PIC time at JSUPT, it depends...
As always there is no clear answer. I would say by Part 61 rules that you as a multi engine pilot could log time as PIC/Dual Received in your personal log book after solo. Your systems training academics and emergency procedures simulators and the 15 syllabus sorties before solo would probably meet the experience to qualify you for a warbird authorization letter from the FAA to own and operate a T-37 if you had the means to purchase one. I base this answer on the way that almost all civilians log time when getting BFRs, rental checkouts, Instrument Proficiency checks and so on. The CFI signs the log book and put you down as PIC and recorded the instruction he or she gave in the dual received column.
Now for all the reasons not to log it:
1. We brief on every ride the Instructor is the PIC
2. You are never really qualified as rated in any aircraft as a student as far as the AF is concerned
3. Unless you are independently wealthy or win the lotto/powerball/megamillions jackpot, you will probably never own a T-37/A-37 type aircraft
4. When it comes time for an interview the safest PIC is purely Part 1, that makes it no questions asked if you can show those numbers
Would I still log time at JSUPT? Yes, definitely!! And for the rest of your military flying career. A logbook can be a great source of stories when it comes time for an interview. It can show a love of flying especially if you do a lot of civilian flying and continue to do so after you are a rated military pilot. That log book is also a training record and a good way to keep track of flight records to make sure they don't suddenly short you for 100 hours or miss key training events that would otherwise ground you. For that reason, I would only log the exact time for each sortie and not put any military conversions into your log book. Those can come later when you fill out an application because each company is different some allow 0.3 per hour, some allow 0.3 per sortie, and some only allow a 0.2 addition. On a more serious note, Part 61 allows for instruction received from military flight instructors in a program to award military pilot ratings to count for the experience requirements for civilian ratings. God forbid it, but if you became disqualified or failed to complete JSUPT for any multitude of reasons, the instruction received at UPT can count for your experience requirements for additional ratings on your civilian ticket. I know personally of two people who have been medically disqualified from military flying (one was completely disqualified from military service) and hold unrestricted FAA Class I medical certificates. Student time is treated as only student time in your military flight records and doesn't amount to much. As such any night time or instrument time you receive as a student is not reflected in your flight records. The syllabus required night time and simulated or actual instrument times are recorded in your gradebook and not your flight records, so in your personal log book, I would record any night, instrument or time with a vision restricting device in the appropriate places as well as keep track of what instrument approaches you flew.
You will get a minimum of 7.5 hours of solo time in the mighty Tweet under the current syllabus. Your first solo will be on your 15th lesson and your instructor will see you through three landings one of which can be the full stop, a go around and something called a pattern breakout and re-entry, the IP will get out with the right engine shut down, tie up the ejection seat and be your crew chief while you restart the right engine and marshall you out for your first solo. You'll have anywhere from 1600 to 1000lbs of gas to get as many patterns and landings as you can, and you will have to plan to be on the ground with 500lbs. Your next solo will be in three or four more rides and will be a local pattern only for you to practice normal pattern procedures in preparation for your mid-phase checkride. After mid-phase you get four solos out of eleven rides to go to the area or stay in the pattern to practice your aerobatics (aileron roll, cuban 8, split-s, cloverleaf, chandelle, immelman, barrel roll, lazy 8, and loops) and normal patterns to get ready for your final contact check. Your final solo will be in formation as soon as you are safe after your 8th lesson and you will fly on the wing of another T-37 with 10 feet of lateral seperation between wingtips at speeds between 240 and 120 Kts and bank angles up to 90 degrees inside and outside the turns.
You can fly the Tweet with only 7 power settings (military, 90%, 82%, 74%, 70%, 57% and Idle), you just have to figure when it is appropriate to use each one of them. Good luck, chairfly and have fun.
For all you nitpickers reading UPT which became JSUPT is about to become JPPT which is the new speak for our new love of the T-6 and means Joint Primary Pilot Training. I wonder if we will pronounce the acronym as as "Jipped". Does anyone remember when Undergraduate Navigator Training almost became Combined Undergraduate Navigator Training before it became Specialized Undergraduate Navigator Training? That was only stopped when someone figured out what that acronym was going to be.