AI Solves 70-Year-Old “Holy Grail” Conjecture in Probability Theory

On August 30, 2026, probability theory master and 2022 Fields Medalist Hugo Duminil-Copin wrote with a touch of sadness:

In our field, the most famous conjecture is likely just a matter of time before it falls beneath the roar of the bulldozer (AI).

He is discussing a "holy grail" problem in probability theory that has puzzled the mathematical community for over half a century—the conjecture on continuous phase transitions in percolation theory.

Guess).

For this problem, this Fields Medalist devoted years of effort, persisting through repeated failures.

But what no one anticipated was that reality turned out to be even more surreal than the prophecy.

Almost simultaneously with the publication of his article, peers suddenly discovered a newly committed code repository on GitHub by Anthropic engineers.

No press conference, no grand nationwide promotion, not even a single post on the official blog.

Lying within is a complete code automatically generated by the large language model Claude and rigorously verified using the mathematical formalization language Lean—the long-standing percolation conjecture, unsolved for years by Fields Medalists, has now been proven by AI.

The news spread, and the global mathematics community was instantly thrown into uproar.

Mathematician Benedikt Jahnel from the Technical University of Braunschweig stated directly in an interview:

If a human had proven this conjecture, they would likely have won a Fields Medal. But now, it’s AI that has crossed the finish line.

AI has already laid the machine proof on the table.

Mathematician Gil Kalai stated directly: "If verified, this would be an extraordinary breakthrough."

The conjecture on continuous phase transitions in percolation theory is known as the "Holy Grail" of probability theory.

In 1957, mathematicians Simon Broadbent and John Hammersley pondered a remarkably down-to-earth yet profoundly deep physical phenomenon: how does liquid actually pass through a porous sponge?

You can think of it like brewing coffee, or oil seeping through cracks in the Earth's crust:

Imagine a vast spatial grid where each intersection is connected by tiny pipes. Each pipe has a probability p of being open and a probability of 1-p of being blocked.

If p is very small, such as only 0.1, most of the pathways are blocked, so water droplets can only seep a short distance before stopping and never penetrate through the entire sponge; if p is very large, such as 0.9, the pathways are extensively connected, allowing water to flow effortlessly and form a vast, interconnected network across an infinite system.

In mathematical terms, the magical threshold at which a "local droplet" transitions into an "infinitely connected network" is known as the critical probability.

This is a phase transition, similar to how water freezes at 0 degrees Celsius under standard atmospheric pressure.

Below the critical point, the probability of forming an infinitely connected water flow

It is absolutely 0; above the threshold,

Then it is greater than 0.

Thus, a "question of the century" that has captivated every probabilist was born:

At the exact moment of reaching the critical point, can an infinitely large connected network form within the system? In other words,

Does it equal zero or not?

If

= 0 means this phase transition is extremely smooth and continuous, like morning mist gradually turning into a fine rain; if it is greater than 0, it means the system will undergo an instantaneous change at the critical point, suddenly erupting into an infinite stream that spans the universe.

This seemingly simple conjecture is the "Holy Grail" of probability theory—

=0 Guess.

To solve this puzzle, mathematicians worked tirelessly for over half a century.

In 1980, mathematician Harry Kesten proved that the critical probability on a two-dimensional square lattice is exactly 1/2, and

=0.

This directly established his master status in the history of mathematics.

In ultra-high dimensions (11 dimensions or higher), each node has an enormous number of neighbors, allowing mathematicians to simplify the problem using tools such as statistical averaging and mean-field theory.

Many years ago, scholars demonstrated that phase transitions are also continuous in spaces of 11 dimensions or higher.

However, the real nightmare lies in the middle—the 3D to 10D spatial grid.

There is no special geometric symmetry in two-dimensional space, nor can statistical smoothing tools for higher dimensions be applied; the topological entanglement between dimensions is extremely complex and chaotic.

Three dimensions are our physical reality, four dimensions are spacetime in relativity, yet these most fundamental dimensions became the "Wall of Sighs" that mathematicians have been unable to overcome for decades.

For decades, countless top scholars have spent their youth, ultimately leaving only a sigh on scratch paper.

Fields Medalist Hugo Duminil-Copin is the most famous of these "wall撞者."

In 2022, Duminil-Copin was awarded the Fields Medal, the highest honor in mathematics, for his groundbreaking work on phase transitions in statistical physics—a career deeply intertwined with percolation theory.

He loved this hypothesis so much that it bordered on obsession.

In the blog post of August 30, Duminil-Copin wrote in an exceptionally beautiful prose style:

A mathematical problem is far more than just a theorem waiting to be proven. It is not only a lighthouse in the night but also a guide for the soul.

Solving it may not immediately open up a new field of mathematics, but its depth and beauty alone are enough to captivate generations.

Duminil-Copin shared emotionally that although he had repeatedly failed to solve the problem of continuous phase transitions in dimensions 3 to 10, the long struggle and climb yielded dozens of unexpected sparks of inspiration from what seemed like discarded drafts.

Many of his later achievements that won him the Fields Medal were merely "byproducts" of failing this main quest.

In the traditional narrative of human mathematicians, the process is considered far more sacred than the result. Along the journey to the summit, there are moments of sudden insight at the edge of a cliff, despair when all paths seem blocked, and countless days and nights of aesthetic refinement. This struggle with the unknown is regarded as the highest symbol of human intellectual dignity.

