{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-05-27-sum-product-conjecture-false-over-reals/","md":"https://postcutoff.com/e/2026-05-27-sum-product-conjecture-false-over-reals/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":"2026-05-27-sum-product-conjecture-false-over-reals","date":"2026-05-27","date_precision":"day","short_title":"Erdős–Szemerédi sum-product conjecture shown false over the reals","deck":"A GPT-5.5 Pro agent re-disproves it in 7 of 8 runs","takeaway":"Inspired by the AI disproof of the unit-distance conjecture, Bloom, Sawin, Schildkraut and Zhelezov proved on 27 May 2026 that the Erdős–Szemerédi sum-product conjecture is false over the real numbers.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"The sum-product conjecture is false for real numbers (arXiv 2605.28781)","url":"https://arxiv.org/abs/2605.28781","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"GPT-5.5 Pro agent disproofs of the sum-product conjecture over R (arXiv 2607.20525)","url":"https://arxiv.org/abs/2607.20525","type":"paper","group":"primary","domain":"arxiv.org"}],"official":2,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"],"title":"Erdős–Szemerédi sum-product conjecture shown false over the reals; a GPT-5.5 Pro agent re-disproves it in 7 of 8 runs","summary":"Inspired by the AI disproof of the unit-distance conjecture, Bloom, Sawin, Schildkraut and Zhelezov proved on 27 May 2026 that the Erdős–Szemerédi sum-product conjecture is false over the real numbers. They built sets A with |A+A| and |AA| ≤ |A|^(2−c). A July 2026 paper (arXiv 2607.20525) showed a GPT-5.5 Pro agent autonomously generated correct disproofs in 7 of 8 independent trials, some with new constructions.","key_facts":["Human paper: arXiv 2605.28781 (27 May 2026), 'inspired' by OpenAI's unit-distance disproof, which used related algebraic-number-theory ideas","AI replication: GPT-5.5 Pro agent, three-stage prompting pipeline, correct disproofs in 7/8 runs; some avoid units by using L^p-type regions of algebraic integers","The 1983 conjecture (max(|A+A|,|AA|) ≥ |A|^(2−ε)) remains open over the integers","AI disclosure (Huang, arXiv 2607.20525, abstract): 'we present a simple agent built on GPT-5.5 Pro. Using a problem-agnostic, three-stage prompting pipeline -- proof-plan proposal, proof construction, and review -- the agent autonomously generated correct proofs that the sum-product conjecture is false over R in 7 of 8 independent trials; in the remaining trial, it identified an unresolved gap in its argument.'"],"key_numbers":[],"tags":["math","additive-combinatorics","erdos","counterexample"],"science":{"field":"mathematics","subfield":"additive combinatorics","problem":"Erdős–Szemerédi sum-product conjecture (over R)","result":"Finite sets of reals with both sumset and product set of size at most |A|^(2−c), disproving the conjecture over R; later reproduced autonomously by an AI agent.","open_since":"1983","ai_system":["GPT-5.5 Pro (in the follow-up)"],"human_role":"Original disproof human-led, inspired by an AI result; follow-up shows autonomous AI rediscovery","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed preprint by specialists (Bloom, Sawin, Schildkraut, Zhelezov, arXiv 2605.28781); the autonomous AI rediscovery is a separate unrefereed preprint (Huang, arXiv 2607.20525); no formal proof"},"status":"pending","shock":""},"body_md":"## What happened\n\nThe number-theoretic idea behind the AI's unit-distance counterexample prompted human experts to attack a second famous Erdős conjecture, which fell within a week. A later study showed the AI could have done it alone.\n\n## Why it matters\n\nIt shows AI ideas spreading into human research and then being reproduced autonomously: a feedback loop between AI and human mathematicians.","disputed":[{"label":"Verification","text":"Unrefereed preprint by specialists (Bloom, Sawin, Schildkraut, Zhelezov, arXiv 2605.28781); the autonomous AI rediscovery is a separate unrefereed preprint (Huang, arXiv 2607.20525); no formal proof","sources":[]}],"related":[{"id":"2026-05-20-ai-disproves-erdos-unit-distance-conjecture","url":"https://postcutoff.com/e/2026-05-20-ai-disproves-erdos-unit-distance-conjecture/","date":"2026-05-20","date_precision":"day","short_title":"OpenAI model disproves Erdős's 80-year-old unit distance conjecture","deck":null,"takeaway":"This is the moment AI crossed from solving competition problems to settling a famous open research conjecture, reshaping debate about AI's role in mathematics.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":9,"official":3,"filed":"2026-09-29","updated":"2026-10-29","orgs":["OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-19","type":"edit","text":"Status confirmed → pending (both papers are unrefereed preprints); added Huang's AI disclosure verbatim from the arXiv abstract"},{"date":"2026-09-29","type":"filed","text":"Created"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-09-29","run":null,"sources_read":null,"updated":"2026-10-19","human_review":null,"version":{"date":"2026-10-19"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":27,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":null,"in_training_data":true},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":57,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":null,"in_training_data":true}],"short_url":"https://postcutoff.com/s/erdos-szemeredi-sum-product"}