A human property (T) proof for high-rank $Aut(F_n)$

Existing property (T) proofs for $Aut(F_n)$, $n\geq 4$, rely crucially on extensive computer calculations. We give a new proof that $Aut(F_n)$ has property (T) for all but finitely many $n$ that is inspired by the semidefinite programming approach but does not use the computer in any step. More specifically, we prove property (T) for a certain extension $\Gamma_n$ of $SAut(F_n)$ as $n\to\infty$.