Mathematics

OpenAI and mathematical problems: when artificial intelligence tackles the frontiers of knowledge

Artificial intelligence enters a new phase of its scientific development: one where its systems no longer merely assist researchers but propose results capable of advancing particularly demanding mathematical domains. On 6 October 2026, OpenAI announced the publication of a set of results from a state-of-the-art internal model, accompanied by manuscripts, methodological documents, and formalization of certain proofs in the computer assistant Lean.

This initiative fits into a broader dynamic. From combinatorial geometry problems associated with Paul Erdős’s work to fundamental questions in mathematical analysis, reasoning models are now assessed on their ability to explore novel avenues, construct proofs, and produce results that can be reviewed by the scientific community.

The stakes extend beyond algorithmic performance alone. The ability to accelerate certain research could alter the costs of innovation, tech companies’ strategies, and the international balance in the most advanced scientific fields.

A distinction remains, however, indispensable: an announced result by a company, a formally verified demonstration, and a theorem accepted by the mathematical community do not correspond to the same level of validation. It is from this distinction that the real scope of these advancements can be evaluated.

The Erdős Distance Problem: An Illustration of New Reasoning Capabilities

A mathematical legacy spanning several decades

The name of Paul Erdős is associated with many problems that have profoundly marked combinatorics, number theory, and discrete geometry. A prolific mathematician of the 20th century, he notably contributed to popularising seemingly elementary questions whose resolution requires particularly sophisticated conceptual tools.

Among them are several issues concerning the distances between points.

Their formulation seems simple: given a set of points, how many different distances can be obtained between their pairs? Or, in another version, how many pairs of points can be separated by a distance exactly equal to a unit?

These questions lead to distinct mathematical problems, even if they belong to the same family of research.

Therefore, it is necessary to distinguish:

  • The problem of distinct distances: determine the minimum number of distinct distances generated by a set of points.
  • The Unit Distance Problem: establishing how many pairs of points can exactly lie at a given distance, typically normalized to a unit.
  • Pinpointed distance issues: studying the distances determined from a specific point in a set to other points.

These questions mobilise combinatorial geometry, counting techniques, and spatial structure analysis.

They are important because they highlight a fundamental difficulty in mathematics: a geometric configuration can be easy to represent, yet present global properties that are extremely difficult to prove.

What OpenAI announced about distances

In the public repository dedicated to its mathematical work, OpenAI presents several results concerning distinct distances and unit distances.

Among the listed results are, for example:

  • An advancement on Erdős’s distinct distances conjecture in dimensions greater than or equal to three, with a lower bound of the order of (n^{2/d}), up to a constant factor for a fixed dimension (d).
  • A result concerning the distances pinned in the plane, which establishes an asymptotic bound for the number of distinct distances determined from almost all points of a set.
  • An improvement on the classic upper bound regarding the number of pairs of points separated by a unit distance in the plane.

These results are documented in the public mathematical repository on GitHub by OpenAI, which gathers the manuscripts and related technical elements associated with the announced work.

They, however, should not be summarized by the general statement that AI “solved the problem of unitary distances after 80 years.” The problems of distances form a family of conjectures and theorems, with varying formulations, hypotheses, and degrees of difficulty.

The scientific interest lies precisely in the nature of the results obtained, in the strength of their demonstrations, and in their relationship with established knowledge.

Why are these problems so difficult?

In an elementary geometric problem, a numerical simulation can quickly suggest an interesting configuration. But observing a behavior across several examples is not enough to prove that it remains true for all sets satisfying the hypotheses.

The challenge lies in converting an intuition into a general proposition.

Mathematicians must notably:

  • Identify the properties that remain invariant when the configuration changes.
  • Establish valid bounds for arbitrary size sets.
  • Demonstrate that the proposed constructions achieve, or approach, the sought-after limits.
  • Reject counterexamples that could invalidate a conjecture.
  • Link the problem to previous results to build a coherent demonstration.

Reasoning models can contribute to this work by exploring many avenues, proposing intermediate lemmas, and reorganizing argument chains.

