AI and Mathematics: Are We seeing the First Signs of Superintelligence?

For a long time, mathematics exposed one of Artificial Intelligence’s most curious contradictions. Models could write convincing texts, summarize documents, and produce reasonably sophisticated code, yet they still stumbled over simple calculations, confused logical steps, and defended wrong answers with apparent confidence. 

This limitation was not merely an inconvenient flaw. Mathematics demands something different from everyday language. Producing a plausible answer is not enough. Each conclusion must follow from the previous ones, the rules must remain valid from beginning to end, and a single error can compromise the entire solution. 

In just a few years, however, the landscape changed dramatically. Systems that once struggled with arithmetic problems began achieving gold-medal performance at Olympiads, proposing new algorithms, identifying formulas in vast datasets, and contributing to the solution of open mathematical questions. Artificial Intelligence stopped merely answering questions about mathematics. It began taking part in the creation of mathematics. 

From struggling with calculations to winning gold 

The first large language models were trained primarily to predict which word or symbol should come next. This approach produced remarkable linguistic fluency, but it did not ensure that a model understood numbers or could maintain a correct chain of deduction across many steps. 

In 2022, researchers still emphasized that even the largest models struggled with mathematical problems requiring multi-step reasoning. Tests at the time revealed fragile systems prone to failure when the same problem was presented in a slightly different way. 

The next leap did not come solely from increasing the number of parameters. Models began receiving specialized reasoning training, more processing time before answering, access to computational tools, and mechanisms for exploring different paths, comparing hypotheses, and reviewing results. 

In 2024, Google DeepMind’s AlphaProof and AlphaGeometry 2 systems reached silver-medal level at the International Mathematical Olympiad. The problems still had to be translated into formal languages, and processing took days. One year later, an advanced version of Gemini Deep Think solved five of the competition’s six problems, scored 35 out of 42 points, and reached gold-medal level. This time, it worked directly in natural language and within the contest’s four-and-a-half-hour time limit. 

OpenAI also reported scoring 35 points, equivalent to a gold medal, with a general-purpose reasoning model. This is therefore not an isolated case, but part of a broader shift in models’ ability to sustain long chains of deduction and produce mathematical proofs. 

This transformation is hard to ignore. In a very short period, we moved from models that failed at school-level problems to systems capable of competing at the level of the world’s best young mathematicians. 

Mathematics is more than calculation 

When people talk about AI and mathematics, it is common to imagine an extremely powerful calculator. Yet the most important advances are not about multiplying numbers faster or solving familiar equations. Computers have been doing that for decades. 

The real leap lies in the ability to decide how to approach an unfamiliar problem. Solving a complex question may require creating intermediate representations, identifying patterns, formulating conjectures, abandoning unproductive paths, and combining ideas that originally belonged to different fields. This is closer to research than to simply applying a formula. 

Systems such as AlphaGeometry illustrate this shift by combining a neural model, capable of suggesting constructions and recognizing patterns, with a symbolic deduction mechanism responsible for applying rigorous mathematical rules. Statistical creativity proposes paths, while logic checks whether they actually work. 

This combination matters because it reduces one of generative models’ greatest weaknesses: the possibility of producing a convincing but incorrect conclusion. In mathematics, code, and science, generating candidates is only the beginning. The real advantage appears when the system can also test, refute, and select the candidates that survive verification. 

Claude’s cycles: when code becomes a mathematical laboratory 

One of the most symbolic episodes of this new phase was documented by Donald Knuth, one of the most influential figures in the history of Computer Science and the author of The Art of Computer Programming. 

Knuth was studying a problem involving the decomposition of a graph into directed Hamiltonian cycles. He had solved a specific case and was looking for a general construction. After working on the problem for weeks, he learned that Claude Opus 4.6 had found a solution to a relevant part of it. In his paper Claude’s Cycles, Knuth described the result as a dramatic advance in automated deduction and creative problem-solving. 

The most interesting aspect was not only the final answer, but how it emerged. Filip Stappers presented the problem to the model and conducted a session involving 31 explorations over approximately one hour. Claude tested simple formulas, attempted brute-force searches, wrote programs, ran experiments, investigated geometric patterns, encountered dead ends, and reformulated the problem several times. 

