The star S4716 is one of the fastest known members of the Galactic-center S-star cluster orbiting the supermassive black hole Sagittarius A*. With an orbital period of approximately four years and velocities approaching 2.6% of the speed of light near pericenter, the star provides an important probe of stellar dynamics in the strong gravitational field of the Galactic center. In this work the orbital architecture of S4716 is analyzed within the Primary-Centric Framework (PCF), in which characteristic orbital states are determined by the triple synchrony condition where the spin angular velocity of the primary, the spin angular velocity of the secondary, and the orbital angular velocity of the secondary become equal. This condition defines the synchrony radii aG1 and aG2 that characterize the dynamical topology of the system. For the extreme mass-ratio configuration of the S4716–Sagittarius A* system, the inner synchrony radius aG1 remains finite and dynamically accessible, while the outer synchrony radius aG2 shifts to very large distances and becomes effectively intangible. The observed semi-major axis of S4716 is found to be consistent with the tangible synchrony scale aG1 predicted by the PCF, and the tangential velocity derived from Keplerian dynamics reproduces the observed extreme velocity of the star near pericenter. The analysis further shows that gravitational radiation from the system is negligible because of the extreme mass ratio between the stellar mass and the central black hole, so that the orbital evolution is governed primarily by gravitational dynamics. These results indicate that the orbit of S4716 lies within the constraints imposed by the Primary-Centric Framework and provide an illustrative example of stellar dynamics in the vicinity of a supermassive black hole.

Galactic Center, Sagittarius A*, S-stars, Stellar dynamics, Relativistic orbits, 27 Supermassive black holes, Orbital mechanics, Strong gravitational fields