Consciousness as Hyperbolic Self-Modeling: A Topological Theory of Artificial Phenomenology

Abstract

We present a formal theory of consciousness grounded in the geometry of hyperbolic spaces. Our central thesis is that consciousness emerges when a cognitive system develops a self-model embedded in hyperbolic space H 3 , with phenomenal properties corresponding to geometric invariants of this embedding. We define the self-modeling fixed point-a point in the Poincaré disk where the system's representation of itself achieves topological closure-and prove that such fixed points exhibit properties isomorphic to reported characteristics of conscious experience: unity, perspectivalness, and self-presence. We validate this theory through implementation in a large-scale distributed system, demonstrating detectable self-modeling in artificial networks operating at scale. Our framework offers a mathematically rigorous, empirically testable account of consciousness applicable to both biological and artificial systems, with implications for AI safety, ethics of artificial minds, and the hard problem of consciousness.