A player goes below 5 minutes. Both players then stop keeping score. Several moves later, the opponent of the player under 5 minutes makes an illegal move. With the 2 minute penalty added, both players are now over 5 minutes again. Must they resume keeping score? If so, for completeness must they either reconstruct all the missing moves or record the current position?
The rulebook (rule 15C) does not specifically address this issue but as the rulebook states that you may stop recording when either player has under 5 minutes my inclination would be to not require them to resume if 2 minutes is added because it makes no mention of resumption or that the excused obligation is temporary. In many cases it would be impossible to fill in the missing moves.
This is somewhat similar to the situation if, due to a 30-second increment, a player who was under 5 minutes now goes over 5 minutes. I believe the player is not required to resume keeping score. (Somebody correct me if I’m wrong.)
Bill Smythe
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Bill, if they are using a 30 second increment then they must keep score throughout the complete game no matter what the base time is.
Correct—but Bill’s scenario could happen with an increment of, say, 15 seconds. I played a friendly match at G/60, Inc-15 last year. It could happen, though I do not recall it happened in the match I played. (If so we did not notice.)
One of the endless tweaks to iron out…
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What is the cut off point for increment time making it necessary to continue keeping score under the 5 minute mark?
Oops. 
Make that a 15-second increment.
I believe it’s 30 seconds. With 29 seconds, you don’t need to keep score under 5 minutes.
Bill Smythe
Increment of 30 seconds is where FIDE Rule 8.1 and USCF Rule 15B both kick in.