What was already known
Borisov’s smallest previously known finitely presented example used twelve relators.
AI involvement
GPT-5.6 Sol Pro developed the HNN-extension and Tietze-reduction construction under human direction. Codex developed the Lean proof. Kegel, Li and Ren checked and reorganized the mathematics.
The paper credits GPT-5.6 Sol Pro with the central HNN-extension and Tietze-reduction construction for the nine-relator result. Kegel, Li and Ren directed and checked it.
- Supporting source (github.com) — exact public declarations and presentation-slot interpretation
Importance
Reduces the known relator count for an explicit group with unsolvable word problem from twelve to nine.
Limitations
The Lean comparison covers the algebraic theorem, not the paper’s later four-manifold deductions or a claim of minimality.
Technical verification
The authors report independent checking of Theorem 1.1's mathematical argument. The cited Lean revision reports a complete proof, a standard-axiom audit and Comparator acceptance for that algebraic theorem.
Independent checking
independent checking and reorganization of the arguments, manual statement checking and final argument review. This assessment concerns Theorem 1.1 within the three-theorem check.
- Checker
- Marc Kegel, Shana Yunsheng Li and Qiuyu Ren
- Scope
- Kegel, Li and Ren give a finite presentation with three generators and nine relators whose word problem is unsolvable. The count is an achieved presentation size, not a minimum.
- Basis
- Reported by the linked checker
Formal checking
complete formal proofs, statement comparison, kernel check and axiom audit. This assessment concerns Theorem 1.1 within the three-theorem check.
- Checker
- Lean 4 kernel, Comparator and the authors' transitive axiom audit
- Scope
- Kegel, Li and Ren give a finite presentation with three generators and nine relators whose word problem is unsolvable. The count is an achieved presentation size, not a minimum.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Theorem 1.1: The named authors independently reorganized and checked the argument and explain the HNN and Tietze route in §§1.3–1.5 and the body of their paper.
Correction and publication history
The nine-relator word-problem construction is distinct from the paper’s two undecidable-triviality families.
Read this published revision