(א) וַיְדַבֵּ֤ר יְהֹוָה֙ אֶל־מֹשֶׁ֔ה אַחֲרֵ֣י מ֔וֹת שְׁנֵ֖י בְּנֵ֣י אַהֲרֹ֑ן בְּקׇרְבָתָ֥ם לִפְנֵי־יְהֹוָ֖ה וַיָּמֻֽתוּ׃ (ב) וַיֹּ֨אמֶר יְהֹוָ֜ה אֶל־מֹשֶׁ֗ה דַּבֵּר֮ אֶל־אַהֲרֹ֣ן אָחִ֒יךָ֒ וְאַל־יָבֹ֤א בְכׇל־עֵת֙ אֶל־הַקֹּ֔דֶשׁ מִבֵּ֖ית לַפָּרֹ֑כֶת אֶל־פְּנֵ֨י הַכַּפֹּ֜רֶת אֲשֶׁ֤ר עַל־הָאָרֹן֙ וְלֹ֣א יָמ֔וּת כִּ֚י בֶּֽעָנָ֔ן אֵרָאֶ֖ה עַל־הַכַּפֹּֽרֶת׃
(1) The LORD spoke to Moses after the death of the two sons of Aaron who died when they drew too close to the presence of the LORD. (2) The LORD said to Moses: Tell your brother Aaron that he is not to come at will into the Shrine behind the curtain, in front of the cover that is upon the ark, lest he die; for I appear in the cloud over the cover.
Questions to consider:
1 - What connection does the Yom Kippur Service have to the untimely deaths of Nadav and Avihu?
2 - What is the underlying purpose of the blood ceremony in the Holy of Holies?
3 - What connection do the People of Israel have with the Holy Temple and Holy of Holies in particular? (parshat chukat)
THE CULMINATION OF THE EIGHT DAYS OF CHANUKKAT HAMISHKAN -- POENA MORTIS??
(א) וַיִּקְח֣וּ בְנֵֽי־אַ֠הֲרֹ֠ן נָדָ֨ב וַאֲבִיה֜וּא אִ֣ישׁ מַחְתָּת֗וֹ וַיִּתְּנ֤וּ בָהֵן֙ אֵ֔שׁ וַיָּשִׂ֥ימוּ עָלֶ֖יהָ קְטֹ֑רֶת וַיַּקְרִ֜יבוּ לִפְנֵ֤י יְהֹוָה֙ אֵ֣שׁ זָרָ֔ה אֲשֶׁ֧ר לֹ֦א צִוָּ֖ה אֹתָֽם׃ (ב) וַתֵּ֥צֵא אֵ֛שׁ מִלִּפְנֵ֥י יְהֹוָ֖ה וַתֹּ֣אכַל אוֹתָ֑ם וַיָּמֻ֖תוּ לִפְנֵ֥י יְהֹוָֽה׃ (ג) וַיֹּ֨אמֶר מֹשֶׁ֜ה אֶֽל־אַהֲרֹ֗ן הוּא֩ אֲשֶׁר־דִּבֶּ֨ר יְהֹוָ֤ה ׀ לֵאמֹר֙ בִּקְרֹבַ֣י אֶקָּדֵ֔שׁ וְעַל־פְּנֵ֥י כׇל־הָעָ֖ם אֶכָּבֵ֑ד וַיִּדֹּ֖ם אַהֲרֹֽן׃
(1) Now Aaron’s sons Nadab and Abihu each took his fire pan, put fire in it, and laid incense on it; and they offered before the LORD alien fire, which He had not enjoined upon them. (2) And fire came forth from the LORD and consumed them; thus they died at the instance of the LORD. (3) Then Moses said to Aaron, “This is what the LORD meant when He said: Through those near to Me I show Myself holy, And gain glory before all the people.” And Aaron was silent.
What are the series of parshiyos that extend from the deaths of Nadav and Avihu up until Achrei Mot?
