A Disciple of Shing-Tung Yau Achieves Another Breakthrough: First Formal Verification of the Proof of the Generalized Poincaré Conjecture.
Disciple of Shing-Tung Yau Achieves Another Milestone, First Formal Verification of the Generalized Poincaré Conjecture!
Summary
Xinzhiyuan reports Just now, there is big news in the mathematics community again. In September, AI’s successful formal verification of the “Poincaré Conjecture” shocked the field. Just now, a top academic team led by Ayush Khaitan, together
Xinzhiyuan Report
Xinzhiyuan Report
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Just now, there’s big news in the mathematics community.
In September, AI’s successful formal verification of the “Poincaré Conjecture” shocked the mathematics community.
Just now, a top academic team led by Ayush Khaitan, together with the NVIDIA Humanfia team, successfully completed the full Lean formalization of the Thurston Geometrization Conjecture.
Hamilton–Perelman’s “century proof” spans hundreds of pages, taking even top scholars years to read.
After this year’s celebrated resolution of the Poincaré Conjecture in September, this time it has finally been turned into lines of code with AI assistance.
4.7 million lines of code, completed in just about two weeks.
This is another milestone in the history of mathematics, and a sign that AI‑assisted higher‑order geometric analysis is entering a “breakneck era.”
AI‑assisted formal mathematics has officially broken into the most profound, uncharted territory of human higher‑order geometric analysis.
If someone told you that a team wrote 4.7 million lines of modern mathematics code in two weeks with not a single error, you would think they were crazy.
But that is exactly what just happened.
Recently, Stanford assistant professor of mathematics Jared Duker Lichtman exclaimed:
Truly astounding work.
Poincaré’s proof has been extended to the full Geometrization Conjecture, and all of this was done entirely in Lean!
The results were released by Princeton mathematician and AI scholar Ayush Khaitan. In his tweet, he introduced this ultra‑powerful cross‑disciplinary lineup.
Take a look at this hardcore team roster:
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Bennett Chow (Zhou Peineng): a disciple of Shing‑Tong Yau, professor of mathematics at UC San Diego, an absolute authority and towering figure in the field of Ricci flow. -
Ziyang Qin and Yuan Liao: the definitive leaders of the project’s formalization work. -
Ayush Khaitan: Princeton scholar and the core driver of the project. -
NVIDIA Humanfia team (Jui‑Hui Chung & Ligeng Zhu): representing the cutting‑edge force of NVIDIA, the world’s leading AI computing powerhouse.
Under the collaboration of this super team, a miracle was born.
Ayush revealed that the proof consists of about 4.7 million lines of code.
Building on the roughly 2 million lines of code previously contributed by Chow, Liao, and Qin, the additional2.7 million lines of code in justabout two weeks were completed!
Currently, this great project has been fully open‑sourced on GitHub.
https://github.com/qinz1yang/differential-geometry
Why does this excite a Stanford mathematician so much?
First, we need to understand the difficulty of Thurston’s geometrization conjecture itself.
Everyone has heard of the “Poincaré Conjecture” .
It asks: if every closed loop in a three‑dimensional space can be contracted to a point, does that space have to be equivalent to a three‑dimensional sphere?
From 2002 to 2003, Russian mathematical prodigy Grigori Perelman burst onto the scene, proving the Poincaré conjecture with three preprint papers, then refusing a million‑dollar prize and retreating into obscurity.
But the Poincaré conjecture is really just the “tip of the iceberg” of a grander blueprint. That blueprint is called the “Thurston Geometrization Conjecture”.
In other words, the Poincaré conjecture teaches you how to recognize a “sphere,” while the Thurston geometrization conjecture provides a framework for the entire three‑dimensional universe and even lists a “periodic table of elements.”
It states that any three‑dimensional manifold can ultimately be cut precisely into a few pieces. More importantly, each piece must belong to one of the eight standard geometric structures!
Back then, Perelman proved the Thurston geometrization conjecture, thereby “incidentally” proving the Poincaré conjecture.
His signature move is called 「Ricci Flow and Surgical Theory」 .
It’s like using a “geometric iron” (Ricci flow equation) to smooth out twisted space. When you encounter a knot that won’t smooth out (a singularity), you perform a “surgical operation” to cut it out, seal it, and then continue ironing.
This theory is extremely profound; even the top few dozen mathematicians in the world took two or three years to fully verify Perelman’s proof.
And today, these great scholars and engineers have packed this pinnacle of human intellect into a computer’s mind, word for word!
The complete Lean proof of Thurston’s Geometrization Conjecture means machines now have the capability to handle advanced differential geometry, topology, and partial differential equations.
