Sharp regularity for degenerate fully nonlinear equations with oblique boundary conditions and Hamiltonian terms

We prove optimal boundary $C^{1,\alpha }$ regularity for viscosity solutions of degenerate fully nonlinear uniformly elliptic equations with oblique boundary conditions and Hamiltonian terms of the form $$\{\begin{array}{ll}\mathrm{\mid }Du{\mathrm{\mid }}^{\gamma }F(D^{2}u)+\varrho (x)\mathrm{\mid }Du{\mathrm{\mid }}^{\sigma }=f(x)& \text{in }\mathrm{\Omega },\\ \beta (x)\cdot Du+\zeta (x)u=g(x)& \text{on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\gamma >0$ and $0<\sigma \le 1+\gamma$. We develop a compactness framework for affine translations, linking the size of the translation to the Hamiltonian structure. This is combined with a boundary improvement-of-flatness argument adapted to oblique boundary data, yielding the optimal boundary regularity.