Score the run
 

Sit.
R3, R2, 2 outs.
Ground ball to F5, R3 and R2 are going on contact.
F5 gloves the ball, runs at and tags the hung out R2 for the 3RD out.
R3 touches HP prior to the "tag" of R2.

BR is 1/2 way to 1B "when the tag of R2 is made", BR kicks the dirt and heads towards his 3B dugout to fetch his glove and hat.

F5 seeing BR stop, now throws to F3.
F3 steps on 1B and appeals; "the BR did not advance and touch 1B".
The advantageous 4th out is: "recognized by the umpires, as the 3RD out, on the BR, before he reached 1B, wipe off the run.

Can't appeal abandonment.

BR cannot abandon his responsibility to run to 1st base. Abandonment is only for runners that are on base already.

The BR has been appealed, this is the 4th out - no run scores.

I don't think so. What about the batter heading for the dugout after a dropped 3rd strike?

It's still the advantageous 4th out. No run scores.

The rule specifically says something like, "A runner is out when .... after reaching first, he abandons his attempt to reach the next base"

Somebody (J/R?) calls the BR leaving the baseline "desertion."

Yep, that's J/R applying the term "desertion" for the B/R.

The runner hasn't safely touched 1B before the 3rd out...no run. I don't understand the advantageous 4th out appeal here...that wouldn't be right would it?

R2 was tagged for the 3rd out. BR never touched 1st so the defense has the right to appeal the BR. This appeal creates a 4th out which negates the run. The 4th out is rarely seen because most teams don't even realize that it exists and many amateur umpires don't realize it either! Anytime a BR gives up on a 3rd out not made by him is subject to appeal

great situation to bring up on here.

This might be a stupid question but what if BR keeps running and touches first base (either not realizing the third out has been made or whatever) after the tag on R2? Is the run still nullified because the touch was after the third out?

No. Score the run. The advantageous 4th out is no longer possible.

well, you wouldn't have anything to appeal for the advantageous 4th out appeal. If BR reaches 1B, then, IMO you have a time play. Did the tag take place after R3 touched HP? If yes, count the run. If not, no run.

I still don't know if this sitch (OP) qualifies as an advantageous 4th out appeal. Since the 3rd out already happened, BR doesn't have an obligation to touch 1st does he?

I agree. Now take this: BR rounds first, but misses it -- then the third out is made by tagging BR as he slides into second. Can BR still correct his error?

Wow, Bob. This one gets heavy.

Can a retired runner correct a baserunning error? Well, he can be appealed, and he's still on the field so it would seem unfair to not let him do so.

But can you have an advantaqeous 4th out when the 3rd out was made by the same player? Would the batter be 0 for 2? (just kidding)

I would let him correct the error. And I stand ready to be corrected.

I don't think that it would be possible for the runner to correct this mistake. Once he rounded first, we (as umpires) assume that he touched the bag until appealed. At this point, the BR has completed his responsibility to touch 1st.

In the OP the BR never bothered to complete his responsibility although he could have tried to do so before entering the dugout. :eek::eek::eek: You're right..... it's getting deep!