In a landmark achievement for artificial intelligence and mathematics, researchers at Peking University have developed an autonomous AI system that solved and formally verified a decade-old open problem in commutative algebra known as Anderson's conjecture. This breakthrough, detailed in a preprint on arXiv, demonstrates the potential of AI to conduct independent mathematical research, processing vast literature, constructing novel counterexamples, and proving results without human mathematical input.

The dual-agent framework, named Rethlas for informal reasoning and Archon for formal verification, operated for approximately 80 hours, costing around $1,400 in API usage, to disprove the conjecture proposed by the late Dan D. Anderson in 2014. This feat underscores Peking University's pivotal role in advancing AI-driven mathematical discovery within China's thriving higher education landscape.

Understanding Anderson's Conjecture

Dan D. Anderson, a former professor at the University of Iowa who passed away in 2022, posed the conjecture in his seminal work on open problems in commutative ring theory. The question centered on Noetherian local rings: Does 'weak quasi-completeness' imply 'quasi-completeness'?

• Quasi-complete ring (R, m): For any decreasing sequence of ideals {A_n} and k ≥ 1, there exists s_k such that A_{s_k} ⊆ (∩ A_n) + m^k.
• Weakly quasi-complete: The condition holds when ∩ A_n = 0, simplifying to A_{s_k} ⊆ m^k.

The conjecture asked if every weakly quasi-complete Noetherian local ring is quasi-complete. Human mathematicians had struggled with this for over a decade, as it required deep insights across ring theory, completions, and analytic irreducibility.

Peking University's AI4Math Team Leading the Charge

Led by experts from the School of Mathematical Sciences, Center for Machine Learning Research, and Beijing International Center for Mathematical Research at Peking University (PKU), the 15-member team—including Haocheng Ju, Guoxiong Gao, and Bin Dong—built this pioneering system. PKU's investment in interdisciplinary AI-math labs positions it as a global leader, alongside peers like Tsinghua University, which ranks top worldwide in AI per CSRankings 2026.

This success highlights China's higher education push: Over 100 universities now offer AI-math programs, with PKU and Tsinghua producing 70% of top AI talent.

The Dual-Agent Framework: Rethlas and Archon Explained

The system comprises two synergistic agents:

1. Rethlas (Informal Reasoning Agent): Mimics a human mathematician. Subagents generate toy examples, search theorems via Matlas (semantic search over 13.6M arXiv statements), decompose subgoals, and propose proofs using GPT-5.4 and Codex.
2. Archon (Formal Verification Agent): Translates to Lean 4 code, grounded in Mathlib (267K theorems). Plan Agent decomposes tasks; Lean Agents execute proofs iteratively.

Workflow: Rethlas discovers informal proof; Archon formalizes and verifies. Tools like LeanSearch ensure accuracy.

Step-by-Step: How the AI Cracked the Conjecture

The autonomous process unfolded as follows:

• Literature Synthesis: Rethlas scanned decades of papers, identifying Jensen's 2006 Corollary 2.4 on completions.
• Counterexample Construction: Built ring T = ℂ[[x,y,z]] / (x² - yz), then local UFD A with trivial generic formal fiber (weak quasi-complete) but quotient A/aA not analytically irreducible.
• Proof Generation: Verified via natural language, then passed to Archon.
• Formalization: Produced 19,448-line Lean 4 project (42 files), formalizing 6 papers, filling gaps like injectivity proofs.

No human math judgment; only PDF downloads for OCR. Read the full technical paper for proofs.

Formal Verification: A Game-Changer in Proof Reliability

Archon generated Lean 4 code passing 'lake build' and Comparator checks, ensuring no hidden axioms. This 80-hour effort equals months of human formalizers. Ablation: Human blueprint cut time 70%, but zero-intervention succeeded.

Performance Metrics and Accessibility

Open-sourced: Rethlas, Archon, results.

Implications for Global Mathematics Research

This automates discovery-to-verification, accelerating progress in algebra and beyond. Unlike LeanDojo (training on Lean), this reasons informally first. Experts hail it as 'substantial automation milestone'.

• Cross-domain synthesis: Algebra + completions.
• Scalable to IMO-level, Olympiads (cf. TongGeometry at PKU).
• Risks: Over-reliance? Hallucinations mitigated by verification.

Peking University and China's Higher Ed AI Push

PKU's math-AI integration mirrors Tsinghua's #1 AI ranking. 2026 sees 20+ uni AI-math centers, backed by NSFC funding. Impacts: Faster PhD proofs, new courses in formal methods.

Trending on X: Posts praise PKU's 'first domestic autonomous math win'.

Global Comparisons and Future Directions

Vs. DeepMind's AlphaProof (IMO silver), this targets research conjectures. Future: Scale Matlas (now 8.5M statements), tackle harder problems. China leads with open-source ethos.

Explore PKU's tools on GitHub for your research.