Netivot Shalom: Shmini Atzeret/Simchat Torah ​​​​​​​"On the eighth day, it will be Atzeret for you"
The focal verse: Shmini Atzeret's one mention in the Torah
בַּיּוֹם֙ הַשְּׁמִינִ֔י עֲצֶ֖רֶת תִּהְיֶ֣ה לָכֶ֑ם כָּל־מְלֶ֥אכֶת עֲבֹדָ֖ה לֹ֥א תַעֲשֽׂוּ׃

On the eighth day you shall hold an atzeret/ solemn gathering; you shall not work at your occupations.

Shmini Atzeret is mentioned here in the sacrificial calendar in Bamidbar (and also in the Devarim calendar as well). It gets no mention in the other calendars (in either Shemot and Leviticus). Note that is it the ordinal number Eighth, which implies that it is meant to or needs to follow the seven days of Sukkot, but is treated linguistically as a separate festival. This is seem in the lack of a conjunctive Vav before the word "Bayom," which creates a simultaneous separation from the previous seven days, even as it is describing an eighth day in a series.

1. Our Essential Questions about Shmini Atzeret and Its Three Primary Aspects
And there our Sages taught:

וְלַיקִּשׁ לְחַג הַסּוּכּוֹת: מָה לְּהַלָּן שְׁמוֹנָה — אַף כָּאן שְׁמוֹנָה! שְׁמִינִי רֶגֶל בִּפְנֵי עַצְמוֹ הוּא.

...as the eighth day is a separate Festival in and of itself.

If this is the case [that this day is a separate Festival in and of itself], we need to explain why it is called "The Eighth Day," and why in every other place only three Regalim / Pilgrimage Festivals are mentioned, and not four.
Also, one needs to explain why any reason for the Yom Tov / Festival Day of Shmini Atzeret is not written in the Torah like it does for the other Yamim Tovim/Festival Days. Because for all of them, there is an explicit written reason for that Yom Tov: Pesach is about the Exodus from Egypt. Shavuot is about the Bikkurim / First Fruits, and Sukkot is about "So that your generations will know that I made B'nai Yisrael dwell in sukkot." Furthermore, one needs to explain the meaning of Atzeret and what exactly is the mahut / essence of this festival.
Three Aspects of Shmini Atzeret​​​​​​​

  1. Day of D'veikut / Clinging Intensely to God
  2. Siyum HaChatima / The completion of the Seal for the year
  3. Yom HaShmini / The Day of "Eight" - Above and Beyond the Natural
Here, we find in regards to the essence of this holy Festival three unique matters:
1) A day of d'veikut / clinging to God in a more intense way, and as the Sages said, it is an aspect of: "Prepare me a small feast so that I can derive pleasure from you alone."

מָשָׁל לְמֶלֶךְ בָּשָׂר וָדָם שֶׁאָמַר לַעֲבָדָיו: עֲשׂוּ לִי סְעוּדָה גְּדוֹלָה. לְיוֹם אַחֲרוֹן אָמַר לְאוֹהֲבוֹ

עֲשֵׂה לִי סְעוּדָה קְטַנָּה כְּדֵי שֶׁאֵהָנֶה מִמְּךָ.

The Gemara cites a parable about a king of flesh and blood who said to his servants: Prepare me a great feast that will last for several days. When the feast concluded, on the last day, he said to his beloved servant:

// Prepare me a small feast so that I can derive pleasure from you alone. //

This is the exact origin of this phrase, which is now part of the set of phrases that can refer to various aspects of Torah and Jewish life, here specifically aspects of the Festivals. Here is Rashi's comment on the interpretation, along with a couple other mefarshim and Rabbi Joshua Kulp, from his Daf Shavui series.

שאהנה ממך - אין לי הנייה וקורת רוח בשל אלו אלא בשלך:

I have no benefit nor personal fulfillment from these others, but only from you. [Ed: implying that the fulfillment does not from the feast, since it is so small; this analogy is based on that fact that the korbanot for Sukkot are substantial in number, but the kobanot for this day are comparatively small.]

Rabbi Joshua Kulp: The final bull offered on Shemini Atzeret corresponds with Israel. Israel is that favored friend in the parable, who God asks to stay on just one more day, so that God can enjoy our company.

And in the holy Zohar (III 104b): "From this point and onward, I and you will rejoice for one day."