At one point, it realized that computational search was finding examples but not providing a general construction. The model itself recorded that it needed to abandon that direction and do “pure mathematics.” Later, it identified a structure that made it possible to create a program valid for different odd values and arrive at a generalizable solution. 

The case is often associated with Claude Code because the entire process involved programming, running scripts, and documenting experiments. More precisely, Knuth’s paper records the use of Claude Opus 4.6 in an agentic investigation workflow. The central point is not the interface that was used, but the model’s ability to use code as a tool for thought. 

It was not a demonstration of perfect autonomy either. The model had to be reminded to record its progress, suffered random errors, lost results after restarts, and failed to make satisfactory progress in some cases. The discovery required human guidance and still needed a rigorous proof. Knuth’s own account makes both the strength and instability of the process clear. 

That is precisely what makes the episode relevant to businesses. AI did not deliver a magical answer after a single command. It acted as a tireless researcher within an environment equipped with code, tests, records, and objective criteria. 

An 87-year-old conjecture overturned by a counterexample 

In July 2026, another episode expanded the debate even further. Mathematician Levent Alpöge announced that, with the contribution of Claude Fable 5, he had found a counterexample to the Jacobian Conjecture, formulated in 1939. 

Put simply, the conjecture stated that certain polynomial maps that pass a mathematical invertibility test should always have a polynomial inverse as well. The counterexample presented has a constant, nonzero Jacobian determinant, yet maps three distinct inputs to the same output. It therefore cannot be inverted. 

Unlike proofs that require hundreds of pages, the refutation could be verified directly through exact calculations. The result was quickly checked independently and has already begun to be used in new mathematical work. It settled the conjecture in dimensions three and above, although the two-dimensional case remains open. 

Not enough details have yet been released to reconstruct exactly how much of the discovery came from the model, from human guidance, or from the combination of both. It is therefore more accurate to say that AI played a decisive role in obtaining the counterexample, rather than claiming that it solved the problem entirely autonomously. 

Even with that caveat, the case is historic. For 87 years, mathematicians primarily sought a proof that the conjecture was true. AI helped explore another direction: finding a specific object that demonstrated it was false. 

This is a natural advantage of modern computational systems. They can search enormous spaces of possibilities, preserve countless hypotheses, and investigate combinations that a researcher might dismiss as unlikely or inelegant. 

AI is also discovering formulas 

Another lesser-known but equally important frontier is the automatic discovery of mathematical formulas. In the traditional Machine Learning approach, a model can learn to predict results with great accuracy without clearly explaining the relationships it has found. It works as a black box: it receives variables, performs millions of internal operations, and returns an answer. 

Symbolic regression takes a different path. Its goal is to find a compact mathematical expression that describes the data. Instead of merely predicting the movement of a system, for example, the algorithm looks for a formula that represents the rules responsible for that movement. 

The LLM-SR method combines the scientific knowledge and programming capabilities of large language models with evolutionary search. AI proposes equation structures, tests those hypotheses against the data, receives feedback, and generates new versions. In experiments involving physics, biology, and materials science, the system found accurate equations that were more efficient than those produced by traditional symbolic regression methods. 

In 2026, researchers introduced Deflex, a system capable of extracting formulas at different scales from complex systems. The approach combines neural networks, symbolic computation, and problem decomposition to find compact relationships in phenomena such as particle dynamics, fluids, and collective motion. In published tests, the method achieved efficiency up to seven times greater than previous techniques. 

Discovering a formula does not automatically mean discovering a new law of nature. Data can contain noise, accidental correlations, and hidden variables. Every expression must be interpreted, tested, and confronted with scientific knowledge. Even so, the shift is profound. AI is beginning to do more than recognize patterns. It can transform some of those patterns into structures that human beings can understand, test, and reuse. 

Algorithms that improve other algorithms 

The application of this reasoning is not limited to theoretical mathematics. Algorithms control logistics routes, data centers, search engines, financial systems, telecommunications networks, chips, and the training of AI models themselves. 

