What was already known
Dogon, Levit and Vigdorovich introduced stability-radius growth and sought an explicit bound for an infinitely presented group. Dogon and Vidick first had an exponential lamplighter bound from greedy marker sets.
AI involvement
After Dogon and Vidick supplied a marker lemma, ChatGPT 5.5 found a polynomial improvement. The authors recognized a broader lemma in prior techniques, generalized the construction and derived the final bound.
The disclosed model step changes the authors’ exponential marker approach to polynomial, while the authors supplied the problem, prior lemma, generalization and final argument.
Importance
The polynomial construction gives the first explicit stability-radius upper bound for an infinitely presented group.
Limitations
The published Lean comparison uses M=ceil(C κ^-20 log(2/κ)) for κ≤1/2, while the paper states a looser κ^-21 bound; their precise bridge merits review. The model’s separate Appendix A lower bound is outside this assessment.
Technical verification
The authors report a completed Lean 4 formalization of Theorem 1.1 and publish a theorem/comparator comparison. It supports the polynomial stability conclusion; the comparator uses M=ceil(C κ^-20 log(2/κ)) for κ≤1/2, whereas the paper states the looser M=ceil(C κ^-21). The build and bridge were not replayed here.
Formal checking
Completed formalization of Theorem 1.1 over 14 partial days and 64 prompts, with public statement comparison and Comparator bridge
- Checker
- Lean 4 formal development prompted by Dogon and Vidick through Aristotle
- Scope
- Polynomial Hilbert–Schmidt stability of the lamplighter group for approximate unitary relations
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The human authors explain the polynomial marker lemma through descriptive-combinatorics coloring and derive tower decomposition and stability from it in §§5–7.
Correction and publication history
The polynomial construction gives the first explicit stability-radius upper bound for an infinitely presented group.
Read this published revision