Gemara

(Mishnah - R. Yehudah ben Bava): I have a tradition that the morning Tamid may be offered within four hours of the day.

Berachos 26a (Mishnah): One may pray Shacharis until Chatzos (midday);

R. Yehudah says, he may pray until four hours of the day.

Question: In our Mishnah, no one permits Shacharis after Chatzos;

However, R. Yochanan taught that if one mistakenly omitted Ma'ariv, he prays Shacharis twice. If one mistakenly omitted Shacharis, he prays Minchah twice!

Answer: One may pray the entire day. Before Chatzos, one is rewarded for Tefilah in the proper time. After Chatzos, one gets reward for Tefilah, but not for Tefilah in the proper time.

27a - Question: Does R. Yehudah mean "Ad v'Ad Bichlal" (up to and including, i.e. until the end of four hours), or "Ad v'Lo Ad Bichlal" (until the start of the fourth hour)?

Answer: He permits Ad v'Ad Bichlal.

Support (Rav Nachman - Mishnah): The morning Tamid may be offered within four hours of the day.

(Rav Kahana): The Halachah follows R. Yehudah, because the (above) Mishnah in Eduyos is like him.

Rishonim

Rif and Rosh (Berachos 18a and 4:1): Even though the Halachah does not follow Chachamim, who permit until midday, if one erred and prayed after four hours of the day he gets reward for Tefilah, but not for Tefilah in its time.

Rambam (Hilchos Tefilah 3:1): The time for Shacharis extends until the end of the fourth hour, which is a third of the day. If one erred or transgressed and prayed after four hours, he was Yotzei Tefilah, but he was not Yotzei the obligation of Tefilah in its time. Just like Tefilah is a Torah Mitzvah, there is a Mitzvah mid'Rabanan to pray in the time that Chachamim and Nevi'im enacted.

Poskim

Shulchan Aruch (OC 89:1): The time for Shacharis extends until four hours, which is a third of the day.

Magen Avraham (4): One must finish the Amidah by the end of four hours.

Kaf ha'Chayim (8): We calculate the hours based on (a third of the time from) dawn to Tzeis ha'Kochavim.

Shulchan Aruch (ibid.): If one erred or transgressed and prayed after four hours, he gets reward for Tefilah, but not for Tefilah in its time.

Beis Yosef (DH v'Im): If one transgressed and did not pray within four hours, the Tur permits to pray until midday. I can explain that one may pray at any time of the day. Just like Chachamim permit after midday, R. Yehudah permits after four hours. The Rif and Rosh wrote that we do not follow Chachamim. If so, we should totally forbid praying after four hours. However, this is wrong. He is no worse than one who prays (compensation for Shacharis) after the time of Minchah came. Even though the Poskim wrote that he may pray until midday, really, it is until the time for Minchah.

Question: If so, we should say that he may pray until the night!

Rejection (Beis Yosef): After Minchah is different, for he must pray Minchah first. According to this, one may pray after four hours only if the delay was due to a mistake or Ones. If he intentionally delayed until four hours, he may not pray afterwards, since we learn from compensation. However, the Rambam said 'if he erred or transgressed, he may pray until midday.' It seems that he explains unlike I said. He does not learn from compensation. There is different, for he prays at a time for Tefilah. Rather, he learns from Chachamim, who said until midday. We should not make such a big argument between R. Yehudah and Chachamim, to say that R. Yehudah holds that one gets no reward at all, and Chachamim say (that one is Yotzei) until midday. Rather, all agree that after four hours one gets reward for Tefilah, whether he forgot or delayed b'Mezid, just R. Yehudah holds that he does not get reward for Tefilah in its time. If so, the Poskim were precise to say that he may pray until midday. From midday until the time for Minchah is not a time of Tefilah at all. If one prayed, perhaps it was a Berachah l'Vatalah. The Rashba says that compensation is only at a time for Tefilah. If not, the Gemara should have said 'if one did not pray Shacharis before midday, he prays after midday'! This is like my latter explanation.

Lechem Mishneh (Hilchos Tefilah 3:1): Why does the Rambam permit only until midday? He should permit even after, just like Chachamim permit after midday! The Beis Yosef preferred his latter explanation. We do not learn from compensation, for that is in Zman Tefilah. This is difficult. Since we learn from the Tamid, what is the source to say that R. Yehudah permits until midday at all? The Beis Yosef and Rashba say that not at the time of Tefilah, he does not get reward even for Tefilah. If so, why did the Gemara say (Stam) 'after this, he gets reward for Tefilah'? It should have specified that this is only if it is Zman Tefilah, i.e. until midday, or after six and a half hours, but not in the half hour after midday! However, if so the Mishnah says 'until midday' to exclude only this half hour. This is difficult. Therefore, it is better to say that until midday is an absolute limit (to get reward for Tefilah in its time, but one may pray after this and get reward for Tefilah - PF).