Their interest therefore lies not just in computational speed. It also resides in their ability to produce reasoning structures that researchers can examine, correct, and potentially reuse.

A unusually large scientific publication

Internal models capable of exploring open problems

On 6 October 2026, OpenAI released a new series of mathematical results produced by an internal state-of-the-art model, which was not presented as a freely accessible system to the public.

The company indicates that it has accompanied this publication with documents to better understand the origin of the results, the model’s reasoning, and the computational methods employed. It also shares formalizations of certain proofs in Lean, as well as information on the attempted problems and estimates of the computational effort.

OpenAI states that the average calculation associated with a result roughly equated to three hours of reasoning with ChatGPT Pro, according to their measurement method.

This indication does not mean that each demonstration was found in three hours of humanly comparable work. It provides an estimate of the computation mobilised within the context of the experiment, and not a universal measure of mathematical difficulty.

The company also intends to progressively improve the presentation of the results, their bibliographic references, and the quality of their presentation, while continuing to engage with independent specialists.

Results that exceed geometry

The public repository covers several branches of mathematics and theoretical computer science. The announced works include, among others, results related to distances, number theory conjectures, algebraic geometry, and optimization questions.

The information communicated around this publication also includes proposals on:

  • A property of the Riemann zeta function in a region of the complex plane.
  • The Unique Games Conjecture, a major problem in theoretical computer science.
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Des bornes relatives à la complexité de la multiplication matricielle.

  • Various open problems in specialized mathematical fields.

These topics do not all exhibit the same level of maturity. Some results are accompanied by computational formalization, while others still require in-depth analysis of their scope, assumptions, and novelty.

The value of such a publication therefore depends less on the number of documents produced than on the researchers’ ability to determine which results are correct, new, and truly significant.

Open publication as a verification tool

The decision to make the manuscripts accessible on GitHub is particularly significant.

A scientific assertion can be evaluated more effectively when researchers have access to the texts, references, and, where possible, the technical elements necessary to reproduce the steps of the demonstration.

OpenAI states that it consulted the independent mathematics and artificial intelligence advisory group of the Institute for Advanced Study to improve its publication practices.

The company also recognizes the need to improve the presentation and documentation of its work. This is important: a difficult-to-read demonstration, one that is insufficiently referenced or poorly explained, can slow down verification, even when it contains a promising idea.

The publication of the results therefore constitutes a starting point for collective scientific work, and not an automatic guarantee of validity.

Lean: when computing enables the verification of proofs

The role of formal proof assistants

One of the most important aspects of these researches is the use of Lean, a language and computational environment designed for formalizing mathematical proofs.

In a traditional demonstration, the reasoning is presented in a mathematical language intended to be understood and verified by specialists. This presentation relies on conventions, previous results, and steps that the reader must be able to reconstruct.

A formal proof assistant imposes a different structure. Mathematical objects, hypotheses, and deduction steps must be expressed in a sufficiently precise logical system for a computer verifier to control the validity of the formalized proof.

The software therefore does not merely assess whether a text appears convincing. It checks that a coded demonstration adheres to the rules of the formal system employed.

This approach presents several advantages:

  • It enables the detection of certain logical errors that might go unnoticed in an informal reading.
  • She imposes a rigorous explicitation of hypotheses and intermediate steps.
  • It facilitates the reuse of formalized results in new demonstrations.
  • It enables the creation of libraries of verifiable proofs for other researchers.

For very complex results, this formalisation can represent considerable work, distinct from the initial mathematical discovery.

What formal verification guarantees — and what it does not guarantee

The presence of a Lean proof is an important element of trust, but its scope must be defined with precision.

A correctly verified proof establishes that the formalized proposition follows from the hypotheses and definitions used in the relevant logical system, provided that the verification environment and its foundations are reliable.

She does not guarantee on its own that the formalisation exactly corresponds to the statement that the authors wished to prove. An error in translation between the initial mathematical problem and its formal representation may still exist.

Similarly, a correct proof does not automatically show that the result is novel, significant, or sufficiently general to address the conjecture in question.

