# Goedel-Architect: Streamlining Formal Theorem Proving with Blueprint Generation and Refinement

> GOEDEL-ARCHITECT, an open-weight Lean 4 framework using global blueprint refinement, achieves 99.2% pass@1 on MiniF2F and 75.6% on PutnamBench at 555x lower cost than prior pipelines.

- **Source:** [arXiv](https://arxiv.org/abs/2606.06468)
- **Published:** 2026-08-22
- **Permalink:** https://picx.dev/p/plsrZn
- **Whiteboard:** https://picx.dev/p/plsrZn/image

## Summary

## Summary

- **GOEDEL-ARCHITECT** is an agentic framework for formal theorem proving in Lean 4 that centers on **blueprint generation and refinement** — a dependency graph of definitions and lemmas building up to the main theorem.
- Using the open-weight **DeepSeek-V4-Flash (284B-A13B)** backbone, it achieves **99.2% pass@1 on MiniF2F-test** and **75.6% pass@1 on PutnamBench** — exceeding prior open pipelines at a fraction of the cost.
- With optional **natural-language proof seeding**, it closes **100% of MiniF2F-test**, reaches **88.8% (597/672) on PutnamBench**, and solves **4/6 on IMO 2025, 11/12 on Putnam 2025, and 3/6 on USAMO 2026**.
- The pipeline is **up to 500× cheaper** than comparable open pipelines: **$0.44 per problem** vs. ~$244 for Hilbert (Table 2).
- The key innovation is a **global blueprint refinement loop** that contrasts with recursive lemma decomposition (used by Hilbert, Seed-Prover), avoiding inefficient loops on dead-end strategies.

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## Introduction and Theoretical Foundation

The paper addresses a critical gap in formal theorem proving: while frontier AI systems (e.g., AlphaProof, IMO gold-medal-level systems) achieve remarkable mathematical performance, they are either **closed-weight** or **prohibitively expensive**. The theoretical foundation rests on:

- **Formal verification** in Lean (Moura & Ullrich, 2021) as a rigorous alternative to informal proof checking.
- **Benchmark landscape**: MiniF2F-test (244 high-school competition problems), PutnamBench (672 undergraduate-level Putnam problems), and fresh competition sets (IMO 2025, Putnam 2025, USAMO 2026).
- **Prior approaches** categorized by the LLM's role at inference:
  - **Non-agentic provers** (Goedel-Prover, DeepSeek-Prover, Kimina-Prover): single-pass proof generation, <15% on PutnamBench.
  - **Agentic provers** (AxProverBase, Numina-Lean-Agent): single LLM interleaves reasoning with Lean compiler calls.
  - **Pipelines** (Hilbert, Seed-Prover, Draft-Sketch-Prove): orchestrate multiple LLM components with decomposition/refinement.

The paper argues that existing pipelines use **recursive top-down decomposition** — which can loop inefficiently on dead-end strategies — and proposes a **global dependency-graph blueprint** as an alternative.

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## Methodology

### 2.1 Blueprint Generation
- Receives the formal statement of the target theorem and emits a dependency graph as a single Lean file.
- Each node is a formally stated definition or lemma; edges are **declared dependencies** (which lemmas a proof may rely on).
- The target theorem is the unique sink; lemma bodies are left unproved.
- The model iterates against the Lean compiler to ensure well-typedness and acyclicity.
- **Optional NL guidance**: a natural-language proof (from a stronger model or official solution) seeds the initial blueprint structure.

### 2.2 Theorem Proving
- Each lemma is dispatched to a **Lean theorem prover** in parallel, seeing only its declared dependencies.
- The prover has access to the Lean compiler and a Mathlib retrieval tool, iterating until closure or budget exhaustion.
- On failure, the prover returns a **structured diagnosis** (what was attempted, where the gap lies).
- If a counterexample is verified, the prover can register a proof of the **negated statement**.

### 2.3 Blueprint Refinement
- The refinement model reads per-lemma traces, marks each as proved/unproved, and emits a revised graph.
- Typical refinements: **decomposing hard lemmas** into helpers, rewiring dependencies, repairing/dropping false statements.
- **Proved lemmas are preserved** (signatures intact) so prior compute is not wasted.
- The loop continues until all nodes are solved or iteration budget is exhausted.