THE LAWS OF KASHRUT
(מו) זֹ֣את תּוֹרַ֤ת הַבְּהֵמָה֙ וְהָע֔וֹף וְכֹל֙ נֶ֣פֶשׁ הַֽחַיָּ֔ה הָרֹמֶ֖שֶׂת בַּמָּ֑יִם וּלְכׇל־נֶ֖פֶשׁ הַשֹּׁרֶ֥צֶת עַל־הָאָֽרֶץ׃ (מז) לְהַבְדִּ֕יל בֵּ֥ין הַטָּמֵ֖א וּבֵ֣ין הַטָּהֹ֑ר וּבֵ֤ין הַֽחַיָּה֙ הַֽנֶּאֱכֶ֔לֶת וּבֵין֙ הַֽחַיָּ֔ה אֲשֶׁ֖ר לֹ֥א תֵאָכֵֽל׃ {פ}
(46) These are the instructions concerning animals, birds, all living creatures that move in water, and all creatures that swarm on earth, (47) for distinguishing between the unclean and the clean, between the living things that may be eaten and the living things that may not be eaten.
THE LAWS OF TUMAH V'TAHARA
(נד) זֹ֖את הַתּוֹרָ֑ה לְכׇל־נֶ֥גַע הַצָּרַ֖עַת וְלַנָּֽתֶק׃ (נה) וּלְצָרַ֥עַת הַבֶּ֖גֶד וְלַבָּֽיִת׃ (נו) וְלַשְׂאֵ֥ת וְלַסַּפַּ֖חַת וְלַבֶּהָֽרֶת׃ (נז) לְהוֹרֹ֕ת בְּי֥וֹם הַטָּמֵ֖א וּבְי֣וֹם הַטָּהֹ֑ר זֹ֥את תּוֹרַ֖ת הַצָּרָֽעַת׃ {פ}
(54) Such is the ritual for every eruptive affection—for scalls, (55) for an eruption on a cloth or a house, (56) for swellings, for rashes, or for discolorations— (57) to determine when they are unclean and when they are clean. Such is the ritual concerning eruptions.
CIRCLING BACK TO THE AFTERMATH OF NADAV AND AVIHU
(א) וַיְדַבֵּ֤ר יְהֹוָה֙ אֶל־מֹשֶׁ֔ה אַחֲרֵ֣י מ֔וֹת שְׁנֵ֖י בְּנֵ֣י אַהֲרֹ֑ן בְּקׇרְבָתָ֥ם לִפְנֵי־יְהֹוָ֖ה וַיָּמֻֽתוּ׃
(1) The LORD spoke to Moses after the death of the two sons of Aaron who died when they drew too close to the presence of the LORD.
(ל) כִּֽי־בַיּ֥וֹם הַזֶּ֛ה יְכַפֵּ֥ר עֲלֵיכֶ֖ם לְטַהֵ֣ר אֶתְכֶ֑ם מִכֹּל֙ חַטֹּ֣אתֵיכֶ֔ם לִפְנֵ֥י יְהֹוָ֖ה תִּטְהָֽרוּ׃
(30) For on this day atonement shall be made for you to cleanse you of all your sins; you shall be clean before the LORD.
Rabbi Yair Kahn (Ram at Yeshivat Gush Etzion), Parashat Acharei-Mot: The Death of Aaron's Sons, and the Priestly Service on Yom Kippur, 2017
In this context, the parashiot relating to the various types of ritual impurity come to emphasize the other aspect of religious experience, and to teach us about the infinite abyss that separates between human reality and Divine reality. Impurity is an inseparable part of ultimate human reality. It accompanies his birth, as well as his death; it is bound up with his eating and his marital relations. Mortal man, mired in impurity, cannot come to God without the laws of purification that God Himself commands. Without fulfillment of the purifying Divine command, mortals – dwelling in their material homes – cannot approach the King. "You shall separate Bnei Yisrael from their impurity, that they shall not die in their impurity, when they defile My dwelling that is among them" (Vayikra 15:31).
etzion.org.il/en/tanakh/torah/sefer-vayikra/parashat-acharei-mot/parashat-acharei-mot-death-aarons-sons-and-priestly
Now that we understand that טהרה is the core principle underlying Yom Kippur, moving to the day itself, what was the key service in the Kodesh HaKodashim on Yom Kippur?