The scientific verification therefore requires several levels of control:

  1. Logical Verification: Are the steps of the demonstration valid?
  2. The fidelity of formalisation: does the coded statement match the problem as announced?
  3. The novelty: does the result exceed the published knowledge?
  4. Mathematical scope: Do the hypotheses and conclusions match the claimed importance?
  5. Peer review: do the experts in the field consider the result to be established?

This distinction is essential to avoid confusing algorithmic power with scientific authority.

A fundamental problem in fluid mechanics

The Navier-Stokes equations describe, among other things, the motion of fluids such as water and air. They occupy a central place in fluid mechanics, engineering, and many scientific applications.

The famous mathematical problem associated with these equations concerns the existence and regularity of solutions in the relevant three-dimensional framework.

In simplified terms, it is about determining whether sufficiently regular solutions can exist under all conditions specified by the problem formulation, or if singularities may appear.

This question is one of the seven millennium problems of the Clay Mathematics Institute. A solution meeting the institute’s criteria could be awarded a million dollars.

The difficulty lies in the complexity of the equations and the interactions between the different scales of fluid motion. A numerical simulation can provide extremely useful results without resolving the general mathematical question.

An OpenAI proposal that must be distinguished from an acknowledged resolution

On 8 September 2026, OpenAI published an announcement titled An OpenAI model proposes a solution to the Navier–Stokes problem.

The choice of the verb “proposes” is important: it refers to a solution proposed by a model, rather than a statement establishing definitively that the millennium problem has been conclusively resolved.

The elements provided around this announcement suggest an intensive use of AI agents in parallel and a significant production of reasoning and manuscripts. These figures should be attributed to their respective sources and cannot replace the examination of the demonstration itself.

At this stage, the existence of an algorithmic proposal should not be confused with a definitive mathematical validation.

To establish an advancement of this nature, it is necessary to determine, among other things, whether the reasoning is correct, whether all the hypotheses are respected, and whether the conclusion adequately addresses the problem as defined by the mathematical community.

The institutional reference to consult is OpenAI’s announcement regarding the proposal of a solution to the Navier-Stokes problem.

The computational power does not replace the proof.

The use of multiple parallel agents can accelerate the exploration of different leads. Each agent can propose approaches, test intermediate hypotheses, or seek contradictions.

This organisation can increase the diversity of attempts and allow for the examination of more resolution paths.

She nevertheless introduces an additional challenge: as the volume of produced reasoning increases, so too does the cost of sorting, consolidating, and verifying them.

Thousands of tracks do not necessarily equate to thousands of discoveries. A large set of manuscripts may contain useful ideas, redundant arguments, errors, or results that are already known.

The central question therefore remains the quality of the evidence obtained.

The future of AI-assisted mathematical research will depend on the ability to combine large-scale exploration, formal verification tools, and specialized human expertise.

ARC-AGI-3: measure intelligence beyond correct answers

Static issues to interactive environments

The progress in mathematics is just one aspect of evaluating artificial intelligence systems. Another question is to determine how much an agent can learn to act in an environment it does not know in advance.

This is the objective of ARC-AGI-3, a benchmark developed by the ARC Prize Foundation.

Unlike tests that require answering a single question or solving an isolated puzzle, ARC-AGI-3 presents interactive environments in which an agent must observe, experiment, understand rules, identify objectives, and adapt its actions based on the feedback received.

The agents do not necessarily have explicit instructions telling them what to do. They must discover what matters in the environment and develop a strategy.

The benchmark aims to assess several capabilities:

  • Exploration: to obtain information proactively rather than relying on provided data.
  • Modelling: to construct a representation of the rules and possible states of an environment.
  • Identifying Objectives: understanding what needs to be achieved without relying on detailed instructions.
  • Planning: organising a sequence of actions in accordance with an objective.
  • Adaptation: revise a strategy when new observations contradict initial hypotheses.

Why are these tests important for general AI

A system can be efficient at known tasks without being able to effectively adapt to a genuinely new situation.

The performance on static exercises can partly reflect the knowledge gained during training, pattern recognition, or the use of specialized strategies.

Interactive environments aim to measure a different capability: learning the rules of a situation from experience, without having all the necessary information in advance.

