Topological complexity two in the homogeneous lower model

In the lower multiparameter model of random simplicial complexes, a simplex is retained when all its nonempty subfaces pass independent Bernoulli trials. We prove that, when every trial has the same fixed probability in $(0,1)$, the resulting complex is path-connected and has reduced topological complexity two with probability tending to one along all ambient sample sizes. Thus three continuous local motion-planning rules are necessary and sufficient. This removes the density-one restriction in the result of Barmak and Farber. The new step is an anti-concentration estimate for the Euler characteristic: after the other variables are exposed, many remaining face trials independently add or remove a single maximal simplex.