Praying first for Hashem's universal kingship

בנוסח התפלה תן פחדך הוא בירור מלכותו ית' על כל העולם.

In the text of the tefillah, "Instill Your awe" is the clarification (birur) of His kingship over all the world.

The opening request of the Rosh Hashanah prayer — that Hashem place His dread upon all creation — is about making His malchus recognized universally, over every nation and creature.

ותן כבוד מיוחד רק לבנ"י שהוא דרך התורה אין כבוד אלא תורה.

And "give honor (kavod)" is unique only to Bnei Yisrael, which is the path of Torah, for "there is no kavod other than Torah."

The second request — "give kavod to Your people" — is reserved for Bnei Yisrael, because true honor is Torah, and the Jewish people's distinct glory is their bond with the Torah.

ועתה בעונותינו הרבים בגוים אין תורה צריכין מקודם לבקש על תיקון מלכות שמים.

And now, due to our many sins, [when we are] among the nations where "there is no Torah," we must first request the rectification of malchus Shomayim (the Kingdom of Heaven).

In galus, scattered among the nations where Torah does not flourish, we cannot begin with our own kavod; we must first pray for the establishment of Hashem's kingship in the world.

ויקוים בנו כמ"ש חז"ל המתפלל בעד חבירו והוא צריך לאותו דבר הוא נענה תחלה דכ' ויתפלל אברהם כו' וה' פקד את שרה כו'.

And there will be fulfilled in us what Chazal said: one who prays on behalf of his fellow while he himself needs that very thing is answered first, as it is written: "And Avraham prayed…" and "Hashem remembered Sarah…"

Chazal's principle: when you daven for another for something you also need, you are answered first. Avraham prayed for Avimelech's household to bear children, and immediately Sarah — who shared that need — was remembered with Yitzchak.

וכמו כן בנ"י שמבקשין תן פחדך על כל מעשיך.

And so too with Bnei Yisrael, who request "instill Your awe upon all Your works."

By praying that Hashem's awe rest upon all of creation — including the nations — Bnei Yisrael are themselves davening for others while needing the same recognition of malchus Shomayim, and so they will be answered first.

[ונ"ל כי ענין מתפלל בעד חבירו כו'.

[And it appears to me that the matter of "one who prays on behalf of his fellow"…

The Sefas Emes opens a bracketed clarification on exactly when this rule of being answered first applies.

הוא כשחושש שנענש חבירו על ידו.

is when he is concerned that his fellow was punished on his account.

He suggests the principle works specifically when one feels responsible for the other's suffering — that his fellow was harmed because of him.

כמו שהי' באברהם שנענש אבימלך ע"י.

As was the case with Avraham, where Avimelech was punished on his account.

Avimelech's household was stricken because of Sarah's having been taken on Avraham's account, so Avraham's prayer for them was a prayer to undo harm he had caused.

אבל בלא"ה למה יתפלל בעד חבירו קודם כיון שצריך לאותו דבר.

But otherwise, why should he pray for his fellow first, given that he himself needs that very thing?

Without such responsibility, it would seem strange to put another's need ahead of one's own pressing need; the rule only makes sense when one bears blame for the other's plight.

וכזה הוא בדברינו הנ"ל שבנ"י חוששין שעל ידיהם נדחו האומות ומתפללין עליהם והם נענין תחלה]:

And so it is in our matter above: Bnei Yisrael are concerned that on their account the nations were pushed away, so they pray for them — and they themselves are answered first.]

Bnei Yisrael feel that the nations' distance from Hashem is in some way bound up with their own chosenness; therefore they daven for the whole world's recognition of Hashem, and by that selfless prayer they merit to be answered first.

וז"ש ז"ל אמרו לפני מלכיות שתמליכוני עליכם.

And this is what Chazal said: "Recite before Me Malchiyos, so that you may crown Me over you."

The Gemara's instruction to say the Malchiyos verses is precisely this avodah of accepting and proclaiming Hashem's kingship on Rosh Hashanah.

ע"י שמתפללין בלב שלם שיחול מלכותו ית' על כל.

Through praying with a whole heart that His kingship take effect over all,

When we sincerely and wholeheartedly daven that Hashem reign over the entire world, not only over ourselves —

זוכין אנו תחלה במלכות שמים:

we merit first to receive the Kingdom of Heaven.

This selfless, universal prayer for Hashem's malchus brings us, who share that very need, to be the first to attain it — fulfilling the principle of one who prays for others.

Summary: On Rosh Hashanah "tein pachdecha" asks for Hashem's kingship over all the world, while "tein kavod" is the unique Torah-honor of Bnei Yisrael; in galus we must first pray for malchus Shomayim universally, and since we share that need, we are answered first — like Avraham who prayed for Avimelech and was answered through Sarah — for by wholeheartedly crowning Hashem over all, we are the first to merit the Kingdom of Heaven.