Torus-grid Vietoris--Rips complexes at half the diameter

For every integer \(k\geq3\), we determine the homotopy type of the closed Vietoris--Rips complex of the \(2k\) by \(2k\) periodic Manhattan grid at radius \(k\). It is \(S^3\vee\bigvee^{4k}S^{2k-1}\), as conjectured by Adams, Adetowubo, Barriga-Acosta, Feng and Sterling. Adjacent pairs of rows and columns give the high-dimensional spheres. Removing one facet from each leaves a complex with the integral homology of a three-sphere. The comparison is proved uniformly through a homotopy calculation for finite cylinders and a cover by coordinate semicircles.