A little-publicized but revolutionary trend is changing the face of mathematical research. The interactive programming language Lean recently gained popularity as a robust environment for formalizing and verifying mathematical proofs. On February 23, researchers announced the formal verification of Maryna Viazovska’s renowned 8-dimensional sphere-packing proof using Lean, marking a significant advancement in the collaboration between artificial intelligence (AI) and human mathematicians. This achievement highlights Lean’s ability to facilitate absolute correctness in mathematical proofs, setting a new standard for rigor in the field.
Lean is not just a programming language. As a proof assistant, it allows mathematicians to write, verify, and refine their proofs with greater rigor using computational methods. Viazovska’s proof relies on highly complex mathematical functions known as quasi-modular forms. It indicates Lean’s potential to make mathematical expressions more accurate and reliable. Her breakthrough in proving that the E8 configuration yields the most efficient packing of spheres in eight dimensions represents a milestone in the ongoing efforts to formalize complex mathematical concepts.
The Role of AI in Proof Verification
Human mathematicians now work with AI in thrilling new ways. The introduction of reasoning agents such as Gauss has changed this collaboration. This cutting-edge technology was the key ingredient that helped to autoformalize Viazovska’s 24-dimensional sphere-packing proof in Lean. Within just two weeks, Gauss processed over 200,000 lines of code, showcasing AI’s capacity to handle intricate mathematical frameworks efficiently.
Sidharth Hariharan, one of the original project researchers. He did indicate, though, his amazement for how quickly they were able to get there.
“When they reached out to us in late January saying that they finished it, to put it very mildly, we were very surprised,” – Sidharth Hariharan
The Formalising Sphere Packing in Lean project, which started in March 2024, hopes to capture these innovations even more fully. It’s proven to be a sandbox for mathematicians to experiment with new proof techniques and derived methods for formal verification. Hariharan underscored just how collaborative this effort was. He said that it produced something of a blueprint for Gauss to autoformalize the proofs from both the 8-dimensional and the even-more-complicated 24-dimensional sphere-packings.
The Collaborative Nature of Mathematical Research
Collaboration has been key to these advancements. Jesse Han is the CEO and co-founder of Math, Inc. His ultimate vision, helped by Lean and AI, is to radically transform how mathematical research is done.
“I think the end result of technology like this will be to free mathematicians to do what they do best, which is to dream of new mathematical worlds,” – Jesse Han
The collaborative work went beyond formalization and included exploring gaps in current best practices. Han remarked on the complexity of the 24-dimensional case:
“And it was actually significantly more involved than the 8-dimensional case because there was a lot of missing background material that had to be brought on line surrounding many of the properties of the Leech lattice, in particular its uniqueness,” – Jesse Han
This collective work has led to impressive results, as noted by Liam Fowl, another key figure in the project:
“These new results seem very, very impressive, and definitely signal some rapid progress in this direction,” – Liam Fowl
The Lean ecosystem produces a culture that enables mathematicians to codify their proofs in a form that’s more formalizable. It motivates them to learn from each other, deepening their comprehension and fostering mutual learning.
The Future of Mathematical Proofs with Lean
Lean’s capabilities extend well beyond sphere-packing proofs. But due to its design as a proof assistant, there’s a much wider application in other areas of mathematics. The language has otherwise provided the infrastructure for mathematicians who wish to have formal verification for their work to do so. Lean provides magical tools to make it easy to subject proofs to rigorous scrutiny. This external “rubber stamp” of validation increases our confidence in potentially controversial mathematical results.
An increasing number of mathematicians are using Lean for their research. We hope that this change will help to deepen the culture of mathematical proof. Hariharan reflected on the progress made since initiating the project:
“Formal verification of a proof is like a rubber stamp,” – Liam Fowl
This pace of development represented a major breakthrough in the field of formal verification. It’s indicative of a much larger change towards a more accessible and collaborative mathematical community.
“We had been building the project’s repository for about 15 months when we enabled public access in June 2025,” – Sidharth Hariharan
This rapid development signals not only progress in formal verification but also indicates a shift towards greater accessibility and collaboration within the mathematical community.