A Dynamical Framework for Cognitive Processes Based on Transformations and Semantic Equivalence

This paper proposes a structural and dynamical framework for modeling cognitive processes within a cybernetic perspective. Cognitive states are represented as elements of a state space evolving through an iterative update rule of the form $$X_{t+1} = \pi\big(F(f(X_t))\big),$$ where $f$ describes internal transformations, $F$ represents interpretative mappings, and $\pi$ enforces semantic equivalence. The model is interpreted as a feedback system integrating transformation, observation, and stabilization. A categorical formulation is introduced to capture compositional structure, while the associated dynamics are analyzed through fixed-point arguments and contraction conditions ensuring stability. To demonstrate the operational character of the framework, a computational illustration is provided, together with a qualitative analysis of the induced dynamics. A concrete linguistic application shows how context-dependent interpretation can be modeled as a trajectory toward a stable semantic class. The proposed approach connects dynamical systems, category theory, and cognitive modeling, and provides a unified representation of cognition as a feedback-driven process evolving toward invariant interpretations.