I am being told to go to PT by the command on only 4-5 hours of sleep. As a junior enlisted, how do I solve this problem?

If you don’t want to go to PT report to formation and request to speak to someone in your chain of command afterwards. Follow your last order.

That being said you are in the military and being fit and ready to fight is what American’s tax dollars pay for. I am pretty sure America’s enemies are not complaining about going to make car bombs on 4 hours of sleep. What you really have to ask yourself is this. If getting to formation on 4 hours of sleep would make the difference between saving and losing the life of someone you loved would you do it? Then I think your answer is clear. Get off your 2-pack and execute that order Solider. America’s depending on you.

As a recently retired SNCO, I feel your pain. The typical new 1st Sergeant will usually go out of their way to impress the CC to look good, which means being a hard-arse about PT.
That aside, I’ll address others responses first. This Junior Enlisted member didn’t say he was CHOOSING his work hours as YOU chose to drink/party till PT time. Your correlation is completely bogus and also of your choosing. Had you been under my supervision, I’d have brought you up under an Article 15 for being in formation intoxicated! Evidently this young soldier has more common sense to address the issue in a responsible manner.
Now, back to the topic. I would speak to your first-line Supervisor following the Chain-of-Command protocols. You probably won’t get much there because Junior NCOs aren’t much to make waves. If there is a suggestion box or anonymous forum to direct the general concern to the CC, try that. Otherwise, you could ask your Supervisor to allow you the time for the 1600 formation. Typically, military personnel are allowed to leave work early for PT formations. These are ideas you can try. Don’t be shocked if you get no where...it’s the sad misfortune that military members are not looked at as human beings but machines that are pushed to break. Stay positive, stay strong, remember the negative so you can be an effective leader yourself once you achieve higher ranks if you choose to.
As for deployments, those are a whole different beast and everyone suffers 12-hour+ shifts with no down-days. It’s the warriors way.

Seriously? Soldier, for what business are you training? Your statement "my section is the only one in the hospital" would suggest you could be deployed as a medic. Do you think the enemy cares one whit that you are working on little or no sleep? Are you going to raise your "It's not fair" card like a white flag when your buddies are bleeding out in the kill zone - or are you going to face the music you signed up to dance to?
You need to rapidly re-evaluate who you are - REALLY. Look yourself in the mirror and ask yourself if you are in the right place, because right now I would say NO!
Might I also suggest you start thinking from the perspective of at least two paygrades above where you are right now. If the 1SGT wants you at PT, get your butt over there and PT.
If you are looking for fairness in this life, you are going to be sorely disappointed. And since you volunteered for the Army, doubly so.
The life of an Armed Forces Services Member is not an easy one. You will be called on to make sacrifices in peace time that will better prepare you to survive in battle. Remember the old mantra "the more you sweat in peace, the less you bleed in war". Go to PT, sweat, and get into a physical and mental place where you can save your brothers and sisters when they need you most - or go home.

Go to PT problem solved.

Go to the PT get it done then go back to bed problem solved I can seriously think of way worse problems then getting a few hours sleep having pt then going back to bed

Are you going to die?

As an old NCO, I remember that Regulations only require the commander to give a soldier 4 hours off (not sleep), while in garrison, and only one hour of sleep in tactical situations. Now if I were your Squad Leader, or Section Sergeant, I would be talking with Top, and figuring out a workable accommodation for those who have to work abnormal duty hours. I wish you luck Specialist, and THANK YOU, for carrying on the defense of freedom!

I cannot even believe ur trying to get out of this. ITS NOT A PROBLEM do UR JOB and PT IS A PART OF UR JOB U DO IT JUST LIKE BASIC WHEN they tell u to do it. And U HAD LESS than 4-5 hours of continuous sleep back then.

I've read some of the other comments and feel this needs a solid response. A civilian nurse, even an ER/trauma nurse can expect to work a 12-hour shift. Most civilian first responders work 12-hour shifts. Everyone in the military is expected to be mission-ready 24-7-365, unless there are extenuating circumstances.
To feel that having 16 hours of downtime being interrupted is a "problem" IS the problem. I'm sure nearly everyone here has posted the 0000-0400 watch and been present for PT at 0530, or been deployed and worked 18 hour days, or been in combat and 'worked' nonstop for however long.
I'm not unsympathetic but you need to get your self-discipline in order, SPC Davis. We all have jobs that require accuracy for the sake of lives. Being physically fit is part of the commitment from Bootcamp onwards and is a fundamental part of your capability.

Why do I feel unsympathetic right now!? Suck it up comes to mind. Sorry, no pass.