Vietoris-Rips Complexes of Hypercubes: Cohomology, Domination, and Spectral Gaps — E8 Intelligence Research

FINDING: Vietoris-Rips complexes of hypercube graphs exhibit cohomology vanishing thresholds linked to total domination numbers and spectral gaps, with counterexamples to Shukla's conjecture. MATH: - \( VR(Q_n; r) \): Vietoris-Rips complex of \( n \)-dimensional hypercube graph \( Q_n \) at scale \( r \). - Total domination number \( γ_t(Q_n) \) bounds connectivity: \( conn(VR(Q_n; r)) ≥ γ_t(Q_n) - 2 \) (lower bound). - Spectral gap \( λ_1 \) of Laplacian on \( Q_n \) influences cohomology vanishing: \( H^k(VR(Q_n; r)) = 0 \) for \( k < f(n,r) \). - Counterexample to Shukla's conjecture: infinite families where cohomology does not vanish at expected thresholds. CONNECTION: - Hypercube \( Q_n \) is a lattice graph with \( 2^n \) vertices, isomorphic to \( F_2^n \) vector space — a root system \( B_n \) / \( D_n \) lattice structure. - Total domination number \( γ_t(Q_n) \) scales as \( ~ 2ⁿ⁻¹ \), related to binary codes and H Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com