A Symplectic Trace Boundary Mechanism for High-Frequency QPO Resonance in Kerr Spacetime

The physical mechanism driving the strict 3:2 frequency ratio in high-frequency quasi-periodic oscillations (HFQPOs) around black holes remains a major open question in astrophysics. This paper presents a rigorous derivation of the frequency selection equation governing these oscillations within resonant Hamiltonian systems. By explicitly mapping continuous Kerr Hamiltonian dynamics to a discrete finite-sample recurrence via a second-order Störmer-Verlet integration, we prove that the universal boundary of dynamical stability strictly occurs at the dominant eigenvalue \lambda=2, yielding the quantized product \nu T=\ln 2. By evaluating the antisymmetric shear operator across the separatrix, we establish an exact topological self-consistency equality between the symplectic stretching and the resonance winding number, \lambda - \lambda^{-1} = p/q. Unit-magnitude eigenvalue constraints necessitate integer quantization of the propagator for higher harmonics, producing a harmonic sequence p/q = (n^2-1)/n that uniquely isolates the 3:2 ratio (n=2) as the dynamically dominant fundamental mode, eliminating the need for ad hoc heuristic parameters. Furthermore, through an exact derivation of the spacetime shear from Kerr effective potentials, the framework produces a parameter-free, falsifiable observational prediction: the fractional rms amplitude scales inversely with the QPO quality factor, A_{rms} \propto 1/Q. Preliminary analysis of archival RXTE data for GRO J1655-40 shows consistency with this predicted Trace-Boundary Slope Factor, lending empirical support to the theory and offering a novel dynamical probe of black hole spin