Or the throwin team commits a throwin violation.

This precise situation has been discussed before (within the last month). Not only would 4 of 5 officials do the "re-do" thing, there would be fewer than 1 out of 10 coaches who would complain (regardless of score) since they know it is the "fair thing to do."

We have a correctable error situation in which a team can get fouled on a non-shooting foul, score on an inbounds play, have the horn sound (with the scorer informing the officials that the previous foul was the SEVENTH team foul, and therefore the 1+1 should have been shot), have the team that just scored shoot and make both ends of the 1+1 for a FOUR (4) point play, YET we cannot correct our own boo-boo in situation of the wrong team inbounding.

I am convinced that the Rules Book writing team wants to keep the "Corrrectable Error" section nice, short or succint. We can only correct something having to do with free throws.

As several have said, the NF Rules Book was never put to a "Logic" Test.

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Who says?

NFHS 2-10 CORRECTABLE ERRORS
ART. 1 Officials may correct an error if a rule is inadvertently set aside and
results in:
a. Failure to award a merited free throw.
b. Awarding an unmerited free throw.
c. Permitting a wrong player to attempt a free throw.
d. Attempting a free throw at the wrong basket.
e. Erroneously counting or canceling a score.

Official incorrectly signals a "touchdown" three, when the shooter is clearly inside the arc. Within the confines of the correctable error time limit, at the next dead ball, timeout, situation, scorekeeper asks the same official, "Was the last shot a two, or a three?". Official replies, "It was a two, I don't know why I made the touchdown signal". Correctable. One point comes off the scoreboard.

So if we have this situation, possession belongs to home... Visitor team throws it in, ball is legally touched in bounds... and 5 seconds in your partner blows the whistle to attempt to correct a non-correctable error.

You discuss this with your partner and agree it cannot be corrected.

Do you then (by rule), give the ball to Visitor team and indicate an inadvertent whistle (assuming visitor team had team control at the POI)?

Let's also say the above scenerio WITHOUT the whistle (because both you and your partner have read this thread so know the rule...). The table, going on auto-pilot, changes the possession arrow (this is more likely in a rec game where there is no clear home or away). You don't realize they changed it until the next jump ball. Is the clock/book official or your personal method of keeping track with definitive knowledge (I switch a card in my pockets) the official ruling?

Yes.

I make sure the arrrow is corrected at the next stoppage.

If I know the arrow is wrong, I'll go with what I know is right. I don't use the "whistle in the pocket" trick.

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Required mechanic here in my little part of the Constitution State.