On the sharp linear convergence rate of the circumcentered--reflection method on subspaces
For two subspaces $U,V\subseteq\RR^n$, the circumcentered--reflection method (CRM) of Behling, Bello-Cruz, and Santos~\citeyearpar{BBS2018} computes the projection onto $U\cap V$ using only the reflections across $U$ and $V$, with known linear-convergence rate equal to the cosine of the Friedrichs angle. We prove that, when CRM is initialized in $V$, it contracts at the strictly smaller rate $ρ_V=(\sin^2θ_p-\sin^2θ_F)/(\sin^2θ_p+\sin^2θ_F)$, where $θ_F\in(0,π/2]$ is the Friedrichs angle, whose cosine we denote by $c_F\coloneqq\cosθ_F$, and $θ_p\in[θ_F,π/2]$ is the largest principal angle between $U$ and $V$. The bound is sharp, attained on an explicit ray in $V$, and optimal among parameter-free single-step iterations. The constant itself is not new: Bauschke, Bello-Cruz, Nghia, Phan, and Wang~\citeyearpar{BBNPW2016} identified it as the optimal rate of the relaxed alternating-projection family and of their adaptive linesearch map $B_T$. Our contribution is that the parameter-free geometric circumcenter attains it as well, via Kantorovich's inequality applied to a single self-adjoint operator on $V$. Restricted to $V$, CRM coincides pointwise with the linesearch maps $A_T$ and $B_T$ from the Gubin--Polyak--Raik framework~\citep{GPR1967}. We further prove $ρ_V<c_F^2$ whenever $θ_F<π/2$, with one-step convergence exactly when $θ_F=θ_p$. Over-reflecting either or both of $R_U$, $R_V$ inside the circumcenter does not help. Going faster than $ρ_V$ universally requires memory: Chebyshev semi-iteration applied to $P_VP_U$ attains a strictly smaller rate, beating $ρ_V$ by a factor at most $2$, attained in the limit $θ_F\toθ_p$.