Quantitative vanishing-regularization limits for degenerate cross-diffusion with changing boundary strata

We study a vanishing-regularization limit for a degenerate cross-diffusion system on the simplex when the active boundary stratum of a smooth reference solution may change in space and time. The mobility $M(u)=\mathrm{diag}u-u\otimes u$ loses rank when a component vanishes, so a fixed relative entropy or an unweighted two-point gradient estimate cannot remain uniformly coercive across moving faces. We introduce a geometric-relative framework that separates the single emerging normal direction from the active tangential subsystem. Its ingredients are a global resolvent geodesic distance, a boundary-oriented rank-one perspective correction, an intrinsic Schur-normal De Giorgi alternative, a mean-free tangential iteration with temporal-flux control, and a terminal coupled Schur compactness argument. These mechanisms yield a non-circular first-exit theorem for smooth approximations and then pass to admissibly approximable entropy weak solutions. If the strong solution stays a fixed distance from the codimension-two skeleton and its vanishing components satisfy $\mathrm{\mid }\mathrm{\nabla }{v}_i\mathrm{\mid }\lesssim v_i^{\alpha }$ and $\mathrm{\mid }\{0<v_i<r\}\mathrm{\mid }\lesssim r^{\sigma }$, then the augmented relative energy is bounded by ${\rho }_{\lambda }={\lambda }^{2\alpha +\sigma }$ if $2\alpha +\sigma <2$, ${\rho }_{\lambda }={\lambda }^2\mathrm{\mid }\log \lambda \mathrm{\mid }$ if $2\alpha +\sigma =2$, and ${\rho }_{\lambda }={\lambda }^2$ if $2\alpha +\sigma >2$. For a hypersurface contact $v_i\asymp d^p$, $p\ge 2$, this gives ${\rho }_{\lambda }=O({\lambda }^{2-1\mathrm{/}p})$ and $\mathrm{\parallel }{u}^{\lambda }-v{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(0,T;L^2)}=O({\lambda }^{1-1\mathrm{/}(2p)})$.