Verifying When Black-Hole Hair Can Be Learned: Physical Identifiability, Generalization, and Observable Complementarity

Scientific machine-learning models can achieve high test accuracy while exploiting interpolation, numerically fragile features, or non-identifiable parameter directions. We study this verification gap in a controlled black-hole inverse problem: recovering the Kiselev parameters $(k,w_q)$ from synthetic leading-eikonal ringdown and observables generated by validated timelike and three-dimensional null-geodesic shooting. We enforce grouped physical validation, fit all learned transformations within training folds, compare random interpolation with blocked-region and four directional extrapolation tests, and analyze the whitened observable Jacobian. At $k=0$, $w_q$ is exactly non-identifiable. Away from this boundary, photon geometry provides a complementary parameter direction: adding it to ringdown improves the median minimum singular value by $4.54$ times, reduces the median condition number by a factor of $2.28$, and lowers grouped $w_q$ normalized mean absolute error from $0.3093$ to $0.0874$. Random splits are optimistic, while conformal coverage degrades under physical holdouts and extrapolation. A uniform 161-phase recomputation shows that typical predictions are stable, with median normalized shift $9.48\times10^{-4}$ and 95th percentile $0.0261$. However, unresolved higher-harmonic features and catastrophic multilayer-perceptron extrapolation outliers reveal unstable tails. Verification therefore requires joint tests of physical identifiability, distribution shift, and numerical convergence rather than accuracy alone. This is a controlled methodological benchmark, not an observational constraint.