Q-quadratic convergence of the centralized circumcentered-reflection method under a relative interior condition

The centralized circumcentered-reflection method (\cCRM) of~\cite{Behling:2024} converges superlinearly to a solution of $\operatorname{find}\;z\in X\cap Y$ when $\inte(X\cap Y)\neq\emptyset$ and the boundaries of $X$ and $Y$ are $\mathcal{C}^1$ hypersurfaces in $\re^n$. Both conditions fail when $\aff(X)=\aff(Y)\subsetneq\re^n$, as in equality-constrained feasibility and spectral matrix problems. We prove that \cCRM\ converges superlinearly when $\aff(X)=\aff(Y)$, $\operatorname{ri}(X)\cap\operatorname{ri}(Y)\neq\emptyset$, and the relative boundaries are $\mathcal{C}^1$ of appropriate relative dimension; and Q-quadratically when the relative boundaries are $\mathcal{C}^2$, with explicit asymptotic constant expressed in terms of the boundary curvatures at the limit point and the local error-bound constant. The case $\aff(X)\neq\aff(Y)$ is identified as open.