Wed Feb 25 10:58:13 UTC 2026: ### AI Formally Verifies Sphere-Packing Solution, Ushering in New Era of Mathematical Certainty

The Story:
In a landmark achievement, a team of mathematicians has successfully used a machine to formally verify the solution to the sphere-packing problem in eight dimensions, for which Ukrainian mathematician Maryna Viazovska won the Fields Medal in 2022. This rigorous verification was achieved using the Lean software and an AI tool called ‘Gauss,’ developed by Math, Inc. The formal verification process ensures that every step of the proof, including those considered “obvious” by human mathematicians, is checked by the machine, minimizing the risk of logical flaws and unstated assumptions.

This development marks a significant step towards making mathematical correctness less reliant on trust and social processes and more on explicit, verifiable mathematics. The success of this project paves the way for future formalization efforts and the development of even more sophisticated AI tools for mathematical research.

Key Points:

  • A team of mathematicians has verified the solution to the sphere-packing problem in eight dimensions using a machine.
  • Maryna Viazovska won the Fields Medal in 2022 for her work on this problem.
  • The verification process used the Lean software and an AI tool called ‘Gauss.’
  • ‘Gauss’ was developed by California-based company Math, Inc.
  • The formal verification ensures that every step of the proof is checked, minimizing the risk of errors.
  • The project team is reviewing and revising Gauss’ code to ensure maintainability and reusability.
  • Other major theorems that have been completely formalized include the four-colour theorem, the prime number theorem, the Feit-Thompson odd order, and the Kepler conjecture.
  • AI is being used to generate high-quality standardized tests and solve complex mathematical problems.
  • Christian Szegedy predicted that models will be mathematically ‘superhuman in almost all respects’ in six months to a year (February 2026).
  • A team led by Siddhartha Gadgil from the Indian Institute of Science won a grant in September 2025 for its work on ‘LeanAide,’ an AI+Lean environment.

Key Takeaways:

  • AI is increasingly playing a crucial role in verifying complex mathematical proofs, enhancing their reliability.
  • The formalization of mathematical proofs is becoming more accessible through AI-powered tools like ‘Gauss’ and ‘LeanAide.’
  • This development signifies a shift towards a more verifiable and less trust-based approach to mathematical research.
  • AI’s capabilities in mathematics are expanding beyond calculation to reasoning, conjecture generation, and even theorem proving.
  • The integration of AI in mathematics has the potential to accelerate discoveries and reshape the landscape of mathematical research.

    Read More

Related News: