Source authenticated

The uniform Turán density of the tetrahedron

$\pi_u(K_4^{(3)}) = 1/2$ , matching the long-known lower-bound construction. The AI contribution is one identity inside the argument rather than the argument, which is why this is recorded as AI-assisted; it is included in the catalog because the identity is a step of the proof, not a matter of exposition.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 10, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: preprint. AI contribution: ai-assisted. 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: The uniform Turán density of the tetrahedron · Uniform Turán density of $K_4^{(3)}$

Confidence: Not scored

Registry verification: unreviewed · preprint · resolved

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Attribution

VibeMathed
registry · event recorded by

ChatGPT 6 Pro
model · ai model contributor · OpenAI

Aristotle
model · ai model contributor · Harmonic

Matija Bucić
human · human collaborator

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: The uniform Turán density of the tetrahedron

specialist registry · passed

Checked here on 22 September 2026 against arXiv:2609.11802: the abstract states the value $1/2$ and that it answers a question of Erdős and Sós from the founding 1982 paper, and the introduction names that paper and the companion question about $K_4^{(3)-}$ . The AI declaration is quoted as written. The mathematics was not checked here. Aristotle's Lean check covers the computational verifier used inside the proof, not the proof as a whole, so this is not a formalisation of the theorem and the entry does not claim one. Twelve days old, no referee.