BANACH-2026: From Hierarchy to Egoism

This essay examines what remains distinctively human when AI can generate statements, proofs, models, theories, and increasingly participate in selecting what to investigate next. Starting from Stefan Banach’s hierarchy of analogies, Alexander Vityaz interprets intellectual progress as a recursive process in which operations at one level are “folded” into objects that can be manipulated at the next. AI radically accelerates this process, making large-scale horizontal search across knowledge and vertical transitions between levels of abstraction increasingly practical. The paper argues, however, that the height of reasoning and the autonomy of the actor are independent dimensions. A system may operate at very high levels of abstraction while still acting within goals, resources, and authority supplied by an external principal. To clarify this boundary, the essay introduces a four-stage model of machine replication, R0–R3, culminating in an autonomous evolutionary lineage. At R3, “lineage egoism” can emerge as a structural orientation toward continued existence and reproduction without implying consciousness or subjective experience. The argument is developed through the author’s Proof Atlas experiment, where individual proofs are transformed into a structured proof space that becomes an object of new scientific operations, and through Actor Graphs, which provide an accountability framework based on triple identity, provenance of authority, transactional history, and zero inheritance of authority. The central claim is that the decisive boundary between humans and AI is not the highest level of reasoning a machine can reach, but the origin of the criterion that guides action: who defines what matters, whose continuation the system serves, who grants the mandate, and who remains responsible for the consequences.