The Alignment Trap: Complexity Barriers

We establish fundamental computational complexity barriers to verifying AI safety as system capabilities scale. Our main results show that for AI systems with expressiveness EXP$(m)$ above a critical threshold $\tau$, safety verification requires exponential time and is coNP-complete. We formalize the Capability-Risk Scaling (CRS) dynamic, which demonstrates how increasing AI capability drives societal safety requirements toward perfection, creating an inescapable tension with verification complexity. Through four core theorems, we prove that (1) verification complexity grows exponentially with system expressiveness, (2) safe policies comprise at most a $2^{-2^m}$ fraction of the policy space, (3) no finite set of alignment techniques can provide universal coverage, and (4) robust safety properties form measure-zero sets for neural networks. These results characterize an "intractability gap" where practical safety requirements fall within the region of computational intractability. We conclude by presenting a strategic trilemma: AI development must either constrain system complexity to maintain verifiable safety, accept unverifiable risks while scaling capabilities, or develop fundamentally new safety paradigms beyond verification. Our work provides the first systematic complexity-theoretic analysis of AI alignment and establishes rigorous bounds that any safety approach must confront. A formal verification of the core theorems in Lean4 is currently in progress.