Gemara

(Rav): One must mention Rosh Chodesh (Ya'aleh v'Yavo) in Birkas ha'Mazon.

(R. Chanina): He need not mention it.

Support (for Rav - Rav Zerika - R. Oshaya's Beraisa): On Rosh Chodesh and Chol ha'Mo'ed, in Shemoneh Esre, one mentions the day in Avodah (Retzei). If he did not mention it, he must pray again. One mentions the day in Birkas ha'Mazon (in Boneh Yerushalayim).

Berachos 49a (Rav Gidal): If one forgot to mention Shabbos (Retzeh, in Boneh Yerushalayim), he says "Baruch Asher Nasan Shabbasos li'Menuchah... Baruch Mekadesh ha'Shabbos";

For (omitting) Yom Tov (Ya'aleh v'Yavo), he says "Baruch Asher Nasan Yomim Tovim... Baruch Mekadesh Yisrael veha'Zemanim."

For Rosh Chodesh, he says "Baruch she'Nasan Roshei Chodashim...."

R. Zeira: I do not know whether or not he says "l'Simchah," and whether the Berachah has a Chasimah (closing Berachah).

49b (Rav Idi bar Avin citing Shmuel): If one omitted Rosh Chodesh in Shemoneh Esreh, he goes back. If one forgot Rosh Chodesh in Birkas ha'Mazon, he does not go back.

Question (Rav Avin): Why is Shemoneh Esreh different than Birkas ha'Mazon?

Answer (Rav Nachman): Shemoneh Esreh is an obligation, so one must repeat it. Birkas ha'Mazon is not an obligation, i.e. one need not eat, therefore he need not repeat it.

Question (Rav Avin): If so, on Shabbos and Yom Tov, when one must eat bread, he should have to repeat it!

Answer (Rav Amram): Yes! R. Shila said so in the name of Rav.

Question: Rav Huna cited Rav to say that if one erred, he says "Baruch she'Nasan..."!

Answer: If he already began ha'Tov veha'Metiv, he must return to the beginning.

Rishonim

Rif (Berachos 36a DH v'Heicha): If he omitted Rosh Chodesh in Birkas ha'Mazon, he says "Baruch she'Nasan Roshei Chodoshim l'Amo Yisrael l'Zikaron." It does not have a Chasimah. If he remembered after beginning ha'Tov veha'Metiv, he finishes the Berachah. He need not return to the beginning, like Rav Nachman taught.

R. Yonah (DH l'Os): After mentioning Asher Nasan Menuchah..., he adds l'Os ul'Bris. These are two matters, so this is like a long Berachah. Therefore, he concludes with Shem u'Malchus (Hash-m's name and kingship), Baruch Atah Hash-m Elokeinu Melech ha'Olam Mekadesh ha'Shabbos.

Rambam (Hilchos Berachos 2:12): If one forgot to mention on Shabbos Kedushas ha'Yom, he says "Baruch Asher Nasan Menuchah... Baruch Atah Hash-m Mekadesh ha'Shabbos."

Rambam (13): If one omitted Ya'aleh v'Yavo on Rosh Chodesh, if he remembered before beginning the fourth Berachah, he says "Baruch she'Nasan Roshei Chodashim l'Amo Yisrael l'Zikaron." He is not Chosem. He then begins and finishes the fourth Berachah. If he remembered after beginning the fourth Berachah, he finishes it and does not go back. The same applies to Chol ha'Mo'ed, Chanukah and Purim. If he forgot the addition for the day, he does not go back.

Kesef Mishneh: The Rambam and Rif hold that since it is a Safek whether or not we mention Simchah and are Chosem, we do neither. The Rambam connotes that we are Chosem the other Berachos with Shem and Malchus. Most Meforshim agree. The Gemara connotes like this. The Safek was only about Rosh Chodesh. This connotes that surely, one is Chosem the others.

Hagahos Maimoniyos (7): Also Avi ha'Ezri says that we are not Chosem, since our Gemara did not resolve this. This is unlike Bahag, who rules that one is Chosem "Baruch Atah Hash-m Mekadesh Yisrael v'Roshei Chodoshim, like the Yerushalmi.

