Nonexistence of a covering of sequential update modes for Rule 37

Show that no covering set of sequential update modes exists for elementary cellular automaton Rule 37; concretely, prove that there is no set S of sequential update orders such that for every initial configuration (in particular for ring lengths n that are multiples of three, the only case where fixed points exist) at least one order in S yields convergence to a fixed point.

Background

Rule 37, like Rule 45, possesses fixed points only when the ring size is a multiple of three, namely (001)^{n/3}, (010)^{n/3}, and (100)^{n/3}. However, simulations conducted for n=6 and n=9 identified specific configurations that failed to converge to a fixed point under any sequential update order, suggesting the absence of a covering.

A covering (as defined earlier in the paper) is a set of sequential update modes with the property that, for every initial configuration, there exists at least one mode in the set that leads to a fixed point. The conjecture asserts such a set does not exist for Rule 37.