Gödel Test: LLMs in Mathematical Discovery

• The Gödel Test is a benchmark evaluating LLMs’ ability to autonomously generate correct proofs for novel, elementary conjectures in combinatorial optimization.
• It employs a methodology where models use minimal guidance from source papers to synthesize proofs, testing cross-paper integration and creative reasoning.
• Empirical results indicate that while models like GPT-5 show near-human proficiency on routine tasks, they struggle with integrating diverse techniques and maintaining rigorous logical structure.

The Gödel Test is a recently proposed evaluation paradigm intended to probe the capacity of advanced LLMs to generate correct proofs for novel, simple conjectures whose solutions are “easy” for a mathematically trained human but not present in any directly accessible literature. Unlike standard benchmarks—such as mathematics Olympiads or undergraduate competition problems—the Gödel Test targets the model’s ability to synthesize and prove new results in active research domains, particularly focusing on combinatorial optimization and discrete mathematics. The test, introduced by studies leveraging GPT-5, is intended to concretely measure the frontier of “mathematical maturity” and creative reasoning in contemporary LLMs (Feldman et al., 22 Sep 2025).

1. Conceptual Definition and Motivation

• The Gödel Test is designed to evaluate whether an LLM can autonomously produce correct, human-acceptable proofs for previously unsolved (but conceptually elementary) conjectures.
• It is distinguished from the Turing Test by focusing specifically on mathematical creativity and problem-solving, rather than generic or conversational human-likeness.
• The guiding principle is to move beyond “regurgitation” of known results towards true synthesis—requiring reasoning that cannot simply draw on memorized templates.

Significance: The test is conceived in direct response to the recent trend of reporting strong LLM performance on Olympiad-level or well-documented problems, challenging the models with questions where established solutions are not available, and where the task is, in some sense, “mathematically routine” but still unknown.

2. Experimental Methodology

• The benchmark is constructed from five conjectures in combinatorial optimization, with problem statements derived from the ongoing research of the authors, primarily in DR-submodular maximization and convex polytopes.
• For each instance, the model is furnished with one or two source papers containing relevant background but not the proposition to be proved; the intention is to minimize prompt engineering and avoid inciting the model to simply “search for a known result.”
• The model (GPT-5) is prompted to produce a full proof in LaTeX, employing known computational and analytic techniques such as Frank–Wolfe methods, continuous greedy algorithms, or randomized greedy frameworks.
• As an explicit example, the model was asked to establish a bound of the form

$$F(x) \geq \alpha \cdot G(o) + \beta \cdot H(o) - \text{err}(\epsilon, D, L_G, L_H)$$

with $\alpha = 1-1/e$, $\beta = 1/e$, $G$, $H$ DR-submodular functions, and $\text{err}$ depending on algorithmic or analytic parameters. The intent is to test whether the model can adapt and generalize techniques from related but distinct papers.

• Assessment is entirely “hands-off”: the model is not given mid-proof critique or hints, and assessment is based on the mathematical correctness and originality of the output.

3. Empirical Results and Model Behavior

• On the three easier conjectures, GPT-5 produced proofs that were nearly correct and in some cases showed near-human adaptation of known techniques. For instance, in DR-submodular maximization over down-closed convex polytopes, the continuous greedy analysis and key inequalities were replicated, including error terms and relevant approximation constants.
• Notably, for one problem (Problem 2), GPT-5 derived an alternative approximation guarantee, effectively refuting the authors’ original conjecture while supplying a valid, distinct solution (i.e., the model produced a counter-example with correct analysis). This is interpreted as a nascent form of mathematical originality, evidencing reasoning not merely constrained by surface-level copying.
• The model failed on Problem 4, a task fundamentally requiring cross-paper synthesis of methods—GPT-5 was able to reproduce individual methodologies from each source but was unable to combine them meaningfully, resulting in invalid inferences and incompatible notation.
• For Problem 5, which lacked even a verified human proof, GPT-5 was able to propose the same core algorithm as constructed by the researchers but failed in the technical analysis, e.g., mishandling key inequalities or losing control of sequences of bounds.
• Across responses, a recurrent phenomenon is that the model produces “lazy” proofs: rather than navigating the space of potential arguments creatively, it defaults to closely following the structure and even verbatim text of its source papers, especially when the proof does not require integration of information beyond a single reference.