Rosh (Berachos 7:23): The Rif says that we are not Chosem the compensation Berachah for Rosh Chodesh. The Ri says that it is no different than Shabbos and Yom Tov, so we are Chosem. Presumably, we do not mention Simchah, for this does not apply to Rosh Chodesh.

Rosh (ibid.): Rav Nachman taught that if one forgot to mention Rosh Chodesh, he does not go back, for he need not eat. This does not mean that one may fast on Rosh Chodesh. Megilas Ta'anis forbids this! Rather, he could eat fruits or vegetables that do not obligate Birkas ha'Mazon.

Mordechai (Berachos 176 DH d'Iy): Even at night, if one remembered before beginning ha'Tov veha'Metiv, he blesses Asher Nasan Roshei Chodoshim. Even though if one forgot Ya'aleh v'Yavo in Shemoneh Esre at night he does not return, this applies only to Tefilah. Here it is one Berachah, so he says it even at night.

Rashba (Berachos 29a DH Kasav): R. Elchanan says that whenever we say that one who erred does not go back for Al ha'Nisim, Anenu or Ya'aleh v'Yavo at Ma'ariv, if he did not yet begin the next Berachah, he goes back, even if he finished this Berachah. In Birkas ha'Mazon, finishing a Berachah is not like starting the next. He does not mean that he repeats the Berachah. Rather, he mentions the day between the Berachos, just like in Birkas ha'Mazon. Tosfos says that he does not go back at all. Presumably, this is correct. One mentions the day between Berachos only for what is like Shabbos and Rosh Chodesh, for which there is a fixed mention, since there is a Korban Musaf. (Tefilas Musaf is totally for its sake - PF). Even though if one totally erred on Rosh Chodesh (began the next Berachah) he does not return, for he was not required to eat bread, if he did not yet start the next Berachah, he mentions it. There is no fixed mention of Al ha'Nisim or Anenu, just one must mention them in every Tefilah. If he passed the place to mention them, he need not mention them. Even though Rosh Chodesh has a fixed mention, since we are not Mekadesh Rosh Chodesh at night, we do not obligate him to go back. Even though in Birkas ha'Mazon, it seems that even at night he mentions it if he did not yet start ha'Tov veha'Metiv, there it is the end of the Berachos mid'Oraisa. In Tefilah, once he finished the Berachah, he may not mention anything in the middle.

Poskim

Shulchan Aruch (OC 188:7): If one forgot to mention Rosh Chodesh, during the day or at night, he says "Baruch she'Nasan Roshei Chodashim l'Amo Yisrael l'Zikaron. He is not Chosem.

Beis Yosef (DH v'Im): The Gemara did not resolve whether one mentions Simchah, and whether or not he is Chosem. The Roke'ach brings from Pesikta Rabasi that one is Chosem. The Rif and Rambam agree that one is not Chosem, so we follow them.

Birkei Yosef (9): The Mechaber clearly rules that one does not say in it Shem and Malchus, unlike Shabbos and Yom Tov. The Acharonim questioned this, for the Gemara was unsure only about being Chosem! This connotes that Rosh Chodesh is like Shabbos regarding the beginning. Some explicitly say that one says Shem and Malchus on Rosh Chodesh. The Kesef Mishneh is difficult. What is his source that one is Chosem with Shem and Malchus? The Rambam mentioned Shem in the Chasimah (for Shabbos), but not Malchus! Also, we never find Chasimah with Malchus. Any Berachah that begins with Shem and Malchus, the Chasimah (if it has one) has only Shem! The Kesef Mishneh holds that the Rambam holds like the Rashba, that there is Shem and Malchus in the beginning. If so, there is no Malchus in the Chasimah. Also, the Gemara did not mention Shem or Malchus in the beginning or Chasimah, so it suffices to equate this to other Berachos. What is the source in the Gemara or Rambam to say that it has Malchus in the Chasimah? The Kesef Mishneh says that most Meforshim say so. Only R. Yonah says so! Shirei Keneses ha'Gedolah questioned the proof from Rosh Chodesh. The Safek is whether or not one is Chosem. This does not teach about mentioning Hash-m's name! Why did the Kesef Mishneh need to infer about Chasimah for Shabbos and Yom Tov? The Gemara explicitly said the Chasimah!

