---
id: "2026-05-27-sum-product-conjecture-false-over-reals"
url: "https://postcutoff.com/e/2026-05-27-sum-product-conjecture-false-over-reals/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-05-27"
date_precision: day
category: science
importance: 4
confidence: high
status: [Event confirmed, Awaiting review]
verification: Preprint only
sources: 2
editor: human editor
human_review: null
version: "2026-10-19"
---

As of: 2026-11-13 13:49 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/2026-05-27-sum-product-conjecture-false-over-reals/

# Erdős–Szemerédi sum-product conjecture shown false over the reals

Full 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

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.'

## What happened

The 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.

## Why it matters

It shows AI ideas spreading into human research and then being reproduced autonomously: a feedback loop between AI and human mathematicians.

## What is disputed or not yet verified

- Verification: 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

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 27 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): in its training data
- Gemini 4 Hydrogen (training cutoff March 2026): 57 days after its cutoff
- Grok 4.7 (training cutoff May 2026): in its training data

## Sources

1. [The sum-product conjecture is false for real numbers (arXiv 2605.28781)](https://arxiv.org/abs/2605.28781) (arxiv.org, paper)
2. [GPT-5.5 Pro agent disproofs of the sum-product conjecture over R (arXiv 2607.20525)](https://arxiv.org/abs/2607.20525) (arxiv.org, paper)

## Changes

- 2026-10-19: Status confirmed → pending (both papers are unrefereed preprints); added Huang's AI disclosure verbatim from the arXiv abstract
- 2026-09-29 (filed): Created

## Related

- 2026-05-20: [OpenAI model disproves Erdős's 80-year-old unit distance conjecture](https://postcutoff.com/e/2026-05-20-ai-disproves-erdos-unit-distance-conjecture/index.md)