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. Standard infinite-time limit models (such as classical KAM theory) fail to capture the transient, finite coherence of these signals. This paper proposes a novel, purely geometric mechanism rooted in finite-sample trace dynamics and symplectic geometry. By mapping the coherence lifetime of the QPO to a Finite-Sample Ratio Estimator (FSRE), we require strict physical selfconsistency between the symplectic eigenvalue structure and the phase-locking necessary to survive Kerr spacetime shear. We demonstrate that the dominant eigenvalue λ = 2 is not a free parameter, but is uniquely determined as the only positive solution of λ - λ⁻¹ = 3/2, normalizing the winding number drift. We prove that this integer boundary represents the dynamically dominant fundamental mode (n=2) of a harmonic sequence defined by primitive Pythagorean triples, rigorously ruling out fractional closures in typical accretion environments. This mathematical rigidity yields a strictly quantized coupling stiffness via νT = ln 2 and generates a direct, falsifiable observational prediction: the fractional rms amplitude (A_rms) scales inversely with the QPO Quality factor (Q). The slope of this relation is determined entirely by the natural logarithm of 2 and the Kerr spacetime shear, possessing zero free parameters. The predicted slope factor Γ(a*) varies by a factor of ~4 across the disputed spin range of GRO J1655-40, decreasing as the resonant orbit approaches the Innermost Stable Circular Orbit (ISCO). This makes the amplitude-coherence relation a highly sensitive, dynamically derived spin discriminator for next-generation X-ray timing missions. Furthermore, we expand this framework using Arnold Tongue scaling and the Lichtenberg binary sequence to definitively explain why the 3:2 mode dominates the astrophysical population, while the fractional 5:3 mode (observed in GRS 1915+105) represents a fragile, period-doubled phase-space boundary accessible only in extreme super-Eddington regimes. Preliminary analysis of archival RXTE data for GRO J1655-40 shows exceptional consistency with the predicted linear A_rms ∝ 1/Q relation, lending robust empirical support to this unified theory.