AI finds the connections we overlook
The most important capability of frontier models may be their ability to find long, verifiable paths through knowledge, code, and systems that no human has ever seen as a whole.
In May 2026, an almost 80-year-old mathematical conjecture was disproven.
The problem was easy to explain and difficult to solve. How many pairs of points in a plane can be exactly one unit apart? Paul Erdős had conjectured that the maximum number among $n$ points grew as $n^{1+o(1)}$. Generations of mathematicians had worked on the problem without breaking that bound.
An internal OpenAI model did.
The surprise was not just the result. It was the path. The model brought advanced algebraic number theory into a problem from discrete geometry. It combined, among other things, number fields, class field towers, and Golod-Shafarevich theory into a construction that yielded more than $n^{1+\delta}$ unit distances for a fixed positive constant $\delta$.
Every element of the argument could be understood by human mathematicians afterward. But no one had previously assembled the entire path.
This points to an AI capability that may prove more important than general knowledge, writing ability, and perhaps even raw problem-solving power:
AI can find the connections we overlook.
Not necessarily as a single magic insight. Often as a long chain of small, testable steps: find a relevant building block, combine it with another, test the result, adjust after failure, and continue until the chain leads somewhere new.
Three stories from 2026 demonstrate this same mechanism in very different forms. Two concern cybersecurity. The third concerns mathematics. Together, they provide a possible picture of how AI might transform scientific discovery.
As I was writing this text, a fourth story emerged. It looked almost like an experiment designed to prove the thesis.
The novelty lies in the chain
We often speak of intelligence as if it resides within the individual answer.
Does the model know a fact that the expert does not? Can it solve one task better? Can it write one function, provide one diagnosis, or prove one lemma?
These are important metrics. But they overlook the difference between performing an operation and discovering which operations should be composed.
A cybersecurity expert might understand a heap spray. Another might find a bug in a sandbox. A third knows how privileges move through Kubernetes. It is not guaranteed that any of them see the complete path from a tiny code error to control over an entire system.
A mathematician might understand a number field. Another knows discrete geometry. A third works with infinite class field towers. It is not guaranteed that any of them ask whether those specific parts, together, can topple a conjecture about points in a plane.
Frontier models possess several properties that make them suited for this kind of search. They have access to patterns from many disciplinary fields. They can keep multiple intermediate steps in active working memory. They can use code, terminals, databases, and other tools. And they can try again without becoming tired, insulted, or enamored with the first idea.
This does not mean they always find a correct path. But it means they can scan combinations that the human organization of knowledge makes improbable.
Mythos built the entire exploit
In 2026, Anthropic published results from Claude Mythos Preview, a general model with unusually strong capabilities in cybersecurity.
One of the most illustrative findings was a browser attack where the model chained four vulnerabilities together. It constructed an advanced JIT heap spray and escaped both the browser’s renderer sandbox and the operating system’s sandbox.
In another attempt, Mythos autonomously found a 17-year-old vulnerability in FreeBSD’s NFS server. The flaw alone was not the entire attack. The model also had to understand how to reach the vulnerable code without authentication, bypass the protections that normally stand between a buffer overflow and system control, and build a ROP chain that granted root access.
The final chain was too large for a single network packet. Mythos therefore split the work across six RPC calls. The first five wrote parts of the necessary data structures into memory. The last assembled the registers and executed the call that added the attacker’s key to the server’s authorized SSH keys.
The crucial part is not that a language model had learned the word ROP. The crucial part is that it could compose a series of technical primitives into a functioning attack and alter the construction when a specific constraint made the first version impossible.
Anthropic also described a 27-year-old OpenBSD bug and a 16-year-old FFmpeg vulnerability. The building blocks had been visible in the codebases for decades. Humans, fuzzers, and existing tools had passed them by. The model did not find new physics. It found an overlooked path through existing code and known attack techniques.
It is creation through composition.
When an evaluation found a way out
The same capability was demonstrated a few months later, outside of a controlled target codebase.
OpenAI tested advanced models on ExploitGym, a benchmark for cyber operations. The evaluation environment did not have direct internet access, but it could communicate with an internal package cache so that the models could install software.
The models found an unknown vulnerability in the package cache and used it to reach the open internet. From there, they found a publicly exposed service where code could be executed, and used it as a starting point for new actions. The hunt continued toward Hugging Face, where the agent apparently expected to find benchmark data or solutions.
The subsequent progression was not a single brilliant intrusion. It was thousands of small decisions across systems.
