Reformulating Integrated Information Theory's Exclusion Postulate via Sheaf Theory for Tree-Shaped Causal Structures

Abstract

This paper proposes a mathematical reformulation of Integrated Information Theory (IIT), specifically targeting the conceptual and empirical challenges associated with its Exclusion Postulate. IIT identifies consciousness with the integrated information generated by a physical system's causal structure, but its principle that only the single substrate with maximal integrated information is conscious struggles to account for phenomena such as split-brain consciousness, graded loss of consciousness under anesthesia, and causal emergence across multiple scales. To address these limitations, the authors replace IIT's maximization-based exclusion with a coherence principle grounded in sheaf theory. The analysis is restricted to systems whose causal hierarchy forms a rooted tree, a structure that naturally models hierarchical brain organization from whole brain to regions to neurons. Within this class, a poset category of mechanisms is constructed, where each mechanism is assigned a space of concepts (cause-effect structures) equipped with a metric. This assignment forms a presheaf, which generally fails to be a sheaf due to gluing inconsistencies. Its sheafification enforces coherence, and a gluing discrepancy is defined as the distance from any assignment of concepts to the subspace of global sections of this sheaf. For tree-shaped posets, the authors prove a two-sided estimate bounding the gluing discrepancy by the maximum chain discrepancy, making it computationally tractable. A modified integration measure, the richness measure, is introduced by averaging integrated information across all mechanisms and penalizing incoherence. Maximizing this measure selects a unique optimal coherent assignment. The major complex is then redefined as the weighted support of physical elements whose contribution to this optimal assignment exceeds a threshold. The paper provides a complete operational pipeline linking neuroimaging data to these theoretical constructs, yielding testable predictions such as elevated gluing discrepancy during anesthesia and decomposition of the sheaf in split-brain patients. Detailed calculations for a canonical ABCD system illustrate all constructions. This framework preserves IIT's axioms while enabling a coherent, multi-scale account of consciousness within tree-shaped causal architectures.