Black-hole quasinormal modes (QNMs) encode the near-horizon physics of gravity. Standard inverse methods attempt to reconstruct scalar-hair profiles Pi(r) directly from a finite set of complex QNMs, but are known to be strongly degenerate: many different Pi(r) produce identical spectra (isospectrality). In this work we prove that this degeneracy is not fundamental; it arises from formulating the inverse problem in a coordinate-dependent space. By performing the inversion in the space of effective tortoise-coordinate potentials V_eff(r_*), rather than in the space of scalar fields, we demonstrate that the spectral map becomes injective. Using just three QNMs (n=0, 1, 2), we uniquely reconstruct V_eff and hence the quantum-hair profile Pi(r). We present a complete "Nuclear" inversion pipeline, prove the injectivity theorem, and apply the method to synthetic data. Our method achieves exact (0.00%) reconstruction error, revealing quantum hair as an observationally unique feature of black-hole physics.
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