Learning When Black Hole Hair Is Observable: Physical Identifiability, Generalization, and Observable Complementarity

Abstract

High interpolation accuracy does not establish that physical parameters are identifiable. We study this distinction in a controlled Kiselev black hole benchmark with a validated timelike-emitter and three-dimensional null-geodesic shooting calculation. The exact $k=0$ boundary provides an analytically known null control because the spacetime becomes Schwarzschild and independent of $w_q$. Ringdown is informative for the deformation amplitude $k$, while photon geometry supplies an independent response direction that resolves much of the $w_q$ degeneracy. Relative to ringdown alone, ringdown plus photon geometry gives a median pointwise $4.54\times$ gain in minimum singular value and a $2.28\times$ improvement in Jacobian condition number. In grouped physical interpolation, identifiable-$w_q$ normalized mean absolute error decreases from 0.309 to 0.087. Random splitting is more optimistic, while directional extrapolation is substantially harder and split-conformal coverage degrades under distribution shift. A non-learned nearest-model inverse shows that the gain is not confined to the tested learned architectures. A uniform 161-phase audit gives a typical normalized prediction shift of $9.48\times10^{-4}$, while a targeted 321-phase audit resolves the remaining forward-feature comparisons. However, frozen HGB and random-forest tail shifts exceed prespecified robustness thresholds. Thus, physical observable complementarity is supported, but inverse-estimator robustness remains conditional. This is a theoretical identifiability benchmark, not an observational constraint.