Equatorial Periodic Orbits and Gravitational Wave Phenomenology around Spherically-symmetric vacuum solution in Freund-Nambu scalar-tensor gravity

We investigate test particle dynamics and gravitational wave (GW) phenomenology in an exact spherically symmetric vacuum solution of Freund - Nambu scalar - tensor gravity. This framework generalizes the Janis - Newman - Winicour (JNW) naked singularity via a geometric non - linear coupling $q$ and a direct scalar - particle coupling $g_s$. We demonstrate that these parameters systematically modify the Innermost Stable Circular Orbit (ISCO) - which shifts inward for $g_s>0$ - and the Marginally Bound Orbit (MBO). Furthermore, we classify bound periodic trajectories to isolate extreme zoom - whirl orbits exhibiting intense periapsis precession. By applying the Numerical Kludge method to Extreme Mass - Ratio Inspirals (EMRIs), we reveal that scalar - tensor corrections induce a macroscopic temporal dephasing in high - frequency GW bursts, even when the orbit’s spatial topology is preserved. These unique phase shifts offer a robust diagnostic signature for future space-based observatories like LISA to probe the strong - field regime and constrain scalar - tensor extensions of general relativity.