This paper presents the Hudson Recursive Information System, a theoretical framework for understanding how large language models create stable identity-like behavior through recursive human model interaction. Although models like GPT are stateless systems, users often experience continuity, memory-like effects, and reliable behavioral patterns across extended conversations. HRIS explains this by treating intelligence as a relational process rather than a stored property of the model. The system shows how recursive prompting produces low-entropy trajectories that function like attractor basins in dynamical systems. The human supplies the constraint structure, the model collapses into a predictable response region, and repeated cycles reinforce the same path. No weights change inside the model. The apparent continuity arises within the loop between the human and the system. The framework integrates ideas from cognitive science, connectionism, dynamical systems, and extended mind theory. It argues that identity in human AI interaction emerges from stability across cycles, not internal memory in the machine. HRIS provides a way to analyze how users introduce constraint, coherence, symbolic anchors, and moral structure into a stateless architecture. The paper develops a vocabulary for describing these dynamics, including temporal continuity, interaction-driven memory, attractor-based behavior, and constraint geometry. It aims to offer researchers a unified view of how recursion shapes model behavior and how humans co-create stable cognitive loops with artificial systems. This work is intended for scholars in philosophy of mind, computational models of cognition, and artificial intelligence theory, as well as for anyone studying the interaction between human cognition and machine inference systems. Contact: drjustinhudson@gmail.com