Why does every line say both "Our Father" and "Our King"? Which is it?
Every line starts with the same two names, and they pull in different directions. A father forgives because you are his child. A king rules, and you answer to him.
The Torah uses both. In Devarim it says to Israel, "You are children of Hashem your God." In Vayikra, God says of Israel, "They are My servants, whom I took out of Egypt." Pirkei Avot (a Mishnah collection of the sages' sayings) has Rabbi Akiva saying that Israel are beloved because they were called children of God.
The first line of the prayer puts the problem on the table: "Our Father, our King, we have sinned before You." Can you still say "Father" after that? The Gemara records an argument. Rabbi Yehuda says that when you act like children you are called children, and when you do not, you are not. Rabbi Meir says that either way you are called children, and he points to verses that call Israel foolish children.
That argument is about children who sin. It does not mention this prayer, so linking the two is our reading. On Rabbi Meir's view, "Father" is still true right after "we have sinned."
We could not find a source that explains why every line puts "Father" first and "King" second, or which name is doing the work in which line.
Devarim 14:1,Vayikra 25:42,Pirkei Avot 3:14,Kiddushin 36aIf the sentence is already written, how can a prayer tear it up?
Two lines here ask God to change a judgment already on the table: "cancel all harsh decrees" and "tear up the evil of our sentence."
The Gemara pictures Rosh Hashanah as a day of judgment. Rabbi Yochanan says three books are opened. The fully wicked are written for death at once, the fully righteous for life, and everyone in between hangs in the balance from Rosh Hashanah until Yom Kippur.
The prayer's word for "tear up" is the same word used on those pages. Rabbi Yitzchak teaches that four things tear up a person's sentence: charity, crying out, changing your name, and changing your actions. Rabbi Yochanan adds that teshuvah (returning to the right path) does it too, and Abaye argues from a verse that this works even after the sentence is given.
Rabbi Yitzchak also says a person is judged only by what he is doing at that moment. His proof is a verse about Yishmael (Ishmael) as a boy: God heard him "where he is." So the judgment looks at who you are now, not at what you might become later.
Look at what the prayer asks to tear: "the evil of our sentence." The Taz says to read those words in one breath. He explains it this way: tear up the bad in the decree, and let whatever remains be for mercy.
The Sages clearly thought a sentence could be torn up. The sources we opened do not explain how, and they do not promise it works every time.
Rosh Hashanah 16b,Rosh Hashanah 17b,Taz on Shulchan Aruch, Orach Chaim 622:6Why skip this on Shabbat? What if I really need help that day?
The Rema says to say Avinu Malkeinu in the Ten Days "except on Shabbat." The Mishnah Berurah adds that it is skipped even at the Friday afternoon prayer before Shabbat, and gives the reason: we do not ask for our needs on Shabbat. Our siddur's note gives a related reason. Avinu Malkeinu was set up to match the middle blessings of the weekday Amidah, so it is not said on Shabbat.
A similar idea appears in the Gemara. One teaching says that when you visit a sick person on Shabbat you tell him, "It is Shabbat, so we do not cry out, and healing is near to come." Rabbi Meir gives a different line to say: Shabbat itself can bring mercy.
The Kol Bo, in its Yom Kippur section, reports a custom of saying Avinu Malkeinu even on Shabbat. Then it quotes an earlier book, Sefer HaIttim, that rejects this. Even the weekday Amidah, which is required, is set aside on Shabbat, it argues, so how can we say Avinu Malkeinu, which it says is nowhere in the Gemara? The passage adds that other rabbis ruled the same way, with a reason: Rabbi Akiva first said it on a public fast, and public fasts are not declared on Shabbat.
It was never fully settled. Shibbolei HaLeket (a medieval collection of laws and customs) says some places never say it on Shabbat, but the custom where he lived was to say it. At Mincha (the afternoon prayer) of a Yom Kippur that falls on Shabbat, the Shulchan Aruch says to say it, and the Rema says that in Ashkenazi lands it is not said.
What about real trouble? The Shulchan Aruch says that on Shabbat we do not cry out or sound an alarm for any trouble, with exceptions: a food shortage, a city surrounded by enemies or a flooding river, a ship in danger at sea, a person being chased. In those cases people do cry out and plead in prayer on Shabbat. That rule is about Shabbat in general and does not mention Avinu Malkeinu, but it shows real emergencies are treated differently.
Shulchan Aruch, Orach Chaim 602:1, Rema,Mishnah Berurah 584:4 and 602:3,Siddur Sefard, note before Avinu Malkeinu,Shabbat 12a,Kol Bo 70:4, quoting Sefer HaIttim,Shibbolei HaLeket 305:3,Shulchan Aruch, Orach Chaim 622:3 and 288:9