The Grimmer--Shu--Wang Certificate and the Drori--Teboulle Minimax Constant-Stepsize Bound for $N\ge 3$

We prove, for every horizon $N\ge 3$, the existence of the strengthened low-rank performance-estimation certificate proposed by Grimmer, Shu, and Wang for the Drori–Teboulle constant-step gradient-descent bound. For each $N\ge 3$, let ${\rho }_N\in (0,1)$ be determined by ${\rho }_N^{2N}(2N{\rho }_N+2N+1)=1$. We show that the GSW certificate equations admit positive vectors $a,b,c,d$ satisfying all residual equations. The proof proceeds through a reduced residual system in the variables $d$, a simplex existence argument for a positive reduced zero, a terminal residual completion identity, and a tail-square convolution argument proving the cumulative margins that force $b>0$ and then $a>0$. Consequently, the GSW low-rank PEP certificate exists for every $N\ge 3$ and yields the Drori–Teboulle upper bound. We also include the one-dimensional quadratic and Huber lower-bound examples for nonnegative steps, while the quadratic example excludes negative constant steps from being optimal. Together these prove the Drori–Teboulle minimax constant-stepsize value over all real constant steps for every $N\ge 3$.