What was already known
Tancer’s earlier Adian–Rabin family had deficiency nine; the new four-generator, eleven-relator family has deficiency seven.
AI involvement
GPT-5.6 Sol Pro found two further Tietze eliminations and identified issues in the Tancer route. Kegel, Li and Ren directed, checked and reorganized the argument; Codex built its Lean proof.
The paper credits GPT-5.6 Sol Pro with the two further Tietze eliminations behind this four-generator, eleven-relator family. Kegel, Li and Ren directed and checked it.
- Supporting source (github.com) — exact public declarations and presentation-slot interpretation
Importance
Improves a fixed-size construction for the undecidable triviality problem.
Limitations
The formal comparison concerns the algebraic family. The paper’s topological applications and optimality of the counts are separate questions.
Technical verification
The authors report independent checking of Theorem 1.3'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 1.3 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 four generators and eleven relators for which deciding triviality is impossible. These are fixed presentation slots, not a minimality result.
- Basis
- Reported by the linked checker
Formal checking
complete formal proofs, statement comparison, kernel check and axiom audit. This assessment concerns Theorem 1.3 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 four generators and eleven relators for which deciding triviality is impossible. These are fixed presentation slots, not a minimality result.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Theorem 1.3: 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 four-generator, eleven-relator triviality family has its own presentation bound.
Read this published revision