Andrei Galiautdinov

We introduce an exact, two-parameter family of static, spherically-symmetric, constant-curvature $Λ$-vacuum solutions within the four-dimensional Starobinsky $f(R)=R+αR^2$ model. When the bare cosmological constant is precisely fine-tuned to $Λ= 1/(8α)$, the scalar curvature is fixed such that the derivative $f'(R)=1+2αR$ identically vanishes, demonstrating that the family represents a pathological $R_0$-degenerate boundary of the viable physical states. This mathematical degeneracy decouples the modified field equations, permitting the existence of an arbitrary $1/r^2$ integration constant in the metric, which functions as a purely geometric, Reissner-Nordström hair mimicker. However, any infinitesimal deviation from this exact boundary instantaneously destroys the degeneracy, rigorously forcing the geometric hair to vanish and collapsing the spacetime back into the standard Schwarzschild-de Sitter family. We provide the exact algebraic derivation of this spacetime and highlight its physical pathologies, including the identically vanishing Wald entropy of the associated black hole horizons, the divergence of the effective gravitational coupling, the resulting backreaction catastrophe, and the onset of severe ghost instabilities. Ultimately, this exact solution functions as a rigorous no-go theorem within the Starobinsky model, pedagogically illustrating the extreme fragility and physical hostility of degenerate, purely mathematical solutions in highly non-linear $f(R)$ gravity theories.