4. Given a right triangle $Rt \triangle A B C$ with the two legs $A C=12$, $B C=5$, and $D$ is a moving point on the hypotenuse $A B$. If the right triangle is folded along $C D$ to form a right dihedral angle $A-C D-B$, and the minimum value of $A B$ is obtained, the plane angle of the dihedral angle $B-A C-D$ is denoted as $\alpha$, then $\tan ^{10} \alpha=$ $\qquad$
4.32.

As shown in Figure 1, before folding, draw $A E \perp C D$ at point $E$, and $B F \perp C D$ at point $F$.
Let $\angle A C D=\theta$.
Then $A E=12 \sin \theta, C E=12 \cos \theta$,
$B F=5 \cos \theta, C F=5 \sin \theta$.
Thus, $E F=|12 \cos \theta-5 \sin \theta|$.
After folding the right triangle along $C D$ to form the right dihedral angle $A-C D-B$,
$$
\begin{array}{l}
|A B|^{2}=|A E|^{2}+|E F|^{2}+|F B|^{2} \\
=(12 \sin \theta)^{2}+|12 \cos \theta-5 \sin \theta|^{2}+(5 \cos \theta)^{2} \\
=169-60 \sin 2 \theta .
\end{array}
$$

When $\theta=\frac{\pi}{4}$, $|A B|$ reaches its minimum value.
In the folded figure, draw $F M \perp A C$ at point $M$, and connect $B M$.

Thus, $B M \perp A C$. Therefore, $\angle F M B$ is the plane angle $\alpha$ of the dihedral angle $B-A C-D$.
Given $|B F|=\frac{5 \sqrt{2}}{2},|F M|=\frac{5}{2}$, in $\mathrm{Rt} \triangle B F M$, $\tan \angle F M B=\frac{|B F|}{|M F|}=\sqrt{2}$.
Therefore, $\tan ^{10} \alpha=32$.