Machines have officially entered the “no‑man’s land” of higher‑order geometry.
If you open the GitHub repository named differential-geometry, you’ll get a hardcore blow.
Link: https://github.com/qinz1yang/differential-geometry
This is a digital Great Wall of modern differential geometry built with machine language.
In Lean, this interactive theorem prover, the computer doesn’t understand what “obviously true” means; it only recognizes the most fundamental logical axioms. To make the machine grasp Thurston’s geometrization conjecture, the team had to start from scratch and define the rules of the entire universe in code.
From the technical details of the repository, the team hardcore filled a large amount of missing foundational infrastructure in Lean’s official math library (Mathlib).
They fundamentally rebuilt Riemannian geometry.
The code defines extremely complex manifolds (Manifold), tangent bundles (Tangent Bundle), tensor fields (Tensor Fields), and Riemannian metrics (Riemannian Metrics).
The team used rigorous generic programming to finally let the machine “understand” what a curved space is.
In traditional mathematical papers, the Ricci flow is nothing more than a partial differential equation .
But in a GitHub repository, a team needs thousands of lines of code to precisely define “time‑dependent metric evolution,” “local computation of the Ricci curvature tensor,” and the highly intricate existence‑boundary conditions for solutions of the PDE.
The most difficult part of Perelman’s proof is undoubtedly the “surgery theory.”
How to implement “clipping” and “stitching” of an abstract space in code?
The team introduced complex topological connectivity and geometric decomposition algorithms into the codebase, turning the handling of geometric singularities into a logical tree that a computer can execute and verify step by step.
The most exhilarating part is this set of numbers: in two weeks, nearly 2.7 million lines of code were added, totaling 4.7 million lines.
In the software engineering world, a 2.7‑million‑line project is enough to keep a seasoned development team of dozens working overtime for an entire year.
Yet in Lean, a language with extremely high logical density where each line requires a long compilation and verification time, completing this in two weeks is absolutely impossible by human effort alone!
How did they do it?
The answer is obvious: the NVIDIA Humanfia team delivered a “dimensionality‑reduction strike” using large models and automated inference.
NVIDIA not only provides the world’s most powerful compute, they are exploring how AI can directly assist cutting‑edge fundamental scientific research.
Although the details of all AI‑generated content have not been publicly disclosed, the massive surge in code volume over a short period suggests an “industrial‑scale” proof pipeline.
Top mathematicians such as Chow, Liao, Qin and others are responsible for drawing the “tactical map,” breaking down massive geometric conjectures into thousands of lemmas and sub‑goals.
The NVIDIA team leverages LLMs and automation scripts to wildly generate massive amounts of low‑level proof code, fill in tedious algebraic derivations, and exhaustively enumerate boundary conditions based on human‑provided context.
Finally, all the code is fed into the Lean compiler for ruthless “logical quality inspection”; if it fails, it’s sent back for rewriting until the green light shines.
This is not only a major event in the mathematics community but also an absolute focal point in the history of AI evolution!
Current large language models still suffer from “hallucinations” in complex, long-chain logical reasoning, and the ultimate weapon to fix these logical flaws is mathematical formalization.
If AI can proficiently generate Lean code to prove something like the “Thurston Geometrization Conjecture,” a pinnacle of human intellect, then the gap to AI “proposing new conjectures and providing proofs” is truly just one step away.
The full formalization of Thurston’s Geometrization Conjecture is far more than a high‑difficulty “code translation”. What it leaves to the world is a valuable digital treasure.
Through the differential-geometry project on GitHub, the team has left humanity a code library of differential geometry that has passed the “absolute truth verification.”
This means that in the future, mathematicians worldwide can directly call the library they have written when studying multi‑dimensional manifolds, the geometric structure of black holes, or even spacetime evolution on a computer!
This is a great project where earlier generations plant trees and later generations enjoy the shade.
From Poincaré’s famous conjecture written over a hundred years ago to Thurston’s grand blueprint of a three‑dimensional universe.
From Perelman crunching numbers in his St. Petersburg apartment to today, scholars such as Ayush Khaitan, Qin Ziyang, Liao Yuan, Zhou Peineng and others collaborating with NVIDIA, using 4.7 million lines of code to make machines achieve a true “epiphany”…
Human pursuit of truth is like a relay race.
AI hasn’t stolen mathematicians’ livelihoods; instead, it has given them a “starship” for exploring the universe.
The pace at which humanity solves problems will unleash an unprecedented acceleration.
Reference:
Edited by: David Aeneas
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