Perturbative vacuum constraints in higher-curvature gravity: Schwarzschild deformations and strong-field observables

Higher-curvature terms modify the action but do not necessarily generate new vacuum geometries on the branch perturbatively connected to GR. We develop a first-order framework for static, spherical vacuum black holes in 4D metric theories with ${\mathcal{L}}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}=R\mathrm{/}2+\lambda \mathrm{\Psi }(R,X,Y)$, $X=R_{\mu \nu }{R}^{\mu \nu }$, and $Y=R_{\mu \nu \rho \sigma }{R}^{\mu \nu \rho \sigma }$. On the coupling-analytic branch, purely Ricci-based terms $\mathrm{\Psi }(R,X)$, analytic in curvature with $\mathrm{\Psi }(0,0)=0$, do not deform a Ricci-flat background under adopted regularity and boundary assumptions. Riemann-dependent terms can, since Kretschmann scalar is nonzero. In areal-radius gauge, we derive model-independent first-order expressions for the horizon shift, ISCO, epicyclic frequencies, periapsis advance, photon sphere, critical shadow impact parameter, Wald entropy, and Hawking temperature. For ${\mathrm{\Psi }}_{\eta }={\mathrm{\ell }}_{\lambda }^{-2+4\eta }{\mathcal{G}}^{\eta }$, treating noninteger powers as phenomenological parametrizations of nonlinear curvature, we obtain a closed first-order Schwarzschild-connected solution. Perturbations decay for $\eta >1\mathrm{/}2$, while fixed-ADM interpretation requires $\eta >2\mathrm{/}3$; at $\eta =2\mathrm{/}3$, sourced $1\mathrm{/}r$ term mixes with asymptotic mass mode. For positive effective coupling, the horizon, ISCO, photon sphere, and critical shadow impact parameter increase for $2\mathrm{/}3<\eta <1$ and decrease for $\eta >1$. On the fixed-ADM branch, mass, Wald entropy, and Hawking temperature satisfy $dM=T_H,d{S}_{\mathrm{W}}$ to first order at fixed couplings. At $\eta =1$, the linear 4D Gauss–Bonnet term leaves local geometry and geodesic observables unchanged but adds a constant topological entropy shift. Finally, an Event Horizon Telescope-inspired shadow-size criterion gives a conservative first-order sensitivity estimate for the effective dimensionless coupling, a geometric consistency test rather than a complete observational constraint.