The BIG-IP is 4000s with TMOS 11.5.2 build 0.0.141
the license is as folow
admin@(BIGIP-01)(cfg-sync Changes Pending)(Active)(/Common)(tmos) show sys license
Sys::License
Licensed Version 11.4.1
Registration key xxxxx-xxxxx-xxxxx-xxxxx-xxxxx
Licensed On 2015/05/07
Service Check Date 2015/05/07
Platform ID C113
Appliance Serial Number xxxxx-xxxxx-xxxxx
I guess the reason is the hardware you have.
I have experience with the 2000s, that after you apply an add-on key, you make it a 2200s.
Apparently, the 4000s look to be the same, as the hardware is the same as 4200v.
A major difference, in both 2000 or 4000 series, is the compression, either been via software or hardware.
I guess that is the reason for the limited license, as you only have software compression, and that is seen as a limitation for AAM.
Anyway, best way to confirm that is talk with F5 sales team.
They can confirm, and also provide you the price of the add-on if is available.
I also did a quick search before post my previous answer, and I don't think there is public information about that.
Because of that, I said is better you talk with F5 sales team, as they must know the answer.