(יד) וְלָקַח֙ מִדַּ֣ם הַפָּ֔ר וְהִזָּ֧ה בְאֶצְבָּע֛וֹ עַל־פְּנֵ֥י הַכַּפֹּ֖רֶת קֵ֑דְמָה וְלִפְנֵ֣י הַכַּפֹּ֗רֶת יַזֶּ֧ה שֶֽׁבַע־פְּעָמִ֛ים מִן־הַדָּ֖ם בְּאֶצְבָּעֽוֹ׃ (טו) וְשָׁחַ֞ט אֶת־שְׂעִ֤יר הַֽחַטָּאת֙ אֲשֶׁ֣ר לָעָ֔ם וְהֵבִיא֙ אֶת־דָּמ֔וֹ אֶל־מִבֵּ֖ית לַפָּרֹ֑כֶת וְעָשָׂ֣ה אֶת־דָּמ֗וֹ כַּאֲשֶׁ֤ר עָשָׂה֙ לְדַ֣ם הַפָּ֔ר וְהִזָּ֥ה אֹת֛וֹ עַל־הַכַּפֹּ֖רֶת וְלִפְנֵ֥י הַכַּפֹּֽרֶת׃ (טז) וְכִפֶּ֣ר עַל־הַקֹּ֗דֶשׁ מִטֻּמְאֹת֙ בְּנֵ֣י יִשְׂרָאֵ֔ל וּמִפִּשְׁעֵיהֶ֖ם לְכׇל־חַטֹּאתָ֑ם וְכֵ֤ן יַעֲשֶׂה֙ לְאֹ֣הֶל מוֹעֵ֔ד הַשֹּׁכֵ֣ן אִתָּ֔ם בְּת֖וֹךְ טֻמְאֹתָֽם׃
(14) He shall take some of the blood of the bull and sprinkle it with his finger over the cover on the east side; and in front of the cover he shall sprinkle some of the blood with his finger seven times. (15) He shall then slaughter the people’s goat of sin offering, bring its blood behind the curtain, and do with its blood as he has done with the blood of the bull: he shall sprinkle it over the cover and in front of the cover. (16) Thus he shall purge the Shrine of the uncleanness and transgression of the Israelites, whatever their sins; and he shall do the same for the Tent of Meeting, which abides with them in the midst of their uncleanness.
FUNCTION OF THE SPRINKLINGS
(טז) וְכִפֶּ֣ר עַל־הַקֹּ֗דֶשׁ מִטֻּמְאֹת֙ בְּנֵ֣י יִשְׂרָאֵ֔ל וּמִפִּשְׁעֵיהֶ֖ם לְכׇל־חַטֹּאתָ֑ם וְכֵ֤ן יַעֲשֶׂה֙ לְאֹ֣הֶל מוֹעֵ֔ד הַשֹּׁכֵ֣ן אִתָּ֔ם בְּת֖וֹךְ טֻמְאֹתָֽם׃
(16) Thus he shall purge the Shrine of the uncleanness and transgression of the Israelites, whatever their sins; and he shall do the same for the Tent of Meeting, which abides with them in the midst of their uncleanness.
KAPPARA ON טומאה
רַבִּי שִׁמְעוֹן אוֹמֵר: כְּשֵׁם שֶׁדַּם הַשָּׂעִיר הַנַּעֲשֶׂה בִּפְנִים מְכַפֵּר עַל יִשְׂרָאֵל בְּטוּמְאַת מִקְדָּשׁ וְקָדָשָׁיו — כָּךְ דַּם הַפָּר מְכַפֵּר עַל הַכֹּהֲנִים בְּטוּמְאַת מִקְדָּשׁ וְקָדָשָׁיו. וּכְשֵׁם שֶׁוִּידּוּי שֶׁל שָׂעִיר הַמִּשְׁתַּלֵּחַ מְכַפֵּר עַל יִשְׂרָאֵל בִּשְׁאָר עֲבֵירוֹת — כָּךְ וִידּוּי שֶׁל פָּר מְכַפֵּר עַל הַכֹּהֲנִים בִּשְׁאָר עֲבֵירוֹת.
