A uniform bound of three on generalized star height (the generalized star-height problem)

Claims Theorem 1.1: for every finite alphabet $\Sigma$ and every regular $L\subseteq\Sigma^*$, $h(L)\le3$, using expressions over $\Sigma$ itself with complement in $\Sigma^*$. Companions prove the weaker bounds 13 and 4 by different constructions. It does NOT exhibit any language of generalized star height greater than one, does not decide whether height one always suffices, gives no algorithm for computing $h(L)$, and bounds neither expression size nor running time.

Claims Theorem 1.1: for every finite alphabet $\Sigma$ and every regular $L\subseteq\Sigma^*$, $h(L)\le3$, using expressions over $\Sigma$ itself with complement in $\Sigma^*$. Companions prove the weaker bounds 13 and 4 by different constructions. It does NOT exhibit any language of generalized star height greater than one, does not decide whether height one always suffices, gives no algorithm for computing $h(L)$, and bounds neither expression size nor running time.