How to handle end of game clock situation
 

5.0 seconds left on the clock
division line throw in
throw in completed into the team's front court
dribble to basket
subsequent shot
ball caroms off rim
still 5.0 seconds left on clock
Timer admits that the clock was never started
officials have no count or definite knowledge of time elapsed

What do you do? Do over? Or, go to rebound and finish from there?

You have definitive knowledge that more than zero seconds came off the clock. A tap for goal is .3 seconds, and you had at least .4 if there was a shot involved. Start with that and get with your partners and decide if 1 second definitely went off, if 2 seconds definitely went off, etc.

If it is less than 5 seconds, put the ball in play at the point of interruption. If more than 5 seconds, the game is over.

And next time everyone should have a count in their head.

Next time everyone should be blowing it dead when they see the clock not starting. Easiest way to fix this.

You still need a count since you will need to adjust clock and inbound the ball from the POI.

Since the ball is in the front court and you don't have a backcourt or closely guarded count .. You can't just have a count!

Yes, you can -- and yes, you should. It's not visible, of course, but each official should have one.

And, with five seconds left, there's plenty of opportunity for one of the two/three officials to see that the clock didn't start.

There are two options supported by rule: you can either blow it dead and adjust the clock and inbound from the POI, or you can blow it dead, leave the clock where is was originally, and inbound from the original location.

Under 5 seconds, blowing this dead to fix, and possibly remove time to then have the team redo the throw in with LESS time is tantamount to a disaster. You are screwed either way, but I would always count. It's a tough spot.

Adjust it with what count? Supported rule count

You have to catch it right away and leave it at 5.0 and go from original spot

or

I think I would ask everyone at the table if there is any alternate timing device that could help us out (not a likely thing to have around). If there’s no other information that can help, sounds like we just have to take the ball to the POI and give it to whoever had it when we stopped play. Then we finish those 5 seconds and GTFO as quickly as possible.

I don't think that is supported by rule (maybe I am mis-understanding it)

The NFHS Mechanics manual, pgae 11: "Remember, an officials count (backcourt, throw-in, etc.) may be used to correct an obvious timing error."

This phrase seems to suggest that the count should be visible, but it does not explicitly say so. I interpret "an official's count" to mean "when an official is counting", which may or may not be visible. Thus, I am allowed to count in my head in this situation and adjust the clock as necessary.

Personally, I have my hand at my side and snap my fingers in rhythm with the clock. That has worked quite well for me judging the release of the last-second shot and correcting a timing mistake like the one in the OP.

I don't know why so many officials take this stance. If I'm counting in my head, then I have "an official's count". Why would you mess things up more by pretending no time elapsed when you (or one of the other officials) knows how much time elapsed?

SECTION 10 TIMING MISTAKES
ART. 2 . . . If the referee determines that the clock malfunctioned or was not started/stopped properly, or if the clock did not run, an official’s count or other official information may be used to make a correction.

no do overs from the original spot. you may have to leave 5 seconds because you have no other definite knowledge but the ball would go in play to wherever it was when you killed it. i haven't gotten my head into the rules like I should just yet but I don't know where you can go back to the original location under the rules...

I agree and think you can have your own count even when not ten second or 5 count etc. But if you are thinking about doing that with 5 left .....communicate with the timer and your partners before ball inbounded. I will say something to the timer and my partners. talking to the timer usually solves any issues but there are timers who might be a little slow on the trigger. one of us has to see that the clock didn't start. right away..(unless you see an immediate scoring play off the inbounds..then id wait....)timing mistakes at end can be nightmare. always communicate..

We've had this discussion numerous times before on this site.
I've always advocated the following:
1. Only the Referee may correct a timing mistake. The Referee may take input from the umpire(s), but must make the final decision.
2. Once the throw-in is touched inbounds, it is over and any stoppage for a timing error now results in a POI throw-in. There are no do-overs under NFHS rules.
3. 2-7-9 lists the officials counts during the contest. Those are what may be used to correct an unnoticed timing mistake.
4. If an official notices that the clock did not properly start or has improperly stopped and does not wish to halt play immediately because a scoring playing is in progress, the official needs to start a visible count and continue it until either halting play for a correction or to signal that time has expired. Having this count be visible is important because it can be verified via video in the event of a complaint.
5. If the clock improperly starts prematurely, prior to the ball being touched inbounds, an official needs to notice and stop play immediately and before the ball is touched inbounds. The time can then be restored and the game resumed from where that the throw-in was located. If already touched, then POI is required and some small amount of time must be deducted for the inbounds touching.
6. If no visible counts were made while the clock was not properly running, then no correction can occur, other than a small deduction for a throw-in catch or touching. This is unfortunate, but officials can't make their best guess nor use non-visible counts which aren't part of 2-7-9. There does not exist a single NFHS ruling or Case Book play in which an official counts in his head and uses that for a correction.