Formalizing pi(4)(S-3) congruent to Z/2Z and Computing a Brunerie Number in Cubical Agda

Brunerie's 2016 PhD thesis contains the first synthetic proof in Homotopy Type Theory (HoTT) of the classical result that the fourth homotopy group of the 3-sphere is Z/2Z. The proof is one of the most impressive pieces of synthetic homotopy theory to date and uses a lot of advanced classical algebraic topology rephrased synthetically. Furthermore, the proof is fully constructive and the main result can be reduced to the question of whether a particular "Brunerie number" beta can be normalized to +/- 2. The question of whether Brunerie's proof could be formalized in a proof assistant, either by computing this number or by formalizing the pen-and-paper proof, has since remained open. In this paper, we present a complete formalization in Cubical Agda. We do this by modifying Brunerie's proof so that a key technical result, whose proof Brunerie only sketched in his thesis, can be avoided. We also present a formalization of a new and much simpler proof that beta is +/- 2. This formalization provides us with a sequence of simpler Brunerie numbers, one of which normalizes very quickly to in Cubical Agda, resulting in a fully formalized computer-assisted proof that pi(4)(S-3) congruent to Z/2Z