The above can be understood on the basis of the explanations in Torah Or1וְיוּבַן זֶה עַל פִּי הַמְּב ֹאָר בְּתוֹרָה אוֹר
that Purim and Yom Kippur share a connection with each other.2דְּפוּרִים וְיוֹם הַכִּפֻּרִים שַׁיָּכִים זֶה לָזֶה.
The Hebrew term for Yom Kippur, Yom Kippurim, means “a day like Purim.”דְּפֵרוּשׁ יוֹם כִּפֻּרִים הוּא כְּמוֹ פּוּרִים
[Moreover, the description of Yom Kippur as a day “like Purim”[וִיתֵירָה מִזּוֹ, דְּמִזֶּה שֶׁיּוֹם הַכִּפֻּרִים הוּא כְּפוּרִים
(only “like Purim”)(בְּכָ"ף הַדִּמְיוֹן בִּלְבָד),
indicatesמוּכָח,
that Purim is loftier than Yom Kippur.]3שֶׁפּוּרִים הוּא נַעֲלֶה יוֹתֵר מִיּוֹם הַכִּפֻּרִים].
A connection can be drawn to the opinion in the Midrash4 that in the Ultimate Future, all the festivals – even Yom Kippur – will be nullified with the exception of Purim.
Among the common factors shared by Yom Kippur and Purim isוּמֵהַשַּׁיָּכוּת דְּיוֹם הַכִּפֻּרִים וּפוּרִים הִיא,
that both feature a lottery.שֶׁבִּשְׁנֵיהֶם הוּא עִנְיַן הַגּוֹרָל.
On Yom Kippur, lots were cast to determine which of the two goats would be sacrificed to G‑d and which would be sent to Azazel.5דִּבְיוֹם הַכִּפֻּרִים הִפִּילוּ גּוֹרָל עַל שְׁנֵי הַשְּׂעִירִים,
And Purim is named for the pur, the lot, cast by Haman.וְגַם פּוּרִים הוּא עַל שֵׁם הַפּוּר הוּא הַגּוֹרָל.
There is another common factor shared by Purim and Yom Kippur:וְעוֹד עִנְיָן בְּהַשַּׁיָּכוּת דְּפוּרִים וְיוֹם הַכִּפֻּרִים,
Both reflect a spiritual plane loftier than G‑d’s name Havayah.6שֶׁשְּׁנֵיהֶם הֵם לְמַעְלָה מִשֵּׁם הֲוָיָ'.
With regard to Yom Kippur, it is written,7 “You shall purify yourselves before G‑d (Havayah).”דִּבְיוֹם הַכִּפֻּרִים כְּתִיב לִפְנֵי הֲוָיָ' תִּטְהָרוּ,
Implied is that purification from sin is attained by accessing a spiritual rung above Havayah.8לְמַעְלָה מֵהֲוָיָ'.
The mitzvos are expressions of G‑d’s will and enable a Jew to bond himself to G‑d and His name Havayah. When a person sins, he severs this connection. It is possible to reconnect to G‑d only by accessing a plane in G‑dliness that is above His revealed will and cannot be limited by a name. See sec. 4 below.
Similarly, with regard to Purim, it is well known9 that the reason G‑d’s name Havayah is not mentioned at all in Megilas Esther10וְגַם בְּנוֹגֵעַ לְפוּרִים יָדוּעַ דְּהַטַּעַם שֶׁלֹּא נִזְכַּר שֵׁם הֲוָיָ׳ בְּכָל מְגִלַּת אֶסְתֵּר
is that on Purim, G‑d’s infinite light that is above His name Havayah was revealed.הוּא לְפִי שֶׁבְּפוּרִים הוּא גִלּוּי אוֹר אֵין סוֹף שֶׁלְּמַעְלָה מֵהֲוָיָ'.
To explain the connection between the two common factors shared by Purim and Yom Kippur,וּבֵאוּר הַשַּׁיָּכוּת דִּשְׁנֵי הָעִנְיָנִים בְּהַהַשְׁוָאָה דְּיוֹם הַכִּפֻּרִים וּפוּרִים
[that both feature a lottery,[שֶׁבִּשְׁנֵיהֶם הוּא עִנְיַן הַגּוֹרָל,
and both access a level above G‑d’s name Havayah]:11וְשֶׁשְּׁנֵיהֶם הֵם לְמַעְלָה מִשֵּׁם הֲוָיָ׳],
The matter can be understood by prefacing the concept that a lottery accesses a level above intellect.12יוּבַן בְּהַקְדִּים דְּעִנְיַן הַגּוֹרָל הוּא לְמַעְלָה מֵהַשֵּׂכֶל.
