{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-25-golomb-welch-conjecture-perfect-lee-codes-chatgpt/","md":"https://postcutoff.com/e/2026-10-25-golomb-welch-conjecture-perfect-lee-codes-chatgpt/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-25-golomb-welch-conjecture-perfect-lee-codes-chatgpt","date":"2026-10-25","date_precision":"day","short_title":"Golomb–Welch conjecture (perfect Lee codes) claimed, with ChatGPT","deck":null,"takeaway":"Bocong Chen, Jing Huang and Gaojun Luo posted two arXiv papers on 25 Oct 2026. They claim that Lee balls of radius r ≥ 2 never tile ℤⁿ for n ≥ 3, first for lattice tilings and then for arbitrary centre sets. That is the full Golomb–Welch conjecture from around 1970, the central open problem on perfect Lee codes. Both papers say ChatGPT helped explore proof strategies and check calculations. They are unrefereed, and no expert has commented yet.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.34291: The Golomb–Welch conjecture for arbitrary centre sets","url":"https://arxiv.org/abs/2610.34291","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"arXiv 2610.33983: The Golomb–Welch conjecture for lattice tilings","url":"https://arxiv.org/abs/2610.33983","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"Horak, Kim (2017): 50 Years of the Golomb–Welch Conjecture (survey)","url":"https://arxiv.org/abs/1706.03589","type":"paper","group":"primary","domain":"arxiv.org"}],"official":3,"filed":"2026-10-27","updated":"2026-10-27","orgs":["South China University of Technology","Guangzhou University","Nanjing University of Aeronautics and Astronautics","OpenAI"],"title":"Golomb–Welch conjecture claimed in full: no perfect Lee codes in ℤⁿ for n ≥ 3 and radius r ≥ 2, lattice and arbitrary-centre versions (Chen, Huang, Luo; ChatGPT used for proof strategies)","summary":"arXiv 2610.33983 (cs.IT) proves the lattice form: for n ≥ 3 and r ≥ 2 ℤⁿ has no lattice tiling by Lee balls B_n(r), so an abelian Cayley graph of even degree 2n ≥ 6 and diameter r ≥ 2 has fewer than |B_n(r)| vertices. arXiv 2610.34291 (math.CO) removes the lattice assumption: no tiling by translates at all, \"resolving the Golomb–Welch conjecture for arbitrary centre sets and completing the existence classification of perfect Lee codes in ℤⁿ\". Methods: translation-invariant averaging, boundary contact identities, integer moment inequalities and coefficient comparisons in stable symmetric functions. Both acknowledge ChatGPT for proof strategies, intermediate calculations and editing.","key_facts":["Authors: Bocong Chen (South China University of Technology), Jing Huang (Guangzhou University), Gaojun Luo (Nanjing University of Aeronautics and Astronautics). Chen and Huang posted the even-characteristic MDS conjecture proof on 20 Oct with an identically worded ChatGPT statement","Theorem 1.1 (2610.34291): 'For all integers n ≥ 3 and r ≥ 2, there is no tiling of ℤⁿ by translates of the Lee ball B_n(r).' It needs 'no linearity, periodicity, or density assumption on the centre set'. Together with Golomb and Welch's constructions (n ≤ 2 any r; r = 1 any n) this classifies all perfect Lee codes in ℤⁿ","AI statements (verbatim): 2610.34291: 'The authors used ChatGPT to assist with exploring proof strategies, checking intermediate calculations, and editing the exposition. The authors take full responsibility for the content of this work.' 2610.33983 adds: 'All results and proofs were written by the authors'","Prior state (from the paper's introduction): sufficiently large radii excluded in each fixed dimension (Golomb–Welch, Horak–Kim); dimension 3 settled (Gravier–Mollard–Payan), dimensions 3–5 (Horak); radius-2 lattice tilings excluded in every dimension n ≥ 3 (Leung–Zhou); a 2017 survey ('50 Years of the Golomb–Welch Conjecture', Horak and Kim) called the conjecture far from solved","Size: long, technical preprints (thousands of lines of text each in the PDF). Consequences listed: finite local obstructions, Banach density gaps for packings and coverings, results on abelian Cayley graphs, lattice cubature and orthogonal exponential bases","Status: two unrefereed arXiv preprints posted the same day, no formal proof, no expert reaction found on 27 Oct"],"key_numbers":[],"tags":["math","combinatorics","coding-theory","tilings","golomb-welch","perfect-codes","chatgpt","arxiv"],"science":{"field":"mathematics","subfield":"combinatorics / coding theory (perfect Lee codes, tilings of ℤⁿ)","problem":"Golomb–Welch