{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-29-petty-equilateral-set-conjecture-dimension-five-tencent-hyprover/","md":"https://postcutoff.com/e/2026-10-29-petty-equilateral-set-conjecture-dimension-five-tencent-hyprover/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":"2026-10-29-petty-equilateral-set-conjecture-dimension-five-tencent-hyprover","date":"2026-10-29","date_precision":"day","short_title":"Petty's conjecture in dimension 5: proof by Tencent's hyprover agent","deck":null,"takeaway":"Petty conjectured that every n-dimensional normed space contains n + 1 points at equal distances. Petty proved dimension 3 and Makeev dimension 4. On 29 Oct Kaiwen Yang (Tongji University and Tencent Hunyuan) posted a proof for dimension 5. Its AI declaration says the proof 'was generated by Tencent Hunyuan's AI agent hyprover on September 29, 2026', and that the author organised, verified and rewrote it. The paper prints the full prompt, which points the agent to Lu–Yang's solution of Banach's isometric conjecture. Few research results in our dataset credit a Chinese lab's agent. It is a preprint, not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.39967: Petty's equilateral-set conjecture in dimension five (Kaiwen Yang)","url":"https://arxiv.org/abs/2610.39967","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-10-30","updated":"2026-10-30","orgs":["Tencent","Tongji University"],"title":"Petty's equilateral-set conjecture claimed in dimension five, with the proof 'generated by Tencent Hunyuan's AI agent hyprover'","summary":"arXiv 2610.39967 (29 Oct 2026), \"Petty's equilateral-set conjecture in dimension five\" by Kaiwen Yang (School of Mathematical Sciences, Tongji University; Tencent Hunyuan), proves that every five-dimensional real normed space contains six equidistant points, using a bordism of equilateral configurations and a homological obstruction. The proof was generated by hyprover, an AI agent of Tencent Hunyuan, on 29 Sept 2026 from a prompt reproduced in Appendix A. Petty's conjecture remains open in dimension six and above.","key_facts":["Claim (abstract): 'Petty conjectured that every n-dimensional real normed space contains n+1 equidistant points. The conjecture was proved in dimensions three and four by Petty and Makeev, respectively. We prove the five-dimensional case.'","AI declaration (verbatim): 'The proof was generated by Tencent Hunyuan's AI agent hyprover on September 29, 2026. The author organized, verified, and rewrote the proof. The prompt was written by the author and is reproduced in Appendix A.'","The prompt (Appendix A) asks the agent to 'Prove that every five-dimensional real normed vector space contains six equilateral points', hints at 'an appropriate differential-topological reformulation' and to 'consult Lu–Yang's A Solution to Banach's Isometric Conjecture for methodological inspiration, not a ready-made argument', and demands that the agent 'Distinguish proved statements, conjectures, failed approaches, and unresolved gaps. Never replace a missing argument with a claim of verification'","Author: Kaiwen Yang, with two affiliations, Tongji University and Tencent Hunyuan; a Kaiwen Yang co-authored the Lu–Yang solution of Banach's isometric conjecture in August (arXiv 2608.13536), which used ChatGPT 5.5/5.6 Pro","No public description of 'hyprover' found on 30 Oct (no Tencent blog, model card or GitHub repository); we know it only from this paper","Verification status: preprint, not refereed, no formal proof; no reaction found on HN on 30 Oct"],"key_numbers":[],"tags":["math","convex-geometry","functional-analysis","normed-spaces","algebraic-topology","tencent","hunyuan","hyprover","china","preprint"],"science":{"field":"mathematics","subfield":"geometry of normed spaces / topological methods","problem":"Petty's equilateral-set conjecture (1971): every n-dimensional normed space contains n + 1 equidistant points","result":"Claimed proof in dimension 5 (dimensions 3 and 4 were known)","open_since":"","ai_system":["Tencent Hunyuan hyprover"],"human_role":"Author wrote the prompt with a methodological hint; the agent generated the proof; the author organised, verified and rewrote it","verification":{"key":"preprint","label":"Preprint only","text":"Preprint; not refereed"},"status":"pending","shock":"A research-level proof credited to an undocumented agent from a Chinese lab, with the full prompt published."