OpenAI · 134-02 · 2026-09-25 · Theoretical computer science
Generalized Star Height at Most Four
preprints/Generalized-Star-Height-at-Most-Four-September-25-2026
Family 134. Generalized star height at most three.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Every regular language over a finite alphabet has a generalized regular expression with at most three nested Kleene stars, allowing union, concatenation and complement over the same alphabet. This establishes an absolute bound independent of automaton size, resolving the uniform-boundedness version of the generalized star-height problem.
Every regular language over a finite alphabet has generalized star height at most four over that same alphabet. We give a complete construction using an affine correction that hides one interval product, a finite clock, and several scales for moving boundaries through periodic words.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 134: Generalized star height at most three. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.