Ax-Prover and Beyond: From Formal Reasoning to Semantic Coherence in Artificial Intelligence

Formal verification, multi-agent architectures, and the future of “Lean-for-Language” reasoning systems

Formal theorem proving has recently emerged as a critical bridge between symbolic reasoning and large-scale language modeling. Ax-Prover, a 2025 multi-agent framework built on the Lean proof assistant and Model Context Protocol (MCP), integrates the creativity of large language models (LLMs) with the logical rigor of formal verification. This paper provides a detailed examination of Ax-Prover’s architecture, performance, and use cases from abstract algebra to quantum mechanics and cryptography and introduces a broader vision:

a Lean-for-Language system capable of verifying the logical consistency of natural-language reasoning, not just mathematics. Such a framework could fundamentally transform how LLMs are trained and evaluated, turning generative AI into provably consistent reasoning systems.

Will LLMs ever graduate?

1. Introduction

Large Language Models (LLMs) have reached impressive levels of reasoning performance.

They can solve mathematical problems, code complex algorithms, and even explain physical theories.

Yet, a fundamental limitation persists: they cannot verify the correctness of their own answers.

In mathematics, this problem was solved through formal proof assistants.

Lean, developed at Microsoft Research, allows mathematical statements to be expressed in a machine-verifiable language.

Every inference step must comply with strict logical rules; if the proof compiles, it is certainly true.

Ax-Prover builds on this principle by integrating Lean with modern LLMs through a multi-agent orchestration framework.

Instead of a single monolithic model, Ax-Prover deploys a network of reasoning agents — each responsible for proving, verifying, or coordinating the process.

The result is a system that combines human-like reasoning flexibility with formal mathematical certainty.

This paper explores Ax-Prover in depth, presents its applications in physics and cryptography, and extends its paradigm to a new domain:

the idea of formal semantic verification for natural language a “Lean-for-Language” model.

2. The Ax-Prover Framework

2.1. Core Architecture

Ax-Prover consists of three specialized agents:

• Prover: A large language model (e.g., Claude Sonnet) that generates and iteratively refines Lean proof code.

• Verifier: A Lean compiler instance that formally checks whether the proof is syntactically and semantically correct.

• Orchestrator: A supervisory controller that manages interactions, analyzes errors, and triggers retries when necessary.

Communication among agents occurs via Model Context Protocol (MCP), which bridges the LLM’s reasoning loop with Lean’s Language Server Protocol (LSP).

Through this interface, the Prover can query Lean’s proof goals, context, and error diagnostics as if they were external tools.

This design forms a closed reasoning loop:

generate → verify → correct → iterate → prove

2.2. Workflow Summary

1. The user provides a theorem or problem statement in Lean syntax.

2. The Prover generates tactics or lemmas using the LLM.

3. The Verifier compiles the Lean file and reports logical errors.

4. The Orchestrator interprets feedback and prompts the Prover for corrections.

5. Once no “sorry” or type errors remain, the theorem is formally certified.

3. Experimental Results and Benchmarks

Ax-Prover was evaluated across four benchmark suites:

Dataset	Domain	Description
NuminaMath-LEAN	General mathematics	Lean-formatted theorems with human annotations.
AbstractAlgebra (AA)	Algebra	Group, ring, and field structures requiring symbolic reasoning.
QuantumTheorems (QT)	Quantum mechanics	Operator theory, Hermitian properties, and commutation proofs.
PutnamBench	Mixed mathematics	Competition-level problems translated into formal logic.

3.1. Quantitative Performance

Under constrained conditions (limited API calls, 25-minute runtime), Ax-Prover achieved the following pass@1 accuracies:

• AbstractAlgebra: 64% (vs. DS-Prover 24%, Kimina 13%).

• QuantumTheorems: 96% overall (100% on easy, 92% on medium difficulty).

• NuminaMath-LEAN: 51% (double that of baseline LLMs).

• PutnamBench: 14% with limited computation ranking third overall, but first in efficiency.

3.2. Key Insights

• High generalization: Performs robustly across diverse domains without domain-specific fine-tuning.

• Efficiency: Achieves competitive accuracy under strict computational budgets.

• Self-correction: Leverages Lean feedback for real-time error recovery.

