Worked solution: Hill-Hopkins-Ravenel proof via equivariant stable homotopy theory (2009)

Combining Browder's reduction (Step 1) with the non-existence theorem just proved (Step 8): framed manifolds of Kervaire invariant $1$ exist only in dimensions $n = 2^{j+1}-2$ for $j\le 6$, i.e. $n \in \{2,6,14,30,62,126\}$, and by explicit prior construction (Step 2) they genuinely exist in the first five of these; only $n=126$ ($j=6$) remained undetermined by Hill-Hopkins-Ravenel's own methods.

This settles, for every dimension outside this six-element exceptional set, several classical questions in differential topology tied to Browder's work: which framed manifolds are framed-cobordant to a homotopy sphere, when the Kervaire manifold admits a smooth structure, and when a Whitehead square is divisible by $2$ (Kervaire-Milnor 1963; Browder 1969).

In December 2024, Weinan Lin, Guozhen Wang, and Zhouli Xu posted a preprint (not yet peer-reviewed) reporting a machine-assisted computation of the relevant Adams $E_2$-page arguing that $h_6^2$ is in fact a permanent cycle, so $\theta_6$ exists after all — closing the one remaining case, subject to that computation's verification.