Let's play a fun little game of Rulebook Lawyer. We can all agree that (by rule) an OBS runner cannot be called out between the bases she is obstructed, on the play that she was obstructed, for anything not listed as an exception. The question at hand is: when does one play end and another begin?
The forum refers to the Appellate Court for this question: literally, appeals. I believe we can all agree as an axiom that all appeals must made on or after the play during which the appealable situation happened, in other words "before the NEXT play". Any appeal, be it live-ball/dead-ball/BOO, must be made before the next pitch legal or illegal (by rule). We can use this to assume that the start of the next pitch is the start of the next play, and since fastpitch is a live-ball game, it is also the end of the previous play. If for some reason an umpire declares TIME, the play has ended for the purposes of base running, but not for appeals. NFHS rules require the plate umpire to "point the ball live" before play resumes.
This raises a new question. When does the pitch occur? It does NOT occur when the ball is thrown by the pitcher. It does not occur when the hands come together. Ignoring pre-pitch violations resulting in an IP, the pitch begins when the pitcher separates her hands to start a legal delivery. Until that point, the pitcher can legally remove herself from the pitching plate by stepping back, and no "next" pitch has occurred.
Thus, as to the first question:
The play is considered over when the hands separate for the next legal pitch, or when an IP is called prior to the hands separating, or an umpire calls TIME.
As to the second question:
An OBS runner leaving early on the next pitch would be called out, as the play on which she was OBS is over, and the next pitch started. However, if the OBS runner was to leave so early that the pitcher had not yet separated her hands, we would have an LBR violation instead. The ball would be dead, and the runner would be returned to her base....unless an umpire had called TIME and/or made awards which were properly touched by runners before the ball was again made live by the plate umpire.
Feel free to pick this semi-TWP analysis apart for purposes of discussion. I'm not an official rules interpreter, but I did stay at a Holiday Inn Express once.
