What was already known
The exact Product Conjecture predicted join structure for every extremizer when a fixed forbidden family has polynomially superlinear surplus. Earlier theorems gave asymptotic structure.
AI involvement
Xu credits GPT-5.6 Sol with the counterexample and initial draft. Xu verified and revised the proof; GPT-6 Astra later formalized it in Lean.
GPT-5.6 Sol is credited with generating the central counterexample; the author performed verification and revision. The source does not establish near-zero human mathematical direction required for AI-autonomous.
Importance
Refutes the exact Product Conjecture and its related decomposition-family prediction.
Limitations
The construction does not refute the weaker claim that some extremizer at each order is a join. The Lean build and audit are source-reported.
Technical verification
The paper reports Xu's verification, and the released Lean package records a successful fresh build and axiom audit of Theorem 3.1 and both corollaries, with source correspondence to the September 6 revision. The build and audit are source-reported.
Formal checking
Theorem 3.1 for the fixed O6 repair obstruction family and its exact non-join extremizer and surplus conclusions. Fresh lake build and Audit.lean proof-interface and transitive-axiom checks
- Checker
- Lean 4 kernel, in the standalone package build reported by Xu
- Scope
- Theorem 3.1 for the fixed O6 repair obstruction family and its exact non-join extremizer and surplus conclusions
- Basis
- Reported by the linked checker
Formal checking
The paper's repaired-obstruction counterexample to the exact Product Conjecture. Author verification and revision of the model-generated manuscript
- Checker
- Chuandong Xu, the human author
- Scope
- The paper's repaired-obstruction counterexample to the exact Product Conjecture
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Xu explains endpoint-injective repair, the extremal formula and the connected-complement witness, and documents the exact distinction from the weak conjecture.
Correction and publication history
Gives a counterexample to the exact Product Conjecture.
Read this published revision