The fourth integral Betti number of a torus-grid Rips complex

We compute the fourth integral Betti number of the closed radius-five Vietoris--Rips complex on the thirteen-by-thirteen torus grid with its periodic Manhattan metric. It is two, giving a counterexample to the integer-rank formula in AIM Problem 531, which predicts twenty-six for this grid. The proof separates a subcomplex with the homotopy type of the torus from the simplices that cannot be lifted to the plane. A signed integer chain certificate computes the resulting relative homology. Its nonzero even boundary coefficients explain the difference from the published mod-two data.