In 1998, Thomas Hales and Samuel Ferguson announced a proof of the Kepler conjecture – a famous problem in discrete geometry that had remained unsolved for over 300 years. What is the most efficient way to cram balls into a given space?

The answer is not hard to guess; it’s just how oranges are stacked in the supermarket. But it has been a remarkably difficult answer to prove, and mathematicians were not satisfied without a proof. We wanted to understand why this was the best possible method, rather than just accepting the fact that it seemed plausible and nobody could find a better method.

The solution of the Kepler conjecture involved a combination of elaborate proofs running over hundreds of pages with huge computer calculations to check many cases.  It was far too complex and lengthy for just about anyone to check thoroughly, and in fact a team of a dozen referees spent years working on it before giving up. Their verdict was that it all seemed to work; they had checked many parts carefully and had found no errors. But they just did not have the time or energy to verify everything comprehensively.

This was an unsatisfactory situation: Hales had completed a magnificent achievement, but it was just too much to verify using the ordinary social processes of mathematics. Would it end up going down in history books with a little footnote indicating that nobody else had the patience to check it all, so we could never be quite sure?

The proof was published in 2005, but the uncomfortable situation remained.

Hales then turned to computers, using the techniques of “formal verification”. Here the entire proof is written out in extraordinary detail using strict formal logic, so that a computer program can verify every step and ensure that there are no gaps and that nothing has been overlooked. Of course we then have to trust the program doing the verification, but a proof checker is far simpler than the solution of the Kepler conjecture. There are several such proof checkers that have been carefully checked themselves and are widely trusted in the community.

Working out a proof at the level of detail where it can be checked by computer is an enormous undertaking, which involves filling in far more detail than is conventionally given in a proof intended to be read by humans (who are much more clever than current proof verification technology). This project took many years and Hales worked with a large team of collaborators, who are the other authors of this paper.

This group of mathematicians finally completed the task a few years ago. This is the resulting paper. It completely resolves the question of whether the proof is valid, and is therefore the definitive solution of Kepler’s conjecture.

Additionally, this is as large and complex a proof of a mathematical theorem as has ever been verified by computer. It is an important milestone along the path to formal verification of all of mathematics.

A Formal Proof of the Kepler Conjecture’ is free to view and download via open access, and is available now through Forum of Mathematics, Pi.

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