Screened scalar hair and the weak-lensing separation of black holes from neutron stars in quadratic f(R) gravity

Quadratic f ( R ) gravity carries a massive scalar degree of freedom, the scalaron, whose finite range λ λ screens its influence on the geometry outside a compact object. We show that this screening severs the exterior of a black hole from that of a neutron star of the same mass. The correction to the Schwarzschild metric is Yukawa-suppressed, falling as e-r/λ e - r / λ instead of polynomially in the coupling, and is therefore invisible to any expansion in powers of that coupling; the exterior is moreover intrinsically isotropic, so that inverting the temporal potential alone, as one does for single-potential solutions, fails to solve the field equations. Applying the Gibbons–Werner Gauss–Bonnet theorem to this geometry, we find the leading deflection angle to be exactly the general-relativistic 4 GM / b , with b the impact parameter, the scalaron contributions cancelling identically in the combination that bends light. The first correction carries the unfamiliar signature λ -1/2b-3/2e-b/λ λ - 1 / 2 b - 3 / 2 e - b / λ , screened beyond the scalaron range and confirmed against direct quadrature to around $$20%$$ 20 % at b=4λ b = 4 λ , improving to $$7%$$ 7 % at b=12λ b = 12 λ . What survives is not a difference of degree but of kind. The ingredients that deliver it — namely, non-analyticity in the coupling, the intrinsically isotropic gauge, the closed-form screened correction and the pressure-weighted scalar charge — are obtained within a single, self-consistent derivation here. A static black hole carries no scalar hair, and lenses precisely as general relativity requires; a neutron star acquires a scalar charge weighted by the pressure supporting it against collapse, and does not. Weak lensing therefore closes as a discriminant of the theory, whilst the horizon, as against a material surface, remains one in principle. The observational advantage lies not in bending angles at large impact parameter but in the strong-field imaging of the photon sphere, and in the stellar interior, where the scalar charge is fixed by the equation of state.