Seven days before Yom Kippur we segregate the Kohen Gadol.

A substitute Kohen is designated.

(R. Yehudah): A substitute wife is also designated.

(Chachamim): There would then be no limit.

Seven days prior to the burning of the Parah Adumah we segregate the Kohen who will burn it.

He is brought to the Lishkah in the Northeast corner facing the Birah, called the Lishkas Beis ha'Even.

Question: Why is it called that?

Answer: Because the activities there were conducted in utensils which cannot become Tamei, such as stone.

Question: Why is this stringency needed?

Answer: Owing to the leniency of T'vul Yom.

We learnt that this leniency became a point of contention with the Tzedukim.

As such, we would make the Kohen Tamei.

This stringency (of the utensils) was instituted to prevent the Parah Adumah from being treated lightly.

Question: Why was the Northeast selected?

Answer: A Chatas (so the Parah is called) is in the North and P'nei Ohel Moed brings us to the East (thus alerting the Kohen to aspects of his upcoming Avodah).

Question: What is Birah?

Answer #1: A place on Har ha'Bayis.

Answer #2: The entire Mikdash is called Birah (based on the Pasuk).

Question: What is the source for a seven day separation?

Answer: It is learned from the seven days of Miluim.

"La'asos" refers to the Parah Adumah.

"Lechaper" refers to the Avodah of Yom ha'Kipurim.

Question: Given that the Pasuk cannot be entirely referring to Parah Adumah; but may it not all refer to Yom ha'Kipurim?

Answer: Gezeirah Shavah of "Tzivah-Tzivah".

Question: But perhaps the connection is only to Yom ha'Kipurim, not to Parah at all?

Answer: "Tzivah" before the performance is the more reliable connection (over a different Tzivah after its Avodah).

Question: Perhaps we connect to the Tzivah of communal offerings (and the Kohen must separate seven days prior).

Answer: "Tzivah" does not match "Tzavoso".

Question: But we find that topical connections are used?

Answer: That is only where no identical word connection is to be found.

Question: Perhaps "Lechaper" refers not to the Avodah of Yom ha'Kipurim but to all Korbanos.

Answer: We would not know which Kohen it will be.

Question: We should require segregation of the entire Beis Av!?

Answer: It is more logical to connect to Yom ha'Kipurim whose time/date is fixed (unlike daily Korbanos).

Question: Then connect to the Korbanos of the Regalim!?

Answer: It is more logical to connect with that which takes place once a year.

Question: Then perhaps we should choose one Regel!?

However we would not know which.

Pesach has the advantage of being first in the Parshah.

Sukos has the most Mitzvos.

Answer: Connect a seven day separation for a one day Avodah with another of the same (not seven for seven).

Question: Then connect to Shemini Atzeres (seven for one)?

Answer: Connect to that which does not have Kedushah preceding it (Yom ha'Kipurim) and not to that which does.

Question: But one which does should all the more so require segregation!?

Answer #1: Hazeh teaches that only a similar day requires segregation.

Answer #2: Shemini Atzeres (the Tafel) could not require more than the (Ikar) Regel itself!

Even according to the opinion that Shemini Atzeres is a separate Regel, that is only regarding PEZER KESHEV.

Regarding Tashlumin, however, Shemini Atzeres still serves Sukos (as we have learned in the Mishnah).