Recursive Differentiation and Trace Mediated Recovery

Abstract This paper establishes three mathematical results for a minimal recursive dynamical system incorporating feedback and retained historical trace. The first theorem identifies a critical threshold at which stable differentiated states emerge. The second demonstrates that retained trace increases recovery from perturbations beyond what equivalent direct feedback can achieve. The third establishes global convergence and a unique recovery boundary under specified positive-determinant conditions. Together, these results provide a mathematical foundation for investigating recursive differentiation, persistence, and history-dependent restoration in nonlinear systems. The model is developed in connection with Recursive Process Monism (RPM), although its mathematical results do not depend on RPM's philosophical interpretation of consciousness. The paper distinguishes established dynamical properties from their possible applications to self-organizing biological and cognitive systems.