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Answer: Earliest Mincha time on our halachic calendars is a half hour after chatzot (astronomical midday). Since the three tefillot relate to morning, afternoon, and night, we would have expected Mincha to begin at chatzot. The reason it does not is critical to answering your question.
One gemara (Yoma 28b) reports that Avraham davened Mincha right after chatzot and wonders why in the Beit Hamikdash they never slaughtered the afternoon korban until a half hour later. One of the gemara’s answers is that Avraham was an expert in knowing when it was chatzot, whereas others wait a half hour to make sure it is afternoon. The Magen Avraham (233:1) learns from here that fundamentally the earliest time for Mincha is chatzot¸ and our practice of waiting a half hour is precautionary.
Another gemara (Berachot 26b) writes categorically that the earliest first starting time of Mincha (Mincha Gedola) is 6.5 hours (i.e., half an hour after chatzot), which corresponds to the earliest afternoon korban of the year (Mincha Ketana is at 9.5 hours into the day, the usual time of the korban). One reconciliation of the gemarot is that Yoma follows the approach that the tefillot were modeled after the patriarch’s tefillot and that Berachot follows the approach that they are modeled after the korbanot, i.e., never before 6.5 hours (Magen Gibborim 233:1).
Conceptually, the Magen Avraham (ibid.) saw the extra half hour as a precaution, according to which we would expect that if one accidentally davened during the half hour he would be yotzei. Yet, he infers from the Shulchan Aruch (Orach Chayim 233:1) that since he views Mincha Gedola itself as b’di’eved, before 6.5 hours must be totally invalid. The Pri Megadim (ad loc.) counters that there could be three levels: optimal- Mincha Ketana; less than optimal- Mincha Gedola; b’di’eved- during the half hour.
Bemare Habazak - Rabbis Questions (651)
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We now turn to your case’s specific details. The fact that you ended SE at the right time does not help. Although a minority opinion holds that if one starts SE at the right time, he is yotzei even if he finishes after the end point, that is because the beginning may have special importance (see Aruch Hashulchan, OC 110:5). The end does not have such importance!
What is helpful is that your case occurred in early winter. It is not clear whether the half hour is a sha’ah zemanit or 30 minutes on the clock (see Sha’ar Hatziyun 233:8), and most calendars are machmir in both directions. Therefore, you started before the end of the 30 minutes, but after half of a winter sha’ah zemanit. Your being a woman might call for slight leniency. Women are obligated in tefilla, but some say that once a day suffices (see Mishna Berura 106:4). Even if you daven Mincha regularly, this doubt can be added to the others.
Most poskim rule that one who davened Mincha during the half hour after chatzot was yotzei (see Ishei Yisrael 27:5; Tefilla K’hilchata 3:39). This is clearer in your case due to the additional indications.
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