Birkei Yosef: With difficulty, I answer for the Kesef Mishneh that the Rambam connotes that one mentions Shem and Malchus and is Chosem with Shem, opposite to the Ra'avad's opinion. The Kesef Mishneh means that the other Berachos have a Chasimah with Shem, and Malchus, but the Malchus is in the beginning. Alternatively, the other Berachos have a Chasimah, and begin with Shem and Malchus, i.e. they are proper Berachos, unlike the Ra'avad. He never meant that there is Malchus in the Chasimah. There is no such Berachah. Clearly, what is written in R. Yonah (that it concludes with Shem and Malchus) is a printing mistake, therefore the Beis Yosef did not mention it. He infers that the other Berachos have Shem in the Chasimah. If not, also Rosh Chodesh would have a Chasimah. There is no reason to omit Chasimah without Shem! R. Zeira did not ask about the beginning, for Rav Gidal did not specify the beginnings of the Berachos. For Shabbos and Yom Tov they have a Chasimah with Shem, so surely they are proper Berachos that begin with Shem and Malchus. Likewise, if the Berachah for Rosh Chodesh has a Chasimah, it begins with Shem and Malchus. The Rambam and Rambam rule like the other possibility, that the Berachah for Rosh Chodesh has no Chasimah. If so, it is not a real Berachah, and has no Shem and Malchus. The Shulchan Aruch rules like this.

Bi'ur Halachah (DH Omer): Also the Pri Megadim was unsure whether it begins with Shem and Malchus. In practice, it seems that one relies on those who explicitly say that it does, i.e. Shibolei ha'Leket, Ri'az, and R. Yerucham. Tosfos Rabbeinu Yehudah says that it is like Berachos on Peros and Mitzvos, which have no Chasimah. I.e. it has Shem and Malchus. The Shitah Mekubetzes connotes like this. Many Rishonim hold that it has even a Chasimah, like Shabbos, i.e. Bahag, the Roke'ach, Ra'avan, the Eshkol according to one text of the Gemara, Tosfos R. Yehudah, the Or Zaru'a citing the Yerushalmi, and Tosfos ha'Rosh. The Tur says that the Rosh held like this, even though the (Piskei) ha'Rosh did not decide the matter. Even the Poskim who say not to be Chosem, not one of them explicitly says that the beginning has no Shem and Malchus, and many explicitly say that it has. The Eliyahu Rabah, Ma'ase Roke'ach, Graz and Chayei Adam say so.

Kaf ha'Chayim (31): The Darchei Moshe (7) added that Safek Berachos Lehakel. We are lenient (not to bless) even when a minority says so. Meforshim argue about also Shem and Malchus, therefore one says it without Shem and Malchus at all.

Shulchan Aruch (ibid.): This if he remembered before beginning ha'Tov veha'Metiv. If he remembered after beginning it, he does not go back, for one need not eat bread and be obligated to bless Birkas ha'Mazon. Chol ha'Mo'ed is like Rosh Chodesh.

Mishnah Berurah (27): Chol ha'Mo'ed is similar, i.e. one does not go back (if he already started ha'Tov veha'Metiv), and if he remembered before starting, he says Baruch Atah Hash-m Elokeinu Melech ha'Olam she'Nasan Mo'adim l'Amo Yisrael l'Sason ul'Simchah. He is not Chosem, like on Rosh Chodesh.

Rema: In any case, perhaps one should say it amidst the ha'Rachaman's, like was explained (187:4) regarding one who omitted Al ha'Nisim. Perhaps one can distinguish. Ya'aleh v'Yavo has occurrences of Hash-m's name, and one should not say them l'Vatalah. It seems that this is the custom.

Magen Avraham (11): This is difficult, for every day we say supplications with names of Hash-m. Why is this worse? Only a Berachah l'Vatalah is forbidden! This requires investigation.

Bi'ur Halachah (DH v'Ein): Bigdei Yesha answered that if one says Ya'aleh v'Yavo with intent to fulfill a Chiyuv, this is worse (than other supplications with Hash-m's name), and there is concern for a Berachah l'Vatalah. He brought supports for this.

Kaf ha'Chayim (34): In any case, one says "ha'Rachaman Hu Yizkerenu l'Tovah v'Yifkedenu li'Vrachah b'Yom Rosh haChodesh (or Yom Chag Ploni) ha'Zeh."