Non-maximal persistence in Vietoris–Rips complexes of torus grids

Adams, Adetowubo, Barriga-Acosta, Feng, and Sterling asked whether every inclusion-induced map in third homology along the Vietoris–Rips filtration of a square torus grid has the largest rank allowed by the Betti numbers of its source and target. We answer the question in the negative over F_2 and describe a systematic mechanism behind the failure. For 3 ≤ k ≤ 7, the inclusion VR(T_{3k−1,3k−1}; k) → VR(T_{3k−1,3k−1}; k+1) induces the zero map on H_3, although the source and target have third Betti numbers 6k−2 and 1, respectively. For k ≥ 8, the same diagonal inclusion has rank exactly two over every field. Published computations give target third Betti number three over F_2 for k = 8, 9, while a signed-chain construction gives a third independent rational target class for every k ≥ 8. For k ≥ 11, a signed projection proves integral third-homology rank at least three and non-maximality over fields of characteristic different from two and three. The paper also gives reproducible finite-field computations, a general boundary-rank formula, an exact certificate for the smallest example, and minimality among square torus grids through side length eight.