However, Anthropic's entry has completely shattered this romantic filter.

Justin Leder, who submitted the proof, is not a leading figure in mathematics and is unknown even within the percolation theory community. He relies on an advanced, unreleased Claude model from Anthropic.

AI does not experience confusion, nor does it understand what aesthetics means.

Faced with this 70-year-old challenge for humanity, AI acted like a bulldozer, pushing through with airtight symbolic logic and formalized language.

In an interview with the media, mathematician Benedikt Jahnel admitted:

This result evokes in me extremely complex emotions.

On one hand, the conjecture was finally proven, and as a mathematician, he felt genuine joy; on the other hand, a profound sense of loss and disillusionment settled over him, as it was AI that delivered the final, decisive kick.

The door to a殿堂 that humanity had waited for over half a century was casually unlocked by AI with a master key.

Many people automatically assume: Is AI just blindly stumbling upon answers through massive computing power, trying every possible option by brute force?

The answer is: Not at all. Exhaustive enumeration alone could never prove a continuous conjecture involving infinite dimensions, even if computed until the end of the universe.

The AI demonstrated a nearly terrifying ability for high-dimensional logical convergence and cross-domain reasoning.

This closed door could only be opened thanks to the final piece laid by anthropologists in the early stages.

In 2024, Gady Kozma from the Weizmann Institute of Science in Israel and Shahaf Nitzan from the Georgia Institute of Technology published a highly groundbreaking paper.

Link: https://arxiv.org/abs/2401.12397

Two mathematicians proved that if a specific, relatively simple algebraic inequality can be verified, then it holds in dimensions 3 through 10.

The guess automatically holds!

This paper instantly gave the academic community a glimpse of victory.

The finish line is right in front of us—just one final step remains: proving that damn inequality.

However, this final step became an insurmountable barrier.

Mathematicians attempted to construct auxiliary functions and find upper and lower bounds, but this inequality proved as slippery as an eel, always causing singularities and breakdowns at critical points in the derivation.

Anthropic's model delivered the final blow right here.

It did not start from scratch but instead keenly leveraged the springboard built by Gady Kozma and Shahaf Nitzan in 2024.

Under the stringent constraints of the Lean proof assistant, the large model invoked a vast array of human-known analytical tools and inequality techniques, constructing a sophisticated, thousands-of-lines-long reasoning loop at the foundational level of formal code—forcefully proving the inequality in a manner no human mathematician had ever envisioned.

Mathematician Ahmed Bou-Rabee of the University of Pennsylvania announced that Anthropic has proven this Fields Medal-level conjecture.

What shocked Ahmed Bou-Rabee even more was the speed of the subsequent evolution.

Ahmed Bou-Rabee, with the assistance of large models, generalized and modified Anthropic's proof code in just one day.

"AI has enabled me to achieve things I never even dared to imagine," said Ahmed Bou-Rabee. "I struggled for eight full years on some research projects with almost no progress, but with AI's help, I am now just one step away from a complete solution."

At the same time, the original author, Professor Kozma, appeared notably restrained, even cold.

In response to inquiries from all sides, he calmly stated:

I will not comment on their claims for now.

We are still waiting for Anthropic to release a “human-readable version” and explain exactly how they achieved it.

When the steam engine was invented, coachmen believed the art of driving had ended; when the camera was born, realist painters lamented the death of art; when Deep Blue defeated Kasparov and AlphaGo swept away Ke Jie, the world wept for Go.

But is Go dead? No.

Although human players can no longer defeat AI, they have gained access to entirely new opening patterns through the AI’s perspective—patterns never imagined in thousands of years—expanding the dimensions of the board infinitely.

The same is true in mathematics. As Duminil-Copin collectively reflected: proving theorems is never the entirety of mathematics.

In the history of human mathematics, the allure of conjectures often lies not in the proof itself, but in the entire revolution of tools born from tackling them—

To prove Fermat's Last Theorem, humanity laid the magnificent foundation of algebraic geometry; to attempt to prove the Riemann Hypothesis, the towering edifice of analytic number theory was erected.

Duminil-Copin spent years struggling to crack percolation theory, but as he put it: “Those failed attempts gave rise to dozens of entirely new ideas, which I applied in other fields and led to many major discoveries I never could have imagined.”

Future mathematicians may no longer need to expend their primary energy on decades-long, intricate algebraic derivations. Instead, their new mission may lie in selecting the most valuable research pathways, directing thousands of reasoning agents to explore the unknown, and interpreting for the human world the fragments of wisdom brought back by machines.

Einstein once said:

The most beautiful and profound emotion we can experience is the sense of mystery. It is the source of all true art and science.

From a simple 1957 hypothesis about sponges absorbing water to being decisively outpaced by AI on GitHub in 2026, humanity spent 70 years proving its greatness—and witnessing the transition of intelligent forms.

Reference materials:

https://www.quantamagazine.org/updates/transformation/

https://www.scientificamerican.com/article/ai-solves-a-holy-grail-problem-from-probability-theory/

https://proofsandprompts.com/2026/08/30/care-for-a-little-more-ai/

https://github.com/anthropics/formal-math/tree/795efb86f191735c5481675763537cfb4ff37e55/percolation

Edited by: David

This article is from the WeChat public account "New Intelligence Yuan" (ID: AI_era), authored by ASI Revelation.