The strong Kervaire invariant problem in dimension 62

Using a Toda bracket computation $⟨{\theta }_4,2,{\sigma }^2⟩$ due to Daniel C. Isaksen [11], we investigate the $45$-stem more thoroughly. We prove that ${\theta }_4^2=0$ using a $4$-fold Toda bracket. By [2], this implies that ${\theta }_5$ exists and there exists a ${\theta }_5$ such that $2{\theta }_5=0$. Based on ${\theta }_4^2=0$, we simplify significantly the $9$-cell complex construction in [1] to a $4$-cell complex, which leads to another proof that ${\theta }_5$ exists.