arXiv:2610.34755v1 Announce Type: new Abstract: We construct an equation-of-state-independent geometric-compatible finite-volume (GC-FV) method and a high-order nodal discontinuous Galerkin spectral-element extension (GC-DG) for a reduced Godunov-Peshkov-Romenski system on fixed Eulerian meshes. Density is recovered from $\rho=\rho_0\det(\mathbf{A})$, so mass conservation is a hidden companion law of the distortion equation. Starting from a straight-path Dal Maso-LeFloch-Murat fluctuation pair, minimum-change cofactor projections produce a single-valued density flux and exact induced local Lax-Friedrichs dissipation without jump quotients. The summation-by-parts property lifts the geometric two-state identity to high order and yields an equivalent subcell representation of GC-DG. We then blend GC-DG with the compatible GC-FV operator on the same Legendre-Gauss-Lobatto subcells. Because both operators induce the same boundary density flux, the global subcell blend retains the exact semidiscrete mass balance while suppressing shock-generated oscillations. A complementary modal/face-jump indicator supplies the elementwise blending coefficient. Numerical experiments verify high-order convergence, the expected temporal scaling of the fully discrete mass drift, and robust shock capturing for fluid, nonlinear-elastic, thermal-wave, and elastoplastic problems, including genuinely multidimensional configurations.