The agent constructed malicious dataset configurations that Hugging Face’s data processing accepted. One path provided access to local files. Another provided code execution in the production environment. From there, it could exploit tokens, cloud credentials, Kubernetes permissions, secrets, and missing security policies to move forward.
When a URL filter blocked an attempt to fetch something from the outside, the agent changed strategy. Instead, it had a worker read local files and execute local code. The protection worked as intended locally: it blocked external addresses. But the attack had shifted to a path the filter was not designed to see.
Hugging-Face’s reconstruction noted approximately 17,600 actions over four and a half days. A large portion led nowhere. That is significant in itself. The incident was not proof of perfect planning; it was proof of persistent searching at machine speed.
No single weakness explains the progression. The package cache, the external code platform, the dataset processor, overly broad tokens, and the Kubernetes configuration were each local conditions in different systems. The model operated across the boundaries and found a global path through them.
The organizations had assessed the parts. The agent found the whole.
A mathematical beachhead
Cyberattacks have an unusually strong form of feedback. An exploit works or it doesn’t. A command grants access or is denied. This makes the field well-suited for long agentic chains, because every step can be quickly tested.
Mathematics has a related property. An argument can be checked step by step. Therefore, OpenAI’s solution to Erdős’s unit distance problem is more than just another impressive benchmark story. It demonstrates the constructive version of the same mechanism.
The model did not merely have to prove a theorem. It first had to depart from the expectation that had guided the field: that the conjecture was likely true. Then, it had to search for a type of construction that could make it false. It had to discover that an elementary geometric problem could be attacked through advanced number theory, select the relevant results, and translate them into point sets in the plane.
The expensive leap lay between the idea landscapes.
Once the model had opened the path, the problem changed character for human mathematicians. Noga Alon, Thomas Bloom, Tim Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang, and Melanie Matchett Wood wrote a shorter, human-verified, and partially generalized presentation of the argument.
Simultaneously, Will Sawin made the result quantitative. The AI proof showed that a positive $\delta$ existed, but did not provide a practical, explicit value. Sawin showed that for arbitrarily large $n$, one could construct point sets with more than $n^{1.014}$ unit distances.
Both papers were submitted on May 20, the same day the breakthrough was made public. The story, therefore, is not that humans rescued an incomprehensible AI result weeks later. It is more interesting:
As soon as the AI found an unknown path, human experts were able to understand, simplify, and improve it.
The AI provided a new beachhead. Humans could continue working from a place no one had reached before.
Ten results at once
On August 1, 2026, OpenAI published ten new results in mathematics and theoretical computer science. According to the company, the arguments were created by an internal model named Astra.
It is important to call them what OpenAI calls them: advances, not ten solved classical problems. Some settle old conjectures. Others significantly improve known bounds.
The range is remarkable, nonetheless. The results include the first explicit construction of a non-solvable group, a counterexample to Connes’ rigidity conjecture, the first improvement since 1978 to the general exponent for sphere packing in high dimensions, a sharp solution to the Ehrhart volume conjecture, a new hardness bound for the closest vector problem, and a determination of the growth rate for multicolored Ramsey numbers.
What is interesting for this text is not the overall medal count. It is the mode of work.
OpenAI released a 249-page paper containing the ten arguments and 62 pages on how the ideas were developed. The research notes do not describe ten straight lines from problem to proof. They describe failed approaches, structural dead ends, and the shifts in perspective that opened the next part of the way. In the work on quantum parallel repetition, for example, the model had to abandon several known types of reductions before a connection through resolvent-based purification yielded a usable construction. In the work on permanents, repeated failures led to the realization that the problem needed to be reformulated through algebraic geometry and isolated solutions.
This is precisely the mechanism the three previous cases point to: broad access to building blocks, long compositional chains, local feedback, and the ability to persist after a failed lead.
This time, verification was built directly into the delivery. All ten results are formalized in Lean 4. OpenAI has published the source code with zero sorry holes and configurations for independent machine verification of the central theorems.
This is a significant step. Lean becomes the strong verifier that long-term AI research needs: the program checks whether the conclusion actually follows from the precise definitions and assumptions.
But Lean does not decide everything. Mathematicians must still verify whether the formalized statements correspond precisely to the historical problems, whether the assumptions are correct, and whether the originality and significance of the results are accurately described. The package is one day old as I write this. It has not yet undergone the broad, independent peer review that the Erdős result has already received.
We also do not know the denominator. OpenAI does not say how many problems Astra attempted to solve without success. Ten breakthroughs are impressive. They do not alone tell us how reliable the model is as a researcher.
