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FOOTNOTES:

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(1) He may be Tovel in order to Daven Minchah. If he has already Davened Minchah and then sees Keri, he may not be Tovel during the day in order to Daven Ne'ilah, because he can push off Ne'ilah until the night and be Tovel at night and then Daven Ne'ilah. (TOSFOS, DH Iy d'Tzali, however, suggests that it is possible that even according to the Rabanan of the Beraisa, if one wants to Daven Ne'ilah during the day, he may be Tovel during the day. The Rabanan are talking about a case where one specifically intends to Daven Ne'ilah at night.)

(2) This is the simple understanding, because otherwise it would be permitted to be Tovel during the day even if one did not need to Daven. However, TOSFOS (DH Kasavrei) says that it could be that these Rabanan also hold that Tevilah b'Zemanah Mitzvah, and when they say that one may not be Tovel during the day if he already Davened Minchah, they mean that one should not be Tovel simply in order to Daven, since one may Daven Ne'ilah at night. But in order to fulfill the Mitzvah of Tevilah b'Zemanah, he may be Tovel.

(3) The Gemara brings another Beraisa (regarding immersing with the name of Hash-m written on one's body) in which we see that Rebbi Yosi holds Tevilah b'Zemanah Mitzvah. If so, even if Rebbi Yosi holds that Ne'ilah could be said at night, one would still be permitted to be Tovel during the day because of the Mitzvah of Tevilah b'Zemanah, just like the Rabanan of the Tosefta (#4 in chart) hold (see footnote 5). However, according to TOSFOS as mentioned in footnote 2, who suggests that the Rabanan and Rebbi Yosi are only discussing being Tovel in order to Daven, Rebbi Yosi of the Beraisa must also permit the Tevilah for a second reason, because Ne'ilah has to be said during the day.

(4) The Gemara says that this Rebbi Yosi is actually Rebbi Yosi b'Rebbi Yehudah, who holds that Tevilah b'Zemanah Lav Mitzvah (Nidah 29b). Otherwise, Rebbi Yosi would be contradicting his own opinion, as expressed in the Beraisa in Shabbos regarding immersing with the Name of Hash-m written on one's body (in which case Rebbi Yosi clearly holds that Tevilah b'Zemanah Mitzvah).

(5) As far as Ne'ilah is concerned, either the Tosefta is discussing a situation where one already Davened Ne'ilah and therefore Tevilah is not necessary in order to Daven (the Gemara's original interpretation), or else Rebbi Yosi b'Rebbi Yehudah of the Tosefta holds that Ne'ilah may be Davened at night (as in footnote #1, above).

(6) Since they hold Tevilah b'Zemanah Mitzvah, the Rabanan permit Tevilah despite the Isur of Rechitzah on Yom Kipur -- even if one does not need to be Tovel in order to Daven (i.e. he already Davened all of the Tefilos before he saw Keri). In this regard, the Rabanan of the Tosefta disagree with the Rabanan of the Beraisa, who hold Tevilah b'Zemanah is not a Mitzvah and therefore he may not be Tovel after Minchah (RASHI DH Ka Savrei, TOSFOS DH Iy d'Tzali)