Lyapunov Exponents and Kelvin-Simplex Descriptors in Three-Body Dynamics: A Complementarity Analysis

We investigate the relationship between the maximum Lyapunov exponent (λₘₐₓ) and the six Kelvin-simplex descriptors introduced in companion works, for a dataset of 300 three-body trajectories spanning six canonical orbit families. Using the shadow-orbit renormalisation method, we compute λₘₐₓ for each trajectory alongside the descriptor vector, and analyse the resulting correlations. Three principal findings emerge. First, the figure-eight and Lagrange families produce statistically indistinguishable λₘₐₓ values (0. 286 ± 0. 051 and 0. 287 ± 0. 047 respectively), despite being perfectly separable in the six-dimensional descriptor space. Second, the six descriptors decompose naturally into two partially orthogonal subspaces: a position subspace Hₘₑₐₙ, Dₘₑₐₙ, Cₚᵣₒₓ whose members are highly mutually correlated (Pearson r = 0. 92--0. 96) and moderately correlated with λₘₐₓ (r = 0. 34--0. 42), and a motion subspace Path, SpecConc whose members are largely independent of the position subspace (Path vs Dₘₑₐₙ: r = -0. 04) and only moderately correlated with λₘₐₓ (r = -0. 31 and -0. 37 respectively). Third, Path -- the dominant descriptor in the Paper 3 classifier at 29. 5% importance -- is the descriptor least predictable from λₘₐₓ. These results demonstrate that the Kelvin-simplex descriptor space is informationally richer than the maximum Lyapunov exponent: it encodes complementary geometric structure about the simplex trajectory that λₘₐₓ does not capture, and it is this additional information that drives the near-perfect classification accuracy reported in 3. We derive an exact entropy production formula from the replicator equation that explains why the position descriptors correlate with λₘₐₓ, and discuss why path length and spectral concentration are fundamentally beyond the reach of local Lyapunov characterisation. v2: Simulation and analysis code (paper4ₗyapunov. py) added to this record.