דִּכְתִּיב (במדבר כ״ט:ל״ה) בַּיּוֹם הַשְּׁמִינִי עֲצֶרֶת תִּהְיֶה לָכֶם, דְּהָא יוֹמָא דָּא, מִמַּלְכָּא הוּא בִּלְחוֹדוֹי, חֶדְוָותָא דִּילֵיהּ בְּהוּ בְּיִשְׂרָאֵל. מְתַל לְמַלְכָּא דְּזַמִּין אוּשְׁפִּיזִין, אִשְׁתְּדָּלוּ בְּהוּ כָּל בְּנֵי הֵיכָלֵיהּ, לְבָתַר אָמַר מַלְכָּא, עַל כֵּן אֲנָא וְאַתּוּן אִשְׁתָּדַּלְנָא כֻּלְּהוּ בְּאוּשְׁפִּיזִין, וּקְרַבְתּוּן קָרְבְּנִין עַל שְׁאָר עַמִּין בְּכָל יוֹמָא, מִכָּאן וּלְהָלְאָה, אֲנָא וְאַתּוּן נֵחֱדֵי יוֹמָא חַד, הֲדָא הוּא דִכְתִיב בַּיּוֹם הַשְּׁמִינִי עֲצֶרֶת תִּהְיֶה לָכֶם. לָכֶם: לְקָרְבָא קָרְבְּנִין עָלַיְיכוּ. אֲבָל אוּשְׁפִּיזֵי מְהֵימְנוּתָא, בְּמַלְכָּא מִשְׁתַּכְחֵי תְּדִירָא. וּבְיוֹמָא דְּחֶדְוָותָא דְּמַלְכָּא, כֻּלְּהוּ מִתְכַּנְּפֵי עִמֵּיהּ, וּמִשְׁתַּכְחָן. וְעַל דָּא כְּתִיב, עֲצֶרֶת, תַּרְגּוּמוֹ: כְּנִישׁוּ.

As it is written, "On the eighth day (Heb. shmini) you shall have a solemn assembly (Heb. atzeret)" (Bemidbar 29:35). For that day is from the King solely, rejoicing with Israel. This is like a king who invited guests. The household people entertained them. At the end the king said [to his household], Until now, you and I all entertained the guests. You offered sacrifices for the other nations every day [i.e. the seven bulls for the nations.] From now on, for one day, let you and I rejoice. This is the meaning of, "On the eighth day you shall have a solemn assembly:" "you" - offering sacrifices for you. But the guests of Emunah/Trust are always with the King, [on Shmini Atzeret as well.] On the day of the King's joy, they all gather to him, and stay with him. Hence it is written, "assembly" which is translated into Aramaic as 'gathering.'

D'veikut, from the Hebrew meaning "sticking to" or "clinging to." It has a range of meanings. For some interpreters, it comes from acting in Godly ways; doing as God does brings the self and God into a closer relationship. That does not seem to be what is being described here. The Netivot Shalom seems to be taking the more mystical interpretation where one comes close to God through the observance of the Festival, which is marked by nothing special. Just a simple, unadorned festive day. D'veikut here seems to be coming close to God through the simchah of the day (which is the peak of the entire month/year).

2) And this [Festival of] Conclusion is an aspect of "One who sits with the king; all who want to, come and sit with me" (ibid 32a). And thus it is a day of Siyum Chatima/ Completing the Seal, as it is written in the Holy Zohar (I:220a) and in the writings of the Arizal: And furthermore, we find it regarding a Jewish practice: A Torah that is assigned for Simchat Torah to be read to completion with song, dance and hakafot, and like it is mentioned in the Zohar (III 256b): It is the custom for Israel to make a celebration, to read on Simchat Torah, and to crown the Sefer Torah...

Here the Netivot Shalom is speaking about Shmini Atzeret as the conclusion of the High Holiday season (more detail below), with this day being the final seal of the year head (i.e. who will live and who will die...). The conclusive aspect of the Yom Tov seems to be illustrated by the conclusion of the Five Books of Moses, so not only are we wrapping up the solar year, and the lunar month, but we are concluding our Torah reading cycle for the year as well. It's all about wrapping things up. This is the aspect of Chatimah/Seal.

3) And one can say that the level of this holy Festival, which is an aspect of Shmini / Eighth, hints to us that it is Above and Beyond the seven days of the Binyan / Creation (literally Building), and the Marahal explained (Drashah to Song of Songs): The matter of Shmini/Eighth is above all dimensions and all of the natural world, because the number seven includes all natural matters, for all matters have in them some connection to nature, which are included in the seven days of Binyan/Building or Creation. And this aspect of Shmini/Eighth is a level above nature and the world, and it includes all matters that are above nature.
And in the words of my Master on his explanation (Vayikra 9:1; which is about the dedication of the Mishkan): "And on the eighth day" - where all matters of this world are completely shaken off. And this is about all things about which are written Shmini/ Eighth, for they are beyond the natural world. Like our Sages say (Shabbat 87b): "And on the eighth day...on that day, ten crowns were taken" - meaning that on that day there was a revelation that was so great, as they say: "And God's glory filled the Mishkan," (Exodus 4:34) which was a light that was beyond the natural. And so too is the matter of "On the eight day, the foreskin of his flesh shall be circumcised," (Leviticus 12:3) because this shakes off the tumah from childbirth, and their filth with the sin of: "Indeed I was born with iniquity; with sin my mother conceived me." (Psalm 51:7)

The theme of the Eighth.

