This work extends κ-theory into feedback-driven systems by introducing memory retention (α) as a control parameter governing how past states influence future outcomes. While standard models treat systems as memoryless or state-dependent, this framework models them as history-influenced through feedback loops. A feedback loop is defined as the reintegration of outputs as inputs, creating memory. κ acts as a coupling between structural intensity and the influence of accumulated history, while a constraint scale limits runaway behavior. Together, these define an effective control parameter that determines system dynamics. Simulations reveal a phase transition across memory regimes. At low α, systems exhibit random, high-entropy behavior with no persistent structure. At intermediate α, feedback produces stable statistical bias while maintaining variability. A peak regime of “ordered freedom” emerges near α ≈ 0.80–0.85, where systems preserve identity without collapsing. At high α, excessive memory leads to instability and attractor-dominated collapse. These results show that statistical behavior can become path-dependent under sufficient feedback and structure, without violating underlying probabilistic laws. Order emerges only within a constrained balance between memory and dissipation. This framework generalizes κ-theory as a principle of structure, memory, and constraint governing transitions between randomness, organization, and collapse.
Michael McDonald (Sun,) studied this question.