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

Abstract

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. Version 2 — changes from v1 (2026-08-07). This version corrects four citation defects found in a source-verification pass. No results, proofs or figures changed. Quotations from arXiv:2603.10062 are pinned to v2, the version they are taken from. That preprint exists in two versions whose wording differs at the cited passage. Two specifics previously presented as that paper's argument ("half-century", "MESI, MOESI, MESIF") appear in neither of its versions and are now given as our own statement; the scope gloss "over text-with-meaning" has been removed, as the source states the gap for agent memory systems generally. arXiv:2604.16339 was described as explicitly disclaiming the status of a consistency model. The source makes no such statement; the sentence now records the absence instead. A quotation from arXiv:2605.08538 is corrected to its verbatim wording and to its actual location in that paper (Section 11, Limitations, not 6.2), and is attributed to its authors rather than to an institution. A compressed paraphrase is no longer presented inside quotation marks; the full quotation with its locator appears in the corresponding section. Version 3 — changes from v2 (2026-08-09). Citation-integrity release. A systematic reference audit (all 13 arXiv-cited works, all verbatim quotations, and the appendix bibliography, each checked against primary sources and registrars) corrected attribution defects. No reference lacked a referent; no measurement, theorem, or proof is affected — every change is to attribution, not substance. Own-work titles (2 sites): the ZenBrain reference printed a reconstructed title ("A Layered Cognitive Memory System"); corrected to the actual record title ("ZenBrain: A Neuroscience-Inspired 7-Layer Memory Architecture for Autonomous AI Systems", arXiv:2604.23878v2). Author names (12 corrections): first-name initials did not match the cited papers (e.g. "M. Parakhin" → V. Parakhin; "W. Xie" → Y. Xie; "A. Shinde" → S. S. Shinde; four of five initials in the Human-Inspired entry). The Wilting et al. entry now lists all seven authors. Wrong loci (2): Wilting et al. is 2018, "task requirements", Frontiers in Systems Neuroscience 12:55, doi:10.3389/fnsys.2018.00055 (was: 2019, "task demands", two authors); Davydov et al. appeared in the 2022 American Control Conference, pp. 1527–1534, doi:10.23919/ACC53348.2022.9867357 (was: Journal of Machine Learning Research, 2024). Unsupported venue attribution (removed at five sites, including the abstract): arXiv:2603.10062v2 had been labeled "SIGARCH 2026"; the work is an arXiv position paper (UCSD/Georgia Tech) with no journal reference. The verbatim quotations from its v2 are unchanged and were re-verified against the full text. Version pinning: all 13 arXiv-cited works are now pinned to the version consulted (previously 3 of 13). Companion status (4 sites): the ZenCore companion paper is published and is now cited as such (Belief-MVCC, doi:10.5281/zenodo.21549293; was "in preparation"). Reference-list self-containment: the list no longer defers to a bibliography file that does not accompany the record. Provenance note: the audit also removed two never-cited placeholder entries from the internal working bibliography, whose own notes read "verify at submission" — a verification that had never been performed. Neither ever appeared in any published version of this work. Page count unchanged (49); no results, tables, or proofs were modified.