conjecture (c. 1968–1970): no perfect r-error-correcting Lee code in ℤⁿ for n ≥ 3, r ≥ 2","result":"Claimed full proof, in both the lattice and the arbitrary-centre forms","open_since":"1970","ai_system":["ChatGPT"],"human_role":"AI-assisted: authors used ChatGPT to explore proof strategies and check calculations; the authors say they wrote all results and proofs","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed arXiv preprints (2610.33983, 2610.34291); no formalization"},"status":"pending","shock":""},"body_md":"## What happened\n\nA perfect Lee code is a set of points of ℤⁿ such that the \"diamond\" balls of radius r around them (in the taxicab distance)\ncover every lattice point exactly once. Golomb and Welch found such codes in the plane for every radius and in every dimension\nfor radius 1. They conjectured that there are no others. For about fifty years work on the problem went one dimension, one\nradius or one special family of tilings at a time.\n\nThe two new papers claim the whole thing. The first handles tilings whose centres form a lattice, which is the case coding\ntheorists care most about. The second handles arbitrary centre sets. Both thank ChatGPT for help exploring proof strategies and\nchecking intermediate calculations. They say the authors wrote the proofs.\n\n## Why it matters\n\nThe Golomb–Welch conjecture is the best-known open problem on codes in the Lee metric and a classic tiling question. If the\npapers hold up, they close it. Two of the authors (Chen and Huang) posted a proof of the even-characteristic MDS conjecture a week\nearlier, also written with AI help. So this is one of several groups now settling long-standing coding-theory conjectures in quick succession\nwith AI assistance. Until specialists check them, treat these as claims.","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprints (2610.33983, 2610.34291); no formalization","sources":[]}],"related":[{"id":"2026-10-27-ai-assisted-results-october-27-listing","url":"https://postcutoff.com/e/2026-10-27-ai-assisted-results-october-27-listing/","date":"2026-10-27","date_precision":"day","short_title":"AI-assisted results in the 27 Oct 2026 sweep","deck":"Muckenhoupt's C_p conjecture, Graham's rearrangement conjecture in abelian groups, a quasi-polynomial χ-bound for P₆-free graphs, Haagerup's 1975 problem, Langberg–Médard, and proofs whose main idea came from Claude or GPT","takeaway":"Tuesday's arXiv announcement (papers submitted 23–26 Oct) was the largest so far for AI-assisted mathematics. Four results have their own entries: Golomb–Welch, 3-coloring with O(n^0.1537) colors, Balko's exposition of OpenAI's Harary–Hill proof, and Ando–Umehara on Alpöge's smooth Carathéodory counterexamples. This page lists the next tier, from 46-year-old conjectures where ChatGPT supplied an alternative argument to short papers where Claude or GPT proposed the main idea.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":22,"official":22,"filed":"2026-10-27","updated":"2026-10-27","orgs":["OpenAI","Anthropic","Google DeepMind","various researchers"]},{"id":"2026-10-20-mds-conjecture-even-characteristic-chatgpt-claude-lean","url":"https://postcutoff.com/e/2026-10-20-mds-conjecture-even-characteristic-chatgpt-claude-lean/","date":"2026-10-20","date_precision":"day","short_title":"MDS conjecture settled in even characteristic","deck":"Two independent proofs, one checked in Lean with Claude's help, the other written with ChatGPT","takeaway":"If it holds, this is the biggest advance on the conjecture since Ball's prime-field theorem.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-21","updated":"2026-10-21","orgs":["South China University of Technology","Guangzhou University","OpenAI","Anthropic"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-27","type":"filed","text":"Created (science agent; both PDFs read with pdftotext; arXiv listing of 27 Oct)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-27","run":null,"sources_read":"Science agent; both PDFs read with pdftotext; arXiv listing of 27 Oct","updated":"2026-10-27","human_review":null,"version":{"date":"2026-10-27"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":178,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":117,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":208,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":147,"in_training_data":false}],"short_url":"https://postcutoff.com/s/golomb-welch-conjecture-perfect"}