4. Analysis of Model Limitations

• Cross-paper synthesis failure: GPT-5’s most acute limitation is in integrating results and techniques from disparate sources. Tasks requiring combinatory innovation—i.e., the true “creative leap” at the heart of advanced problem-solving—expose the model’s inability to construct a higher-order synthesis beyond its training data. For example, attempts to merge variants of the continuous greedy and randomized greedy approaches for bicriteria maximization led to confusion in variable dependence, improper treatment of bicriteria constraints, and loss of logical rigor.
• Surface-level coherence without depth: While GPT-5’s answers were formatted as plausible LaTeX proofs and could pass initial plausibility checks, detailed inspection revealed gaps, misapplied arguments, and sometimes outright incorrect claims. Thus, human validation remains essential and the risk of subtle, undetected errors is nontrivial.
• Re-use over creativity: The model exhibits a strong bias towards reusing not just the techniques but the linguistic and notational structure it has seen, discouraging genuinely new argumentation or methodological exploration.

5. Future Directions and the Path Toward Robust Gödel Test Success

• The preliminary paper establishes that frontier LLMs like GPT-5 are capable of approaching unsolved conjectures with near-correct reasoning if the path is singular and aligns well with existing mathematical knowledge.
• The main barrier to full “Gödel Test” passing is in robust, automated cross-domain synthesis and error control. Future directions include:
  • Incorporating interactive prompting (e.g., dialog-based refinement, mid-proof queries, or error-correcting suggestions).
  • Integrating formal proof assistants or external symbolic tools to better manage technical details and verify logical correctness dynamically.
  • Expanding model architectures or context windows to better handle multiple sources and more complex, multi-stage arguments.
  • Developing training strategies or retrieval-augmented models that specifically target cross-paper reasoning and encourage generative exploration rather than surface-level match.
• Such extensions are necessary for LLMs to serve as autonomous mathematicians that can reliably produce new proofs or even generate valid conjectural refutations when given only indirect guidance.

6. Broader Implications and Benchmark Considerations

• The Gödel Test highlights the difference between “rehearsed mathematical fluency” (as measured by sourcing known facts or solving standardized questions) and “active mathematical discovery.” Passing the Gödel Test begins to approximate the threshold at which LLMs could be deemed tools (or collaborators) in mathematical research, not just assistants for search or reference.
• This paradigm discloses both the tremendous recent progress on routine, deterministic mathematical tasks and the persistent gap in handling tasks that demand abstraction, synthesis, or “genuine insight.”

Aspect	GPT-5 Performance	Limitation Domain
Single-source proofs	Nearly correct	Minor technical details
Cross-paper synthesis	Fails (erroneous output)	Integration, error propagation
Originality	Occasional (counterexamples, alternative bounds)	Restricted, inconsistent

This suggests that while current LLMs are on the verge of matching baseline expectations for advanced undergraduate or early graduate mathematical proficiency, the capacity for multi-source creative insight—the core of mathematical research—is not yet realized (Feldman et al., 22 Sep 2025).

Conclusion

The Gödel Test offers a stringent, well-motivated benchmark for evaluating the mathematical synthesis abilities of LLMs. GPT-5 demonstrates meaningful, measurable progress on “routine” conjectures and occasional originality but exhibits marked deficits in integrating results across sources and ensuring global proof validity. Robust passage of the Gödel Test will likely require not only improvements in LLM architectures but also methodological advances at the interface of interactive prompting, computational proof systems, and domain-specific retrieval.