Delayed Shacharis

Rambam Tefillah (3:1) rules תפילת השחר - מצותה, שיתחיל להתפלל עם הנץ החמה; וזמנה, עד סוף שעה רביעית שהוא שליש היום. ואם עבר או טעה והתפלל אחר ארבע שעות עד חצי היום - יצא ידי חובת תפילה, אבל לא יצא ידי חובת תפילה בזמנה: שכשם שמצות תפילה מן התורה, כך מצוה מדבריהן להתפלל אותה בזמנה שתיקנו לנו חכמים ונביאים.  There are different interpretations as to the understanding that one may pray shacharis after 4 hours until chatzos.  Either this is the law of tashlumin, a law that R' Yehuda that holds one may pray only untill 4 hours agrees בדיעבד there is a concept of praying until chatzos, (see Beis Yosef siman 89,) or it is a new law of a cheftah of prayer שלא בזמנו (see shiur of R' Willig.)  

According to the peshat the Beis Yosef (siman 89) favors and the peshat of the Lechem Mishna there is a concept of prayer after 4 hours even according to R' Yehuda.  Where does that come from?  If R' Yehuda limits the time of the prayers to when the tammid was actually offered why would there be room to daven until chatzos? 

As noted in the past, the Rambam rues both of tefillot being in place of korbanot and that they were established by the avot. The rule of praying until 4 hours is is tefillah is patterned after the korbanot and the times Chazal gave to pray are patterned after the korbanot.  However, the actions of the avot exit to give three basic times to the framework of prayer, morning, afternoon and night.  Hence, even though the timeframe of tefillah to correspond to the korban is lost after the 4^th hour of the day, the fact that it is still morning allows one to pray as a tefillah corresponding to the prayer of Avraham.  That aspect remains to allow one to pray שלא בזמנו and still get a kium of tefillah to parallel that of Avraham.