DeepSeek‑V4 Powers Formal Math Proofs with 500× Cost Savings, Setting New Records

Recent AI breakthroughs have begun to overturn long‑standing mathematical problems: OpenAI’s internal reasoning model disproved Paul Erdős’s 1946 unit‑distance conjecture, and Fields Medalist Timothy Gowers called such AI‑generated proofs a historic milestone, while fellow Fields Medalist Terence Tao warned that the field is moving from a "proof‑scarcity" era to a "proof‑surplus" era, creating a verification crisis.

In response, a Princeton Language and Intelligence (PLI) team released the paper Goedel‑Architect: Streamlining Formal Theorem Proving with Blueprint Generation and Refinement , employing the open‑source large model DeepSeek‑V4‑Flash as its core. The system’s name honors Kurt Gödel, reflecting the team’s Princeton heritage.

On the standard PutnamBench suite (672 problems), the previous leading open‑source pipeline Hilbert (driven by Google Gemini 2.5 Pro) incurred roughly $170,000 in API costs, whereas Goedel‑Architect completed the same evaluation for $294—a cost reduction of about 500×. Moreover, Goedel‑Architect achieved a higher pass@1 rate of 75.6% compared to Hilbert’s 70.0%.

The key innovation is the blueprint concept: before any proof attempts, the system constructs a directed acyclic graph that enumerates all required definitions and lemmas and their dependencies. Each node represents a precise lemma, and edges encode prerequisite relationships, giving a global view of the proof strategy.

During execution, the blueprint enables parallel processing: each lemma node is dispatched to a Lean prover that only sees its own statement and upstream dependencies. Successful proofs are marked green, failures blue, and lemmas disproved by reverse reasoning red. Failures are not dead ends; they trigger a structured diagnostic report.

Two failure‑handling paths are defined. First, if a lemma is false, the system records the counterexample and rewrites the lemma in the next iteration—illustrated by a Putnam 1989 problem where an auxiliary lemma about binary multiplication was incorrectly phrased, leading to a counterexample at n=5. Second, if a true lemma exceeds the token budget, the prover suggests a decomposition strategy; for a Putnam 1985 problem about quintic polynomial roots, the suggested case split into four sub‑cases allowed the next iteration to prove each sub‑lemma and solve the original problem.

Proven nodes persist across iterations, so the process resembles a gradually completing puzzle rather than restarting from scratch each time.

Additional benchmarks show Goedel‑Architect solving 242 of 244 MiniF2F-test problems (99.2% pass@1), matching the strongest open‑source systems, and, with natural‑language proof scaffolding from a larger model (e.g., Gemini 3.1 Pro), solving the remaining two IMO‑level problems. For nine particularly hard problems requiring non‑local structure, the system failed without language assistance after multiple runs, but succeeded on all nine when the scaffold was provided.

Ablation experiments moving the Hilbert pipeline onto the same DeepSeek‑V4‑Flash backbone yielded only 84.4% on MiniF2F, while Goedel‑Architect reached 99.2% on the identical backbone. On a 200‑problem Putnam subset, a tool‑integrated single‑agent approach achieved 54.5% versus Goedel‑Architect’s 76.0%, using fewer tokens per problem.

These results demonstrate that the performance gains stem primarily from the pipeline design—global blueprint generation and iterative refinement—rather than solely from a superior model. Goedel‑Architect provides an open‑source, ultra‑low‑cost infrastructure that brings formal theorem‑proving capabilities previously limited to expensive closed‑source systems within reach of the research community, and offers a trustworthy foundation for verifying AI‑generated mathematical claims.