Exact Avoid-or-Compress Extrema for Dense Reachable Images: Defects Four and Five

Abstract

Let $S$ be a reachable image of a strongly connected synchronizing automaton and $\varnothing\ne A\subsetneq S$. We determine the maximum shortest avoid-or-compress length over binary and arbitrary finite alphabets for $m=n-|S|=4$ and $m=5$. The exact values are $\min(n-|A|,11)$ and $\min(n-|A|,14)$, respectively. Uniform binary constructions attain both bounds. For the upper bounds, short-pair graphs and actual-image constraints reduce all parameters to finite necessary domains. Exhaustive checks exclude the domains, apart from a single relaxation eliminated by a multipartite overlap lemma. The largest calculation covers 3,206,988 domains.