Why do we say השישי in the שיר של יום?
(יג) מזמור ליום ששי הַיּוֹם יוֹם הַשִּׁשִּׁי בְּשַׁבַּת קוֹדֶשׁ, הַשִּׁיר שֶׁהָיוּ הַלְוִיִּם אוֹמְרִים עַל הַדּוּכָן:
I originally thought it had something to do with the השישי mentioned in the 6 days of creation, but did not understand the connection.
(לא) וַיַּ֤רְא אֱלֹהִים֙ אֶת־כָּל־אֲשֶׁ֣ר עָשָׂ֔ה וְהִנֵּה־ט֖וֹב מְאֹ֑ד וַֽיְהִי־עֶ֥רֶב וַֽיְהִי־בֹ֖קֶר י֥וֹם הַשִּׁשִּֽׁי׃ (פ)
(31) And God saw all that He had made, and found it very good. And there was evening and there was morning, the sixth day.
דְּאָמַר רֵישׁ לָקִישׁ, מַאי דִּכְתִיב: ״וַיְהִי עֶרֶב וַיְהִי בֹקֶר יוֹם הַשִּׁשִּׁי״, ה׳ יְתֵירָה לָמָּה לִי? — מְלַמֵּד שֶׁהִתְנָה הַקָּדוֹשׁ בָּרוּךְ הוּא עִם מַעֲשֵׂה בְרֵאשִׁית וְאָמַר לָהֶם: אִם יִשְׂרָאֵל מְקַבְּלִים הַתּוֹרָה — אַתֶּם מִתְקַיְּימִין, וְאִם לָאו — אֲנִי מַחֲזִיר אֶתְכֶם לְתוֹהוּ וָבוֹהוּ.
Reish Lakish said: What is the meaning of that which is written: “And there was evening and there was morning, the sixth day” (Genesis 1:31)? Why do I require the superfluous letter heh, the definite article, which does not appear on any of the other days? It teaches that the Holy One, Blessed be He, established a condition with the act of Creation, and said to them: If Israel accepts the Torah on the sixth day of Sivan, you will exist; and if they do not accept it, I will return you to the primordial state of chaos and disorder. Therefore, the earth was afraid until the Torah was given to Israel, lest it be returned to a state of chaos. Once the Jewish people accepted the Torah, the earth was calmed.
This week I think I found the answer
(כט) רְא֗וּ כִּֽי־יְהוָה֮ נָתַ֣ן לָכֶ֣ם הַשַּׁבָּת֒ עַל־כֵּ֠ן ה֣וּא נֹתֵ֥ן לָכֶ֛ם בַּיּ֥וֹם הַשִּׁשִּׁ֖י לֶ֣חֶם יוֹמָ֑יִם שְׁב֣וּ ׀ אִ֣ישׁ תַּחְתָּ֗יו אַל־יֵ֥צֵא אִ֛ישׁ מִמְּקֹמ֖וֹ בַּיּ֥וֹם הַשְּׁבִיעִֽי׃
(29) Mark that the LORD has given you the sabbath; therefore He gives you two days’ food on the sixth day. Let everyone remain where he is: let no one leave his place on the seventh day.”
יום הששי is not only referencing the 6th day of creation but it is referencing the giving of the Manna. Every Friday once must make sure to prepare for Shabbat
מי שטרח בערב שבת יאכל בשבת
One who takes pains on Shabbat eve will eat on Shabbat
I also found in Thursday's Zohar portion of Hok l'Yisrael (Parshat B'Shalah)
אָמַר רִבִּי אֶלְעָזָר, מַאי דִּכְתִּיב יוֹם הַשִּׁשִּׁי, וְלָא אִתְּמַר הָכִי בְּכָל שְׁאַר יוֹמִין. אֶלָּא הָכִי אוּקְמוּהָ, הַשִּׁשִּׁי. דְּאִזְדַּוְּוגָא (דאזדמנא) בֵּיהּ מַטְרוֹנִיתָא. לְאַתְקְנָא פָּתוֹרָא לְמַלְכָּא, וּבְּגִין כָּךְ, אִשְׁתְּכָחוּ בֵּיהּ תְּרֵין חוּלָקִין, חַד לְיוֹמֵיהּ, וְחַד לְתִקּוּנָא, בְּחֶדְוָותָא דְּמַלְכָּא בְּמַטְרוֹנִיתָא.
The original יום השישי of the 6 days of creation also includes the idea that one must prepare the meal on Friday for Shabbat dinner.
Now the question remains why do we reference the six days of creation when we say
(ג) מזמור ליום ראשון הַיּוֹם יוֹם אֶחָד בְּשַׁבַּת קוֹדֶשׁ, הַשִּׁיר שֶׁהָיוּ הַלְוִיִּם אוֹמְרִים עַל הַדּוּכָן: