Betti Numbers of Vietoris-Rips Complexes of Hypercube Graphs in Degrees 7 and 15 on a Single Laptop

The Vietoris-Rips complexes of hypercube graphs under Hamming distance were recently studied by Galetto, Montaño, and Wellner (arXiv:2606.20784), who describe their homology as representations of the hyperoctahedral group. The one complete homology computation in that work, the 64-vertex complex X(6,4) with 2,932,100,732 simplices and top dimension 21, required approximately 18 hours and at least 800 GB of memory on a university cluster, and the authors describe their approach as impractical for n ≥ 8. This paper reports the same Betti numbers, β7 = 239 and β15 = 14, computed on a single laptop in 14 minutes with a peak of 17.5 GB, and computes the Betti numbers of X(8,3) on 256 vertices directly, β4 = 351 and β7 = 1,120, in under 90 seconds each. All eleven computations reported match the published theorems. The Euler characteristic of the enumerated complex, −252 = 1 − 239 − 14, confirms independently that the object computed is the one in the literature. Computations use Ordinal Locality Filtration, introduced in Paper I of this series, which consults no metric and no threshold. Companion to Persistent Homology from H0 Through H4, and at 500,000 Points.