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

The centralized circumcentered-reflection method (\cCRM) of Behling, Bello-Cruz, Iusem, and Santos~\cite{Behling:2024} is known to converge superlinearly for the feasibility problem $\mathrm{find}z\in X\cap Y$ under a ${\mathcal{C}}^1$ smoothness assumption on the boundaries of $X$ and $Y$. We sharpen this to a quantitative rate: when the boundaries are ${\mathcal{C}}^2$ near the limit point $\bar{z}$, \cCRM\ converges Q-quadratically, with an asymptotic constant $2\max ({\kappa }_X,{\kappa }_Y)\mathrm{/}\omega$ governed by the boundary curvatures ${\kappa }_X,{\kappa }_Y$ at $\bar{z}$ and the local error-bound modulus $\omega$. The estimate matches Newton-type second-order behavior even though \cCRM\ uses only projections and circumcenters, and numerical experiments on equality-constrained and spectral feasibility problems exhibit the predicted quadratic rate, with \cCRM\ reaching machine precision in a handful of steps where alternating projections and Douglas–Rachford take many. The argument is local and does not require $X\cap Y$ to have nonempty interior in ${\text{re}}^n$: it suffices that the sets share an affine hull $L=\text{aff}(X)=\text{aff}(Y)$ and meet with nonempty relative interior, which is the natural setting for equality-constrained and spectral feasibility problems, where the classical full-dimensional hypothesis necessarily fails. A ${\mathcal{C}}^1$ version of the argument recovers and extends the superlinear rate of~\cite{Behling:2024} to this lower-dimensional regime. The case $\text{aff}(X)\mathrm{\ne }\text{aff}(Y)$ is identified as open.