Last July, Maryna Viazovska made headlines. Her pioneering work on sphere packing won her the 2014 Fields Medal, one of the highest honors a mathematician can receive. The sphere packing problem, a long-standing question in mathematics, explores how densely identical spheres can be arranged in n-dimensional space. Even before her proof, Viazovska made truly spectacular contributions by solving various complex packing problems in eight and twenty-four dimensions. Her research has opened up new horizons in the field of mathematics.
During her work at IBM, she found that the E8 lattice has the densest packing configuration in eight dimensional space. At the same time, the Leech lattice really sparkles in twenty-four dimensions. After her groundbreaking achievement, Viazovska was understandably interested in the verification process. She said she wanted to make her proofs more formal, motivated by her interest and not need.
The Role of Formal Verification
Sidharth Hariharan was very active in helping to formalize Viazovska’s proofs. For his proof, he used Lean — a programming language and proof assistant — created for verifying complex mathematical concepts. This creative new approach seems like an important step forward that brings the promise of formal verification much closer to reality. Today, mathematicians can produce proofs that a computer checks for full correctness.
The intersection of Viazovska and Hariharan’s work serves as a reminder that human intelligence often augments artificial intelligence. By applying Lean, Hariharan found that he could systematically chop up Viazovska’s elaborate arguments into more bite-sized, checkable pieces. Interestingly, this process improved the proofs quite a bit. It produced a human-readable “blueprint” that shows the different pieces that fit into the puzzle of Viazovska’s eight-dimensional proof.
“These new results seem very, very impressive, and definitely signal some rapid progress in this direction,” – Liam Fowl
The autoformalization process yielded significant insights. In fact, Hariharan’s study of the eight-dimensional case showed it to be more complicated than previously expected. As Jesse Han noted, “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.”
Advancements in AI-Assisted Proof Verification
Since 2022, progress in AI-assisted formal proof verification has surprisingly taken off. Math, Inc. has created a new kind of reasoning agent, Gauss by name, intended to augment mathematicians in their research pursuits. Gauss has already shown great power in narrowly autoformalizing Viazovska’s proofs.
Gauss did accomplish the remarkable task of autoformalizing the whole twenty-four-dimensional case. Even more impressively, he single-handedly built this incredible project—over 200,000 lines of code—in just two weeks! This achievement illustrates the transformative impact that AI can have on the conduct and verification of mathematical research and proofs.
“We had been building the project’s repository for about 15 months when we enabled public access in June 2025,” – Sidharth Hariharan
The use of AI in the mathematics verification process is a great example of how technology can help solve big problems. By employing reasoning agents like Gauss, researchers can focus on more creative aspects of mathematics rather than getting bogged down in tedious verification tasks.
“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
Collaboration Between Humans and AI
The partnership between Viazovska and Hariharan is just one of many examples in which humans and AI are working together to accomplish incredible things. The formalization of Viazovska’s proofs highlights not only her exceptional contributions but the evolving role of AI in mathematics.
Hariharan noted the fruitful nature of their collaboration, stating, “So it was a pretty fruitful collaboration.” After all, the recent advances in AI-assisted formal proof verification seem to indicate an exciting new era of mathematical inquiry. Mathematicians such as Viazovska are making new strides in their discipline. As they figure it out, AI will play a key role—one that has already begun—to help formalize and verify complex proofs.
“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
Beyond delivering verification, autoformalization is the key to unlocking these following transformative developments for software development. It’s a watershed AI moment for bringing AI into mathematical fields. As technology continues to evolve, mathematicians are increasingly readying themselves for innovative partnerships with AI. These collaborations will enhance their capacity to innovate new frontiers in their discipline.