(Ref AC 61-65H solo endorsements 61.87(n) and 61.87(p)). When should I give 61.87(n) vs 61.87(p)? At first, it seems clear (according to AC 61-65H) that 61.87(n) is for the first 90 calendar-day period and 61.87(p) is for each additional 90 calendar-day period. However, when I look up the actual regs, it says: &quot;(n) Limitations on student pilots...&quot; and &quot;(p) Limitations on flight instructors...&quot;
So, now it seems like (n) should be the only 90 day endorsement. (p) are just the rules the flight instructor has to follow (every time). So don't (n) and (p) always go together? And, is this idea that we start with (n) and then use (p) if 90 days go by...incorrect? Dare I say, should there be an AC 61-65I that corrects this? And, what should the endorsement say?
I see the NAFI guide sets up the question the same way the FAA does (initial vs additional 90 days), but they just use (n) for both of them.
Remember the ACs are guides, not laws, as opposed to the FARs.
14 CFR §61.87(n) reads:

Limitations on student pilots operating an aircraft in solo flight. A student pilot may not operate an aircraft in solo flight unless that student pilot has received an endorsement in the student’s logbook for the specific make and model aircraft to be flown by an authorized instructor who gave the training within the 90 days preceding the date of the flight.

14 CFR §61.87(p) reads:

Limitations on flight instructors authorizing solo flight. No instructor may authorize a student pilot to perform a solo flight unless that instructor has—
(1) Given that student pilot training in the make and model of aircraft or a similar make and model of aircraft in which the solo flight is to be flown;
(2) Determined the student pilot is proficient in the maneuvers and procedures prescribed in this section;
(3) Determined the student pilot is proficient in the make and model of aircraft to be flown; and
(4) Endorsed the student pilot’s logbook for the specific make and model aircraft to be flown, and that endorsement remains current for solo flight privileges, provided an authorized instructor updates the student’s logbook every 90 days thereafter.

The upshot of which is that (n) is, technically the only requirement for the student to possess for initial authorization for solo flight by a student pilot. (p) is also applicable to the endorsement that must be met by the instructor for initial endorsement, but must be met for additional 90 day solo endorsements by the student.
Here’s the thing, as there is no legal requirement for exactly HOW TO PHRASE an endorsement; they can technically be written by a CFI, provided said endorsement conforms with §61.87, you’re perfectly welcome, as a flight instructor, to give an initial endorsement referencing both (n) and (p) if you desire.