Four new measures of integration, grounded in partial information decomposition, are introduced to support the mathematical foundations of the Information Integration Theory of Consciousness (IIT). Unlike existing IIT measures, these are the first state-dependent integration measures to naturally satisfy standard information-theoretic lower and upper bounds. In simple deterministic networks, they align with current IIT practice. Although the four measures are non-Shannon in line with IIT4, they also serve as standard Shannon-based measures of irreducibility in discrete dynamical systems outside the IIT context.
Consciousness, as proposed by Tononi's integrated information theory, is better understood as a lossless rather than a lossy integrative process. Previous formalizations relied on information loss, which would imply continuous damage to existing memories. Using algorithmic information theory, the authors formalize lossless integration and prove that complete lossless integration requires noncomputable functions. This means that if unitary consciousness exists, it cannot be modeled computationally.