---
id: "2026-08-04-period-index-conjecture-false"
url: "https://postcutoff.com/e/2026-08-04-period-index-conjecture-false/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-08-04"
date_precision: day
category: science
importance: 4
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/2026-08-04-period-index-conjecture-false/

# Alexander Perry disproves the period-index conjecture

Full title: Alexander Perry disproves the period-index conjecture (Colliot-Thélène, 2001); a flawed ChatGPT example was the starting point

Alexander Perry posted counterexamples to the period-index conjecture for Brauer classes: d-dimensional varieties (d ≥ 3) with a Brauer class of period 2 and index 2^d (arXiv 2608.03684; v1 4 Aug, v2 3 Sep 2026). His AI disclosure says the starting point was a flawed ChatGPT example of the right general shape. Perry found the correct construction himself (a generic K3 surface with a symplectic group action), with ChatGPT helping on literature and lattice calculations.

## Key facts
- Period-index conjecture: stated by Colliot-Thélène in 2001; predicts the index of a Brauer class on a d-dimensional variety divides period^(d−1)
- Result: failures in dimension 3 over Q and F_p(t), and in all dimensions d ≥ 3 over C and F_p((t)); the lower bound on the index uses a motivic obstruction, namely the non-existence of integral Hodge or Tate classes satisfying an equation mod 2
- Perry proposes a corrected period-index conjecture with corrections at small primes and proves its Hodge-theoretic counterpart
- AI disclosure, §1.7: prompted to use the paper's Hodge-theoretic strategy, ChatGPT 'produced a flawed example of a threefold X and Brauer class α' that 'had some features of our final solution'; 'Drawing inspiration from this failed example, the author eventually discovered Theorem 1.3'; ChatGPT then 'assisted with literature searches and lattice theory calculations'. The paper was written by the author
- Status: preprint, not peer-reviewed

## What happened

Perry, who had earlier worked with de Jong on the Hodge theory of the period-index problem, asked ChatGPT to look for a counterexample along that strategy. The model's example was wrong, but its shape pointed Perry to a correct construction built from K3 surfaces with symplectic group actions.

## Why it matters

The period-index conjecture was a central open question on Brauer groups of function fields. It is also an example of an AI error being productive: the model gave the idea and a human supplied the correct mathematics.

## What is disputed or not yet verified

- Verification: Preprint by an expert in the field; not peer-reviewed

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 96 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 35 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 126 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 65 days after its cutoff

## Sources

1. [arXiv 2608.03684: The period-index conjecture is false for motivic reasons (Perry)](https://arxiv.org/abs/2608.03684) (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)