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## Empirical Validation / Results

### Table 1: Benchmark Comparison

| Model               | MiniF2F-test | PutnamBench | IMO 2025 | Putnam 2025 | USAMO 2026 |
|---------------------|--------------|-------------|----------|-------------|------------|
| Goedel-Prover-V2    | 92.6% @ 1024 | 13.0% @ 184 | -        | -           | -          |
| LongCat-Flash-Prover| 97.1% @ 72   | 41.5% @ 118 | -        | -           | -          |
| Seed-Prover         | 99.6%        | 50.4%       | 5/6      | -           | -          |
| Seed-Prover 1.5     | -            | 87.9%       | 5/6      | 11/12       | -          |
| Hilbert             | 99.2%        | 70.0% @ 1840| -        | -           | -          |
| Numina-Lean-Agent   | -            | -           | -        | 12/12       | -          |
| AxProverBase        | -            | 54.7% @ 1   | -        | -           | -          |
| **Goedel-Architect**| **99.2% @ 1**| **75.6% @ 1**| -       | -           | -          |
| **Goedel-Architect (+ NL)** | **100%** | **88.8% @ 4** | **4/6** | **11/12**   | **3/6**    |

### Table 2: Cost Comparison on PutnamBench

| Metric          | Goedel-Architect | Hilbert* | AxProver |
|----------------|------------------|----------|----------|
| Total spend    | $294             | ~$163k   | $8,467   |
| Avg. cost / Q  | $0.44            | ~$244    | $12.60   |
| Avg. cost / solved Q | $0.21       | —        | —        |
| Avg. cost / unsolved Q | $1.14     | —        | —        |

### Key Findings
- **MiniF2F**: 242/244 at pass@1 (99.2%). The two remaining problems (IMO 1984 P6, IMO Shortlist 2007 Algebra P6) are closed with NL guidance — the **first Lean prover to close all 244 problems**.
- **PutnamBench**: 75.6% at pass@1 (exceeding Hilbert's 70.0% at pass@1840 and AxProverBase's 54.7%). With NL guidance: 88.8% (597/672).
- **IMO 2025**: Solves 4/6 (P1, P3, P4, P5), matching Seed-Prover 1.5 on all non-geometry problems (P2 is geometry-handled by Seed-Prover with a dedicated engine).
- **USAMO 2026**: 3/6 (P1, P4, P6) — contamination-free since it postdates all training cutoffs.
- **Compute scaling**: Solve rate grows roughly log-linearly with refinement iterations (Figure 2): from 200 problems (29.8%) at iteration 0 to 508 (75.6%) at iteration 16.

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## Theoretical and Practical Implications

### Theoretical Contributions
- **Blueprint-as-graph** vs. **recursive decomposition**: The dependency-graph structure allows global refinement, parallel proof attempts with shared context, and avoids dead-end loops that plague recursive strategies.
- **Diagnostic channels**: The paper introduces two novel signals for refinement:
  - **Negated sub-lemmas**: When the prover verifies a counterexample, the disproof (with reflection) is fed into the next revision step — converting dead ends into localized, machine-checked edits.
  - **Forfeited proofs**: When a node runs out of budget, the prover writes a structured post-mortem (statement-wrong vs. proof-too-hard) that proposes a decomposition plan — turning a failed monolith into a provable sub-graph.

### Practical Implications
- **Open-weight, open-pipeline** accessibility: Both backbone and pipeline are publicly available, democratizing formal theorem proving.
- **Cost efficiency**: ~$0.44/problem vs. ~$244 for Hilbert — a **555× reduction** in compute cost.
- **Contamination-free evaluation**: USAMO 2026 (post-training-cutoff) validates generalization capability.
- **Scaling behavior**: The log-linear scaling with refinement iterations suggests **predictable performance gains** from additional compute — useful for resource planning.

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## Conclusion

GOEDEL-ARCHITECT establishes a **new Pareto frontier** for formal theorem proving: state-of-the-art performance for open-weight systems at a fraction of the cost of proprietary pipelines. The key insights are:

1. **Blueprint-centric approach**: A global dependency graph (rather than a recursive tree) allows parallel proof attempts, shared context, and global refinement.
2. **Structured failure signals**: Both negated sub-lemmas and forfeited proofs provide actionable diagnostics that feed directly into the next refinement iteration.
3. **Natural-language guidance is decisive** for non-local structure problems (cyclic sums, parity/divisibility chains, tile-counting) — where deriving the lemma graph from the formal statement alone is the bottleneck.

### Future Directions
- Extending the blueprint approach to more complex mathematical domains (e.g., geometry, where specialized engines are still needed).
- Reducing the dependency on natural-language guidance for the hardest problems.
- Investigating further compute scaling behavior beyond iteration limits.
- Exploring the potential of the open-weight backbone with additional fine-tuning or RL training.

**Bottom line**: GOEDEL-ARCHITECT demonstrates that **efficient, open-source formal theorem proving** can rival closed-weight frontier systems — a significant step toward democratizing rigorous AI-generated mathematics.

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