--- id: "2026-10-28-laczkovich-lipschitz-mapping-problem-chatgpt-gaspar" url: "https://postcutoff.com/e/2026-10-28-laczkovich-lipschitz-mapping-problem-chatgpt-gaspar/" as_of: "2026-10-30T13:42:00+01:00" date: "2026-10-28" date_precision: day category: science importance: 4 confidence: medium status: [Awaiting review] verification: Preprint only sources: 1 editor: human editor human_review: null version: "2026-10-30" --- As of: 2026-10-30 13:42 CET. Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/ Canonical page: https://postcutoff.com/e/2026-10-28-laczkovich-lipschitz-mapping-problem-chatgpt-gaspar/ # Laczkovich's Lipschitz mapping problem: claimed answer, ChatGPT key step Full title: Attila Gáspár claims a positive answer to Laczkovich's 1991 Lipschitz mapping problem in every dimension; the key proposition is 'entirely due to ChatGPT' (26 Oct) arXiv 2610.38834 (28 Oct 2026), "Solution of Laczkovich's Lipschitz mapping problem" by Attila Gáspár, proves that for every measurable A ⊆ ℝ^d with positive measure there is a 1-Lipschitz map f: ℝ^d → ℝ^d with f(A) having non-empty interior, so A maps onto a cube. This was Laczkovich's 1991 question, known for d = 2 (Preiss; Matoušek's constant version; Jones via Uy). The paper also answers Kolmogorov's problem on contractions onto polygons with small measure loss for Jordan measurable sets. The decisive proposition was produced by ChatGPT two days before posting. ## Key facts - Claim (abstract): 'In 1991, Laczkovich asked whether every measurable set of positive measure in ℝ^d can be mapped onto a cube with a Lipschitz map. We show that the answer is positive for every d.' - AI disclosure (verbatim): 'The proof of Theorem 1.8 was obtained using ChatGPT on 26 October. In particular, the proof of Proposition 2.8, including the statement and proof of Lemma 2.7, is entirely due to ChatGPT. This proof was rewritten by the author to make it easier to understand. The proof of Proposition 2.9 was also written with the help of ChatGPT.' The lower bound for γ(C) (Prop. 2.5) and the consequences of equality in Theorem 1.8 were the author's own, from early 2026, without AI - On the model's limits (verbatim): 'When asked to give lower bounds for the value of γ for specific sets, ChatGPT did not come up with this lower bound, though it could prove it given the statement.' - Prior state (paper): d = 1 is elementary; d = 2 by Preiss, then Matoušek (an absolute constant c with a 1-Lipschitz surjection A → [0, c]² when λ(A) = 1, via Erdős–Szekeres) and Jones (via Uy's theorem); 'until now, the case d ≥ 3 was open'. The proof uses the author's theory of generalized strips (proximal maps of convex functions) - Also answers Question 4.2 of an earlier paper: an ε-perturbation of the identity maps A into finitely many hyperplanes; and Kolmogorov's problem for Jordan measurable sets - Verification status: arXiv preprint, posted two days after the AI step; not refereed, no formal proof; no expert reaction or HN thread found on 30 Oct ## What happened Miklós Laczkovich asked in 1991 whether any measurable set of positive volume can be squeezed by a Lipschitz map onto a full cube. The answer was known to be yes in the plane, through Preiss and later Matoušek, but open in three dimensions and above. Attila Gáspár posted a proof for every dimension on 28 Oct. He states that the main inequality's proof came from ChatGPT on 26 Oct and that one proposition and its lemma are entirely the model's; he rewrote them and supplied the lower bound and the framework himself. ## Why it matters It is a classic question of geometric measure theory with a clear statement, and the paper separates the AI's step from the author's earlier work in unusual detail, including what the model failed to find. It was posted two days after the AI step and still needs checking. ## What is disputed or not yet verified - Verification: Preprint; not refereed ## Your AI and this story - GPT-6 Astra (training cutoff April 2026): 181 days after its cutoff - Claude Opus 5.5 (training cutoff June 2026): 120 days after its cutoff - Gemini 4 Argon (training cutoff March 2026): 211 days after its cutoff - Grok 4.7 (training cutoff May 2026): 150 days after its cutoff ## Sources 1. [arXiv 2610.38834: Solution of Laczkovich's Lipschitz mapping problem (Gáspár)](https://arxiv.org/abs/2610.38834) (arxiv.org, paper) ## Changes - 2026-10-30 (filed): Created (science agent; arXiv abstract and PDF read, AI disclosure quoted) ## Related - 2026-10-30: [AI-assisted results, 30 Oct: arXiv announcement of 29–30 Oct](https://postcutoff.com/e/2026-10-30-ai-assisted-results-october-30-listing/index.md)