Why Did Perelman's Proof Take Years to Review? Lex Fridman on the Slow Philosophy of Mathematical Verification

The Collective Verification Problem Behind a "Lone Genius"

On the Lex Fridman podcast, when the guest brought up the story of Perelman's proof of the Poincaré conjecture, he raised a sharp observation: Perelman posted his paper on arXiv, the whole world was stunned, but the actual verification process took years.

This wasn't because the proof was wrong, but because of how mathematics is understood—our path to truth is deliberately designed to be slow, cautious, and precise. Perelman himself traveled around giving talks, using three-hour sessions to explain his ideas: the first hour on what he was doing, the next two hours with audience members pointing at Lemma 412 on page 247 saying "I don't understand how this follows," and then he'd work it out live for half an hour. The entire proof lived in his head, and what was written on paper was the best he could do, with the rest of the work left to the community—line by line, section by section, careful review until everyone agreed: yes, this is a genuine proof of the Poincaré conjecture.

This process reveals a frequently overlooked fact: arriving at a solution is only the first step; verifying it is the longer collective endeavor.

The Poincaré conjecture is a fundamental proposition in topology about the characteristics of three-dimensional spheres, proposed by French mathematician Henri Poincaré in 1904. It asserts that any simply connected, closed three-dimensional manifold is homeomorphic to the three-dimensional sphere. In 2000, the Clay Mathematics Institute listed it as one of seven "Millennium Prize Problems," with a one-million-dollar prize. Between 2002 and 2003, Russian mathematician Grigori Perelman posted three preprints on arXiv, using the Ricci flow method introduced by Richard Hamilton to sketch a complete proof of the conjecture. However, because the proof was highly condensed and omitted a great deal of detail, the mathematical community spent years holding seminars and writing detailed explanatory papers before finally confirming its correctness. Perelman declined the Fields Medal in 2006 and the Millennium Prize in 2010, and his maverick style further reinforced the public image of the "lone genius."

The "Slowness" of Mathematics Is an Institutional Design

The guest pointed out that compared with other fields—circuit design, aircraft design, protein folding, astronomy—mathematics still retains the romantic image of the "lone genius." But many fields have long passed the threshold where individuals can push forward alone and must rely on billion-dollar collaborative projects. What about mathematics? Mathematicians themselves are extremely independent and don't like being directed or told what to research; this independence is an intrinsic need.

However, when crises arrive—for example, when journals and conferences are overwhelmed—the community still needs organization. The guest admitted: "Never let a good crisis go to waste." The journal system is overloaded, conferences are overloaded, and now there's the new task of reviewing AI-generated papers. Some AI-generated papers are important, but getting community members to volunteer for review work is very difficult. He explicitly argues: finding new models is not about replacing the old ones, but about relieving pressure.

The Influx of AI Papers and the Search for Verification Models

The conversation mentioned OpenAI's improvements to the fast Fourier transform and the phenomenon on Twitter of people接力 improving constants. The guest said he isn't even on Twitter and doesn't know how to regard this fragmented accumulation of improvements. But he believes we need new organizational models to sort out the current state of these improvements.

The traditional publishing model is: you work hard on a draft, post it on arXiv, and that's your version. You can update a second version, a third version, but the draft belongs to you—the community won't come along and annotate "I think this argument could be written differently" or "this could be simplified here." However, for certain types of work, perhaps we need to allow crowdsourced evolution of understanding. The guest expressed full support for this.

Lean is an interactive theorem prover developed by Microsoft Research, and also a programming language that allows mathematicians to write definitions, theorems, and proofs in a computer-checkable form. Its core advantage is that any proof that compiles in Lean has passed strict logical checking by the kernel, thereby providing an extremely high degree of correctness assurance. In recent years, the Lean community has launched the Mathlib project, aimed at formalizing a large amount of the foundations of modern mathematics, and it now covers tens of thousands of theorems. For mathematical research, Lean formalization can serve as a verifier of "technical building blocks": once certain lemmas or computational steps in a paper are formalized, reviewers can trust those parts to a limited extent, thereby focusing their energy on the more creative core arguments. This offers a possible path for alleviating the verification pressure brought by the influx of AI papers.

The Value of Research Talks and New Possibilities for Lean Formalization

The guest shared why he likes giving research talks: you can temporarily set aside the messy complexity in a paper—the details you know must be filled in but aren't really excited about—and clearly explain the core idea and the special cases that best illustrate why the idea is powerful. When someone asks about Lemma 57, he may not remember it at all.

This leads to a deeper question: must a paper be a behemoth designed for complete formalization? Today, we can trust Lean formalization to some extent to handle certain technical building blocks, which in turn allows us to redesign how we write—putting more energy into communicating ideas rather than locking everything down in a paper proof.

Implications for the Research Ecosystem in the AI Era

The value of this conversation lies in turning Perelman's individual case into a universal lens. AI is mass-producing mathematical proofs and engineering improvements, but the community mechanisms for verifying these results are still designs from decades ago. The "slowness" of mathematics has its reasons—it prevents erroneous conclusions from being treated as truth. But when AI increases output speed by orders of magnitude, slow verification becomes a bottleneck.

The direction the guest proposes is not to overturn peer review, but to supplement it: crowdsourced verification, limited trust in Lean formalization, more flexible version control and annotation mechanisms. These ideas are not yet fully formed, but the cautious openness in the conversation is exactly the attitude the research ecosystem most needs in the face of AI's impact.