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

We study 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. For the diagonal inclusion VR(T3k−1,3k−1; k) → VR(T3k−1,3k−1; k+1), we prove that for 3 ≤ k ≤ 7 the map is zero over F₂, while the third Betti numbers of the source and target are 6k−2 and 1. For every k ≥ 8, the same map has rank exactly two over every field. A signed secondary-product construction gives a third independent target class over Q, yielding rational non-maximality for all k ≥ 8; together with the published F₂ target dimensions for k = 8,9, this gives seven consecutive F₂ counterexamples. For k ≥ 11, an integral projection gives non-maximality over fields of characteristic different from 2 and 3. We also prove that T8,8 at scales 3 → 4 is the smallest counterexample among square torus grids of side length at most eight. The source package contains the associated finite-field certificates, signed-chain constructions, and reproducibility scripts.