An agent must be able to:

  • Test a hypothesis through action.
  • Observe the consequences of this action.
  • Distinguishing a useful result from an ineffective attempt.
  • Modify your representation of the problem.
  • Planify in multiple steps despite limited feedback.

The 100% score, according to the benchmark definition, corresponds to a performance level that enables success in all games as effectively as humans.

This threshold on its own does not constitute proof of general intelligence. It provides a measurable reference within a precise set of environments, with defined evaluation rules.

The aim of ARC-AGI-3 is therefore to make certain discrepancies between human capabilities and those of artificial agents more observable, rather than sticking to general statements about the intelligence of models.

From mathematical resolution to autonomous learning

Mathematical research and interactive benchmarks assess different, yet complementary, abilities.

In a mathematical research, a system can explore proofs, manipulate abstract concepts, and propose formalizable results.

In an interactive environment, it must gather information about a situation, infer its rules, and choose actions whose consequences are not all known in advance.

The first approach places particular emphasis on the production and verification of reasoning. The second is concerned with adaptation, interactive learning, and planning.

Their closer relationship allows for questioning a fundamental issue: how far can a system transfer its skills to problems it has not encountered in this form during its training?

The response remains dependent on the tasks, models, and evaluation methods. Progress in one category should not be automatically interpreted as a general mastery of all others.

Economic consequences: towards an acceleration of research and development

The scientific research as a competitive advantage

The ability to produce more quickly usable results can become an important economic advantage.

Fundamental research feeds many applied fields, even when the link between a theoretical discovery and its commercial applications is not immediate.

Advances in mathematics, optimization, or theoretical computer science can, depending on their nature, contribute to improving algorithms, reducing certain computational constraints, or opening new research avenues.

Companies capable of transforming these results into reliable tools could obtain several advantages:

  • Reduce the time necessary for exploring certain hypotheses.
  • Accelerate the design of prototypes and models.
  • Automate some of the repetitive research tasks.
  • Explore more tracks before mobilising experimental resources.
  • Improve the productivity of scientific and technical teams.

These benefits are, however, neither automatic nor uniform. A mathematical result can be fundamental without having an immediate industrial application. The transition from a theoretical proof to an operational system may require years of development, additional investments, and experimental validation.

Optimisation, networks and cybersecurity: possible applications

Several sectors could, in the long term, benefit from systems capable of better solving complex problems.

Optimisation of Networks. Advanced mathematical methods are used in the design of communication networks, traffic management, and resource allocation. Progress in optimisation could contribute to improving some of these operations, provided that theoretical results translate into effective methods for real systems.

Geospatial and satellite communications. Positioning, coverage, and resource allocation issues rely on geometric models and optimization algorithms. Work on distances may shed light on some related problems, but they do not directly provide a ready-to-use solution for organizing a satellite constellation.

Robotics. Trajectory planning, coordinating multiple robots, and decision-making in complex spaces mobilize mathematical methods. Agents capable of exploring and adapting their strategies could improve certain design and control processes.

Cybersecurity. Mathematics and optimization play a role in the analysis of security systems, anomaly detection, and protocol evaluation. However, an advancement in combinatorial geometry does not alone guarantee better protection against cyberattacks.

The key factor is the transfer of knowledge: theoretical advancements become economically relevant when they can be integrated into a system, measured under realistic conditions, and exploited at an acceptable cost.

A new scientific computing economy

Advanced reasoning models introduce a new variable into the research economy: the possibility of buying more compute to explore a problem more broadly.

This approach may alter the distribution of resources among researchers, engineers, and IT infrastructures.

She can also benefit larger organisations with significant computing centres, high-quality data, and teams capable of designing robust validation procedures.

The total cost is not limited to the generation of results. It also includes:

  • The calculation necessary for exploring hypotheses.
  • Human time devoted to reviewing demonstrations.
  • The formalization and verification of results.
  • The independent reproduction of experiments.
  • The integration of discoveries into operational applications.

A system that produces thousands of unreliable suggestions quickly may turn out to be less useful than a more modest system that generates a smaller number of verifiable results.

