We prove that the element \(h_6^2\) is a permanent cycle in the Adams spectral sequence. As a result, we establish the existence of smooth framed manifolds with Kervaire invariant one in dimension 126, thereby resolving the final case of the Kervaire invariant problem.
Combining this result with the theorems of Browder, Mahowald–Tangora, Barratt–Jones–Mahowald, and Hill–Hopkins–Ravenel, we conclude that smooth framed manifolds with Kervaire invariant one exist in and only in dimensions \(2, 6, 14, 30, 62\), and \(126\).
On the Last Kervaire Invariant Problem
Note on this blueprint. This blueprint is adapted from the paper On the Last Kervaire Invariant Problem by Weinan Lin, Guozhen Wang, and Zhouli Xu [ . Compared to the original paper, a “Prerequisites” section has been added, covering foundational concepts (spectral sequences, stable homotopy theory, Adams spectral sequence, etc.) that are not yet formalized in Mathlibv4.28. These additions provide the necessary infrastructure for autoformalization of the full proof in Lean 4.