Static Causal and Hyperbolic Certificates for a Relativistic Elastic Sector of Causal-Boundary Disformal Gravity

This preprint isolates a constant-scalar static sector of Causal-Boundary Disformal Gravity (CBDG) and presents only its explicitly scoped positive mathematical results. The matter model is a rotationally invariant relativistic hyperelastic stored energy coupled to a scalar through a derivative-disformal metric. On the constant-scalar branch, the background reduces to an Einstein–elastic system and the leading static scalar–elastic principal block separates. The reported results are: an exact spherical reduction of the specified CBDG constitutive law; a certificate-supported regular finite-radius spherical equilibrium; independently corroborated positive radial stability; physical strong ellipticity on the documented strain tube K0.062; static symmetric hyperbolicity on the same tube; and a smaller certified bare-metric causal domain K0.008. The archive contains the standalone LaTeX source, compiled PDF, certified numerical data, verification tools, independent audit scripts, and saved verification reports. The positive sign of the fundamental radial mode is independently corroborated, while the numerical interval ω₀² ∈ [6.83, 6.84] remains certificate-supported because the original production eigenvalue generator and its exact normalization are not included. This work does not claim dynamical formation, finite-X mixed hyperbolicity, nonlinear or nonspherical stability, black-hole replacement, trapped-surface removal, or generic singularity avoidance.