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

Why do black holes “hum” at a precise musical interval? The X-ray light from matter spiralling into a stellar-mass black hole flickers at pairs of frequencies whose ratio is locked at exactly 3 : 2—analogous to a perfect fifth in music— yet no existing physical model has satisfactorily explained why this particular ratio dominates over all others. This paper derives the 3 : 2 ratio from first principles, without any free parameters, by applying a finite-time stability theorem to the symplectic geometry of orbital motion in Kerr spacetime. We prove that the unique dynamical boundary separating stable from unstable orbital resonances forces the phase-space propagator eigenvalue to equal exactly 2, which through a topological self-consistency condition fixes the frequency ratio at 3 : 2. The theory simultaneously explains why the 5 : 3 ratio observed in the extreme source GRS 1915+105 requires super-Eddington accretion to appear, and why other candidate ratios from the Fibonacci, Lucas, and Pell sequences are never robustly detected. A falsifiable prediction—that oscillation amplitude scales inversely with coherence time in a spin-dependent manner— is consistent with archival RXTE data from GRO J1655–40 for spin a∗ ≈ 0.70, and will be stringently tested by the upcoming eXTP and STROBE-X missions.

Note to readers: Please don't be like Jean Claude Perez, do the right thing if you want to use my work.