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
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
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
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
Checking a formal statement does not establish that it matches the original problem.
Human understanding
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
Provides a distinct certificate-based proof of the matching 1/2 upper bound.
Read this published revision