---
id: "2026-09-05-i3322-bell-inequality-infinite-dimensions"
url: "https://postcutoff.com/e/2026-09-05-i3322-bell-inequality-infinite-dimensions/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-09-05"
date_precision: day
category: science
importance: 3
confidence: high
status: [Event confirmed, Awaiting review]
verification: Preprint only
sources: 1
editor: human editor
human_review: null
version: "2026-10-04"
---

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/2026-09-05-i3322-bell-inequality-infinite-dimensions/

# The I3322 Bell inequality needs infinite dimensions

Full title: The I3322 Bell inequality needs infinite dimensions: Pál–Vértesi conjecture (2010) proved from an approximate proof by GPT-5.5 Pro

Andrea Coladangelo (University of Washington) proved the Pál–Vértesi conjecture that no finite-dimensional quantum strategy reaches the maximal violation of the I3322 Bell inequality, while an infinite-dimensional one does (arXiv 2609.06038, 5 Sep 2026). I3322 is one of the simplest Bell inequalities. His AI statement says 'An approximate proof (containing all of the main ideas) was produced by GPT 5.5 Pro'. He then substantially revised it for completeness and correctness, with help from GPT-5.6 Sol.

## Key facts
- Conjecture: Pál and Vértesi (Physical Review A, 2010); I3322 has three questions and two answers per party, the simplest Bell scenario where this phenomenon could occur
- Previously a 5-question, 3-answer correlation with this property was known (Coladangelo and Stark, Nature Communications 2020)
- Proof tool: a Bellman-function-style certificate from two families of inequalities that bounds the value of any finite-dimensional strategy
- AI usage statement: 'An approximate proof (containing all of the main ideas) was produced by GPT 5.5 Pro after some rounds of interaction with the author. That first approximate proof was much shorter, and arguably quite difficult to parse by a human'; the final text is the author's revision, with help from GPT 5.6 Sol
- Status: preprint (78 pages)

## What happened

Coladangelo, who had earlier found a larger Bell scenario with the same property, worked with GPT-5.5 Pro on the simplest candidate, I3322. The model produced a terse approximate proof containing the key ideas. The author expanded it into a 78-page rigorous paper.

## Why it matters

It settles a basic question about how much entanglement quantum correlations can require, in the smallest possible Bell scenario. It is also one of the clearest 2026 cases of an AI supplying the core idea in quantum foundations.

## What is disputed or not yet verified

- Verification: Preprint; not peer-reviewed

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 128 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 67 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 158 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 97 days after its cutoff

## Sources

1. [arXiv 2609.06038: The I3322 Bell inequality requires infinite dimensions (Coladangelo)](https://arxiv.org/abs/2609.06038) (arxiv.org, paper)

## Changes

- 2026-09-30 (filed): Created

## Related

- 2026-09-30: [Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help](https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/index.md)