A Mathematical Model of Consciousness: From Physical Axioms to the Brain-Network Matrix and the Sleep-Wake Spectrum

This work develops a closed mathematical model of consciousness grounded in five physical axioms: physical computability, the physical nature of information, the necessity of a physical mechanism for self-observation, the requirement of recursive structure for self-reference (to at least the third order), and the classical limit of brain dynamics via decoherence. From these axioms, a minimal three-variable model (sensory input x, instinctual drive y, and self-referential consciousness z) is first constructed, yielding a cubic-to-linear bifurcation that distinguishes conscious from unconscious states. The model then integrates ΦΦ as a dynamical variable and embeds the full dynamics into a 16-region brain connectivity matrix, producing a 22-dimensional dynamical system. Finally, the model is extended to the full sleep-wake spectrum through component-wise control of self-reference (β1,β2,β3), spatially structured endogenous input capturing PGO waves, and state-dependent dynamical complexity, differentiating seven states from wakefulness through NREM stages to REM, lucid REM, and anesthesia. The model predicts that lucid REM should exhibit elevated integrated information (Φ) relative to ordinary REM (ratio ≈1.2), a consequence of the partial restoration of third-order self-reference. All results are numerically verified with provided open-source code. No similar work combining these elements into a single closed dynamical framework has been identified.