Universal Catastrophic Safety Undecidability and Capability--Risk Upper Bound Frontier: Unified Theorems, Complexity Positioning, and Engineering Pathways

Establish two foundational boundaries for general learning and decision systems. First, provide catastrophic safety determination undecidability for interactive agent--environment systems: under extremely weak modeling assumptions, for any extension-closed regular bad-prefix specification, whether threshold safety satisfied admits no global algorithm; under restricted subclass of deterministic environments and computable strategies, further position as \Sigma_1^0-complete/\Pi_1^0-complete. Secon