OpenAI’s Next-Generation AI Solves 10 Fields Medal-Level Math Problems

OpenAI still has more up its sleeve!

The internal model Astra, just demonstrated by Ultraman, has made significant breakthroughs on 10 mathematical challenges at once!

This 249-page PDF is extremely groundbreaking throughout the entire mathematics community.

Paper: https://cdn.openai.com/pdf/ten-proofs-oai.pdf

Proof: https://openai.com/index/ten-advances-in-mathematics/

Github open-source Lean proof: https://github.com/openai/ten-proofs

Mathematician, Fellow of the American Mathematical Society, and Distinguished Professor at Rutgers University Alex Kontorovich was left speechless, offering only two exclamation marks.

This is a historic turning point—significant both in the field of mathematics and on the journey toward AGI.

To put it simply: if these results withstand scrutiny from the entire academic community, then today’s single round of releases could represent the largest single-day leap in this field in modern history!

Claude Fable 5 even stated directly: "According to the Fields Medal criteria, any one of them would be sufficient to win the award!"

Most astonishingly, OpenAI solved these ten challenges using just $2,000.

In May, OpenAI announced a counterexample to the Erdős unit distance conjecture discovered by AI.

It has now been confirmed that this was cracked by Astra, OpenAI’s next-generation model, which Altman is currently demonstrating to the U.S. Congress.

Now, OpenAI has shared the latest breakthroughs from Astra on a problem that has seen no progress for at least a decade.

These questions span a wide range of fields, including high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice-based cryptography, and extremal combinatorics.

As soon as the news broke, the math community erupted!

Among these achievements, the most notable may be Astra's resolution of Gromov's non-sofic group problem, posed in 1999.

A mathematics PhD from Caltech, Qian Xuesen’s alma mater, said: “This is a Fields Medal-level achievement!”

According to Epoch AI's OpenMath scoring criteria, GPT-5.6 Sol Pro and Fable 5 Max conclude:

Most results have received high recognition from peers and have been rated as "Major Advance."

Only the third item has the potential to break through boundaries and is expected to be one of the top mathematical achievements of the year, earning the designation of "Breakthrough."

The third result is also a counterexample:

OpenAI Astra constructed an infinitely presented non-sofic group, refuting the conjecture that "all countable groups are sofic."

Thomas Bloom, a Royal Society University Research Fellow and mathematician at the University of Manchester, stated directly: This breakthrough is more significant than the previous disproof of the unit distance conjecture by OpenAI.

Moreover, the total cost to generate proof of these 10 breakthrough achievements is less than $2,000 when calculated at Sol API pricing, averaging just $200.

In other words, solving a scientifically valuable conjecture is roughly equivalent to a graduate student’s weekend stipend.

And all of this was merely an unexpected byproduct of evaluating an unreleased model.

Another point, which people might overlook if they don't pay close attention:

These 10 guesses are the result of OpenAI's selection!

Noam Brown, the primary architect of OpenAI's reasoning model, has made a strong statement directly.

He frankly stated that OpenAI has indeed attempted other difficult problems but has not yet successfully solved a Millennium Prize Problem similar to the Riemann Hypothesis.

But more importantly, during testing, computational power is far from capped—even million-dollar global challenges could be solved.

It’s both moving and sorrowful: will mathematics still be the pride of the human mind?

This 249-page paper shook the mathematics community not because of its calculations, but because AI, like a master mathematician, simultaneously delivered dimension-reducing proofs and disproofs across fields such as geometry, algebra, and group theory.

Achieving multiple groundbreaking advancements across various fields at once may be unmatched by anyone else—this alone is impressive. Today, we’ll focus on three key issues.

In 1999, Russian mathematician Mikhail Gromov, a recipient of the Abel Prize—often regarded as the Nobel Prize of mathematics—introduced the concept of sofic groups.

Sofic, from Hebrew meaning "finite".

In simple terms, a sofic group is an infinitely large, complex group whose local multiplication table can be perfectly approximated by finite permutations.

You can think of it as "any infinitely complex 3D model can be perfectly rendered using a finite number of pixels (voxels)."

The question arises: Are all countable groups sofic groups?

