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 proposes a novel, purely geometric mechanism rooted in finite-sample trace dynamics. By mapping the coherence lifetime of the QPO to a Finite-Sample Ratio Estimator (FSRE) and requiring self-consistency between the symplectic eigenvalue structure and the 3:2 resonance, we show that the dominant eigenvalue λ = 2 is uniquely determined—the only positive solution of λ - λ⁻¹ = 3/2. This yields a strictly quantized coupling stiffness via νT = ln 2 and a direct, testable observational prediction: the fractional rms amplitude (A_rms) scales inversely with the QPO Quality factor (Q), with a slope determined by the natural logarithm of 2 and the Kerr spacetime shear. The predicted slope factor Γ(a*) varies by a factor ∼4 across the disputed spin range of GRO J1655–40, decreasing with increasing spin as the resonant orbit approaches the ISCO, making the amplitude-coherence relation a potential spin discriminator for next-generation X-ray timing missions.