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Answer: There are several points of machloket/safek that impact this topic.
The concept of mentioning me’ein hameora (=mehm – about the special day) in a beracha acharona (classically, in Birkat Hamazon) after the day is over because one started the meal during the day is the subject of a machloket Rishonim (see Beit Yosef, Orach Chayim 188). The Shulchan Aruch has an apparent contradiction on the matter (see OC 188:10; OC 271:6), and the strongest explanation is that out of doubt, it is appropriate to say mehm based on either the beginning of the meal or the time of the bentching (Magen Avraham 271:14). How bad it is to mention something from a wrong day is complicated (see Ginat V’radim OC, I:28, Mishna Berura 108:38).
There are two explanations of why to follow the beginning of the eating on Shabbat. One is that when one ate on Shabbat, he became obligated in mehm, and that obligation does not cease (see Mishna Berura 188:32). Another is that even though Shabbat is over, it is still a time that could be Shabbat if he had laudably extended it further (Magen Avraham 188:18 in the name of the Shelah). The second idea, related to extending the day, does not apply to Rosh Chodesh. The first idea does apply, but the comparison is imperfect because the obligation of mehm on Rosh Chodesh is weaker, as there is no obligation to eat and therefore one who skips Ya’a’leh V’yavo, does not have to repeat Birkat Hamazon (Shulchan Aruch, OC 188:7). Nevertheless, the Shulchan Aruch (ibid. 10) rules that one who started a meal during Rosh Chodesh mentions mehm when bentching at night.
Does this concept also apply to Al Hamichya? Mehm is less important in Al Hamichya, as we rule that if one forgot it, even on Shabbat and Yom Tov, he need not repeat Al Hamichya (Mishna Berura 208:58). There are few reasons why not. For one, it is not unanimous that one ever has to do mehm in Al Hamichya (Sha’ar Hatziyun 208:60). Also, this type of non-bread eating does not typically fulfill required eating (ibid.). Finally, mehm might have been instituted only as a preferable recitation (ibid.). Therefore, the case for there being a need to say it after the day is weaker. Shevet Hakehati (V:42) argues that only important eating on a special day enables mehm to extend to the night and therefore eating that gets only Al Hamichya does not warrant it. There is room to disagree, as one way or the other, there was an obligation or expectation to recite mehm in this beracha acharona. As mentioned above, erring on the side of saying is not a major problem, as we say mehm when there is a safek, and in Al Hamichya it contains only a few "innocuous words," which are not a beracha (see Ginat Veradim ibid.) and do not even include Hashem’s Name.
In your specific case, there was a reason to specifically not have added mehm (even if it had been a bread meal) – the fact that you already davened Ma’ariv, officially ushering in the new day. The Magen Avraham (188:17) says that fundamentally after eating makes one obligated in Birkat Hamazon with mehm, whenever you bentch, the special day should be mentioned. But once ushering in the new day with Ma’ariv, it looks like a contradiction if one makes a recitation that is a throwback to the previous day. Contradictions are problematic, not just unnecessary. There is significant logic to say that this only applies to day changes that have significant halachic consequence, like when Shabbat enters or ends, and not to the end of the day of Rosh Chodesh (see Machatzit Hashekel to 188:17). On the other hand, davening Ma’ariv any day has some halachic significance regarding ending halachot of the previous day (see Rama, Yoreh Deah 196:1). In any case, the Mishna Berura (188:34) clearly states that Ma’ariv of the evening after Rosh Chodesh precludes saying mehm of Rosh Chodesh.
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