Dark Matter as Topological Knots in the Internal S^3 Fibre of the Zhang Effective Metric

The nature of dark matter remains one of the most profound mysteries in modern physics. While weakly interacting massive particles (WIMPs) and axions are leading candidates, no definitive experimental signature has been found. In this paper, we propose that dark matter consists of stable topological knots (Skyrmions) in the internal S³ fibre of the 15-dimensional manifold within the rotational-order geometric unification framework. These topological defects are protected by a conserved topological current, rendering them absolutely stable. By performing dimensional reduction from a 15D non-linear sigma model, we derive the dark matter mass M_DM ~ (f_4/e_4)B. The internal S³ fibre is a strict gauge singlet, so the Wess-Zumino-Witten term vanishes, rendering the knots electromagnetically invisible. The relic abundance is naturally set by the Kibble-Zurek mechanism during the early-universe phase transition. We show that with a natural correlation length, the Kibble-Zurek mechanism produces topological knots with a mass around the electroweak scale, perfectly matching the observed cosmological dark matter density Ω_DM ≈ 0.26 without invoking new fundamental scalar fields.