Halakhah for Rosh Hashanah 1:2
בְּאַרְבָּעָה פְרָקִים הָעוֹלָם נִדּוֹן, בְּפֶסַח עַל הַתְּבוּאָה, בַּעֲצֶרֶת עַל פֵּרוֹת הָאִילָן, בְּרֹאשׁ הַשָּׁנָה כָּל בָּאֵי הָעוֹלָם עוֹבְרִין לְפָנָיו כִּבְנֵי מָרוֹן, שֶׁנֶּאֱמַר (תהלים לג) הַיּוֹצֵר יַחַד לִבָּם, הַמֵּבִין אֶל כָּל מַעֲשֵׂיהֶם. וּבֶחָג נִדּוֹנִין עַל הַמָּיִם:
The world is judged in four periods: On Pesach, for produce [From the Torah's stating: Bring the omer before Me on Pesach so that I bless the grain in the fields for you, we derive that we are judged for produce on Pesach]; on Shevuoth, for the fruits of the trees [(This is derived) from the Torah's stating: Bring before Me the two breads on Shevuoth so that I bless the fruits of the trees for you. (The wheat plant is called a "tree," viz. (Genesis 2:17): "But from the tree of knowledge, good and evil, etc." This, according to the view that the tree from which Adam ate was a wheat plant.]; on Rosh Hashanah, all who have entered the world pass before Him as sheep, [which are made to pass through a narrow entrance to be tithed, two not being able to pass as one], viz. (Psalms 33:15): "He fashions their hearts as one; He considers all of their deeds." [The intent is: "The Fashioner sees their hearts all together and comprehends all of their deeds. For even though they pass before Him one by one, they are all "comprehended" in one "glance."]; and on Succoth, they are judged for water. [(This is derived from the Torah's stating: Pour water before Me on the festival (of Succoth)].
Shulchan Arukh, Orach Chayim
One needs to juxtapose “redemption” [i.e. the last blessing of the Sh'ma - "Ga-al Yisrael"] to “prayer” [i.e. the Amidah]. And one should not interrupt between them, even with "Amen" after “Ga-al Yisrael”, and not for any verse other than “Hashem Sefatai” [Psalms 51:17, the introductory verse for the Amidah]. Gloss: And there are those who say that it is permitted to respond Amen after “Ga-al Yisrael”, and so we practice (Tur). And there are those who say that this requirement to juxtapose “redemption” to “prayer” is only on a weekday or Yom Tov, but on Shabbat one does not need to. (Meaning, that the reason that we require to juxtapose “redemption” to “prayer” [during the week] is because it is written “God will answer you in a day of distress,” [Psalms 20:2] and juxtaposed to it [is written], “Let the words of my mouth be desire … and my redeemer.” [Psalms 19:15], but Shabbat is not a time of distress. And in my humble opinion, it seems [that the reason] this is not so on Yom Tov is because they are days of judgement, as we learned in Mishnah Rosh Hashanah 1:2, “On Passover [the world is judged] on grain…”) (Hagahot Ashir"i [Rabbi Yisrael MiKrems] on Chapter 1 of Berachot; and Kol Bo, Laws of Shabbat; and Mahari"l, the Laws of Yom Tov) And it is good to be stringent unless in a place that needs to do such (Tur).