Scientific productivity should therefore be assessed based on confirmed discoveries, rather than solely on the volume of text produced, the number of agents involved, or the computational power mobilized.

Geopolitical challenges: artificial intelligence as a strategic scientific infrastructure

A competition that transcends commercial products

Artificial intelligence has become an international domain of competition in which technological companies, universities, and states seek to strengthen their research capabilities.

Models capable of assisting scientific discovery may present a particular strategic interest, as they are likely to intervene in many sectors: energy, materials, engineering, health, computing, and defence.

A significant advance in automated reasoning can therefore have a value that extends beyond its original domain.

She can contribute to strengthening a company’s position in the research ecosystem, attract researchers and capital, or improve access to cutting-edge scientific tools.

However, it is necessary to distinguish three forms of advantage:

  • The scientific advantage: the ability to achieve new and verifiable results.
  • Industrial advantage: the ability to transform these results into useful products or processes.
  • Strategic Advantage: the ability to deploy these technologies sustainably, master the necessary infrastructure, and control sensitive uses thereof.

These dimensions are linked, but they do not necessarily progress at the same rate.

The role of infrastructure and access to computing

The development of advanced models requires computational resources, energy, technical skills, and significant investments.

Organizations with powerful infrastructure can explore more avenues and train more specialized systems. However, this does not come with a guarantee of permanent scientific superiority.

The quality of the data, the relevance of the methods, the ability to detect errors, and collaboration with specialists remain crucial.

The question of access becomes central. If the best research tools are reserved for a small number of actors, the scientific capacity gaps between institutions and countries could widen.

Inversely, publishing results, opening up certain proof libraries, and sharing methods can promote the dissemination of knowledge.

The balance between openness and protection of strategic capabilities thus represents a major challenge for scientific policy.

Between technological sovereignty and international cooperation

Countries may seek to develop their own models, computing infrastructure, and mathematical formalization skills in order to reduce their dependence on a few suppliers.

This strategy can promote the emergence of national or regional research ecosystems.

She does not mean, however, that scientific autonomy should lead to isolation. Mathematics rests on a collectively accumulated body of knowledge and on the possibility of independently verifying results regardless of their origin.

A discovery does not become any more robust simply because it is produced by a dominant company or a national laboratory. Its validity depends on the demonstration, its conformity to definitions, and its examination by specialists.

Geopolitical competition can stimulate investments, but it also risks encouraging spectacular announcements at the expense of scientific prudence.

The publication of verifiable work, the exchange of results, and cooperation between institutions are therefore essential elements for maintaining the credibility of research.

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The risk of large-scale error

Reasoning models can generate complex and表面上合理的论据，同时在关键步骤中引入错误。

In a mathematical demonstration, a single incorrect implication can invalidate the entire reasoning. Therefore, the length of a text or the sophistication of its notation do not guarantee its validity.

The increase in attempts can even make selection more difficult if the verification procedures do not progress at the same pace.

Researchers must be able to identify fragile hypotheses, known results, and passages that require additional proof.

Formalization errors and withdrawn results

The history of scientific publications highlights that errors can appear at all stages of the research: problem formulation, hypothesis selection, reasoning, transcription, or verification.

The human-readable text provided in the publication from OpenAI mentions the withdrawal of three manuscripts following an error in algebraic geometry. This episode, as reported in the initial elements, highlights the importance of a transparent review process.

It is not enough to produce a demonstration; you also need to have mechanisms in place for correcting errors, documenting successive versions, and clearly signaling results that should no longer be considered valid.

The AI-assisted scientific publication therefore needs to integrate correction and retraction as normal components of research work.

The question of originality

Another challenge relates to the novelty of the results.

A model can retrieve a published proof, combine known ideas, or propose an argument similar to a previous work without correctly identifying all sources.

The verification of originality requires a thorough comparison with scientific literature, as well as rigorous documentation of references.

This requirement becomes even more important when systems generate substantial volumes of manuscript.

The value of a discovery does not depend solely on the validity of its reasoning, but also on its real contribution to the state of knowledge.

The role of mathematicians in transformed research

The automation of certain reasoning tasks does not mean the disappearance of human work.

Mathematicians bring a knowledge of the field, an understanding of the links between problems, a capacity to evaluate the interest of a result, and a responsibility in its presentation.

Their role could evolve towards more problem formulation, research guidance, evidence control, and interpretation of results produced by artificial systems.

This transformation is not guaranteed: it will depend on the reliability of the tools, laboratory practices, and the conditions under which researchers will examine the results.

Effective collaboration requires that models make their reasoning sufficiently accessible for evaluation, rather than producing conclusions that are impossible to audit.

Towards a new artificial intelligence-assisted scientific method?

From the conversational assistant to the research agent

The first widespread applications of AI have primarily popularized text generation, document synthesis, and programming assistance.

Reasoning systems and autonomous agents aim for a more ambitious objective: decomposing a problem, selecting strategies, executing operations, evaluating results, and restarting when hypotheses fail.

In a research environment, this capability can be used to explore conjectures, seek counter-examples, generate candidate proofs, or construct verifiable formalizations.

The progression depends nevertheless on several elements:

  • The quality of the initial problem formulation.
  • The ability to distinguish a promising path from a dead end.
  • Access to calculation and verification tools.
  • The memory of previously obtained results.
  • The ability to revise a strategy following a failure.
  • Independent evaluation of the conclusions.

The challenge is to build systems that do more than just accumulate answers; they must be able to leverage the information gained to improve their approaches.

The challenge of reproducibility

A scientific discovery must be examinable by other researchers.

In the case of AI, this entails documenting as many versions of the models, assumptions, tools used, calculation procedures, and available proof elements as possible.

Complete transparency can be difficult to achieve when proprietary models are involved. However, it remains possible to publish demonstrations, formalizations, references, and verification protocols.

Reproducibility does not necessarily mean that another researcher will find exactly the same result using the same method. It means that the result can be examined, controlled, and, where conditions permit, independently confirmed.

This requirement is crucial for transforming experimental performance into lasting scientific knowledge.

Has scientific discovery become an industrializable activity?

The results announced by OpenAI invite us to consider a research endeavor in which part of the intellectual exploration can be entrusted to automated systems.

Such an evolution could modify the organization of laboratories, funding priorities, and methods of evaluating scientific teams.

But research is not limited to the production of demonstrations. It also requires the formulation of relevant questions, the construction of conceptual frameworks, the confrontation of hypotheses with observations, and the interpretation of consequences.

Automation could therefore shift some tasks rather than eliminate the entire scientific workforce.

The key question will be whether these systems allow for obtaining more reliable knowledge at a reasonable cost, and whether researchers have the means necessary to control their outcomes.

Conclusion: a significant advance, but not yet a scientific revolution without reservations

The mathematical work announced by OpenAI in October 2026 illustrates the rapid evolution of artificial intelligence systems towards more complex research tasks. The results related to Erdős numbers, proposals on other open problems, and the use of proof assistants like Lean demonstrate that it is now possible to combine algorithmic exploration, mathematical reasoning, and formal verification.

Their scope must nevertheless be evaluated result by result. A spectacular announcement, a convincing manuscript, and a scientifically accepted proof represent different stages.

The economic and geopolitical stakes are real, but they depend on the ability to transform discoveries into reliable, reproducible, and useful methods. The potential applications in optimization, robotics, engineering, or cybersecurity do not automatically follow from a theoretical advance: they require additional work in validation and adaptation.

Benchmarks like ARC-AGI-3 serve to remind us that mathematical reasoning, interactive learning, and general intelligence are not interchangeable concepts. Excelling in one area does not demonstrate mastery of all problems.

The deepest transformation might ultimately concern the scientific method itself. If models become capable of producing increasingly complex results, the community will need to strengthen verification tools, publication transparency, and correction procedures.

The true indicator of progress will not be the number of problems an artificial intelligence claims to have solved, but the number of discoveries that the scientific community can examine, confirm, and integrate sustainably into human knowledge.

To delve deeper: the reference documentary sources

Cet article est rédigé par Blaise Grenier, Journaliste reporter d'images et auteur. Découvrir le parcours de l'auteur et la charte éditoriale »