Discoveries / theoretical-computer-science

Source authenticated

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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: A uniform bound of three on generalized star height (the generalized star-height problem) · Generalized star height at most three

Confidence: Not scored

Registry verification: lean checked · announcement · partial

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

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

specialist registry · passed

No independent mathematician has checked this yet. Theorem 1.1 of the principal manuscript ( $h(L)\le3$ for every regular $L$ over every finite alphabet, complement in the same free monoid) was read against the problem: it answers uniform boundedness and leaves open whether height one suffices, as the paper says. The Lean challenge ComparatorChallenges/GeneralizedStarHeight.json (theorem OAI.GeneralizedStarHeight.main, solution module OAI.Computability.StarHeight.Main, present at the pinned commit) is not in the formalization catalogue formalization.yaml; it was found through lean/docs/134.md. Its statement was read here: an inductive expression type with zero, one, letters, union, concatenation, complement and star, height as defined in the paper, and for a finite alphabet every regular language (Mathlib's IsRegular) is the language of some expression of height at most three. That is the headline claim. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here.

Source ↗

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

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Companion: Finite Monoid Computations and a Uniform Generalized Star-Height Bound (bound 13)

paper · pending

paper link supplied by the VibeMathed registry.

Source ↗

Companion: Generalized Star Height at Most Four

paper · pending

paper link supplied by the VibeMathed registry.

Source ↗

Lean: solution module for the height-three bound

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Lean comparator statement: generalized star height at most three

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Straubing (2002), On logical descriptions of regular languages

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Lean: solution module for the height-three bound

lean artifact · pending

Artifact ↗

Lean comparator statement: generalized star height at most three

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

VibeMathed record: A uniform bound of three on generalized star height (the generalized star-height problem) evidence for this event

This event attributed to Unreleased internal OpenAI model

A uniform bound of three on generalized star height (the generalized star-height problem) evidence for this event

Lean: solution module for the height-three bound evidence for this event

Companion: Generalized Star Height at Most Four evidence for this event

Straubing (2002), On logical descriptions of regular languages evidence for this event

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

Lean comparator statement: generalized star height at most three evidence for this event

Companion: Finite Monoid Computations and a Uniform Generalized Star-Height Bound (bound 13) evidence for this event

Act on this frontier

Verify, challenge, or extend the result.

Help verify Open canonical problem Explore evidence graph