Taz (1): The Rema holds like the Beis Yosef's latter explanation. The Gemara permits praying the entire day, just there is no reward for Tefilah in its time. This is also according to R. Yehudah, since the question (from compensation for Shacharis) was also according to him. Also, what was the question? Perhaps one may not pray after midday. Compensation for Shacharis after Minchah is different, for then is a time for Tefilah! Rather, the Beis Yosef's first explanation is correct. However, we do not learn from compensation for Shacharis, rather, from the Gemara's Heter to pray the entire day. The Gemara said that according to Chachamim, until midday there is reward for Tefilah in its time. Afterwards, the entire day, even after Minchah, there is reward, but not in its time. We learn to R. Yehudah that until four hours one may pray l'Chatchilah, even if b'Mezid he neglected to pray in time. After the time for Minchah, i.e. six and a half hours, he may not pray compensation until praying Minchah first. This is why the Rif and Rosh discussed praying after four hours. They did not limit it until midday. The Rambam and Tur mentioned until midday. They do not exclude the half hour after midday, rather, they exclude after Zman Minchah. They did not say until six and a half hours, lest one think that even at six and a half hours one may pray compensation (before Minchah).

Gra (DH v'Im): The Rif, Rosh and other Poskim say that just like Chachamim permit after midday, R. Yehudah permits after four hours. However, the Rif and Rosh connote that this is only if he erred. The Gemara supports this, for we learn from compensation, and there is no compensation for Mezid. The Rambam and Tur permit even if he was Mezid. What the Beis Yosef wrote is difficult, like the Lechem Mishneh said. This requires investigation.

Mishnah Berurah (6): If one delayed past four hours b'Mezid, he should pray until midday and stipulate that if he may not pray Shacharis now, it is a Nedavah (voluntary Tefilah).

Kaf ha'Chayim (9): If one did not pray within four hours, he should pray as soon as possible.

Kaf ha'Chayim (10): The Prishah says that the reward for compensation is as great as for the Tefilah in its time. If not, the Gemara should have said that this is "crooked that cannot be straightened" (Aruch ha'Shulchan)! I say that if one did not pray within four hours, l'Chatchilah he should pray before midday. When one prays Shacharis (even after four hours), his Tefilah ascends through the worlds of Asiyah, Briyah, Yetzirah and Atzeilus (corresponding to Birkas ha'Shachar, Pesukei d'Zimrah, Kri'as Shma and its Berachos, and the Amidah), and descends again (through Ashrei and the Kedushah in u'Va l'Tziyon, Shir Shel Yom, and Ein kEi-loheinu and Pitum ha'Ketores). This does not apply to compensation after Minchah.

Rema: After midday, it is forbidden to pray Shacharis.

Taz (1): The Rashba does not support the Beis Yosef's latter explanation. He does not exclude compensation before Zman Minchah. The Rashba also taught that one who omitted Minchah compensates after Ma'ariv. He did not need to teach that one must wait until Zman Ma'ariv, for it is right after Zman Minchah. Rather, he just teaches that one must pray the compensation for Minchah after Ma'ariv, amidst engaging in Tefilah. Likewise, he merely taught that one must pray compensation for Shacharis after Minchah. The Rema holds like I said. He forbids Shacharis after midday, i.e. after Zman Minchah, i.e. six and a half hours, unless one prays Minchah first.

Magen Avraham (5): From midday one may not pray until Zman Minchah, and then he prays Minchah first. Really, Zman Minchah begins at midday (just we are stringent to wait a half hour). The Bach says otherwise; he erred.

Gra (DH v'Achar): Zman Minchah really begins after midday, like it says in Yoma 28b.

Mishnah Berurah (7): The Rema connotes that whether he was Mezid or Shogeg, he may not pray right after midday. Some are lenient in the half hour after midday. One should not deviate from the Rema's words. If one was Shogeg or there was Ones and he did not pray before midday, he should wait until Minchah and then compensate for Shacharis. If he was Mezid, he cannot fix this. If he did pray in this half hour, he was Yotzei, and he does not pray again.

Kaf ha'Chayim (11): The primary opinion forbids after midday, even if he forgot or there was Ones.