The Defense Trilemma: Why Prompt Injection Defense Wrappers Fail?

We prove that no continuous, utility-preserving wrapper defense-a function $D: X\to X$ that preprocesses inputs before the model sees them-can make all outputs strictly safe for a language model with connected prompt space, and we characterize exactly where every such defense must fail. We establish three results under successively stronger hypotheses: boundary fixation-the defense must leave some threshold-level inputs unchanged; an $ε$-robust constraint-under Lipschitz regularity, a positive-measure band around fixed boundary points remains near-threshold; and a persistent unsafe region under a transversality condition, a positive-measure subset of inputs remains strictly unsafe. These constitute a defense trilemma: continuity, utility preservation, and completeness cannot coexist. We prove parallel discrete results requiring no topology, and extend to multi-turn interactions, stochastic defenses, and capacity-parity settings. The results do not preclude training-time alignment, architectural changes, or defenses that sacrifice utility. The full theory is mechanically verified in Lean 4 and validated empirically on three LLMs.

Large Empirical Case Study: Go-Explore adapted for AI Red Team Testing

Production LLM agents with tool-using capabilities require security testing despite their safety training. We adapt Go-Explore to evaluate GPT-4o-mini across 28 experimental runs spanning six research questions. We find that random-seed variance dominates algorithmic parameters, yielding an 8x spread in outcomes; single-seed comparisons are unreliable, while multi-seed averaging materially reduces variance in our setup. Reward shaping consistently harms performance, causing exploration collapse in 94% of runs or producing 18 false positives with zero verified attacks. In our environment, simple state signatures outperform complex ones. For comprehensive security testing, ensembles provide attack-type diversity, whereas single agents optimize coverage within a given attack type. Overall, these results suggest that seed variance and targeted domain knowledge can outweigh algorithmic sophistication when testing safety-trained models.