{"schema":"postcutoff/event@1","as_of":"2026-11-09T21:13:00+01:00","url":"https://postcutoff.com/e/2026-11-09-nill-reflexive-polytopes-conjecture-disproved-gpt-6-1-sol/","md":"https://postcutoff.com/e/2026-11-09-nill-reflexive-polytopes-conjecture-disproved-gpt-6-1-sol/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":"2026-11-09-nill-reflexive-polytopes-conjecture-disproved-gpt-6-1-sol","date":"2026-11-09","date_precision":"day","short_title":"Reflexive polytopes conjecture disproved, GPT-6.1 Sol","deck":null,"takeaway":"Johannes Hofscheier and Benjamin Nill (arXiv 2611.08819) give counterexamples to Nill's own 2004 conjecture that a d-dimensional reflexive polytope has at most 6^{d/2} vertices, in every dimension d ≥ 8, and show the count can be doubly exponential in d.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"Hofscheier, Nill: Reflexive polytopes can have doubly exponentially many vertices (arXiv 2611.08819)","url":"https://arxiv.org/abs/2611.08819","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-11-09","updated":"2026-11-09","orgs":["OpenAI","various researchers"],"title":"Nill's 2004 conjecture on vertices of reflexive polytopes disproved with GPT-6.1 Sol: doubly exponentially many vertices possible","summary":"Johannes Hofscheier and Benjamin Nill (arXiv 2611.08819) give counterexamples to Nill's own 2004 conjecture that a d-dimensional reflexive polytope has at most 6^{d/2} vertices, in every dimension d ≥ 8, and show the count can be doubly exponential in d. \"These results were found using GPT-6.1 Sol.\" The examples can be checked with standard polyhedral software.","key_facts":["Conjecture (Nill, 2004): at most 6^{d/2} vertices; false for every d ≥ 8","Counterexamples in dimensions 8 and 9, built from an explicit 3-dimensional lattice polytope with 21 points; checkable with standard software","Large d: vertex (equivalently facet) count can be doubly exponential, as large as theoretically possible; the number-theoretic step originally used OpenAI's solution of Erdős's short Egyptian fractions conjecture, now replaced by Bloom and van Doorn's human-written Hexagon paper","AI statement: 'The example, the lemmata and the general result were found in middle of October 2026 using GPT-6.1 Sol based upon prompts with ideas by the first author originally aimed at proving rather than disproving Conjecture 1.1'"],"key_numbers":[],"tags":["math","combinatorics","polytopes","toric-geometry","gpt-6-1-sol","chatgpt","arxiv"],"science":{"field":"mathematics","subfield":"discrete geometry","problem":"Nill's conjecture: a d-dimensional reflexive polytope has at most 6^{d/2} vertices","result":"False for every d ≥ 8; the number of vertices can be doubly exponential in d","open_since":"2004","ai_system":["GPT-6.1 Sol"],"human_role":"AI found the example, lemmas and general result from prompts with the first author's ideas; the conjecture's author is a co-author","verification":{"key":"preprint","label":"Preprint only","text":"Preprint; explicit examples checkable by computer"},"status":"pending","shock":""},"body_md":"## What happened\n\nIn [arXiv 2611.08819](https://arxiv.org/abs/2611.08819) (9 Nov 2026), Hofscheier and Nill report that GPT-6.1 Sol, prompted with an approach meant to prove the conjecture, produced counterexamples in dimensions 8 and 9 and a general construction. The authors write that the doubly exponential growth is \"rather surprising\". Reflexive polytopes correspond to Gorenstein toric Fano varieties.\n\n## Why it matters\n\nThe conjecture's own author confirms the disproof, and the low-dimensional counterexamples are explicit, which makes the claim easy to check. Unrefereed.","disputed":[{"label":"Verification","text":"Preprint; explicit examples checkable by computer","sources":[]}],"related":[{"id":"2026-11-09-ai-assisted-results-november-9-listing","url":"https://postcutoff.com/e/2026-11-09-ai-assisted-results-november-9-listing/","date":"2026-11-09","date_precision":"day","short_title":"AI-assisted results, 9 Nov: arXiv and Hexagon","deck":null,"takeaway":"The 9 Nov sweep listed 175 new arXiv papers mentioning AI. Own entries: the matroid secretary conjecture (strong form, GPT Astra) and Nill's reflexive polytope conjecture disproved with GPT-6.1 Sol.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":44,"official":44,"filed":"2026-11-09","updated":"2026-11-09","orgs":["OpenAI","Anthropic","Google DeepMind","various researchers"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-11-09","type":"filed","text":"Created"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-11-09","run":null,"sources_read":null,"updated":"2026-11-09","human_review":null,"version":{"date":"2026-11-09"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":193,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":132,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":223,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":162,"in_training_data":false}],"short_url":null}