This paper asks whether different kinds of observations can uniquely pick out two parameters that describe a simple “hairy” black hole model. The authors build a controlled test case where one parameter vanishes to give an exact null control. They then show that adding photon-trajectory observables to the usual ringdown data makes the parameters much easier to separate. At the same time they emphasize that practical recovery can still fail under extrapolation or when an estimator is frozen, so the result is methodological not observational.

The model is a two-parameter Kiselev spacetime. One parameter k sets a deformation strength. When k = 0 the spacetime is exactly the normal Schwarzschild black hole and all observables are independent of the other parameter (called w_q in the paper). That exact case gives a firm test of identifiability: if a diagnostic finds information about w_q at k = 0 it is spurious. To probe the observables the team simulated a timelike emitter orbiting the hole, solved full three-dimensional photon (null) geodesics to a distant observer, and recorded phase-resolved quantities like redshift, impact parameter, sky position and arrival times. They sampled a grid of 121 systems and compressed each phase series into a fixed feature vector (means, amplitudes and low-order Fourier coefficients).

The authors compared two forward observable sets. Ringdown features come from quasinormal modes tied to the unstable circular photon orbit. Photon-geometry features come from the simulated photon paths and arrival information. To test identifiability they used Jacobian diagnostics — the Jacobian is the matrix of how small changes in parameters change the observables — and scalar measures derived from it. A larger minimum singular value and a better (smaller) condition number mean the observables respond in more independent ways to the parameters and so the inverse problem is easier. They also tested learned inverse estimators, a simple nearest-model (non-learned) inverse, grouped and directional evaluation splits to probe extrapolation, and audits that re-evaluate predictions at many emission phases.