What is the reason for prostrating four times in Vadakalai tradition versus prostrating one time in Thengkalai tradition? Why is there a difference in applying Tiruman for both sects?

Swami Desikan has proved based on Pancha-Rathra Agama that one has to prostrate four times; similarly, he has quoted in his Satcharitra Raksha, based on evidence from Pancha-Rathra Sastra on the length and breadth of a Tiruman.
Based on his texts, Vadakalai sect follows this practice. The tradition of Thengkalai can be known from the relevant Acarya-s of that sect.