Convergence as Algebraic Closure: Operator Algebra as Memory

Memory in artificial cognitive systems is almost always implemented as storage: records written, indexed, and retrieved. We argue that this architecture is wrong in principle and propose the alternative: memory is the current state of a living algebra. Experience does not produce records to be stored; it produces operators to be accumulated. The algebra of accumulated operators, not a database, is the memory of the system. We instantiate this principle in the exceptional Lie algebra e8, where 240 root vectors provide a maximally non-degenerate operator basis and the unique Killing form provides a canonical convergence criterion. The central result is the Memory Convergence Theorem: the generated subalgebra of accumulated operators is consolidated if and only if the total Killing norm has stabilized. This transforms memory convergence from an empirical judgment into a structural mathematical fact. We prove the theorem in full, analyze its implications for the dream cycle, and derive the multi-timescale cascade that connects tick-rate operator accumulation to geological-rate structural memory.