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.

Key Contributions

• Proves defense trilemma: continuous, utility-preserving wrapper defenses cannot achieve complete safety on connected prompt spaces
• Establishes three impossibility results under successively stronger assumptions: boundary fixation, ε-robust constraint, and persistent unsafe region
• Provides mechanically verified proofs in Lean 4 with ~350 theorems and empirical validation on three LLMs

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