Maryna Viazovska caused quite a stir this past summer in the world of mathematics. It was Mary that showed how to pack spheres more efficiently, in either 8-dimensional or 24-dimensional space. Her work showed that the symmetric arrangement known as E8 gives the densest packing in eight dimensions. In parallel, her work with other mathematicians uncovered that the Leech lattice was actually the optimal solution to packing in 24 dimensions. This monumental achievement has revived the interest in the sphere-packing problem. This astonishing question asks how tightly we can pack parallel spheres (or other shapes) in n-dimensional space.
The sphere-packing problem has deep historical roots, with famous solutions in low dimensions. Even in two-dimensional space, the hexagonal arrangement is the optimum solution. At the same time, pyramid stacking provides the simplest and most effective form of three-dimensional packing. Viazovska’s stunning results in higher dimensions are a perfect reflection of the sophistication and beauty of contemporary mathematics.
The Role of AI in Mathematical Proofs
The combination of recent advancements in artificial intelligence (AI) — tools that can deal with exponentially complex problems — and mathematical verification could help us revolutionize tolling processes. One such tool, Gauss, developed by Math, Inc., acts as a reasoning agent that blends traditional natural-language reasoning with fully formalized reasoning. As Jesse Han, CEO and cofounder of Math, Inc., explains, Gauss is an incredibly powerful tool for literature searches. It produces Lean code and aids in fast proof verification.
Lean is a programming language specifically designed for the purpose of proof assistance. It gives mathematicians the tools to produce proofs that a computer can check with perfect confidence. Our collective project to Formalise Sphere Packing in Lean started in March 2024. Even more importantly, it’s gotten Sidharth Hariharan and his team on board to get some fresh collaboration going. Even so, for the 8-dimensional case, they had already posted an outline of a proof before Viazovska’s acknowledgment.
“It’s a particular kind of language model called a reasoning agent that’s meant to interleave both traditional natural-language reasoning and fully formalized reasoning.” – Jesse Han
Gauss’s powers had a real test when it autoformalized the 8-dimensional sphere-packing proof in roughly five days. Impressively, it even detected and corrected an embarrassing typo in the final published paper! The tool successfully autoformalized Viazovska’s intricate 24-dimensional proof — which stretches to more than 200,000 lines of code — in just two weeks.
A Collaborative Endeavor
The provable AI achievements that have been achieved so far highlight the powerful partnership between our human mathematicians and these technologies. Sidharth Hariharan reminded us of the need to recognize human contributions that set the stage for Gauss’s achievements. He painted the picture of AI and human mathematicians working together as a promising partnership, noting that mathematics is all about collaboration.
“So it was a pretty fruitful collaboration.” – Sidharth Hariharan
Harian’s team had already been working on their project repository for about 15 months before it went public in June 2025. The rapid advance achieved by Gauss caught everyone by surprise—even Hariharan.
“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
This successful autoformalization of these proofs marks a turning point for both autoformalization itself and AI-human collaboration. As mathematicians address ever more sophisticated problems, tools such as Gauss will be the new normal for ensuring complicated proofs are correct.
The Future of Mathematical Research
The consequences of these massive improvements go beyond just being able to verify what’s happening. Jesse Han imagines a world in which technology frees more mathematicians to realize their full creative potential. He hopes that innovations such as Gauss will inspire mathematicians to imagine entirely new mathematical landscapes. This progress allows them to escape the tiresome and limiting shackles of uncomfortable verification processes.
“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
Fellow mathematician Liam Fowl was one of many to express these sentiments about the importance of formal verification to math. He added that formal verification serves as a “rubber stamp,” meaning that proofs can be trusted because they withstand the test of rigorous validation techniques.
“Formal verification of a proof is like a rubber stamp.” – Liam Fowl
As mathematicians and developers of AI technologies work together closely, the possibility for amazing breakthroughs is just beginning. These movements enhance our appreciation of difficult abstract mathematical ideas. They broaden the horizons of what’s possible when human imagination meets AI technology.