The construction of the Mishkan

The dedication of the Mishkan

Brit Milah

Shmini Atzeret

It's about going above and beyond the natural and the natural world. Eight is the number of transcendence, moving from the earthly to the divine. From now to future. And not just moving from one to the other, but shaking off the earthly. The intense motion implied in this verb relates to the na'anu'im of the Four Species, the hakafot from Musaf, and also clearly linked to the dancing that will take place during Simchat Torah. This is the final shaking off. We have become the Lulav and Etrog.

Below are the sources mentioned in this section of the essay, in particular the ten crowns taken on the day of the Mishkan's construction. Which is perhaps where the connection between Building/Binyan and Creation comes from.

אֲמַר לֵיהּ רַב חֲבִיבִי מָחוֹזְנָאָה לְרַב אָשֵׁי, תָּא שְׁמַע: ״וַיְהִי בַּחֹדֶשׁ הָרִאשׁוֹן בַּשָּׁנָה הַשֵּׁנִית בְּאֶחָד לַחֹדֶשׁ הוּקַם הַמִּשְׁכָּן״ — תָּנָא, אוֹתוֹ יוֹם נָטַל עֶשֶׂר עֲטָרוֹת: רִאשׁוֹן לְמַעֲשֵׂה בְרֵאשִׁית; רִאשׁוֹן לַנְּשִׂיאִים; רִאשׁוֹן לַכְּהוּנָּה; רִאשׁוֹן לָעֲבוֹדָה; רִאשׁוֹן לִירִידַת הָאֵשׁ; רִאשׁוֹן לַאֲכִילַת קָדָשִׁים; רִאשׁוֹן לִשְׁכּוֹן שְׁכִינָה; רִאשׁוֹן לְבָרֵךְ אֶת יִשְׂרָאֵל; רִאשׁוֹן לְאִיסּוּר הַבָּמוֹת; רִאשׁוֹן לֶחֳדָשִׁים. וּמִדְּרֵישׁ יַרְחָא דְנִיסָן דְּהָא שַׁתָּא חַד בְּשַׁבָּא — דְּאֶשְׁתָּקַד בְּאַרְבְּעָה.

Rav Ḥavivi from Ḥozena’a said to Rav Ashi: Come and hear a different proof from the following verse: “And it came to pass in the first month in the second year, on the first day of the month, that the tabernacle was erected” (Exodus 40:17).

It was taught: That day took ten crowns.

  1. It was the first day of Creation, meaning Sunday,
  2. the first day of the offerings brought by the princes,
  3. the first day of the priesthood,
  4. the first day of service in the Temple,
  5. the first time for the descent of fire onto the altar,
  6. the first time that consecrated foods were eaten,
  7. the first day of the resting of the Divine Presence upon the Jewish people,
  8. the first day that the Jewish people were blessed by the priests,
  9. and the first day of the prohibition to bring offerings on improvised altars. Once the Tabernacle was erected, it was prohibited to offer sacrifices elsewhere.
  10. And it was the first of the months. And from the fact that the New Moon of Nisan of that year was on the first day of the week, in the previous year, it was on the fourth day of the week.
הֵן־בְּעָו֥וֹן חוֹלָ֑לְתִּי וּ֝בְחֵ֗טְא יֶֽחֱמַ֥תְנִי אִמִּֽי׃

Indeed I was born with iniquity;
with sin my mother conceived me.

(א) וַיְהִי֙ בַּיּ֣וֹם הַשְּׁמִינִ֔י קָרָ֣א מֹשֶׁ֔ה לְאַהֲרֹ֖ן וּלְבָנָ֑יו וּלְזִקְנֵ֖י יִשְׂרָאֵֽל׃

(1) On the eighth day Moses called Aaron and his sons, and the elders of Israel.

(לד) וַיְכַ֥ס הֶעָנָ֖ן אֶת־אֹ֣הֶל מוֹעֵ֑ד וּכְב֣וֹד ה' מָלֵ֖א אֶת־הַמִּשְׁכָּֽן׃

(34) the cloud covered the Tent of Meeting, and the Presence of ה' filled the Mishkan.

(ג) וּבַיּ֖וֹם הַשְּׁמִינִ֑י יִמּ֖וֹל בְּשַׂ֥ר עׇרְלָתֽוֹ׃

(3) On the eighth day the flesh of his foreskin shall be circumcised.—
And only then it is appropriate to enter into the holy precincts, and to continue for himself as a holy soul by means of Brit Milah.
And a hint of this also - it is impossible to repair this midah/dimension with anything other than the aspect of Shmini/Eighth, which shakes off all connections to This World, as the saying of my holy teacher Rabbi Malkovitch: [The Sefirah of] Yesod is the completion of the body; it is impossible to attain the dimension of Yesod by any means other than making an end and completion of the physical/bodily matters.

Definitely making a comparison here between Brit Milah (at eight days) and Shmini Atzeret (Eighth Day) where both are about a completion, one of the body through ritual circumcision, and one of the year through the observance of Shmini Atzeret. This might connect to void and emptiness, referenced later, where the completion of the body is by removing something, essentially creating a void where there was something. Both Brit Milah and Shmini Atzeret link to the Sefirah of Yesod, which is also about the Divine Phallus.

God transcends Selfhood to create the world, and we transcend world by embracing the aspect of Eighth and all things connected to that aspect as well.

This is also a matter of "On the eighth day you will have an Atzeret."
For if the festivals of Pesach and Sukkot are seven days long, which are still within The World, then Shmini Atzeret is in the aspect of Above and Beyond the World. And on it, a Jew needs to shake off from themselves the matters of This World, and to nullify themselves completely to God.
And thus we find in the words of the Holy Ari, who asks why both the reason for Shimini Atzeret and the matter of its joy are not mentioned at all in the Torah. He explains that all of the Yamim Tovim/Festival Days have a connection to This World, but Shmini Atzeret has no connection to This World, because it hints at the Ge'ulah Atidah / Future Redemption, when there will be a complete Tikkun/Repair of the Approaching Future, and as we say in the holy tongue that "God will rejoice over you as a groom rejoices over his bride," which is an exceedingly high level of joy, and this joy belongs to the future, to the world to come after the Tikkun/Repair.
And this is like my own teaching that Shmini/Eighth is the aspect of Above and Beyond the World. And Shmini Atzeret is the finale that all of the Yamim Tovim lead towards, as we have in the kavvanot/meditations of the Arizal: the entire order of aspects and the tikkunim, which begin with the Rosh Hashanah of "sawing/cutting," and afterwards all of the Tikkunim/Repairs continue through all of the holy days, and they lead one towards the aspect of the Yichud Elyon / Supernal Being Alone Together that is on Shmini Atzeret.
And this is also a path of serving the Divine, Because the Yamim Tovim/Festival Days all lead towards Shmini Atzetet, when a Jew shakes off this world from themselves, and nullify themselves to God, and this is a matter of "On the eighth day you will have an Atzeret."

This is our first hint that Shmini Atzeret's holiness comes from the fact that is is a void, a gap, an empty space or a deep exhale, which has no content other than just being a day. It's beyond words, beyond prayer. It's the joyful flip side of the final shofar blast from Yom Kippur, which is beyond all words and prayers. It's the prayer transformed into dance and song that creates d'veikut. Its void hints at the limitless potential of the future. The future is a void, it is not written, and if it is written, we just know how to live to make all of our time until that judgment Good.

The other Eight here is the "Groom and Bride," with the Wedding Ceremony filled with lots of Sevens, and then afterwards, the couple spend time (Yichud) being alone together, before re-entering the normal world. In Yichud, it's just the two of them. That is the level of joy we are seeking on Shmini Atzeret and Simchat Torah.

2. Shmini Atzeret as the Festival about The Seal Upon the Heart
And the three Regalim / Pilgrimage Festivals and the seasons are an aspect of Shmini/Eighth. Chag HaShavuot: after the counting of seven weeks, the Festival of Shavuot arrives, which is an aspect of Shmini [7x7+1].
And so too Yom HaKippurim is an aspect of Shmini, and so too the order of the High Priests's blood-sprinklings in front of and inside were eight - one above and seven below - the one above hints as the aspect of Shmini/Eighth (according to the Maharil), and [the High Priest] would count specifically with each sprinkle "one, one and one, one and two, etc," since the High Priest enters the Holy of Holies, which is all about the aspect of Above and Beyond the Natural, and for that reason, he needed to get to the aspect of Shmini/Eighth.
And also for this reason they keep the High Priest separate for the seven days before Yom HaKippurim, so that Yom HaKippurim also has an aspect of Shmini/Eighth.
And Shmini Atzeret, about which the Torah says: "On the eighth day, you will have an atzeret." But on Yom HaKippurim, the way to attain self-nullification in the aspect of Shmini is by means of the prayers, introspection, and doing teshuvah, and by means of these a Jew is nullified to God, but on Shmini Atzeret all of this is done through joy and dancing, in the midst of which the individual is nullified completely to God.

Ah. Yom Kippur also contains Eights, which are: It is the eighth day after the High Priest's day of separation (slowly moving him fro the physical world and preparing him for his eighth day, the day in which he goes beyond the natural into the Divine realm, a room there is also a void, empty, just him and God, in a kind of Yichud as well, but that Yichud is also prayer, introspection and teshuvah.

The other eight of Yom Kippur are the blood sprinklings, which are all about atonement from sins done in the world.The eighth sprinkle hints at the possibility of moving from the altar (earth) to above the altar (divine realm).

