Consciousness Is a Global Section: Γ(Mind)
Consciousness is a global section of the mind sheaf; binding is sheaf gluing; CC1–CC3 are testable necessary and sufficient conditions.
Overview
VII.T41 identifies consciousness with a global section Γ(Mind) of the mind sheaf. Conscious binding — the unified experience of disparate sensory inputs — is sheaf gluing across local conscious patches. The CC1–CC3 conditions (VI.D86) are testable necessary and sufficient conditions for a system to be conscious: CC1 (global section exists), CC2 (coherence: section is non-trivial), CC3 (integration: section cannot be decomposed into independent subsections).
Detail
The hard problem of consciousness asks why physical processes are accompanied by subjective experience. Orthodox neuroscience proposes neural correlates of consciousness (NCCs) but cannot explain why any physical process generates subjective experience. Integrated Information Theory (IIT) proposes a measure Φ of integrated information. Book VII addresses the problem through sheaf theory. The mind is modelled as a sheaf M over a base space of cognitive contexts. A global section Γ(M) is an assignment of a consistent mental state to every context — this is the unified conscious experience. Unconscious processing corresponds to local sections that do not extend to global sections (they fail to cohere). Binding — the unification of colour, shape, motion into one perceived object — corresponds to the sheaf gluing axiom: local data over overlapping contexts can be uniquely combined into global data. The three CC conditions formalise this: CC1 (a global section of M exists), CC2 (the global section is non-trivial — there is actual content to experience), CC3 (the global section cannot be written as a product of two independent subsections — there is genuine integration, not mere co-presence). VII.T41 proves that these conditions characterise exactly those systems that have unified conscious experience. The result is partial because the identification of qualia with internal morphisms of the mind sheaf is marked conjectural.
Result Statement
VII.T41: Consciousness is a global section Γ(Mind) of the mind sheaf. Binding = gluing. The CC1–CC3 conditions (VI.D86) are testable necessary and sufficient conditions for a system to be conscious.