OpenAI Claims Breakthroughs on Multiple Millennium Math Problems

It has only been two days since OpenAI announced it cracked the Millennium Prize problem, the Navier-Stokes equations, in 88 hours.

Before everyone had even recovered from the shock, an even more explosive revelation emerged.

Today, a leak on X sparked a wildfire across the internet:

OpenAI is rumored to be close to verifying the second Millennium Prize problem—the Hodge conjecture.

This is certainly not baseless.

This morning, a feature article in The New York Times directly released official confirmation from OpenAI:

Over the past five years, the internal team has made substantial progress on another Millennium Prize Problem and is preparing to announce it publicly.

These days, the rumors circulating in the market have completely gotten out of control.

OpenAI and its rival Anthropic appear to be in a wildly intense secret race, both launching their final push toward the third great challenge, the "BSD Conjecture," with some work already entering the verification stage.

If all of this is true, it means that within a single month, humanity will witness three Millennium Prize Problems—unsolved for decades by mathematicians—being solved one after another by AI.

Recent intensive mathematical breakthroughs have repeatedly sent the same message to the world—

This September, AGI no longer seems like a distant concept—it is rapidly evolving into ASI with a force that surpasses the upper limits of human intelligence.

The singularity has arrived.

Further reading:

In 2000, the Clay Mathematics Institute established seven Millennium Prize Problems, each with a $1 million reward.

The problems span fluid dynamics, topology, number theory, algebraic geometry, computational theory, and quantum field theory; solving any one of them would be sufficient to reshape the modern mathematical landscape.

In 26 years, humanity has solved only one of them.

In 2003, Russian mathematician Perelman proved the Poincaré conjecture, then declined the prize and withdrew from academia. The remaining six problems have since remained unsolved.

The breakthrough point occurred the day before yesterday.

OpenAI announced that its internal model invoked approximately 10,000 agents over 88 hours to solve the Navier-Stokes equations, releasing a 166-page paper and verified Lean-formalized code.

Immediately afterward, the focus shifted rapidly to the BSD conjecture and the Hodge conjecture, for which humanity had already made substantial progress and was on the verge of a breakthrough.

Previous explorations in the academic community have tended to suggest that the Hodge conjecture may not hold.

To disprove it, only one valid counterexample is needed—

And this is precisely the blind spot where AI, leveraging brute-force computational power, can precisely intervene—the very "obvious" and "note that" phrases that human mathematicians often casually write on the blackboard.

What exactly are these two legendary conjectures, targeted by AI, asking?

The Hodge Conjecture, proposed by British mathematician William Hodge in 1950 at the International Congress of Mathematicians, belongs to the field of algebraic geometry.

The core issue lies in the correspondence between geometry and algebra.

Mathematicians often study a class of high-dimensional geometric objects known as "algebraic varieties," which are formed by the solution sets of systems of multivariate polynomial equations—dimensions far exceeding the limits of human vision and spatial intuition.

To analyze these shapes, topological tools break down complex forms into "building blocks" of different dimensions.

During the decomposition process, a series of topological features (i.e., Hodge classes) are abstracted to characterize the underlying hidden structure within the shape.

The core question of the Hodge Conjecture is: Can these abstract topological structures be entirely constructed by piecing together tangible, concrete segments defined by algebraic polynomial equations (algebraic cycles)?

Or, in simpler terms, does every natural topological "texture" in a higher-dimensional space necessarily correspond to a specific algebraic "recipe"?

There is significant disagreement in the academic community regarding this conjecture. Some scholars firmly believe it to be true, while others are inclined to search for counterexamples—no consensus has been reached to date.

Special cases in fewer than four dimensions have long been proven, but the vast realm of four dimensions and higher remains a deep frontier.

The BSD conjecture stands for the Birch and Swinnerton-Dyer conjecture.

It was conjectured in the 1960s by British mathematicians Bryan Birch and Peter Swinnerton-Dyer using the EDSAC-2, an early computer at the University of Cambridge, and remains central to number theory.

An elliptic curve is a geometric object defined by a specific cubic equation and holds a dominant position in number theory.

The key breakthrough Wiles used to finally prove Fermat's Last Theorem was elliptic curves, and elliptic curve cryptography is now widely deployed across the modern internet. EC C) This is also based on this.

One of the core challenges surrounding elliptic curves is determining how many rational solutions a given curve has. This measure of the size of the solution set is called the "rank," and computing it is extremely difficult.

The BSD conjecture has built a delicate bridge.

It holds that the number of rational solutions of an elliptic curve is fully encoded in a corresponding L-function. The analytic properties of the L-function at a specific point—such as whether its value is zero or the order of its zeros—precisely correspond to the structure and number of rational solutions.

If this conjecture is proven, the millennium-old problem of congruent numbers will be solved at the same time.

Reviewing the current progress on the millennium problem, the situation is overwhelming:

Solved: Poincaré Conjecture (2003, Perelman)

Claimed solution: Navier-Stokes equations (announced as solved by OpenAI in September 2026, pending verification by the Clay Mathematics Institute)

Rumors are mounting: Hodge Conjecture, BSD Conjecture

No breakthrough迹象: Riemann Hypothesis, P vs NP, Yang-Mills Gauge Theory and Mass Gap

Just a week ago, there were six unsolved mysteries on this list; now, there may be only three left unresolved. All the major changes have been compressed into these few short days.

In 2018, OpenAI documented in its charter the classic definition of AGI: a highly autonomous system that outperforms humans at most economically valuable work.

In July 2024, this goal was broken down into five levels (Dialogue, Reasoning, Agency, Innovation, Organization); by the end of the year, with the release of the o1 model, the official self-assessment progressed to the second level.

In 2025, state-of-the-art models achieved gold-medal-level performance at the International Mathematical Olympiad (IMO); by year-end, OpenAI used GPT-5.2 to prove a new formula in particle physics, marking AI’s natural evolution from an auxiliary tool to a research collaborator.

Last month, GPT-6 Astra was unveiled, and Greg Brockman publicly declared:

Welcome to the AGI era.

Along with the recent collapse of the Navier-Stokes equations and the ensuing alarm over several subsequent conjectures, people have realized a frightening truth: every time technology crosses a threshold, the external benchmark for measuring AGI is automatically raised.

Natural language interaction is no longer rare, complex long-range reasoning is commonplace, and winning competitions has become standard.

When AI begins to systematically and directly tackle the Millennium Problems that have puzzled top scholars for decades or even centuries, mathematics—the pinnacle of human intellectual achievement—seems to be rapidly becoming a benchmark for testing cutting-edge large models.

The world’s top mathematicians, who once believed they sat at the pinnacle of human intelligence, should now seriously consider their place in the coming storm of superintelligence.

Reference materials:

https://www.nytimes.com/2026/09/10/science/tristan-buckmaster-openai-math-navier-stokes.html

https://x.com/synthwavedd/status/2097971881596916185

This article is from the WeChat public account "New Intelligence Yuan," authored by ASI Revelation, edited by Marco, Moses, and Peach.