Betti-three and Betti-four pattern for the 5k family

Determine whether, for every k≥3, the complex VR(T_{5k,5k},2k) has beta_3=1 and beta_4=10k^2.

Background

The paper reports computational confirmation for 3≤k≤5. The proposed formulas describe a persistent third-dimensional class together with quadratic growth of the fourth Betti number at the corresponding scales.

References

Is it true that for $k\geq 3$, the complex $VR{T_{5k,5k}{2k}$ has $\beta_3(VR{T_{5k,5k}{2k})=1$ and $\beta_4(VR{T_{5k,5k}{2k})=10k2$?

— Vietoris-Rips complexes of torus grids  (2502.07134 - Adams et al., 10 Feb 2025) in Question 12, Section Conclusion