Stationary-weight witnesses at eight states and positive uniform stationary bias

Determine whether a synchronizing strongly connected automaton on eight states has a proper stuck subset S with Friedman's centered stationary functional B_e(S)≥0, and determine whether B_e(S)>0 can occur for a stuck subset at equal letter weights.

Background

The stationary-vector functional B_e assigns a centered stationary weight to subsets. Unlike the defect-based certificate B, B_e can be nonnegative—and even positive—on stuck subsets, so it does not generally certify extension within n−1.

Exhaustive searches found no such subset for n≤7, while explicit examples exist for n=9 and beyond under some conditions. The paper therefore isolates the eight-state case and the possibility of a strictly positive value at equal letter weights as unresolved questions.