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.

While prior work has focused on projecting adversarial examples back onto the manifold of natural data to restore safety, we argue that a comprehensive understanding of AI safety requires characterizing the unsafe regions themselves. This paper introduces a framework for systematically mapping the Manifold of Failure in Large Language Models (LLMs). We reframe the search for vulnerabilities as a quality diversity problem, using MAP-Elites to illuminate the continuous topology of these failure regions, which we term behavioral attraction basins. Our quality metric, Alignment Deviation, guides the search towards areas where the model's behavior diverges most from its intended alignment. Across three LLMs: Llama-3-8B, GPT-OSS-20B, and GPT-5-Mini, we show that MAP-Elites achieves up to 63% behavioral coverage, discovers up to 370 distinct vulnerability niches, and reveals dramatically different model-specific topological signatures: Llama-3-8B exhibits a near-universal vulnerability plateau (mean Alignment Deviation 0.93), GPT-OSS-20B shows a fragmented landscape with spatially concentrated basins (mean 0.73), and GPT-5-Mini demonstrates strong robustness with a ceiling at 0.50. Our approach produces interpretable, global maps of each model's safety landscape that no existing attack method (GCG, PAIR, or TAP) can provide, shifting the paradigm from finding discrete failures to understanding their underlying structure.