Gödel Test for AI Proofs

• Gödel Test is a benchmark evaluating AI's ability to generate valid proofs for novel, unsolved mathematical conjectures, emphasizing routine research-level reasoning.
• It focuses on problems in combinatorial optimization and submodular maximization that require integrating techniques from multiple research papers without relying on memorized proofs.
• Results with GPT-5 demonstrate promising routine reasoning while also revealing challenges in synthesizing multi-source mathematical concepts and novel proof methods.

The Gödel Test is a proposed benchmark designed to evaluate whether advanced AI models, specifically LLMs, can generate correct proofs for simple yet previously unsolved conjectures in advanced mathematical domains—most notably, combinatorial optimization and submodular maximization (Feldman et al., 22 Sep 2025). Unlike traditional math competition problems—crafted for high-school or undergraduate levels and often focused on challenge and puzzle-solving—the Gödel Test targets the capacity of AI to perform "routine reasoning" and make discoveries characteristic of early-stage research, particularly when proofs must be synthesized from multiple sources and are not present in existing literature.

1. Definition and Conceptual Motivation

• The Gödel Test is constructed as an evaluation protocol where an AI model is asked to prove or disprove mathematically simple, yet genuinely open conjectures, typically those that would be considered accessible to advanced undergraduate or entry-level graduate students working in the field.
• Conjectures are crafted to be concrete and grounded in current research, but they do not appear in published literature. The test intentionally avoids problems that are standard, widely solved, or "^^^^1^^^^" puzzles; instead, it probes the model’s ability to reason about novelty within a tractable search space.
• Unlike automated theorem-proving in well-axiomatized domains, the Gödel Test often demands reasoning that synthesizes techniques and results from multiple areas or papers, thereby emulating authentic research-level problem-solving.

2. Methodological Framework

• Each problem presented as part of the Gödel Test is accompanied by one or two source research papers from which the conjecture arose, but the model does not receive the conjecture statement directly from these sources. The setup is intentionally non-spoonfeeding: the AI is required to interpret the context, understand the relevant mathematical structures, and generate a plausible proof without seeing the solution or proof strategy in the training data.
• Problems are selected to have a well-delineated conjecture and established body of approaches, allowing expert evaluators to judge solution correctness and logical progression unambiguously.
• Human experts rigorously assess every solution step, carefully scrutinizing the AI's inferences, adaptations of known proofs, handling of technical lemmas, and logical cohesion.

3. Performance of GPT-5 on the Gödel Test

• On three of five selected simple conjectures from combinatorial optimization, GPT-5 produced proofs that were judged "nearly correct" by the evaluation panel. An illustrative case includes adaptation of a Frank–Wolfe-type analysis to demonstrate a guarantee such as

$$F(x) \ge \alpha \cdot G(o) + \beta \cdot H(o) - \mathrm{err}$$

with typical parameter values $\alpha = 1 - 1/e$ and $\beta = 1 - 1/e$ (fully monotone scenario) or $\beta = 1/e$ (non-monotone split).

• On Problem 2, GPT-5 generated a different approximation guarantee than the authors conjectured, which after expert checking, both refuted the original conjecture and provided a valid, alternative solution. This indicates that GPT-5 is capable of nontrivial adaptation and, on occasion, genuine mathematical insight.
• GPT-5 failed to solve Problem 4, which required explicit synthesis of proof strategies from two distinct source papers. For Problem 5, the model reproduced the correct algorithm outlined by the experimenters, but faltered in executing the critical analysis, highlighting the increased challenge when deeper integration or novel analysis is demanded.
• Throughout, GPT-5 leveraged LaTeX-style formulas and mathematical notation common to expert-level combinatorial optimization (e.g., iterates like

$$(1 - \varepsilon) S(f, i-1) + \varepsilon \cdot M(f, i-1) = S(f, i),$$

and structural decompositions in proof steps).

4. Error Modes and Limitations

• The evaluation consistently found that GPT-5's strongest performance was in problems amenable to direct adaptation of existing proof structures, provided in cited source papers. When the reasoning required "staying close" to a single proof template or a classic inequality, the model could assemble near-correct arguments with only minor technical errors.
• GPT-5 exhibited visible limitations in synthesizing ideas that spanned multiple, independently authored papers, particularly when the proof demanded combining disparate lemmas or conceptual frameworks into a coherent whole. This boundary aligns closely with the lack of persistent multi-document synthesis observed in LLMs across other scientific disciplines.
• For the hardest problems, the model’s inability to manage nontrivial analysis or explore alternative proof trajectories—beyond direct adaptation of known techniques—was evident.

5. Analysis and Implications

• Passing the Gödel Test requires an AI to demonstrate capacity for elementary research-level reasoning: adapting familiar proof templates to novel conjectures, combining multiple lines of prior work, and going beyond mere pattern matching to produce logically valid, original arguments for as-yet-unresolved mathematical questions.
• The results suggest GPT-5 is approaching human-level proficiency on routine mathematical proof construction in advanced (but structurally familiar) areas, occasionally displaying independent discovery (e.g., generating a valid counter-approximation or refuting an incorrect conjecture).
• However, models are still consistently challenged when required to perform multi-source synthesis—coherently integrating ideas from the literature that were never explicitly combined before.

Problem Type	GPT-5 Result	Key Limitation
Easy/single-source conjecture	Nearly correct	Minor technical errors
Multi-source synthesis required	Failed	Inability to integrate proofs
New but structurally familiar	Occasional originality	Lacks deeper exploration

6. Future Directions and Benchmark Refinement

• The paper proposes that prompt strategies requesting more explicit stepwise reasoning, integration with formal verification tools (proof assistants, computer algebra systems), and expanding the tested conjecture pool over more domains may further improve and clarify future Gödel Test results.
• Improving the Gödel Test as a benchmark entails more precise calibration of problem difficulty and "novelty," systematic expansion to other mathematical disciplines, and better characterization of what constitutes "human-like discovery" for AI systems.
• The authors anticipate that as AI models advance, passing the Gödel Test—across many instances and with increasing independence from training data—will become a clear marker of progress in machine mathematical reasoning, distinguishing between pattern-matching ability and true integrative synthesis.

7. Significance for AI and Mathematical Practice

• The Gödel Test fills a gap between benchmark-driven competition problem-solving (e.g., MATH or MiniF2F datasets) and unrestricted mathematical discovery, focusing on "easy but new" conjectures that cannot simply be memorized or retrieved from extant corpora.
• Success on the Gödel Test would indicate a frontier AI model’s capacity for mathematical generalization, transfer of techniques, and genuine insight—critical prerequisites for machines to be useful mathematical collaborators in research disciplines.
• The observed limitations, particularly with multi-source synthesis, signal ongoing challenges for current LLMs and motivate further research into architectures, tool integration, and knowledge representation that bridge the gap between human mathematical creativity and automated reasoning.

In summary, the Gödel Test represents a rigorous, well-motivated protocol evaluating an AI’s ability to construct correct proofs for previously unsolved but accessible mathematical conjectures. Experimental results with GPT-5 demonstrate encouraging progress in routine reasoning and the occasional emergence of original argumentation, but also reveal clear deficits in multi-document synthesis and creative proof adaptation (Feldman et al., 22 Sep 2025). The benchmark thus provides a concrete metric for progress toward genuinely collaborative AI in mathematical research.