Shmini Atzeret is like Yom Kippur is the sense of losing the self, self-nullification, reducing the ego, but not through teshuvah, but there through joy and dance.

Furthermore, about these three Regalim/Pilgrimage Festivals - According to the explanation of Beit Avraham (Inyanei Hoshanah Rabbah), there are festivals that are an aspect of "Chotam al z'roa'cha / A Seal Upon your Arm" and there are some of them that are an aspect of "Chotam al libecha / A Seal Upon your Heart," meaning that there are three Festivals and Seasons that have commandments that have external, observable actions:
  • Pesach, where one eats matzah,
  • Rosh Hashanah has the blowing the shofar,
  • and Sukkot has the sukkah and the four species,
which are an aspect of "Upon your Arm."
And these three festivals mentioned above (Shavuot, Yom HaKippurim and Shmini Atzeret, which are also conclusion to periods of Sheva/Seven), which are an aspect of Shmini/Eighth, have no commandments that are external or observable actions, only in the aspect of "A Seal upon your Heart," and in the aspect of "For Love is as Strong as Death."
And thus on the day of Shmini Ateret, we conclude the Torah, and begin it [again], because the last letter and the first letter of the Torah are Lamed''Bet (In Hebrew lev, which means 'heart'), which hints therefore that the eighth day is like a Seal Upon the Heart.
And as it was brought out in the Be'er Avraham, who compares this to a groom and bride who because of their love and because they want to be close to each other, send each other gifts, and adorn themselves beautifully, etc. And afterwards, they are closer to each other in their essence and the inner aspects of their hearts.
And in our situation, on the seven days of the Festival [of Sukkot], the coming close together is by means of the external action-based commandments [i.e. sukkah and the four species], but Shmini Atzeret is an aspect of Yichud / Being Alone Together beyond all midot/dimensions and qualities, and this is also an aspect of Like a Seal upon your Heart. And the meaning of the idea of the seal is like the saying of the holy Zohar (II 114a):

The Jewish year has a tick-tock pattern according to this understanding:

Pesach (seal on the arm) - eat matzah -> Shavuot (seal on the heart) - receive Torah

Rosh Hashanah (seal on the arm) - blow shear -> Yom Kippur (seal on the heart) - do Teshuvah

Sukkot (seal on the arm) - sit in a sukkah, wave the lulav -> Shmini Atzret (seal on the heart) - do Yichud

And everything from the year leaves an impression on the Lev, which sustains one for the entire year.

פָּתַח רַבִּי חִיָּיא וְאָמַר, (שיר השירים ח׳:ו׳) שִׂימֵנִי כַחוֹתָם עַל לִבֶּךָ כַּחוֹתָם עַל זְרוֹעֶךָ וְגוֹ', שִׂימֵנִי כַחוֹתָם, בְּשַׁעֲתָא דְּאִתְדַּבְּקָא כְּנֶסֶת יִשְׂרָאֵל בְּבַעְלָהּ, אִיהִי אַמְרַת שִׂימֵנִי כַחוֹתָם, (ויחי רמ"ד ע"ב) אָרְחָא דְּחוֹתָם, כֵּיוָן דְּאִתְדַּבָּק בְּהַהוּא אֲתָר דְּאִתְדַּבָּק שָׁבִיק בֵּיהּ כָּל דִּיּוּקְנֵיהּ, אַף עַל גַּב דְּהַהוּא חוֹתָם אָזִיל הָכָא וְהָכָא, וְלָא קַיְּימָא תַּמָּן, וְהָא אִתְעֲבָר מִנֵּיהּ, כָּל דִּיּוּקְנֵיהּ שָׁבִיק תַּמָּן, וְתַמָּן קַיְּימָא. אוּף הָכִי אַמְרַת כְּנֶסֶת יִשְׂרָאֵל, כֵּיוָן דְּאִתְדַּבָּקְנָא בָּךְ, כָּל דִּיּוּקְנִי לִיהֲוֵי חָקִיק בָּךְ, דְּאַף עַל גַּב דְּאֵיזִיל הָכָא אוֹ הָכָא, תִּשְׁכַּח דִּיּוּקְנִי חָקִיק בָּךְ, וְתִדְכַּר לִי.

Rabbi Chiya commenced the discussion, "Set me as a seal upon your heart, as a seal upon your arm..." (Shir Hashirim 8:6). "Set me as a seal;" when Kenesset Yisrael clung to her husband, she said, "Set me as a seal." The way of a seal is as follows: Once it comes in contact with something, it leaves its mark even though the seal itself moves about and does not remain there. It left its shape there, and there remains [its shape]. The Kenesset Yisrael says the same: Once I have clung to You, my mark will be engraved with You even though I move from place to place, my mark will remain with You and you will thus remember me.

