At what orbital speed does our best black hole math break down?

Physicists compared the analytic post-Newtonian approximation against a high-precision numerical relativity simulation of two merging black holes, tracking how well each successive correction order performs. Higher orders keep reducing errors for orbital speeds below about 45% of light speed, with the incomplete 6PN expansion performing best—but the improvement is non-monotonic, meaning the old trick of stopping at the first local minimum underestimates where the series is actually headed. Near the innermost stable orbit, around v~0.5c, the series loses the plot entirely, setting a clear benchmark for where analytic calculations must hand off to numerical methods.