Failure of a pairwise four-step bound for restarted Anderson acceleration

Exact diagonal examples refute a conjectured pairwise formula for four-step residual amplification in restarted Anderson acceleration. With $A=I-M$, $\alpha(v)=v^TAv/v^TA^2v$ and $R(v)=M[I-\alpha(v)A]v$, the residual after four steps is $R(R(v))$. For $M=\mathrm{diag}(1/10,1/2,3/5)$, where both $M$ and $I-M$ are positive definite, and $v=(1,1,1)^T$, exact rational substitution gives $\|R(R(v))\|_2^2/\|v\|_2^2=1920682/21289638243>1/14641$, the square of the conjectured maximum $1/121$. A positive-definite one-parameter family gives an unbounded ratio of actual amplification to the conjectured expression. These finite-step statements do not resolve the separate asymptotic convergence-factor conjecture.