Norm-Continuous Semigroups and the Spectral Mapping Theorem for Unbounded Generators of Stochastic Cognitive Evolution in Non-Separable Hilbert Spaces: Establishing a Rigorous Operator-Theoretic Foundation for Phenomenological Decoherence

Mathematical Applications of Science Fiction We present a comprehensive operator-theoretic framework for modeling stochastic cognitive evolutionwithin the context of non-separable Hilbert spaces, denoted HΨ. By constructing a Gelfand-triplestructure Φ ⊂ HΨ ⊂ Φ∗, we define the unbounded generator G of a C0-semigroup T (t) governing thetime-evolution of cognitive states. The central contribution of this monograph is the rigorous derivationof a Spectral Mapping Theorem for G , overcoming the traditional failure of such theorems for non-analytic semigroups in infinite dimensions. We introduce the concept of Phenomenological Decoherenceas a projection-valued measure associated with the essential spectrum of G . Through a series of tenfundamental lemmas and five major theorems, we prove that the asymptotic stability of the cognitivesemigroup is determined by the spectral bound s(G ), and that the non-separability of the underlyingspace is a necessary condition for the emergence of robust qualia. We conclude with applications totrans-universal constant variation and speculative engineering of synthetic consciousness.