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I see this world mentioned vaguely but I can't find where Olam HaPirud/Peyrud/Peirud or alma de-peruda is located in the hishtalshelut.

I read today's Tanya:

such service remains below in this “World of Separation,”1 in the level called the externality of the worlds2 (as opposed to the Sefirot which are the internal aspect of the Worlds);

I left the footnotes in here because I will reference them later. What is this externality of the world's?

In the one hand, the Lubavitcher Rebbe says:

Note of the Rebbe, Rabbi Menachem Mendel Schneerson, of righteous memory, “See further, beginning of ch. 40: it may also be in the World of Yetzirah. This requires to be understood since the Alter Rebbe says here, ‘remains below.’ See also Kuntres Acharon.”

And I think the Raya Mehemna supports this, which pritzker leaves out as being outside the strata of Zohar:

The third world [Yetzira] is beneath the two others and separation [between good and bad] is found in it.

And yet it says just a few lines later:

The first world [he can grasp] is called the World of Separation and man is found there and also not found there.

This seems to align with another footnote by the Lubavitcher Rebbe on the same Tanya:

The Rebbe notes: “Accordingly, the word ‘Worlds’ — in the plural — is now understood, [referring to the Worlds of] Asiyah and Yetzirah.”

But the first world that he may grasp may actually be the third world (Atzilut is 1, Briah is 2, and Yetzirah 3).

But on the other hand, Reishit Chochmah from Neirot says Briah:

The state in which Adam and Eve were originally attached to each other back to back, alludes to Malchut in a fuller state affecting the entire creation as a Partzuf – Stature. Their separation and relating face to face, as well as their previous state of attachment, parallel the union and separation of the world of Briah – Matronita, which is the world of separation – Olam HaPirud, with that of Atzilut. Furthermore, there are many profound concepts which are alluded to in these matters.

If Briah is correct, Atzilut would jive more Olam HaYichud.

Nevertheless, the Lubavitcher Rebbe says:

שלא תהי' בבחי' נפרד מאלקות, כמו בעולם הבריאה, התחלת עולם הפירוד, אלא כמו בעולם האצילות, עולם האחדות

Briah is the beginning of Olam HaPirud. Now I have another question: what is Olam HaAchdut? How is different from Olam Ha-Yichud?

I also found this from Vayetze:

160) It is written, And she stopped [Hebrew: stood], since up to Judah, who is Malchut, she stood in one unification. From here downbelow Malchut de Atzilut, in the world of separationit is the three worlds of BYA of separation. Hence, are the two sons that she bore after Judah Issachar and Zebulon separation because they were born after Malchut?

Below Malchut d'atzilut is Briah.

But in Zohar, 1:158a, it is written:

... the sefirot] are called mountains of separation, as it is written, From there it divides (Gen. 2:10), mountains of division.

The Pritzker commentary writes that these forces (which is my second question, are these the nifradim?) are not included in the unified realm (Olam HaYichud?) of the sefirot. This would seem to support Olam HaYichud as the pnimiut of the olomot.

Are the externality of all three worlds of BYA collectively Olam HaPirud?

כבוד2 מלכותך יאמרו וגבורתך ידברו להודיע לבני האדם גבורותיו וכו'3, הנה מלכות עיקר בנינה מן הגבורות, דהיינו שיהי' יכול להיות מבחינת א"ס בי"ע עולם הנפרדים צריך להיות הסתרים וצמצומים, כי בבחינת אצילות איהו וגרמוהי חד4 כמשל כחות הנפש המתפשט במחדומ"ע וכו', כמו כן בבחינת אצילות אפילו הכלים הכל הוא בחינת אלקות, אבל בבי"ע עולם הפירוד אפילו מלאכים אשר צבא השמים לך משתחוים5 אעפ"כ הם בחינת יש ונפרד דבר חוץ בעל בחירה [וענין זה הוא דבר חידוש, שהרי מפורש בתניא6 שהמלאכים אינם בעלי בחירה, וכאן מבאר שהם בעלי בחירה7, ומביא ראי' ע"ז מזה ש]אפקוהו למט"ט8 וכו', לכן צ"ל מקודם בנינה מהגבורה שיוכל אחר כך ע"י הסתרים וצמצומים להיות בי"ע, דאם לא כן הי' הכל בבחינת אין סוף וכו', אך אם לא היו הגבורות ממותקו' הי' יכול להיות נפרדים לגמרי, לכן אברהם אתקין לכורסיי'9, וע"י שהגבורות ממותקים יוכל להיות ביטול בעולם הנפרדים צבא השמים לך משתחוים.

Where is this world in the chain of things? And what are nifradim?

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Sefer Shaare Ora, first Shaar. "Below the name A.D.N.I., (א-ד-נ-י), is the world of perud". The hataot Maor Yosef says.

הם העולמות שתחת מלכות דאצילות, והם עולם בריאה יצירה עשיה

Olam haperud is the set of worlds Bria, Yetsira, Asia, they are below malchut deatsilut, the lower expression of world of atsilut.

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I’m not sure I followed your entire question, but from a basic understanding Olam Hapirud refers to the bottom 3 olamos of Beria, Yetzira, and Asiyah. Atzilut, although the beginning of the Seder hishtalshelut, is still intimately connected to the Ein Sof and thus is still viewed as having absolute unity. The root of אצילות is אצל, meaning next to, it is still close to the divine unity. It is only after that world that we begin to see any differentiation I.e. Pirud. See the first chapters of the book Inner Space by Rabbi Aryeh Kaplan (actually written by his students) particularly pages 24-25.

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