The strong Kervaire invariant problem in dimension 62

Using a Toda bracket computation $\langle \theta_4, 2, \sigma^2\rangle$ 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.