Stability Radius of the Lamplighter Group
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Statement
Dogon, Levit and Vigdorovich asked for an explicit upper bound on the stability radius of an infinitely presented group. The lamplighter group provides the first: explicit polynomial bounds on both its Hilbert-Schmidt stability rate and its stability radius, obtained through approximately invariant measures and an effective marker construction.
Context
A question from a recent paper on quantitative group stability, with a real but small readership in approximate representation theory.
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The disclosure separates the mathematics from the formalization. The authors had an exponential bound with a greedy marker construction; on being given the marker lemma, ChatGPT 5.5 produced the polynomial improvement, which is the paper's headline. The authors then recognized that the polynomial marker lemma follows from known descriptive-combinatorics techniques and holds for general group actions. The model also supplied the statement and proof of the Appendix A lower bound. Separately, after the paper was complete, Aristotle auto-formalized the main theorem in Lean over 64 prompts and roughly 14 partial days.
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