Generally, a person makes decisions intellectually or emotionally. He does what he understands is best for him or feels most attracted to. Making a decision by casting lots implies that he is not relying on his understanding or emotions. On an obvious level, doing so relegates the decision to mere chance. From a deeper perspective, however, the fact that one relies on the lottery implies that he is going beyond his mind and his heart,
For (by and large,)13 a lottery is employed when both the two matters are the same to him, and thus the person has no intellectual or emotional motivation to select one over the other.שֶׁהֲרֵי גּוֹרָל הוּא (בִּכְלָל) בִּשְׁנֵי דְּבָרִים שָׁוִים
[As in the instance of the two goats that were offered on Yom Kippur, which were identical in appearance, size, and value.]14[וּכְמוֹ שְׁנֵי הַשְּׂעִירִים דְּיוֹם הַכִּפֻּרִים שֶׁהָיוּ שָׁוִים בְּמַרְאֶה וּבְקוֹמָה וּבְדָמִים],
For when one is better than the other,דְּכַאֲשֶׁר אֶחָד מֵהֶם הוּא טוֹב יוֹתֵר,
a lottery is not relevant. It is obvious that the better one should be selected.אֵין מָקוֹם לְגוֹרָל,
When, however, the two objects are equal,וְדַוְקָא כְּשֶׁשְּׁנֵי הַדְּבָרִים הֵם שָׁוִים,
since intellectually it is impossible to decide which one to select,כֵּיוָן שֶׁמִּצַּד הַשֵּׂכֶל אִי אֶפְשָׁר לְהַכְרִיעַ בְּאֵיזֶה מֵהֶם לִבְחוֹר,
one relies on a lottery that transcends intellect.סוֹמְכִים אָז עַל הַגּוֹרָל שֶׁלְּמַעְלָה מֵהַשֵּׂכֶל.
Similar concepts apply Above.וְעַל דֶּרֶךְ זֶה הוּא לְמַעְלָה,
A lottery is above Seder HaHishtalshelus, the chainlike framework of existence.דְּגוֹרָל הוּא לְמַעְלָה מֵהִשְׁתַּלְשְׁלוּת
[Chochmah, wisdom, represents the beginning of Seder HaHishtalshelus,[דְּחָכְמָה הִיא רֵאשִׁית הַהִשְׁתַּלְשְׁלוּת,
and a lottery, which transcends Chochmah, transcends this framework.]וְגוֹרָל שֶׁהוּא לְמַעְלָה מֵהַחָכְמָה הוּא לְמַעְלָה מֵהִשְׁתַּלְשְׁלוּת].
Based on these concepts, the common factor shared by Purim and Yom Kippur –וְעַל פִּי זֶה, הַהַשְׁוָאָה דְּפוּרִים וְיוֹם הַכִּפֻּרִים,
that both feature a lottery –שֶׁבִּשְׁנֵיהֶם הוּא עִנְיַן הַגּוֹרָל,
is that both reveal G‑d’s infinite light that transcends Seder HaHishtalshelus.הוּא – גִּלּוּי אוֹר אֵין סוֹף שֶׁלְּמַעְלָה מֵהִשְׁתַּלְשְׁלוּת.
On this basis, we can understand the connection between the two common factors shared by Purim and Yom Kippur:וְזֶהוּ הַקֶּשֶׁר דִּשְׁנֵי הָעִנְיָנִים בְּהַהַשְׁוָאָה דְּיוֹם הַכִּפֻּרִים וּפוּרִים,
that both feature a lottery and both transcend G‑d’s name Havayah.שֶׁבִּשְׁנֵיהֶם הוּא עִנְיַן הַגּוֹרָל וְשֶׁשְּׁנֵיהֶם הֵם לְמַעְלָה מֵהֲוָיָ',
For G‑d’s name Havayah represents the structure of Seder HaHishtalshelusכִּי הֲוָיָ' הוּא הִשְׁתַּלְשְׁלוּת
(the Yud of that name is identified with Chochmah, the Hei with Binah, etc.).(יו"ד חָכְמָה ה' בִּינָה וְכוּ'),
The concept that Purim and Yom Kippur are above G‑d’s name Havayahוְזֶה שֶׁיּוֹם הַכִּפֻּרִים וּפוּרִים הֵם לְמַעְלָה מֵהֲוָיָ'
stems from the fact that Purim and Yom Kippur are both above the structure of Seder HaHishtalshelus,15 reflectingהוּא שֶׁהַגִּלּוּי דְּיוֹם הַכִּפֻּרִים וּדְפוּרִים הוּא לְמַעְלָה מֵהִשְׁתַּלְשְׁלוּת,
the concept of a lottery.עִנְיַן הַגּוֹרָל.

Summary

Purim and Yom Kippur share a connection, as reflected by the fact that the name Yom Kippurim can be interpreted to mean “a day like Purim.” Among the common factors shared by Yom Kippur and Purim is that both feature a lottery. A second common factor: Both reflect a spiritual plane loftier than G‑d’s name Havayah.
To explain the connection between these two concepts: A lottery is employed when two matters are the same for a person, neither possessing an advantage over the other. Therefore, he does not select one or the other because of an intellectual decision. Rather, he steps beyond his intellect and allows the matter to be decided by the lottery.
Similarly, in a spiritual sense, a lottery represents a level above Seder HaHishtalshelus, the chainlike framework of existence. Both Purim and Yom Kippur feature a lottery and both transcend the structure of Seder HaHishtalshelus which has its source in G‑d’s name Havayah.