What was already known
Gordon’s earlier fixed-size family used thirteen relators.
AI involvement
The paper credits GPT-5.6 Sol Pro with the main HNN and Tietze construction. Kegel, Li and Ren directed and checked the work; Codex developed the Lean formalization.
The paper credits GPT-5.6 Sol Pro with the central HNN and Tietze route used for this two-generator, ten-relator family. Kegel, Li and Ren directed and checked it.
- Supporting source (github.com) — exact public declarations and presentation-slot interpretation
Importance
Lowers the known relator count for this two-generator undecidable-triviality family to ten.
Limitations
The Lean comparison covers the stated algebraic family, not the paper’s separate four-manifold consequences or minimality.
Technical verification
The authors report independent checking of Theorem 1.4'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 1.4 within the three-theorem check.
- Checker
- Marc Kegel, Shana Yunsheng Li and Qiuyu Ren
- Scope
- Kegel, Li and Ren give an effective family of finite presentations with two generators and ten relators for which deciding triviality is impossible. 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 1.4 within the three-theorem check.
- Checker
- Lean 4 kernel, Comparator and the authors' transitive axiom audit
- Scope
- Kegel, Li and Ren give an effective family of finite presentations with two generators and ten relators for which deciding triviality is impossible. 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 1.4: 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 two-generator, ten-relator triviality family has its own presentation bound.
Read this published revision