When A1 pulls the ball back out (towards T), that is what I think was labeled as 'retreating'. At that point B1 hangs back and relinquishes his LGP (he's roughly 10 feet off of A1). There is no maintaining LGP at that point. It is then B1's attempt to again establish (or re-establish) LGP upon A1 driving to the basket, to which he could not do, thus the block call.

Really? Where is there anything in the rules that says a defender loses LGP if either player backs away?

He retreated about 6-7 ft from the defender and started a new path.

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So LGP exists forever?

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Move A1 and B1 from the perimeter to the post area. A1 is faced up towards the goal about a foot from B1, who has both feet on the floor with his/her torso facing A1. A1 backs up so the distance between them is 7-8 feet but B1 doesn't move. Did B1 do anything to lose LGP?

B1 didn't do anything to lose LGP on his end (move forward, etc.) but A1's path is now backward, which means that B1 is no longer "by definition" guarding A1.

Oh course this argument is pointless cuz we don't care whether B1 is legal until contact occurs.

A1's path can also be considered to be the line between A1 and the basket.

Is that supported by any rule or case play or by the dictionary definition of the word "path"?

As I said, this line of reasoning is pointless anyway.

Path isn't defined in the rule book as far as I know. I believe it to mean both the line in which a player is moving AND the line in between a player and where they want to go (the basket).

The case plays largely talk about path as the direction a player is actually moving.

However, imagine a stationary A1 and a stationary B1 where A1 fakes B1 into the air such that B1 jumps perfectly straight up. A1 then, after A1 is airborne, moves into B1. Did B1 have LGP? Why or why not? Was B1 entitled to verticality (noting that verticality only is granted with LGP)?

Alternately, imagine A1 moving across the top of the key with B1 sliding across the top of the key as well. B1 had two feet down momentarily and remains between A1 and the basket but is never in the path A1 is traveling. A1 suddenly changes direction towards the basket contacting B1's chect while B1 still had one foot off the floor. LGP or not?

Since "path" isn't defined in the rule book the next best thing is the dictionary. One of the definitions is "the area in front of someone...that is moving." So while A1's direction in my example is backwards, his/her path would still be towards the goal if they're still facing B1/the basket.

Stop the clip at the 2 second mark. B1 is not in A1's path when A1 starts his drive diagnoally.