Yom Kippur purifies through judgment itself

ואיתא במד' ואנכי איש חלק חלק ה' עמו ד"א חלק שכל השנה מתלכלך בחטאים ובא יוהכ"פ ומטהר אותם ע"ש.

And it is taught in the Midrash on "And I am a smooth man (ish chalak)" (Bereishis 27:11): "the portion (chelek) of Hashem is His people" (Devarim 32:9); another interpretation of "chalak" — that all year long one becomes soiled with sins, and Yom Kippur comes and purifies them; see there.

The Midrash plays on the word "chalak" (smooth) in two ways: first as "chelek" (portion) — Bnei Yisrael are Hashem's own portion; and second, hinting that though one becomes "dirtied" with sins all year, Yom Kippur arrives and washes them clean.

והכל ענין אחד משום שחלק ה' עמו לכן אין החטא בהם בעצם רק במקרה ויכולין להתטהר מהם ובנ"י דומים ללב שאינו מקבל מום כדכ' כולך יפה רעיתי ומום אין בך בעצם.

And it is all one matter: because "the portion of Hashem is His people," therefore the sin is not within them essentially (b'etzem) but only incidentally (b'mikreh), and they can be purified from it. And Bnei Yisrael are likened to the heart, which does not take on a blemish, as it is written, "You are entirely beautiful, My beloved, and there is no blemish in you" (Shir HaShirim 4:7) — essentially.

The two interpretations are one: precisely because Bnei Yisrael are Hashem's intrinsic portion, sin can only cling to them externally, by accident, never penetrating their essence. Like the heart — which the Gemara says is the one organ that cannot become blemished for a korban — their core remains "entirely beautiful, with no blemish."

כמ"ש שחורה אני ונאוה.

As it is written, "I am black, but I am beautiful (sh'chorah ani v'navah)" (Shir HaShirim 1:5).

The outer surface may appear "black," dirtied by sin, yet underneath it is "beautiful" — the blackness is only on the surface, not in the essence.

כאהלי קדר ויכולין להתכבס להיות כיריעות שלמה.

"Like the tents of Kedar" — and they can be laundered (l'hiskabes) to become "like the curtains of Shlomo" (Shir HaShirim 1:5).

The verse compares Bnei Yisrael to the dark, weathered tents of Kedar that can be washed clean to become like the beautiful curtains of Shlomo. The "dirt" is washable precisely because it is external — and "Shlomo" hints at sheleimus (wholeness) and at the Beis Hamikdash.

וזה מתקיים ביוהכ"פ שנעשין טהורים לכן אח"כ בסוכות פורס השי"ת סוכת שלומו עלינו ומתקיים כיריעות שלמה.

And this is fulfilled on Yom Kippur, when they become pure (tehorim); therefore afterwards on Sukkos, Hashem spreads over us the sukkah of His peace (sukkas shlomo), and "like the curtains of Shlomo" is fulfilled.

The "laundering" happens on Yom Kippur, when Bnei Yisrael are made tahor. Then on Sukkos, Hashem spreads His "sukkas shalom" over us — fulfilling "like the yerios Shlomo" (curtains of Shlomo / of peace), the closeness that follows purification.

והנה יוכ"פ הוא גמר הדין והוא בחסד כמ"ש רז"ל ורב חסד ואמת תחלה ואמת ובסוף ורב חסד לכן ר"ה בחי' אמת ויוכ"פ ורב חסד.

And behold, Yom Kippur is the conclusion of the judgment (gemar ha'din), and it is with chesed, as Chazal said on "abundant in kindness and truth (rav chesed v'emes)" (Shemos 34:6): at first "truth," and at the end "abundant in kindness." Therefore Rosh Hashanah is the aspect of "emes" (truth) and Yom Kippur is "rav chesed" (abundant kindness).

Yom Kippur seals the judgment, and it does so through chesed. Chazal note the order in Hashem's attributes: "emes" comes first, then "rav chesed." So Rosh Hashanah, the opening of judgment, embodies "emes" (strict truth), while Yom Kippur, the sealing, embodies "rav chesed."

כי חסד הוא לפנים משורת הדין.

For ordinary chesed is "beyond the letter of the law" (lifnim mi'shuras ha'din).

Plain chesed means going past strict justice — setting the law aside, as it were, to act with mercy.

אבל ורב חסד הוא גודל חסדיו של הקב"ה שיכול לעשות חסד עפ"י המשפט עצמו.

But "rav chesed" (abundant kindness) is the greatness of the kindnesses of the Holy One, Blessed is He — that He can perform chesed by means of the judgment itself.

"Rav chesed" is a higher order: not chesed that overrides justice, but chesed that works through the very mechanism of strict judgment — mercy that emerges from within the law itself.

זה נקרא ורב חסד מה שאינו יכול להשיג זאת רק הקב"ה כל יכול וכמ"ש לטהר חוסיו בדין.

This is called "rav chesed" — something that only the Holy One, Blessed is He, the All-Powerful, can achieve; as it is written, "to purify those who take refuge in Him through judgment (ba'din)."

Only the all-capable Hashem can accomplish this paradox — to be merciful through justice. The piyut's phrase "to purify those who trust in Him ba'din" expresses exactly this: He cleanses His people by means of the judgment, not in spite of it.

עפ"י דין ממש מטהר את בנ"י ביוהכ"פ שנתברר בדין כי אין החטא בהם בעצם כנ"ל:

By strict judgment itself, He purifies Bnei Yisrael on Yom Kippur — for it is clarified through the judgment that the sin is not within them essentially, as above.

The very process of din on Yom Kippur reveals the truth that sin never reached the essence of Bnei Yisrael — it was only external, incidental. Thus the judgment itself becomes the instrument of their purification: "rav chesed" through "din."

Summary: Because Bnei Yisrael are Hashem's intrinsic "chelek," sin can only cling to them externally, never staining their essence — like the heart that takes no blemish, or the tents of Kedar that wash clean to become Shlomo's curtains. Yom Kippur "launders" them, leading to the sukkas shalom of Sukkos. And while Rosh Hashanah is "emes" (strict truth), Yom Kippur is "rav chesed" — the unique Divine kindness that purifies Bnei Yisrael through the judgment itself, which reveals that their sin was never essential.