LLM2Cert verifies 81 of 105 stability and safety certificates

Language models have learned to write code and prove theorems. Now researchers are trying to apply them to an engineering problem where mistakes are costly: confirming that a dynamical system stays stable and safe. The authors of the LLM2Cert preprint (arXiv 2610.30524) built a framework in which an LLM proposes certificates and formal verifiers check them. On DynaCertBench, a new benchmark of 105 tasks, the system verifies 81, or 77.1%. That is 17.1 percentage points more than one-shot generation without feedback (about 60%).

Why certificates remain a bottleneck

Lyapunov functions and barrier certificates provide guarantees in state space. The former show that a system is stable; the latter prove that it will not enter an unsafe region. The difficulty is that such functions must first be found and then rigorously verified. For nonlinear, high-dimensional systems, this remains an open problem.

Existing pipelines check candidates in three ways: numerical solvers (sum-of-squares programs), interval methods and SMT solvers. All three share a set of problems. They require relaxations, they are conservative, or they scale poorly as the dimensionality and nonlinearity of the model grow.

The authors also point to a side effect. Standard test suites consist mostly of low-dimensional polynomial systems chosen to fit what solvers can handle, not what engineering needs. The benchmarks reflect what the tools can do, not what is needed in practice.

DynaCertBench and obligation routing

The first part of the work concerns the test suite. DynaCertBench contains 105 tasks:

• 80 non-redundant systems selected from the literature;
• extensions of those systems with higher-dimensional variants and variants with control constraints;
• new nonlinearly coupled systems;
• four task classes: stability, region of attraction, forward invariance and safety.

The second part is LLM2Cert itself. The model generates a candidate certificate, and the framework then decomposes it into proof obligations. Each obligation is routed down one of two paths: to an SMT solver or directly to a Lean Proof-Repair agent. Lean serves here as a strict formal checker, and the agent repairs the proof if it fails to go through. Feedback from the verifiers delivers most of the gain: without it, a model that produces a single candidate solves noticeably fewer tasks.

The numbers worth keeping in mind

• DynaCertBench: 105 tasks, 80 of them from the literature.
• LLM2Cert verifies 81 of 105 tasks, or 77.1%.
• Gain over a one-shot proposal without feedback: 17.1 percentage points.
• Of the 81 certificates found, 22 were verified in cases where the All-SMT approach, which sends everything to an SMT solver, leaves the question open.

The last figure is arguably the most telling. It is almost a quarter of the certificates in the set that pure SMT cannot reach. Hybrid routing closes some of the tasks that SMT cannot handle on its own.

The paper was prepared by Zhipeng Cao, Peiyang Song, Nikolaus Vertovec, Weiru Cui, Zhihao Yang, Mingkang Liu, Luxia Lin, Bai Xue and Alessandro Abate. Code and data are published in the FormalCert-AI/llm2cert repository.

Who stands to benefit first

The practical impact is expected wherever control of nonlinear systems requires formal guarantees: robotics, autonomous vehicles, power grids and avionics. Today, engineers have to select a certificate by hand or through SOS programs, which hit a wall as dimensionality grows. The "LLM proposes, solver and Lean verify" scheme shifts the creative part to the model and leaves trust with formal verification. Even if the model makes a mistake, an incorrect certificate will not pass verification.

For researchers, DynaCertBench itself is useful: it lets them compare new synthesis methods on tasks closer to real systems than the usual low-dimensional polynomials. The open code makes it possible to reproduce the results and plug in a different model. That leaves 24 of the 105 tasks that LLM2Cert did not solve, and they show where the method's limits lie.