Why Large Language Models Cannot Be Certified for Safety-Critical Systems

Large language models (LLMs) are moving rapidly into domains governed by functional-safety certification: aviation, road vehicles, medicine, and critical infrastructure. Certification regimes such as IEC 61508, DO-178C, and ISO 26262 combine system-level risk targets with process-, traceability-, configuration-, and evidence-based obligations, in mixes that differ by regime but share a demand for verifiable, bounded behavior. This review synthesizes three literatures that rarely meet: statistical learning theory on hallucination, empirical measurements of LLM error rates in high-stakes domains, and the standards and regulatory documents that define certification. The convergent finding is that the mismatch between the two worlds is structural rather than incidental. Impossibility results establish a nonzero error floor for calibrated probabilistic generators under stated conditions; reported benchmark metrics in legal, medical, and agentic tasks are not directly convertible to certification targets, and no published deployment supplies the system-level hazard and exposure model that a compliance demonstration would require; and every major mitigation family (retrieval augmentation, guardrails, formal verification, uncertainty quantification, and neurosymbolic hybrids) is documented to narrow, but not close, the gap. We formalize error compounding over execution horizons, tabulate the documented limits of each mitigation, and examine why plausibility cannot substitute for assurance. Two coherent exits emerge: statistical acceptance criteria in standards for bounded tasks, and architectures that confine the LLM to an untrusted-proposer role inside a deterministic, independently verifiable execution envelope. Certifying the envelope rather than the model is, on the current evidence, the most defensible path consistent with both the mathematics and the standards.