A Mathematical Distinction Between Consciousness, Proto-Awareness, and Conscious Awareness

This technical note accompanies the preprint “Towards a Unified Framework for Consciousness” (https://doi.org/10.5281/zenodo.18908549) It presents the mathematical formulation of the CPA Distinction, which separates three levels of conscious phenomena: fundamental consciousness (C), proto-awareness (P), and conscious awareness (A). Within this framework, consciousness is treated as a fundamental and invariant background capacity C = f, while proto-awareness emerges through the organized interaction of matter, energy, time, and information P = Φ(m,e,t,i). Conscious awareness arises only when proto-aware systems exceed a critical integration threshold P_t, after which awareness scales nonlinearly with increasing integration. The document formalizes the relationship C → P → A and explores its conceptual connections to computational complexity, information theory, thermodynamics, and global versus local descriptions in physics. This supplementary note provides the mathematical structure and theoretical foundations referenced in the main preprint.