Two scale black holes with a phenomenological suppression profile: extremality, null critical curves, and test field spectra

We study an asymptotically flat two scale black hole metric that supplements the Hayward mass profile with a phenomenological short distance suppression factor. The Hayward and suppression lengths control the core deformation and near center effective mass contribution, respectively. We derive the exact extremal boundary and its two parametric branches, identify the nonuniform zero suppression limit, and determine the effective Einstein source. For nonzero suppression, the Ricci and Kretschmann scalars vanish at the center while orthonormal curvature components remain bounded. The metric admits a local \(C^4\) Cartesian extension and the effective source is \(C^2\) . The weak, dominant, and strong energy conditions are characterized over the surveyed domain. We calculate the exterior null critical curve and fundamental \(\ell _m=2\) scalar and electromagnetic test field modes. Third order WKB and time domain results are compared with an independent Chebyshev pseudospectral calculation. Across the validation set, the largest differences relative to the pseudospectral frequencies are \(0.22\%\) for WKB and \(0.48\%\) for time evolution. A logarithmic response analysis quantifies the local conditioning of the fixed background diagnostic map and shows that the selected optical and spectral quantities remain strongly correlated.