• Reproducibility: Produces formal proofs that can be publicly verified and re-compiled.

4. Applications: From Mathematics to Quantum and Cryptography

4.1. Quantum Reasoning

The QuantumTheorems benchmark marks a milestone:

for the first time, a formal-proof system successfully handled quantum mechanical theorems.

Ax-Prover can, for example:

• Prove that Hermitian operators have real eigenvalues.

• Verify the commutation properties of non-commuting operators.

• Establish equivalence between two quantum circuit representations.

This breakthrough enables:

• Formal appendices in physics papers: verified proof files can accompany publications.

• Model verification: formal validation of quantum algorithms and noise models.

• Educational use: interactive quantum proof archives for teaching and exploration.

4.2. Cryptography and Security Proofs

The framework also extends naturally to cryptography.

Ax-Prover can formalize and verify theorems describing security properties of protocols such as the correctness of digital signatures or key-exchange schemes.

Benefits include:

• Automated verification: logical correctness of security claims can be formally proven.

• Error detection: subtle logical flaws in complex protocols are exposed by Lean.

• Transparency: external auditors can reproduce and verify formal security proofs.

4.3. Knowledge Maintenance and Education

Using the community library Mathlib, Ax-Prover can detect broken dependencies, update outdated definitions, and assist in maintaining large formal-mathematics repositories.

For education, it serves as an interactive mentor guiding students through step-by-step formal reasoning.

5. Beyond Formal Logic: The “Lean-for-Language” Vision

5.1. From Mathematical to Semantic Consistency

While Lean guarantees truth within mathematical logic, LLMs often fail to maintain semantic coherence in natural language.

A “Lean-for-Language” system could bring similar formal rigor to textual reasoning.

Possible semantic axioms include:

• Causal statements must remain directionally consistent.

• Character behavior and context must not contradict earlier assertions.

• Temporal and factual dependencies must be preserved.

A Semantic Verifier could analyze generated text, flag contradictions, and prompt the model for revision mirroring how Lean handles logical errors.

5.2. Implications for LLM Training

Today’s LLMs learn by imitation, not validation.

A Lean-for-Language layer could transform this process:

1. Consistency-aware learning: Models internalize why an answer is correct, not just what it is.

2. Automated feedback: Logical and semantic inconsistencies trigger direct correction loops.

3. Data purification: Contradictory examples can be detected and removed automatically.

This shifts model improvement from data volume to reasoning quality a crucial step toward reliable AI.

6. Discussion

Ax-Prover demonstrates that machine reasoning can be both creative and formally verifiable.

The next logical step is to expand this paradigm beyond mathematics.

A Lean-for-Language framework would allow LLMs to verify the coherence of their narratives, arguments, and analyses, leading to AI systems that reason with self-awareness and accountability.

Future research directions include:

1. Semantic logic modeling: Formal definitions of causality, temporal order, and discourse coherence.

2. Layered verification systems: Sentence-, paragraph-, and document-level consistency checks.

3. Human-AI collaboration: Interactive proof-like environments for writing, argumentation, and scientific review.

7. Conclusion

Ax-Prover marks a turning point in the evolution of AI reasoning systems.

By combining LLM creativity with Lean’s rigor, it transforms mathematical problem-solving into a transparent, reproducible process.

Its success in quantum physics and cryptography proves that formal verification can extend well beyond pure mathematics.

Looking ahead, a Lean-for-Language system could do for natural language what Lean did for mathematics:

make reasoning verifiable.

In such a future, artificial intelligence will not only generate knowledge it will be able to prove that it is true.

References

1. Ax-Prover: A Multi-Agent Framework for Lean-based Theorem Proving with MCP Integration, arXiv: 2510.12787v1

2. Li et al., A Survey on Deep Learning for Theorem Proving, arXiv: 2404.09939

3. DeepSeek Research, REAL-Prover: Retrieval-Augmented LLM Formal Reasoning, arXiv: 2505.20613v1

4. AWS, Introducing Strands Agents 1.0: Production-Ready Multi-Agent Orchestration Made Simple, aws.amazon.com

5. Microsoft Research, The Lean Theorem Prover and Mathlib Documentation, leanprover-community.github.io