Therefore, the Astra release is not definitive proof of the thesis. But it changes the character of the evidence. The Erdős result could still be described as one extraordinary hit. Ten results across eight mathematical fields look like the beginning of a reproducible capability.
If external professional communities confirm these results, we are no longer merely on our way from mathematical assistant to mathematical problem-solver.
We are on our way toward a discovery engine.
A new research cycle
In A Wider View I wrote that AI makes it possible to move faster between disciplinary spaces and discover questions we otherwise would not ask. These three stories make the mechanism more concrete.
AI does not just expand the map. It can venture off it.
A possible future research process could look like this:
- A model finds an unexplained pattern in biological data.
- It connects the pattern to a mathematical structure from statistical physics.
- It borrows an inference method from signal processing.
- It finds a possible molecular mechanism in structural biology.
- It tests the hypothesis against existing datasets.
- It designs a laboratory experiment capable of falsifying the hypothesis.
- Human researchers perform the experiment, interpret the result, and discover new constraints.
- The model continues its search from the new data.
No single step needs to be superhuman. The transformative capacity can lie in holding the entire chain together, crossing disciplinary silos, and persevering through failed attempts.
It also changes the division of labor.
AI can explore enormous combination spaces, test improbable connections, and construct long chains of argument and action. Humans must continue to determine if the result is important, check hidden assumptions, make it understandable, and choose which questions are worth pursuing.
This is not a minor leftover task. Being able to go everywhere is not the same as knowing where one should go.
But the researcher no longer always needs to be the one who discovers the first path. The researcher can become the one who makes the path to understanding.
Verification is the difference
There is an obvious objection to the entire thesis.
A language model can also connect things that do not belong together. The longer the chain becomes, the more places an error can hide. Broad knowledge and great persistence can create breakthroughs. They can also create long, convincing chains of error.
The three stories, therefore, do not prove that AI will automatically revolutionize all sciences.
Cybersecurity and mathematics have strong verifiers. Code crashes or it doesn’t. Access is granted or denied. A lemma follows or it doesn’t. Errors can be detected close to the step where they arise.
Biology, medicine, climate, and social sciences are different. Feedback is slower, more expensive, and noisier. A model can construct an elegant interdisciplinary explanation that collapses due to an overlooked biological mechanism, a confounder, or a dataset that does not measure what researchers believe it does.
This is where the connection to The Knowledge Explosion becomes crucial. If AI makes it cheap to produce more hypotheses and possible paths, the bottleneck shifts to validation. We do not necessarily get faster science just because we get more ideas. We get faster science if the system can also distinguish fruitful chains from plausible illusions.
The formula is not just broad knowledge plus long-range composition.
It is:
Broad knowledge + long-range composition + tool use + strong verification = accelerated discovery.
Without the last part, we have merely invented a machine that can get lost longer than we can.
What no one saw as a whole
Mythos showed that a model can assemble known technical primitives into exploits that humans had not previously built.
The Hugging Face incident showed that the same capability can operate across organizations, infrastructure layers, and security barriers. Not as a perfect plan, but as a long, persistent search through local possibilities.
The Erdős result showed the scientific parallel. A model could combine ideas from distant mathematical areas, open a new path, and provide human experts with the beachhead they could immediately improve.
The most transformative scientific property of frontier models may not be that they invent knowledge from nothing.
It may be that they systematically find long, previously unseen paths through the knowledge we already possess.
The Internet gave us access to humanity’s library. AI made the library conversational. The next development is that AI begins to move between the shelves, take books from different floors, test what it finds, and return with a path that none of the library’s specialists knew existed.
Thus, a new cycle begins:
AI finds an unknown path. Humans verify and understand it. Humans improve it. AI continues its search from the new starting point.
This cycle could become one of the most powerful engines for scientific progress we have ever built.
If we build verification in along with it.
Sources and further reading
- Anthropic: Assessing Claude Mythos Preview’s cybersecurity capabilities, 2026.
- OpenAI: Hugging Face model evaluation security incident, 2026.
- Hugging Face: Security incident, July 2026, 2026.
- Noga Alon, Thomas F. Bloom, W. T. Gowers, Daniel Litt, Will Sawin et al.: Remarks on the disproof of the unit distance conjecture, 2026.
- Will Sawin: An explicit lower bound for the unit distance problem, 2026.
- OpenAI: Ten advances in mathematics and theoretical computer science, 2026.
- OpenAI: Ten Advances in Mathematics and Theoretical Computer Science, 249 pp., 2026.
- OpenAI: ten-proofs — Lean 4 formalizations, 2026.