Asymptotic Bounds of Algorithmic Irreducibility in Hyper-Recursive Universal Simulators: Deriving the Information-Theoretic Horizon of Trans-Finite Computational Anomalies

Mathematical Applications of Science Fiction We present a rigorous derivation of the asymptotic bounds governing algorithmic irreducibility in the context of hyper-recursive universal simulators operating at transfinite ordinal levels. By extending classical Kolmogorov complexity to the hyperarithmetical hierarchy, we establish the existence of an Information-Theoretic Horizon—a boundary beyond which computational anomalies appear as irreducible topological defects in the Baire space of programs. We construct a Universal Simulator Monad acting on the category of hypercomputable sets and prove that the density of algorithmic anomalies converges to a noncomputable measure related to a generalized Chaitin constant. Furthermore, we demonstrate that for any hyper-recursive system there exists a critical ordinal at which the compressibility of the system’s state vector undergoes a phase transition, making the system indistinguishable from maximum-entropy noise relative to lower-order oracles. These findings have deep implications for the physics of information in multiversal cosmologies and for the theoretical limits of simulation.