Subs
 

A1 is shooting the 2nd of 2 FT...he misses the second shot. There is 5.5 seconds on the clock. After the miss B1 grabs rebound but his butter fingers knock the ball out of bounds. The clock operator fails to start the clock after missed FT. The C comes to the table to fix the clock and take time off clock since it did not start properly.

B2 is at the table and wants to sub in.

Question: Can B2 enter the game?

NFHS and NCAA ruling on this play?

Why wouldn't a sub be allowed after an OOB call in any rule set?

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Yes.

+1. The fact that the C makes a timing correction has absolutely, positively nothing to do with substitutions here.

On a separate note, I'd be interested to know what kind of definite knowledge/official information the C had with which to make a timing correction. I mean, maybe he started a five second count on the rebound, but I doubt it. Even in NCAA with a monitor I'm not sure if there's much to change here. It's one thing to put time back on after the clock A) should have stopped or B) started when it shouldn't have. It's a different thing to take time off that never came off in the first place (absent some kind of definite knowledge, of course).

Rule Reference
 

NFHS 3-3-1d.

Does the Rule In NCAA, with under 60 seconds left and play is stopped to correct a timing issue impact allowing subs to enter the game?

The NCAA rule deals with monitor reviews and even then it's not that the substitution isn't permitted at all:

Not related to this but falls in line with substitutions. B scores with 10 seconds left in the game they use there final timeout. Substitute B2 for defensive purposes. On the throw-in A throws the ball OOB not touched. B2 comes to the table to re-enter is he allowed?

No. The saying is "have to sit a tick, don't have to play a tick". The clock does not run during such a throw-in violation and a player cannot reenter the game without time coming off.

Play was stopped for the OOB, *then* there was a timing correction. Allow the subs.

If the ball didn't go OOB and C blew the whistle to start the clock, then the rule would come into play.

B3 fouls A1 with 6.3 seconds remaining in the 4th quarter and the score 62-60 in favor of Team A. B7 subs in for B3. A1 goes to the line for 1-1. The first FT is missed and B4 attempts to rebound, but muffs the ball OOB. The official notices that the timer failed to start the clock and goes to the table. Team B's coach sends B3 to the table to report and enter as a substitute. Is B3 allowed to enter if the referee does not make a correction to the clock? Is B3 allowed to enter if the referee does alter the clock?

I vote "Yes" in both cases.

The clock (should have) started, even if less than .1 ran off and so there was no change to the visible display.

I vote yes in both cases as well.

I was going to use this same verbiage.

I asked my question due to the wording of this rule.

3-3-4: A player who has been replaced, or directed to leave the game shall not re-enter before the next opportunity to substitute after the clock has been started properly following his/her replacement.