Universal Catastrophic Safety, Undecidability,\\ and Capability--Risk Frontier\\ in Computational Universe

In previous axiomatic and geometric series works on ``computational universe'' U_{comp} = (X,T,C,I), we have constructed discrete complexity geometry, discrete information geometry, control manifold (M,G) induced by unified time scale, and proposed time--information--complexity joint variational principle on joint manifold E_Q = M \times S_Q, while proving equivalence between physical universe category and reversible QCA computational universe category under unified time scale. However, essential limitations regarding ``catastrophic safety'' and ``capability--risk frontier'' still lack unified computational--geometric--logical framework. This paper proposes within computational universe framework a ``universal catastrophic safety'' theory, connecting it with undecidability and geometric structure of capability--risk frontier. We first formalize catastrophic safety as path property: given catastrophe set C_{cat}\subset X, so-called ``universal catastrophic safety'' means universe evolution paths starting from all allowed initial states never enter C_{cat}. Under this setting, we define universal catastrophic safety decision problem, and prove at computational universe level: this decision problem is undecidable in most general case, i.e., there exists no algorithm that can give correct ``forever safe/possibly catastrophic'' verdict for all computational universes and catastrophe specifications. Second, we model catastrophic safety and capability--risk duality as two types of functionals on computational universe: capability functional Cap evaluates success probability or performance of certain tasks, risk functional Risk evaluates probability or expected loss of reaching catastrophe set C_{cat}. We define capability--risk frontier as Pareto boundary of all realizable strategy (Cap,Risk) pairs under given computational universe and task set, and under constraints of unified time scale and complexity geometry, characterize this frontier as class of ``reachable region boundary'' on control manifold (M,G) and strategy space. We further prove several key results: (1) Universal catastrophic safety verification problem in computational universe is at least as hard as halting problem, thus undecidable; (2) Any algorithmic safety filter attempting to be ``correct for all strategies'', if required to terminate and give verdict for all strategies under unified time scale, necessarily produces unavoidable ``false negative/false positive regions'' on capability--risk plane; (3) Under unified time scale, geometric optimization problem of capability enhancement and risk control can be written as constrained variational problem on joint manifold, where safety constraints naturally form non-recursively separable reachable region, thus capability--risk frontier cannot be algorithmically completely computed in general case. Finally, we connect undecidability of catastrophic safety with previous topological complexity and causal diamond structures: within causal diamond, catastrophe conditions can be viewed as local boundary conditions, but when diamond scale tends to infinity, ``whether there exists some path violating catastrophic safety'' corresponds to problem of whether certain class of closed loops on configuration complex X are contractible, thereby inheriting previously established topological undecidability results. This paper provides systematic foundation for subsequent construction of ``geometric shape of capability--risk frontier'', ``catastrophic safety consensus geometry of multi-agent systems'', and ``safety--capability--undecidability triangle relationship under unified time scale''.