An Operator Algebra of Cognitive Memory Consolidation: Layered Composition, Lyapunov Stability, and the Cooperative-Survival Theorem

We develop an operator-algebraic account of memory consolidation in layered cognitive systems. Consolidation steps are modelled as operators on a memory state space; layering corresponds to composition, and the resulting algebra admits closure conditions under which a layered system remains well-behaved. We give a Lyapunov-style stability argument for repeated consolidation and prove a cooperative-survival theorem characterising when jointly applied mechanisms retain information that each mechanism alone would lose. The framework is intended as a theoretical root for empirical work on long-term memory in artificial agents: it states what composition can and cannot buy, independently of any particular implementation.