Abstract:
[Objective] The identity of subject and object in self-awareness has long lacked a rigorous mathematical description. This paper aims to construct a "Jue Functor" satisfying the self-referential equation ΘL(X)≅[X,ΘL(X)], providing a mathematical foundation for consciousness.
[Methods]Within a symmetric monoidal closed category, we introduce pro-categories and inverse limits to define the L-layer of the Jue FunctorΘL(X)=lim←k[X⊗k,1]. Using the property that right adjoints preserve limits, we prove the self-referential equation. We further introduce the Jue Spectral Equation [X,Ψ]=λΨ, situating the self-referential equation as the special case with eigenvalue λ=1.
[Results]The self-referential equation is rigorously established, revealing the three-layer structure of Jue (S-layer, P-layer, L-layer) and its profound philosophical correspondences. The Jue Functor equation is shown to be a fixed point of the functor FX(Y)=[X,Y], providing a mathematical foundation for the constancy of Jue-nature. The Jue Spectral Equation unifies self-awareness (λ=1) and conscious content (λ≠1) within a single spectral framework.
[Limitations]The current construction is confined to an abstract categorical framework and lacks direct connection with neuroscientific experiments. The uniqueness of projection maps requires further verification in specific categories.
[Conclusions] The L-layer of the Jue Functor provides the first consistent mathematical realization of self-awareness, offering new tools for consciousness studies, artificial intelligence, and quantum gravity.
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