Generalized Star Height One for Computations via Run Padding

This paper was motivated by OpenAI's recent release of mathematical preprints, where they proved every regular language over a finite alphabet has GSH at most three.

In short, we investigate a complementary height-one phenomenon. We show that padding maximal letter-runs in finite-state computations yields languages of generalized star height at most one whenever each letter acts with at most one nontrivial cycle. We construct a common star-free generator, establish a converse for this representation, and obtain explicit examples involving the nonsolvable group A₅. We also investigate limitations of certain single-star expressions and show how, for suitable permutation actions, padding can be performed entirely over the original alphabet.

I've added some interesting open questions at the end.

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