},"body_md":"## What happened\n\nIn a normed space, distances are measured by an arbitrary norm instead of the Euclidean one. Petty asked whether such a space of\ndimension n always has n + 1 points that are all at the same distance from each other, as the vertices of a regular simplex are in\nordinary space. It was proved for dimensions 3 and 4.\n\nKaiwen Yang, of Tongji University and Tencent Hunyuan, posted a proof for dimension 5 on 29 Oct. The paper says Tencent's AI agent\nhyprover produced the proof a month earlier from a prompt the author wrote and printed in full, and that the author verified and\nrewrote it.\n\n## Why it matters\n\nMost AI-maths results so far name OpenAI, Anthropic or Google models. This one names a Tencent agent that has no public documentation\nyet, which suggests Chinese labs run similar research agents. The prompt is published, which is rare. The result itself awaits checking.","disputed":[{"label":"Verification","text":"Preprint; not refereed","sources":[]}],"related":[{"id":"2026-10-30-ai-assisted-results-october-30-listing","url":"https://postcutoff.com/e/2026-10-30-ai-assisted-results-october-30-listing/","date":"2026-10-30","date_precision":"day","short_title":"AI-assisted results, 30 Oct: arXiv announcement of 29–30 Oct","deck":null,"takeaway":"The 30 Oct sweep listed 374 arXiv papers that mention AI. We skimmed all their AI statements and read about 35 PDFs. Six results got their own entries: Das–Fox's Murty–Simon proof (GPT-5.6 Sol found the bound), Colbrook–Stepaniants's spectral isometry proof (found by GPT-6 Astra, Lean), the Θ(log k log log k) ratio of greedy for online Steiner forest (GPT Astra, Lean), Gáspár's Laczkovich solution (key proposition by ChatGPT), Ren's pure-semisimplicity counterexample (GPT-6 Astra, Lean) and Petty's conjecture in dimension 5 (Tencent's hyprover agent). PNNL's OpenConjecture study also has one. This listing has the rest, among them a single-author ChatGPT 6 Pro proof of the Alon–Seymour–Thomas separator conjecture, b₄ ≤ 7 for Borsuk's problem and Fiedorowicz's conjecture from a Claude idea.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":42,"official":42,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Anthropic","Google","Tencent","various researchers"]},{"id":"2026-08-28-tencent-hunyuan-hy4-preview","url":"https://postcutoff.com/e/2026-08-28-tencent-hunyuan-hy4-preview/","date":"2026-08-28","date_precision":"day","short_title":"Tencent open-sources Hunyuan Hy4 preview (770B MoE, 1M+ context)","deck":null,"takeaway":"Tencent joins DeepSeek, Moonshot, Alibaba and Zhipu in shipping ~1T-class open-weights models, deepening the Chinese open-model ecosystem.","category":"open-source","category_label":"Open source","importance":3,"confidence":"medium","status":{"key":"partly","labels":["Partly confirmed"]},"sources":4,"official":0,"filed":"2026-09-29","updated":"2026-10-29","orgs":["Tencent"]},{"id":"2026-08-13-banach-isometric-conjecture-solved","url":"https://postcutoff.com/e/2026-08-13-banach-isometric-conjecture-solved/","date":"2026-08-13","date_precision":"day","short_title":"Banach's isometric conjecture (1932) completed in the real case with key steps from ChatGPT 5.5/5.6 Pro; complex and quaternionic cases follow five days later","deck":null,"takeaway":"Banach's conjecture is one of the oldest questions in the geometry of normed spaces, and Gromov's even-dimensional solution had left the remaining odd cases open.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-09-30","updated":"2026-09-30","orgs":["Xinbao Lu","Kaiwen Yang","Antonio Acuaviva","Tomasz Kania","OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-30","type":"filed","text":"Created (science agent; arXiv abstract and PDF read, AI declaration and prompt quoted)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-30","run":null,"sources_read":"Science agent; arXiv abstract and PDF read, AI declaration and prompt quoted","updated":"2026-10-30","human_review":null,"version":{"date":"2026-10-30"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":182,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":121,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":212,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":151,"in_training_data":false}],"short_url":"https://postcutoff.com/s/petty-conjecture-dimension-5"}