Rabbi Shimon says: Just as the blood of the goat presented inside atones for Israelites who sinned with the ritual impurity of the Temple and its consecrated objects, so too, the blood of the High Priest’s bull atones for the priests who sinned with the ritual impurity of the Temple and its consecrated objects. And just as the confession over the scapegoat atones for Israelites with regard to all other transgressions, so too, the confession over the bull atones for the priests with regard to all other transgressions, and therefore the priests do not require atonement through the scapegoat.
What is going on here? How is this possible? What are the Biblical mechanics generating טומאה within the the Holy?
(יג) כׇּֽל־הַנֹּגֵ֡עַ בְּמֵ֣ת בְּנֶ֩פֶשׁ֩ הָאָדָ֨ם אֲשֶׁר־יָמ֜וּת וְלֹ֣א יִתְחַטָּ֗א אֶת־מִשְׁכַּ֤ן יְהֹוָה֙ טִמֵּ֔א וְנִכְרְתָ֛ה הַנֶּ֥פֶשׁ הַהִ֖וא מִיִּשְׂרָאֵ֑ל כִּי֩ מֵ֨י נִדָּ֜ה לֹא־זֹרַ֤ק עָלָיו֙ טָמֵ֣א יִהְיֶ֔ה ע֖וֹד טֻמְאָת֥וֹ בֽוֹ׃
(13) Whoever touches a corpse, the body of a person who has died, and does not cleanse himself, defiles the LORD’s Tabernacle; that person shall be cut off from Israel. Since the water of lustration was not dashed on him, he remains unclean; his uncleanness is still upon him.
Prof. Patrick Miller, The Religion of Ancient Israel, 2000, pg. 151

So we see there is an inextricable link between Am Yisrael and the status of the Holy. And because of the sins of Am Yisrael, the Holy is "מטמא" in some way -- requiring purging and cleansing -- accomplished through the ritual sacrifice and sprinkling of the blood of the chatat.
As proof of this concept of purging of impurity, the very sacrifice of the פרה אדומה (which we referred to above) incorporates a ritual very familiar to us now learning the Avodah on Yom Kippur...
(ד) וְלָקַ֞ח אֶלְעָזָ֧ר הַכֹּהֵ֛ן מִדָּמָ֖הּ בְּאֶצְבָּע֑וֹ וְהִזָּ֞ה אֶל־נֹ֨כַח פְּנֵ֧י אֹֽהֶל־מוֹעֵ֛ד מִדָּמָ֖הּ שֶׁ֥בַע פְּעָמִֽים׃
(4) Eleazar the priest shall take some of its blood with his finger and sprinkle it seven times toward the front of the Tent of Meeting.
This is touched upon by the Gemara as well...
שבעה הזאות שבפרה מעכבות זו את זו ז' הזיות שעל בין הבדים שעל הפרכת שעל מזבח הזהב מעכבות זו את זו:
With regard to the seven sprinklings of the blood of the red heifer that the priest sprinkles opposite the entrance to the Sanctuary (see Numbers 19:4), failure to sprinkle each prevents fulfillment of the mitzva with the others. With regard to the seven sprinklings of the blood of the bull and goat of Yom Kippur that are sprinkled on the Ark between the staves (see Leviticus 16:14–15), the seven sprinklings that are sprinkled on the Curtain separating the Sanctuary and Holy of Holies, and the sprinklings that are sprinkled on the golden altar on Yom Kippur, and from all other inner sin offerings, failure to sprinkle each prevents fulfillment of the mitzva with the others.
The sprinklings of the blood of the פרה אדומה must be in line-of-sight to the Holy of Holies itself.
הָכִי נָמֵי מִסְתַּבְּרָא דְּרַבִּי אֱלִיעֶזֶר בֶּן יַעֲקֹב הִיא, דִּתְנַן: כׇּל הַכְּתָלִים שֶׁהָיוּ שָׁם הָיוּ גְּבוֹהִין, חוּץ מִכּוֹתֶל מִזְרָחִי, שֶׁהַכֹּהֵן הַשּׂוֹרֵף אֶת הַפָּרָה עוֹמֵד בְּהַר הַמִּשְׁחָה וּמְכַוֵּון וְרוֹאֶה כְּנֶגֶד פִּתְחוֹ שֶׁל הֵיכָל בִּשְׁעַת הַזָּאַת הַדָּם.
