If I'm doing a CONUS PCS with a 31 May RNLTD, how early am I permitted to arrive at my new duty station?

So you posted in Jan 18 and you aren't due to report to your operational unit until the end of May 18. If you've completed all of the training schools you were scheduled for after Pilot Training, then something is not correct. First read your orders carefully as SSG(P) Lauren Manuel suggested. Next contact your admin section (what I used to know as the student squadron orderly room) to confirm details of any additional training you may need. (Long ago, we went TDY enroute to our PCS location to attend Advanced Pilot Training, Tactical Training, and Basic Survival Training. Some also went to Water Survival Training and Jungle Survival Training before reporting in to first operational unit.) If you're truly done with training, then you should report as soon as possible. Ask the admin folks to coordinate a change to your reporting date. If you have a sponsor at your gaining unit, contact him or her. If all this fails, contact your gaining unit Chief Pilot or Operations Officer. Don't be put off by them being Maj or Lt Col. They most likely will have a handle on the staffing in their unit and can direct you to the right person in the Squadron to coordinate a reporting date change. Finally, if you can't get a straight answer anywhere else, try the Air Force Personnel Center. Get in contact with the resource manager for your AFSC. He or she should know the normal schools for a person in your AFSC and where the reporting date was generated from. You sitting around for 5 months won't help the Air Force, your new unit, or you.

Your orders will tell you when you must report and if early report is authorized and how early you can report.