  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 $n\to\infty$.
