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The uniform Turán density of the tetrahedron

Matija Bucić

In Bucić’s alternative proof that the uniform Turán density of the three-uniform tetrahedron K₄³ is 1/2, ChatGPT 6 Pro found the finite sum-of-positive-quadratic-forms identity and proposed the tensor projection inequality used in the proof.

Sources for this record

What was already known

Rödl gave the 1/2 lower bound. Kielak and coauthors obtained an independent computer-assisted upper bound without AI-assisted arguments before Bucić’s model-assisted proof was complete.

AI involvement AI-centered

Bucić guided the discussion and developed the reduction. ChatGPT 6 Pro found the retained identity and suggested the tensor inequality; programs generated the positive-form expansion.

The retained certificate identity and tensor inequality came from the model, within a human-guided proof strategy and human-authored mathematical reduction.

Importance Notable

Provides a distinct certificate-based proof of the matching 1/2 upper bound.

Limitations

Bucić supplied the reduction and hand-checked the exact verifier. A Lean audit checked the verifier, not the large identity directly. The independent proof has separate authorship and priority for the density value.

Technical verification Formal (Exact integer certificate computation; Lean audit of verifier code + human)

The supplied exact verifier checks identity (6) and positivity of seven integer matrices, the finite step completing the written reduction. Bucić hand-checked the verifier and reports a Lean audit of it; Lean did not directly check identity (6).

Formal checking

The supplied exact verifier checks identity (6) and positivity of seven integer matrices, the finite step completing the written reduction. Bucić hand-checked the verifier and reports a Lean audit of it; Lean did not directly check identity (6).

Checker
Supplied exact verifier, as reported by Matija Bucić
Scope
Identity (6) and positive definiteness of its seven integer matrices
Basis
Reported by the linked checker

The supplied exact verifier checks identity (6) and positivity of seven integer matrices, the finite step completing the written reduction. Bucić hand-checked the verifier and reports a Lean audit of it; Lean did not directly check identity (6).

Checker
Matija Bucić
Scope
Correctness of the finite-certificate verifier used in the tetrahedron proof
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Partial

Bucić explains the palette reduction and why certificate nonnegativity gives the bound, but explicitly says computer assistance obscures the underlying mathematical reasons for the identity.

Correction and publication history