Google DeepMind’s AlphaEvolve combines Gemini models with automated evaluators and an evolutionary process. The models propose programs, the evaluators run and score each alternative, and the best solutions are modified to create new generations. This process has already been used to discover faster matrix multiplication algorithms, find solutions to open mathematical problems, and improve components used in Google’s own data centers, chip designs, and AI training. 

Here, AI is not simply writing code according to a human specification. It is looking for better ways to perform a task by reducing unnecessary operations, simplifying rules, or discovering strategies that developers had not considered. 

It is an early form of automated algorithm engineering. Humans define the problem, the constraints, and the success metric. The machine explores, tests, and improves solutions at a scale that would be impossible for a traditional team. 

Are we already in the age of superintelligence? 

The results are extraordinary, but they do not yet allow us to conclude that superintelligence has been achieved. An Artificial Superintelligence would not merely be better than humans in a competition or a specific field. It would have to outperform individuals, the best teams, and even large human organizations across a very broad spectrum of cognitive tasks, with a high degree of autonomy, adaptability, and reliability. A Google DeepMind report describes ASI as a system more intelligent and cognitively capable than large organizations composed of human beings. 

Today’s models display islands of superhuman performance. They may find a novel mathematical construction and, shortly afterward, lose context, execute a program incorrectly, or require guidance to abandon an unproductive strategy. 

Mathematics offers an especially favorable condition for AI: many answers can be verified objectively. A program either works or it does not. A counterexample either satisfies the conditions or it does not. A formal proof is either accepted or rejected by the verifier. 

The business and social worlds are more ambiguous. Information may be incomplete, goals may conflict, and the consequences of a decision do not always appear immediately. Being brilliant at mathematics does not automatically mean understanding people, markets, organizations, and risks. 

We are therefore not yet facing general superintelligence. We are, however, seeing concrete signs that specialized superhuman capabilities are beginning to connect. 

ai can already propose, program, test, deduce, and review. as these abilities begin to operate for longer periods, with less supervision, and across different fields, the boundary between tool and researcher becomes less distinct. 

What mathematics teaches businesses about ai 

The main lesson from these advances is not that every organization should hand its problems to a chatbot and wait for an extraordinary discovery. The most relevant cases emerge when AI operates within the right structure. It needs access to the right data, tools to execute its ideas, objective evaluation criteria, safe environments for experimentation, a record of previous attempts, and professionals capable of interpreting the results. 

That is what happened in the problem described by Knuth. It is also how AlphaEvolve and formula-discovery systems work. The model generates hypotheses, but value emerges when those hypotheses enter a cycle of execution, measurement, refutation, and improvement.

Many business challenges are, in essence, mathematical or algorithmic problems. How should resources be allocated? What is the best production sequence? How can infrastructure costs be reduced? How can demand be forecast? How can anomalies be detected? How should routes, schedules, inventory, prices, and teams be organized? 

AI can explore solutions faster, identify hidden relationships in data, and generate algorithms adapted to each company’s context. However, turning a discovery into an operational solution still requires integration, software architecture, security, observability, governance, and maintenance. 

The intelligence that stops merely answering and starts discovering 

The history of computing has always been closely tied to mathematics. Now, that relationship is entering a new phase. Machines that were created to execute instructions are beginning to contribute to the creation of the very instructions, formulas, and strategies that will be executed. 

This does not mean the end of mathematicians, scientists, or engineers. It means a radical expansion of what these professionals can investigate. A researcher can coordinate dozens of simultaneous attempts. An engineer can test alternatives that were previously economically unfeasible. A company can transform years of accumulated data into new models, rules, and opportunities. 

With 30 years of experience, Visionnaire is following this transformation as a Software and AI Factory, helping organizations integrate models, agents, data, and systems into solutions that can be applied to the business. The challenge is not simply gaining access to the most powerful AI. It is building the environment in which it can research, test, learn, and generate value safely. 

We may not yet be living in the age of superintelligence, but we have already entered an era in which some of humanity’s most difficult questions are no longer investigated exclusively by human beings.