A universal geometric mechanism for chaos-bound violations in black hole spacetimes

Violation of the Maldacena-Shenker-Stanford (MSS) chaos bound has been observed in various black hole spacetimes, but its physical origin remains unclear. In particular, it is uncertain whether these violations arise from modifications of general relativity or reflect a more fundamental feature of black hole spacetimes. In this work, we systematically investigate the instability of circular geodesics across a broad class of black hole solutions in Einstein, scalar-tensor, and higher-curvature gravity. We show that the violations are governed by the relative behavior of unstable circular orbits and the horizon structure in near-extremal regimes. When the relevant orbit remains outside the horizon as the surface gravity vanishes, the instability scale persists, and the chaos bound can be violated. On the other hand, as the orbit approaches the degenerate horizon, the instability becomes suppressed by the associated divergent gravitational time dilation, ultimately leading to saturation of the bound. Motivated by these results, we propose a geometric conjecture that determines the applicability of the MSS bound directly from the photon-sphere and horizon structure of the spacetime. Our findings identify a universal geometric criterion that governs the applicability of the MSS bound in black hole spacetimes, revealing a fundamental constraint on extending the quantum chaos bound to classical gravitational settings.