"So help me God"
The Enlistment oath and officer's Oath of Office both contain this phrase. Normally, it is not required to be said if the speaker has a personal or moral objection, as is true of all oaths administered by the United States government. However, a change in October 2013 to Air Force Instruction 36-2606 made it mandatory to include the phrase during Air Force enlistments/reenlistments. This change has made the instruction "consistent with the language mandated in 10 USC 502".
Tough call OCU. It doesn't seem to be a "religious test" so it would appear that Article VI does not decide the question.
I suggest the Supreme Court would have to determine if AFI 36-2606 is Constitutional.