Semi-Autonomous Formalization of the Vlasov-Maxwell-Landau Equilibrium

arXiv cs.AI / 3/18/2026

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Key Points

A complete Lean 4 formalization of the Vlasov-Maxwell-Landau equilibrium was achieved using an AI-assisted reasoning loop involving Gemini DeepThink, Claude Code, Aristotle, and the Lean kernel.
The process was supervised by a single mathematician over 10 days at a cost of $200, with zero lines of code written by hand.
The development is public, documenting 229 human prompts and 213 git commits, and detailing AI failure modes and successful strategies such as abstract/concrete proof split and adversarial self-review.
The formalization was completed before the final draft of the corresponding math paper, showcasing a rapid, reproducible workflow for AI-assisted mathematical research.

Abstract

We present a complete Lean 4 formalization of the equilibrium characterization in the Vlasov-Maxwell-Landau (VML) system, which describes the motion of charged plasma. The project demonstrates the full AI-assisted mathematical research loop: an AI reasoning model (Gemini DeepThink) generated the proof from a conjecture, an agentic coding tool (Claude Code) translated it into Lean from natural-language prompts, a specialized prover (Aristotle) closed 111 lemmas, and the Lean kernel verified the result. A single mathematician supervised the process over 10 days at a cost of \$200, writing zero lines of code. The entire development process is public: all 229 human prompts, and 213 git commits are archived in the repository. We report detailed lessons on AI failure modes -- hypothesis creep, definition-alignment bugs, agent avoidance behaviors -- and on what worked: the abstract/concrete proof split, adversarial self-review, and the critical role of human review of key definitions and theorem statements. Notably, the formalization was completed before the final draft of the corresponding math paper was finished.