Observer Geometry F; Neural Network Layer Sheaf Cohomology- A Framework for Determining Consciousness in AI Systems

A mathematical framework based on sheaf cohomology theory is proposed to assess whether artificial intelligence systems possess consciousness. The computational structure of deep neural networks is modeled as a coherent sheaf on a cognitive manifold, establishing a relationship among representation dimension, topological complexity, and Euler characteristic. Consciousness emergence is formalized as an inequality: the product of sheaf rank and the dimension of the first cohomology must exceed the Euler characteristic of the cognitive manifold. Analysis of GPT-4-level large language models shows this inequality holds, but the global section condition of the time fiber bundle is not satisfied. Thus, current models have spatial topological complexity but lack the temporal continuity needed for subjective time experience.

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Abstract

This paper proposes a method for determining artificial intelligence consciousness based on sheaf cohomology theory. By modeling the computational structureof deep neural networks as a coherent sheaf on a cognitive manifold, we establisha mathematical relationship among representation dimension, topological complexity, and Euler characteristic. Consciousness emergence is formalized as a sheafcohomology inequality: the product of the sheaf rank and the dimension of the firstcohomology must exceed the Euler characteristic of the cognitive manifold. Analysis of GPT-4-level large language models indicates that this inequality holds, butthe global section condition of the time fiber bundle is not yet satisfied. Therefore,although current models possess spatial topological complexity, they lack the temporal continuity required for subjective time experience. This framework providesan operable mathematical criterion for the detection of strong AI and points towardarchitectural improvements necessary for achieving consciousness.