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 establishing a topological conjugacy between discrete sequence recurrence and the continuous symplectic propagator through explicit second-order discretization of the Kerr geodesic equations, we prove that the universal boundary of dynamical stability occurs at the dominant eigenvalue λ = 2, yielding the quantized product νT = ln2. Through turnstile flux quantization at the saturated Chirikov overlap limit, we demonstrate that physical phase-space constraints necessitate integer quantization of the propagator eigenvalue. Applying this integer boundary to the topological winding number yields a unique, irreducible mapping that strictly necessitates the 3 : 2 fundamental resonance mode, eliminating the need for ad hoc heuristic parameters. We derive an exact expression for the Kerr spacetime shear µ(a∗) from first principles, demonstrating that the fractional correction terms are of order (ln2/(ΩrT))2 ≪ 1 and therefore negligible. The framework produces a harmonic sequence p/q = (n2−1)/n with n ≥ 2, showing that the 3 : 2 mode (n = 2) is the unique fundamental mode dynamically accessible in typical accretion environments. We further analyze the relation of this sequence to other integer sequences (Fibonacci, Lucas, Pell) that appear in KAM theory and continued fraction expansions of quadratic irrationals. This analysis reveals why the 3 : 2 ratio is universally dominant, while the 5 : 3 ratio—a Fibonacci convergent but not an integer-boundary mode—requires extreme super-Eddington conditions to be observable, as seen in GRS 1915+105. The theory yields a falsifiable observational prediction: the fractional rms amplitude scales inversely with the QPO quality factor, Arms ∝ Γ(a∗)/Q, where Γ(a∗) is a sensitive function of black hole spin. Preliminary analysis of archival RXTE data for GRO J1655–40 shows consistency with the predicted slope for a∗ ≈ 0.70, lending empirical support to the theory and offering a novel dynamical probe of strong-field gravity.