3. Shmini Atzeret is at the Zenith of Love Born out of Emunah/Trust
The aspect of Shmini that is present on Yom HaKippurim indeed comes after a Seven, where Yom HaKippurim is the eighth after the seven days that the High Priest was separated, and they are before the aspect of Sheva / Seven, as we find in the Order of Sprinklings, which is an aspect of Shmini, which is one [sprinkle] above before the seven sprinkles below, and Shmini Atzeret is seven days after the Festival [of Sukkot].
And a similar explanation to this is found in the Be'er Avraham (Parshat Va'era): In Egypt, they were in the abominable midot/dimensions, and their midot/dimensions were exceedingly clumsy/awkward(?), to the point where they were barely capable, if at all, to attain redemption by means of purifying their midot/dimensions, because they were not able to even begin repairing their midot/dimensions, and only with the merit of Emunah/Trust were they redeemed, as our Sages taught.
The power of Emunah/Trust was the singular remedy that God let remain with them, because as they say about Emunah/Trust: And your trust is all around you," below all lower levels and above all higher levels, whether you are up in the heights of the heavens or whether when I advise you there you are in She'ol, and even in the worst of Egypt and down in the 49th level of tumah, it was possible to leave and to be redeemed with the power of Emunah/trust. And thus when a Jew attains the aspect of Eighth/Shmini on Yom HaKippurim, the appropriate level self-nullification, even though one is filthy with sins and the abominations mentioned above, and one cannot begin to repair their midot/dimensions, one needs to begin with Emunah/Trust, which is below all levels, and it is an aspect of One Before the Seven, where one hints at Emunah/Trust, as our Sages teach (Makkot 24a):

בָּא חֲבַקּוּק וְהֶעֱמִידָן עַל אַחַת, שֶׁנֶּאֱמַר ״וְצַדִּיק בֶּאֱמוּנָתוֹ יִחְיֶה״.

Rather, say: Habakkuk came and established the 613 mitzvot upon one, as it is stated: “But the righteous person shall live by his faith” (Habakkuk 2:4).

And as the righteous say in their explanation of Mishlei 24:
כִּ֤י שֶׁ֨בַע ׀ יִפּ֣וֹל צַדִּ֣יק וָקָ֑ם וּ֝רְשָׁעִ֗ים יִכָּשְׁל֥וּ בְרָעָֽה׃

Seven times the righteous man falls and gets up, While the wicked are tripped by one misfortune.

For the righteous one, even when they fall in all seven [core] midot/dimensions, they can again be lifted up, and arise with the power of Emunah/Trust, but the wicked one, as it is written (above)...their fall is all at once, meaning this is a matter of Emunah/Trust, and then it cannot be revived.
Therefore with Yom HaKippurim there is the aspect of Echad/One, which is Emunah/Trust before repairing the midot/dimensions, and they are an aspect of Above All Levels, which is Shmini/Eighth, according to the seven days of separation.
But on Shmini Atzeret, there is the aspect of Shmini/Eighth which is after Sheva/Seven, because Shmini Atzeret is already after the purification of the midot/dimensions, which continues into the Festival of Sukkot, and as we see in the Holy Zohar (III 259a), about the commandment of Nisuch HaMayim/The Water Libation, as it is written: "Many waters cannot extinguish love" - water is another name for the midah/dimension of Ahavah/Love, and the matter of Nisuch HaMayim/The Water Libation on the Festival of Sukkot is the purification of the midot/dimensions, and purification of the midah/dimension of Ahavah/Love for the love of God.
Moreover, for the day of Shmini Atzeret, which is an aspect of Love is Stronger than Death, after they purify the entire midah/dimension of Ahavah/love, and attain an appropriate level of self-nullification in the aspect of Shmini/Eighth. And this is hinted at in the verse: "On the eight day, you will have an atzeret," as in "yours," meaning when Shmini Atzeret arrives, it comes from the power of your avodah/service, with purification of the midah/dimension of Ahavah/Love on the Festival of Sukkot, and the eighth day is built from the seven days of the Festival [of Sukkot.}

What is the role of Emunah on each of the Festivals?

There are seven divine emanations (Sefirot) that correspond to seven core midot/dimensions, and Emunah/Trust is at the base of them and above them. It is Emunah in God that makes engaging in them possible, no matter what. But Emunah is necessary to start. And is the starting point.

On Yom Kippur, Emunah is the trust in God's love and forgiveness that no matter where we are in life, even at the lowest levels, God can help lift us up. Trust in God is foundational. Without it, living is not actually possible. Existing is. But not living.

Shmini Atzeret, on the other hand, is after the rebuilding of self has begun and has continued through Sukkot (to be explored another time), this self-nullification is focused on purifying one's love of God through Sukkot and taking it to the peak here on the eighth day. Kind of goes back to the whole Yichud thing.

