9

In the Musaf of Rosh Chodesh, we mention that Hashem has given us "חוקי ראש חודש". What are these Chukim?

6

Nitei Gavriel, in the introduction to the second volume of his Laws of Niddah, quotes the Sefer Imrei Yehudah Al Hatorah. (It starts here, but the part that's relevant to us is here).

The Imrei Yehudah Al Hatorah (Tazria pg. 89) explains that we say:

‫ כי בעמך ישראל בחרת מכל האומות‬ ‫וחוקי ראשי חדשים להם קבעת‬

For you have chosen your nation Israel from among all the nations, and established for them the laws of the new months.

This refers to the laws of family purity, which follow a monthly cycle. By G-d establishing the the laws of family purity, He has separated us from the nations. When G-d sees that we put aside our love for our wife because of our love for G-d, who commanded us to separate from our wives while they are in a state of impurity, He loves us.

So, we are saying in the Musaf Prayer that it is because of the laws of family purity that G-d chose us over all the other nations.

The Nitei Gavriel also says that the Darchei Moshe (417:1) says something similar, but I didn't check it out.

4

The laws of Kiddush HaChodesh, when to make 2 days or 1 day Rosh Chodesh, and when to make a leap year.

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