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You don't get a medical discharge from failing PT tests. You get a medical discharge when your doctor notes certain limitations in your medical records and then recommends separation
based on your inability to meet standards. It's not hard to get a medical discharge if your doctor actually pushes for it and your commander is inclined to grant it.
If you just fail PT tests, your discharge
would be for PT test failing, a performance discharge. It would likely be General rather than Honorable, so you'd lose your GI Bill benefits. This is also easy to get, because you just have to fail to meet the standard. Getting an honorable in this situation would be very hard.
If you just left the country, never to return, you'd wouldn't be pursued in any way. They'd likely just give you an Other than Honorable discharge
which would stop you from getting any benefits. It wouldn't be considered a criminal act, so it wouldn't effect your travel. Now, technically they could list you as AWOL
, not discharge you and purt a warrant out for your arrest. This was flag you for any attempt to reenter the U.S.