Hyper-Invariant Subspaces and Spectral Asymptotics of Unbounded Cognitive Operators: Weak-Star Convergence of Integrated Information Functionals in Rigged Hilbert Spaces

Mathematical Applications of Science Fiction We present a rigorous operator-theoretic framework for quantifying the asymptotic stability of cognitive structures within infinite-dimensional state spaces. By modeling consciousness as a densely defined, unbounded operator T acting on a Rigged Hilbert Space (Gelfand Triple) Φ ⊂ H ⊂ Φ×, we extend the classical Integrated Information Theory (IIT) to the continuous spectrum regime. We introduce the class of Cognitive Operators, characterized by non-vanishing spectral measures over partitions of unity. The central contribution of this monograph is the proof of the existence of hyper-invariant subspaces for these operators under specific spectral decay conditions. Furthermore, we establish the weak-star convergence of a sequence of Integrated Information Functionals {Iₙ} in the dual space of the trace-class operators B₁(H). We demonstrate that the limit of this sequence corresponds to a maximal Φ-structure, implying a theoretical upper bound on the coherence of post-human cognitive architectures. The proofs utilize advanced methods from complex analysis, spectral theory of non-self-adjoint operators, and cohomological algebra.