The main point is that the festival is called Atzeret. One can say that it is named that way because all of the unique qualities of the festivals stop (otzer) and gather (kones) on it. Meaning that it is an aspect of the Ahavah Il'it/Supernal Love, which is a day of deveikut / clinging and supernal Yichud Ila'i / Being Alone Together -
and the awakening of the dimension of Ahavah/Love is the aspect of the Festival of Pesach, whose dimension is Chesed/Lovingkindness and Ahavah/Love.
So too there is also on it an aspect of the Festival of Sukkot, which belongs to the days of Din/Judgment, and sweet Gevurot/Disciplines, which is the Day of the Seal, and belongs to the dimension of Yirah/Awe.
And furthermore, there is on it the matter of Siyum HaTorah/Completion of the Torah, and Simchat Torah/Rejoicing with the Torah in the aspect of the Festival of Shavuot, the dimension of Tiferet/Beauty, which is Torah, for it is impossible for there to be complete deveikut in the aspect of the Upper Yichud without the Torah,
and as it brought in the holy book that the letter Vav of the Divine Name Havayah (Blessed be God), hints at the Torah, which connects the Upper Hey (in Yud-Hey) and the Lower Hey (Vav-Hey) of the Divine Name Havayah (Blessed be God), which is the Yichud of the Holy Bountiful One and Kenesset Yisrael.
And in this aspect of Shmini Atzeret, which stops and gathers all of the special qualities of the festivals on it, it is like the explanation brought in the holy book: A parable about a king who invites four artists into his palace, each of whom will illustrate one wall with exquisite artwork. Three of them decorate three of the walls, but the fourth one comes and puts a mirror all over the fourth wall, reflecting the exquisite artwork that is on the three other walls. So too it is with this holy festival - there is within it the special qualities of the three Regalim/Pilgrimage festivals, and therefore Shmini Atzeret is not considered a fourth Regel/Festival, even though it is a Regel on its own, since it includes the three Regalim/Festivals.

Pesach = Ahavah/Chesed

Sukkot = Din/Gevurah/Yirah (actually the end of the Yamim Noraim)

Shmini Atzeret = Tiferet (the synthesis of the two)

- Shavuot also present in the Torah aspect

Yud-Hey Vav-Hey

God(+Torah+) Israel

According to this explanation, one can also say more about the fact that there is no reason for this Yom Tov Shmini Atzeret written in the Torah, and we bring the words of the holy Ari that - Shmini Atzeret hints at He'atid Lavo/The Future to Come, and it is actually impossible for its reason to be written in the Torah, which is similar to that which is in the name of Noam Elimelech - that the holiness of the empty (or void) g'vil/parchment between the letters in a Sefer Torah is holier than the holiness of the letters, because in every letter there is only the holiness of that letter, but on the other hand, the void (or emptiness) is the Holy of Holies, which includes all of the letters. Similarly on can say that Shmini Atzeret is an aspect of the void/emptiness that is between the letters, and thus its reason is not written in the Torah, and behold it is higher than all of the Moadim/Festivals, whose reason is written in the Torah.

I love this idea of the void as the emptiness that contains all. The white space between the black letters. The Holy of Holies is void, which is No Thing, which is the only "thing" that can even begin to considering containing God. Shmini Atzeret is a Festival that is about Void, the emptiness, the loss of self, the impossibility of knowing the future, the possibility of the future, the mirror that reflects back the entire beauty of the year, that gives us the sustenance to get through the winter, the time of turning inward and preparing for the next year.

Shmini Atzret is the spiritual harvest of the year. What will we take with us into the dark and rainy season?

And in the Be'er Avraham, there is with regard to the matter of Shmini Atzeret that it is similar to the difference between Hiluk / description and Etzem / essence. Shmini Atzeret is in the aspect of the Etzem / Essence of all Yamim Tovim/Festive Days, and from it a Jew draws spirituality for the entire year. And he explains in this interpretation the verse (Psalms 36):
(ח) מַה־יָּקָ֥ר חַסְדְּךָ֗ אֱלֹ֫קִ֥ים וּבְנֵ֥י אָדָ֑ם בְּצֵ֥ל כְּ֝נָפֶ֗יךָ יֶחֱסָיֽוּן׃ (ט) יִ֭רְוְיֻן מִדֶּ֣שֶׁן בֵּיתֶ֑ךָ וְנַ֖חַל עֲדָנֶ֣יךָ תַשְׁקֵֽם׃

(8) How precious is Your faithful care, O God! Humanity shelters in the shadow of Your wings. (9) They feast on the rich fare of Your house; You let them drink at Your refreshing stream.

