  The three-body problem (3BP) poses a longstanding challenge in physics and
celestial mechanics. Despite the impossibility of obtaining general analytical
solutions, statistical theories have been developed based on the ergodic
principle. This assumption is justified by chaos, which is expected to fully
mix the accessible phase space of the 3BP. This study probes the presence of
regular (i.e. non chaotic) trajectories within the 3BP and assesses their
impact on statistical escape theories. Using numerical simulations, we
establish criteria for identifying regular trajectories and analyse their
impact on statistical outcomes. Our analysis reveals that regular trajectories
occupy up to 32% of the phase space, and their outcomes defy the predictions of
statistical escape theories. The coexistence of regular and chaotic regions at
all scales is characterized by a multi-fractal behaviour. Integration errors
manifest as numerical chaos, artificially enhancing the mixing of the phase
space and affecting the reliability of individual simulations, yet preserving
the statistical correctness of an ensemble of realizations. Our findings
underscore the challenges in applying statistical escape theories to
astrophysical problems, as they may bias results by excluding the outcome of
regular trajectories. This is particularly important in the context of
formation scenarios of gravitational wave mergers, where biased estimates of
binary eccentricity can significantly impact estimates of coalescence
efficiency and detectable eccentricity.
