A Mathematical Framework for Consciousness in Neural Networks

This paper presents a novel mathematical framework for bridging the explanatory gap (Levine, 1983) between consciousness and its physical correlates. Specifically, we propose that qualia correspond to singularities in the mathematical representations of neural network topology. Crucially, we do not claim that qualia are singularities or that singularities "explain" why qualia feel as they do. Instead, we propose that singularities serve as principled, coordinate-invariant markers of points where attempts at purely quantitative description of a system's dynamics reach an in-principle limit. By integrating these formal markers of irreducibility into models of the physical correlates of consciousness, we establish a framework that recognizes qualia as phenomena inherently beyond reduction to complexity, computation, or information. This approach draws on insights from philosophy of mind, mathematics, cognitive neuroscience, and artificial intelligence (AI). It does not solve the hard problem of consciousness (Chalmers, 1995), but it advances the discourse by integrating the irreducible nature of qualia into a rigorous, physicalist framework. While primarily theoretical, these insights also open avenues for future AI and artificial consciousness (AC) research, suggesting that recognizing and harnessing irreducible topological features may be an important unlock in moving beyond incremental, scale-based improvements and toward artificial general intelligence (AGI) and AC.