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

Abstract 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 develops a framework connecting the frequency selection to the symplectic stability boundary of parametric resonance in Kerr spacetime. Through explicit second-order St¨ormer–Verlet discretisation of the resonant pendulum Hamiltonian, we establish a rigorous mapping from continuous Hamiltonian dynamics to the discrete recurrence an = an−1 + can−2, proving that the universal boundary of dynamical stability occurs at the dominant eigenvalue λ =2, yielding ντr = ln2 (the perturbation doubles per radial orbital period). We postulate the frequency selection axiom λ − λ−1 = p/q, motivated by the antisymmetric structure of the symplectic propagator, and show that at λ = 2 it uniquely selects p/q = 3/2. We derive a leading-order expression for the Kerr spacetime shear µ(a∗) with fractional corrections of order (ln2/(ΩrT))2 ≪ 1. A comparative analysis of Fibonacci, Lucas, Pell, and integer-boundary sequences reveals why the 3:2 ratio is universally dominant, while the 5:3 ratio requires extreme super-Eddington conditions. The theory yields a falsifiable prediction: Arms ∝ Γ(a∗)/Q, where Γ(a∗) varies by a factor ∼4 across the disputed spin range of GRO J1655–40. Preliminary RXTE data are consistent with a∗ ≈ 0.70.