So too, it is reasonable to conclude that the mishnayot in tractate Middot are in accordance with the opinion of Rabbi Eliezer ben Ya’akov, as we learned in a mishna there: All the walls that were there surrounding the Temple Mount were high except for the Eastern Wall, as the priest who burns the red heifer stands on the Mount of Olives, where the red heifer was slaughtered and burned, and directs his attention and looks toward the entrance of the Sanctuary when he sprinkles the blood.
UNDERSTANDING THE FUNCTION OF THE KORBAN CHATAT
Prof. Patrick Miller, The Religion of Ancient Israel, 2000, pg. 114

Rabbi Shlomo Zuckier, PhD also cites Prof. Milgrom and helps us understand that what is taking place with the sprinklings is a type of purging of the impurity of the People of Israel.
Rabbi Shlomo Zuckier, PhD, Doctoral Dissertation, Flesh and Blood: The Reception of Biblical Sacrifice in Selected Talmudic Sources in Comparative Context (2020), pp. 67-68
Jacob Milgrom offers a study of the term כ.פ.ר throughout its biblical appearances, distinguishing various uses. On his analysis, the primary (both in the sense of ‘original’ and ‘major’) meaning of the term is the ritual one of “purging,” itself stemming from a term for wiping and parallel to Akkadian and other Mesopotamian literature and its related term kupurru. In this system, blood is wiped (or sprinkled) on the altar to purge it of the miasma that it acquired through certain Israelite sins, thus undoing the sin through a process of purgation.
What is this symbiotic relationship between Am Yisrael and the Mikdash?
I believe that is the mystery behind the seemingly incongruent pronoun at the end of this famous verse:
(ח) וְעָ֥שׂוּ לִ֖י מִקְדָּ֑שׁ וְשָׁכַנְתִּ֖י בְּתוֹכָֽם׃
(8) And let them make Me a sanctuary that I may dwell among them.
On this pasuk, the Malbim wrote a treatise called רמזי המשכן.
וזאת היתה הכונה במלאכת המשכן והמקדש אשר נבנה אח"ז בירושלים בעיר שחוברה לה יחדו שם נתאחדו כל נפשות האומה הישראלית ויעמדו כאיש אחד פרטי הכולל בנפשו כל ההארות והרוחניות הנמצא בזהר האלהי המאיר לעולם כולו בכבודו באשר אליו יפנו וינהרו מכל מקומות פזוריהם ובכל תפלותיהם ומעשיהם בקדש דרך הנה פניהם כמ"ש (פ"ד דברכות) העומד במזרח מחזיר פניו למערב וכו' נמצא כל ישראל מכוונים לבם למק"א, ונשלם בזה הקומה השלמה הנפשיית המתיחסת לקומה הרוחניית אשר משם אצילות נשמותיהם, ועז"א ועשו מקדש ושכנתי בתוכם, ר"ל בתוך נשמותיהם שהתאחדו במקדש הזה לקומה שלמה וגויה כוללת...
כי המשכן שהוא הגויה הכללית האוספת בפנימותה כל הנפשות של העם כולו אשר יתאספו בתוך גויה זו כאלו הוא איש אחד שהמשכן הוא גוף האיש ונפשות כל ב"י המקובצות בו הוא רוחו ונשמתו...
(א) צריך לכוין נגד ארץ ישראל ודין הרוכב או יושב בספינה ובו ט סעיפים:בקומו להתפלל אם היה עומד בח"ל יחזור פניו כנגד ארץ ישראל ויכוין גם לירושלים ולמקדש ולבית קדשי הקדשים היה עומד בא"י יחזיר פניו כנגד ירושלים ויכוין גם למקדש ולבית קדשי הקדשים. היה עומד בירושלים יחזיר פניו למקדש ויכוין גם לבית קדשי הקדשים היה עומד אחורי הכפורת מחזיר פניו לכפורת:
(1) When he gets up to pray if he was standing outside the Land of Israel, he should return his face toward the Land of Israel and focus also on Jerusalem and to the Temple and to the Holy of Holies. If he was standing in Jerusalem he should return his face to the Temple and focus also to the Holy of Holies. If he was standing behind the (Ark) he should return his face to the (Ark)....