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AI can now produce correct solutions to difficult research-mathematics problems, but the October 14, 2026 First Proof results show that the strongest performance depends on expensive models, substantial computing access and a technically sophisticated multi-agent setup. The benchmark demonstrates powerful proof generation—not autonomous mathematical research—and exposes a sharp gap between frontier and open-weight systems.
What the third First Proof batch tested
A controlled proof-writing exam
First Proof is an independent project designed to evaluate AI on research-level mathematics transparently. In its third batch, five systems received 10 previously selected problems in geometry, topology, statistics and dynamics. Each system had 24 hours, no human assistance and anonymized submissions, so reviewers could judge the work without knowing which system produced it.
Thirty expert reviewers assessed mathematical correctness, exposition and attribution of prior work. The scoring also introduced negative marks, making a careful abstention preferable to a confident but incorrect proof.
What the exam does—and does not—measure
The problems represent the final, well-specified proof stage of research. First Proof does not test whether an AI can choose an important question, invent a useful mathematical concept, decide which result matters or develop an agenda over years of work. Those remain outside the benchmark’s scope.
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Results: a wide spread in performance
| System | Correct solutions | Citation quality | Exposition | Openness and setup | Third-batch run cost |
|---|---|---|---|---|---|
| ProofCouncil | 9 of 10 | Adequate on 7 of 10 | Negative average exposition scores on one-third of responses | Coordinates several high-end models; requires paid model access and programming expertise | $1,888 |
| Direct GPT-6 Astra run | 7 of 10 | Adequate on 4 of 10 | System-specific result not stated | Direct run; additional harness complexity not stated | $36 |
| Open-weight Kimi K3-based entrant | 0 of 10 | Major errors found in every submission | Major errors found in every submission | Open-weight model; other setup details not stated | Not stated |
Every problem was solved correctly by at least one system when citation and exposition issues were set aside. That finding indicates that the task was within reach of at least one configuration for each problem, while the table shows how much system design and verification affected the submitted answers.
Why the result supports a “pay-to-play” description
The cost is in the whole system, not just a model
ProofCouncil’s $1,888 run cost was measured against $36 for the direct GPT-6 Astra run in this batch. The previous batch reported costs of $3,200 and $117, respectively. These are run costs, not a universal price for every researcher: they reflect the particular systems, model calls and computing used in each evaluation.
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The University at Buffalo account of the release describes ProofCouncil as coordinating several high-end models. Building and operating that harness requires technical programming expertise and paid access to multiple models. In other words, the advantage is tied to an orchestration layer and compute budget as well as to the underlying model.
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Compute access can become a research barrier
First Proof’s official release links additional computational access with better performance and warns that unequal access could become a barrier to basic mathematical research. Adam Sikora, a mathematics professor at the University at Buffalo, said, “At present, that makes this kind of setup much less accessible to most academic researchers.” The benchmark therefore measures an access gap as well as a capability gap.
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Correctness is only one part of a usable proof
Citations and attribution
Only four of the 10 direct GPT-6 Astra solutions met the reviewers’ standard for adequate citations. ProofCouncil raised that number to seven of 10. A mathematically correct argument can still be unsuitable for publication or collaboration if it fails to identify prior results, misattributes a technique or leaves the reader unable to check its claims.
Exposition and safe uncertainty
Even the leading ProofCouncil configuration received negative average exposition scores on one-third of its responses. The negative-scoring rule matters here: a system that admits uncertainty can be preferable to one that writes a polished-looking but false proof. The available results do not report a system-by-system abstention rate, so First Proof cannot establish which configuration was safest when it lacked a reliable solution.
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Why open-weight systems are far behind in this test
The open-weight Kimi K3-based entrant solved none of the 10 problems, and reviewers found major errors in every submission. Mohammed Abouzaid, a Stanford mathematician and First Proof team member, described the disparity plainly: “The gap between the open-weight models and frontier models is very large in research mathematics—much larger than in other domains.”
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That result should not be read as proof that open weights can never catch up. This batch compares one open-weight entrant with frontier-model configurations under a demanding proof-writing protocol; it does not isolate training data, inference budget, tool use or engineering quality as separate causes. It does show that, on this exam and at these budgets, open access did not deliver comparable performance.
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Lauren Williams, a Harvard mathematician and First Proof team member, summarized the entrant’s showing more bluntly: “It really fell flat on its face.”
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The benchmark’s answer is no
First Proof provides evidence that AI can generate research-level proof attempts and, in the best configuration, solve nine of 10 selected problems. It does not show independent mathematical discovery in the broader sense. The benchmark supplies the questions, constrains the time and relies on experts to grade the outputs.
Humans still set direction and meaning
Sikora said, “I think we are in a process of developing a new and very powerful culture of mathematical research in which humans continue to play the leading role.” In that model, people choose important questions, develop concepts, judge significance and seek understanding, while AI accelerates parts of the technical work.
Abouzaid also said, “The models have become very good,” adding that testing frontier models on this exact task “might not be that relevant in the future.” That warning points to a moving benchmark: as systems improve, the hard question may shift from producing a proof to deciding which problems deserve attention and how the result changes mathematics.
What researchers and readers should take from the exam
- Capability is configuration-dependent. A direct frontier-model run scored 7 of 10, while a costly multi-model harness scored 9 of 10.
- Proof quality is multidimensional. Correctness, citations, exposition and responsible abstention all affect whether an answer is useful.
- Access matters. The strongest setup required paid access to multiple models, programming skill and substantial compute.
- Open weights remain uncompetitive here. The Kimi K3-based entrant failed every problem in this batch, but the test does not prove that outcome is permanent.
- Human judgment remains central. Selecting questions, interpreting significance and checking arguments are not capabilities established by this benchmark.
The practical verdict
First Proof’s third batch shows that AI-powered mathematical proof is already credible at the research frontier, but the best results currently look like an expensive infrastructure project rather than a universally available tool. Until high-end model access, compute and verification become more affordable—and systems improve their citations and exposition—research mathematics will remain, in an important sense, pay to play.
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