Finite-Precision Reconstruction of Rotating Quantum Black-Hole States

We study whether finite scattering data can reconstruct the effective quantum state of a rotating, charged, near-extremal black hole and whether the resulting inference can be trusted. The unknown parameters are the excitation energy E_i, the inverse Schwarzian breakdown scale C = E_brk⁻¹, the horizon angular velocity Ω_H, and the horizon electric potential Φ_H. The near-horizon response depends on the shifted frequency ω_eff = ω − mΩ_H − qΦ_H and on Schwarzian and U(1) gauge-sector spectral factors. We show that neutral channels alone are insensitive to Φ_H and reduce the inverse problem to rank three, whereas non-collinear neutral and charged channels can make the four-parameter map locally full rank. Frequency optimization and a fifth charge-sensitive channel reduce the dimensionless Jacobian condition number from 34.8 to 18.6 at the benchmark point and preserve full rank throughout a ±25% local parameter ensemble. For independent 3% response uncertainties, a local Fisher analysis gives approximate relative 1σ errors (20.3%,18.3%,5.2%,5.9%) for (E_i,C,Ω_H,Φ_H). Nonlinear injection-recovery tests agree at 1%, begin to depart from Gaussian expectations near 3%, and show rapid degradation of E_i and C at 5–10%. More seriously, coherent 1–3% calibration or model shifts can produce several-to-tens-of-percent parameter bias while leaving an excellent fit. A five-channel design can accommodate at most one unconstrained calibration nuisance in addition to the four physical parameters; a seven-channel extension can be made full rank for three selected nuisance modes, but self-calibration significantly degrades precision. The central lesson is that structural identifiability, finite-precision reconstructibility, and reliable state inference are distinct requirements. The results are restricted to an idealized near-horizon effective response with known channel labels and controlled transfer functions; they are not a detector forecast. Charge labels m and q denote angular-momentum and electric-charge transfer carried by a channel. With a neutral initial state, the final gauge charges have the opposite sign under the source convention q_f = q_i − q. Because the reduced benchmark density depends quadratically on the final charges, this sign choice does not alter the reported numerical results; it is stated explicitly to keep the implementation aligned with the primary-source conservation convention.