Cohomology Vanishing in Hypercube Vietoris-Rips Complexes — E8 Intelligence Research

FINDING: Vietoris-Rips complex of hypercube graph \( Q_n \) exhibits cohomology vanishing thresholds linked to total domination number and spectral gap, with infinite families countering prior conjectures. MATH: - \( VR(Q_n; r) \): Vietoris-Rips complex of \( n \)-hypercube graph, scale parameter \( r \). - Cohomology vanishing threshold: \( r \approx \lfloor n/2 \rfloor \) or related to domination number \( \gamma(Q_n) \). - Total domination number \( \gamma_t(Q_n) = 2^{n-1} \) for \( n \geq 2 \) (known). - Spectral gap of \( Q_n \): \( \lambda_2 = 2 \) (Laplacian eigenvalues: \( 2k, k=0..n \)). - Counterexample families: specific \( n, r \) where cohomology does not vanish as Shukla predicted. - No explicit constants like 0.618 appear; instead, thresholds scale with \( n \) and binary structure. CONNECTION: - Hypercube \( Q_n \) is a lattice graph (cubic lattice in \( n \)-dim binary space), related to root system \( B_n \) and \( \mathbb{Z}_2^n \) group. - Vietoris Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com