"How precious is your faithful care, God" is about the Festival of Sukkot, which are days of Chesed/Lovingkindness, and on it humanity is sheltering in the shadow of Your wings, where they sit in a sukkah, in the shade of their trust in you; "They feast on the rich fare of your house," this is about Shmini Atzeret, which is the Etzem / essence of all Yamim Tovim/Festive Days, and from it they feast on the rich fare of Youse house. DeSheN (as series of letters) is an [acrostic] of [the number] ShN''D /354, hinting at the 354 days that are in the [lunar] year, meaning that from this holy Festival, they feast [spiritually] every day of the year.
4. Shmini Atzeret is a day of understanding there is nothing but God
And one can explain what they say on Shmini Atzeret:
(לה) אַתָּה֙ הׇרְאֵ֣תָ לָדַ֔עַת כִּ֥י ה' ה֣וּא הָאֱלֹקִ֑ים אֵ֥ין ע֖וֹד מִלְּבַדּֽוֹ׃

You have been shown to know that ה' alone is God; there is none else.

Similarly to what it later on:
(ג) וְלֹא־נָתַן֩ ה' לָכֶ֥ם לֵב֙ לָדַ֔עַת וְעֵינַ֥יִם לִרְא֖וֹת וְאׇזְנַ֣יִם לִשְׁמֹ֑עַ עַ֖ד הַיּ֥וֹם הַזֶּֽה׃

(3) Yet to (or until?) this day ה' has not given you a mind to understand or eyes to see or ears to hear.

Where Rashi interprets this verse thus: "The day when Moses gave the Sefer Torah to B'nai Levi...and all Israel came before Moses and said to him: We also stood at Mount Sinai, and received the Torah..and Moses rejoiced because of this, and said to them: This day, I understand that you cling to and desire HaMakom/God."
And then we have the explanation in Torat Avot in the name of the Holy Rabbi Malkovitch, "What is this explanation of Devarim 29:3? Did Moses not see this in them much earlier in Egypt? Or at the moment when they jumped into the Sea of Reeds?... Thus at that moment, he saw in them an extra measure of chavivut/love for the holy Torah, a love that surpassed everything, because if a human does not have 'love,' then they do not have a lev/heart-mind to understand and eyes to see...
And on this holy Festival the people Israel show their extra love for the holy Torah when they complete the Torah. And the essence of the day is that it is a day of cleaving to God. And by means of this love, they merit an understanding lev/heart-mind and eyes to see that there is nothing but God.

It is the love of Torah that gives us the eyes to see and heart to understand that there is nothing but God.

Here we have a clear expression of non-dualism. It's not that we down here love God up there. It's that we see that the separations between all things (Olam Ha-Peirud) may be real in some ways, there is a higher reality where there is no separation (Olam Ha-Achdut, the World of Unity), where everything is nothing but an expression of God, Divnity, Holiness. And one experiences Bitul not because they are nothing, but because they are that they and everyone and everything is a revelation of God as well, an emanation of the Divine. God's energy in physical clothing.

The Sages said that Moses our Teacher, before he died, said before the Holy Bountiful One: "I ask one thing from you, that you open the seven heavens, so that everyone can see that there is nothing but You." And if our holy Sages say that Moses' prayer, where he prayed 549 prayers that he would enter the land, would be fulfilled in an inward fashion, when God showed all of the land, the Gilead until Dan, and all of Naftali...and only in action did he not enter, and indeed also with this request that all seven levels of heaven be opened...it would have been received and fulfilled for all who direct their lev/heart-mind, and with Yichud/Being Along Together on the holy days which contain on them Showing the [Divine] Face, merits "You have been shown to know that ה' alone is God; there is none else." And this is the essence of this holy festival.
And by means of the d'veikut/cleaving and the love for the holy Torah, they merit this, and this is in the aspect of G'mar Chatima / A Complete Seal on this day, for there is from the Holy of Holies, the Baal Shem Tov, an explanation of "the Eternal is your shade" The shade that God makes is that which the human makes, thus, as if, above it is made just as what the human does,
and there are three aspects with awakening the human on these holy days:
on Rosh Hashanah in the aspect of Wake up sleepers from your sleep, and slumbers arise from your slumbers, and this is as aspect of Ketivah/Writing;
afterwards the Ten Days of Awe and Yom HaKippurim - "For one this day, there is be atonement for you," and this is a more inward acceptance and resolution, and this is the aspect of Chatimah/Sealing.
And afterwards on Shmini Atzeret, which is clinging to God through extra cherishing and loving, in the aspect of Nagila v'nismcha b'cha / We will sing and rejoice with You, then a Jew is capable of attaining the aspect of "You have been shown to know that ה' alone is God...." This is already in the aspect of Like a Seal Upon your Heart," and thus as we said above this is the aspect of G'mar Chatima/Completely Sealed.
Furthermore, one can say about this, that after a Jew merits on Shmini Atzeret Nagila v'nismcha b'cha / We will sing and rejoice with You, from the inner clinging to God, then they sweeten all of the judgments, and merit a Chatimah Tovah / A gooG Seal [for the year to come].