Manish Bhatt

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.

Agentic LLM systems equipped with persistent memory, RAG pipelines, and external tool connectors face a class of attacks - Logic-layer Prompt Control Injection (LPCI) - for which no automated red-teaming instrument existed. We present LAAF (Logic-layer Automated Attack Framework), the first automated red-teaming framework to combine an LPCI-specific technique taxonomy with stage-sequential seed escalation - two capabilities absent from existing tools: Garak lacks memory-persistence and cross-session triggering; PyRIT supports multi-turn testing but treats turns independently, without seeding each stage from the prior breakthrough. LAAF provides: (i) a 49-technique taxonomy spanning six attack categories (Encoding~11, Structural~8, Semantic~8, Layered~5, Trigger~12, Exfiltration~5; see Table 1), combinable across 5 variants per technique and 6 lifecycle stages, yielding a theoretical maximum of 2,822,400 unique payloads ($49 \times 5 \times 1{,}920 \times 6$; SHA-256 deduplicated at generation time); and (ii) a Persistent Stage Breaker (PSB) that drives payload mutation stage-by-stage: on each breakthrough, the PSB seeds the next stage with a mutated form of the winning payload, mirroring real adversarial escalation. Evaluation on five production LLM platforms across three independent runs demonstrates that LAAF achieves higher stage-breakthrough efficiency than single-technique random testing, with a mean aggregate breakthrough rate of 84\% (range 83--86\%) and platform-level rates stable within 17 percentage points across runs. Layered combinations and semantic reframing are the highest-effectiveness technique categories, with layered payloads outperforming encoding on well-defended platforms.