An Eight-Vertex Counterexample to Initial Two-Step Stationarity in Majority Dynamics

Unrefereed preprint. We exhibit an explicit connected eight-vertex, thirteen-edge, all-odd-degree graph and an initial configuration for synchronous neighbour-majority dynamics such that no vertex satisfies the two-step relation at every nonnegative time. This is the negation of the all-time initial-stationarity question in Section 6 of Benjamini, Zakharov and Zhukovskii (2026). It does not refute the existence of an even-time 0-stable vertex. Reproducible exhaustive calculations show vertex-minimality within the all-odd-degree model, edge-minimality among eight-vertex graphs, and one equivalence class of graph–initial-state pairs under graph relabelling and global sign reversal at (n,m)=(8,13). The finite witness and the computational evidence are included. Novelty and priority are not established; this work has not been peer reviewed. All rights reserved. Code and evidence: https://github.com/yonezaemon1-hub/no34-initial-two-step-stationarity Research Inquiries — Yonezu Research LabContact: yonezaemon1+research@gmail.com