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So, the way I interpret all this:

 

My son and his ex girlfriend have joint 50/50 custody.

Despite if he has his son less that 183 days, ie 180 days, as long as she agrees to give him the rights to claim him as his dependent, he can claim head of household, and other child credits/exemptions.  He does make more money.

 

Is this in line with what you are saying