This is not a niche technical detail. The properties of sofic groups impact an entire landscape of mathematics, including sofic entropy theory, ergodic theory of dynamical systems, and operator algebras. If the answer to this question is “no,” it would mean that there exist groups fundamentally unapproximable by finite structures—requiring a complete reevaluation of the entire theoretical framework.

Over 27 years, countless top mathematicians attempted to construct counterexamples, none of which succeeded.

Astra's answer is: Construct an infinitely presented non-sofic group.

OpenAI Astra directly pulled an existing structure from the mathematical codebase: "the unit group of a binary Leavitt algebra," and presented an impeccable proof: this group cannot be approximated by finite permutations!

To demonstrate this, the AI operated with extreme force, merging the Kun-Thom extended graph theory with the well-known Thompson's group V, forcibly generating a logical contradiction.

It’s like humans are still searching everywhere for substances that can’t be pixelated, while AI simply points to a higher-dimensional Rubik’s Cube on the table and says, “Stop looking—it’s this one. Let me prove it to you.”

Complete construction. With reasoning. With details.

More importantly: it has been formally verified using Lean 4, accompanied by a machine-verifiable certificate.

This is the watershed moment. The Lean 4 certification means that every step of reasoning has been formally verified by a machine, leaving no room for approximation based on "feeling right."

Mathematician Elliot Glazer immediately confirmed the news was true, calling it "the most significant AI-assisted mathematical achievement to date."

Imagine you have a cardboard box—how would you fit the most oranges inside?

In three dimensions, humans only figured it out a few hundred years ago through the Kepler conjecture. But in higher dimensions, the problem becomes a nightmare.

In 2022, mathematician Maryna Serhiivna Viazovska received the Fields Medal, the highest honor in mathematics, for solving the sphere packing problem in 8 and 24 dimensions.

Highlight: It directly shatters the "intelligence ceiling" in this field since 1978.

But note that she solved for a "specific dimension."

If the dimension tends toward infinity, what is the maximum density?

Since two Soviet mathematicians established a limit in 1978, the world's top mathematicians have made no progress for a full 46 years, unable to optimize even a few decimal places.

And this time, AI casually walked into this dead end.

It not only provides a completely new proof but also directly and precisely calculates the exponential decay rate of the Cohn-Elkies linear programming bound, first surpassing the 1978 boundary!

It’s like humans have been climbing this mountain for half a century and made no progress—AI simply flies straight to the summit in a helicopter and paves the path up along the way.

In 1982, Fields Medalist and founder of noncommutative geometry Alain Connes proposed the famous "rigidity conjecture": for a certain class of exceptionally special groups, the von Neumann algebras they generate are as unique as fingerprints.

For decades, mathematicians have been wandering through the maze of this conjecture.

As a result, OpenAI’s new model not only escaped the maze but also dismantled it.

It not only proves that Connor is wrong, but also provides an overwhelmingly decisive refutation: instead of merely finding a single counterexample, it directly constructs a “countably infinite family of groups.”

These groups are non-isomorphic to each other (completely different in structure), yet they generate identical von Neumann algebras!

This is like Connes once claimed that “no two snowflakes have the same internal atomic structure,” but AI didn’t just find two—it unleashed a blizzard, where each snowflake has a unique appearance, yet all share identical core algebraic structures.

"Godfather of AI" Hinton predicts:

In the next 10 to 20 years, AI may even create new mathematics that humans cannot understand.

And OpenAI's results this time make that timeline look overly conservative.

Beyond rigorous mathematical proofs, OpenAI’s next-generation model, Astra, knows how to leverage conditional probability to solve quantum entanglement games and how to use polynomial differentiation to establish computational complexity lower bounds.

Solving this type of problem requires genuine reasoning, and OpenAI Astra possesses an exceptionally profound intuitive and constructive ability in pure mathematics. OpenAI has also specifically released the complete derivation process this time:

https://cdn.openai.com/pdf/reasoning-walkthroughs.pdf

This may be a clear example: AI is smarter than the best human mathematicians.

To the mathematical community, this paper is nothing short of the arrival of a "wall-breaker" for mathematicians.

Reference materials:

https://cdn.openai.com/pdf/ten-proofs-oai.pdf

https://openai.com/index/ten-advances-in-mathematics/

https://x.com/stalkermustang/status/2083485500250198453

Edited by: David

This article is from the WeChat public account "New Intelligence Yuan," authored by ASI Revelation.