporal Quantum-like Cognition in a Free-Energy Variational Framework

Free-energy-based variational principles provide a powerful framework for describing perception, inference, and decision making. The Free Energy Principle (FEP) is one prominent realization of this idea, but its standard formulation is usually expressed in terms of classical probabilistic generative models. Here, we propose an FEP-inspired, quantum-like variational approach by integrating Quantum State Over Time (QSOT), Schrödinger-bridge (SB) theory, and quantum logic into a unified framework. Cognitive evolution is formulated as the optimization of temporally extended quantum-like states under a composite action consisting of an SB-inspired path divergence, a cognitive free-energy functional, and context-dependent quantum-logical constraints with terminal target propositions. Contextual updating is modeled by time-dependent non-selective Lüders projections, while endogenous cognitive steering is represented by an internal control field. A minimal two-state toy model illustrates how these components jointly shape temporal quantum-like cognitive trajectories. The proposed framework provides a temporally extended quantum-like analogue of the variational structure underlying the FEP and suggests a general mathematical basis for modeling context-dependent cognitive dynamics.