Eidolons don't get a BAB of +6/+1All of there attacks are at full BAB... Witch would be +6
There max attacks/ round at BAB of +6 is 5 natural attacks
But here he is using Weapons with the Multiwepon Feat
So with 4 arms, he would only get 4 attacks
My mistake is saying "Primary" instead of Main Hand and Off Hands
Main Hand +9
Off Hands +6
Any creature using natural attacks wouldn't get additional attacks from BAB. Eidolon have limitations on natural attacks, they however do not have a limitation per level on attacks with weapons.
The eidolon is using weapons and using multi weapon fighting. They get the additional attacks from BAB.
You probably need to go read the combat section of the book to brush up on your rules. You are just flat out incorrect with your post.