---
id: "2026-11-03-mixed-volume-fpras-gurvits-conjecture-disproved-gpt-6-astra"
url: "https://postcutoff.com/e/2026-11-03-mixed-volume-fpras-gurvits-conjecture-disproved-gpt-6-astra/"
as_of: "2026-11-09T21:13:00+01:00"
date: "2026-11-03"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: Preprint only
sources: 1
editor: human editor
human_review: null
version: "2026-11-03"
---

As of: 2026-11-09 21:13 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/2026-11-03-mixed-volume-fpras-gurvits-conjecture-disproved-gpt-6-astra/

# FPRAS for mixed volumes, ideas from GPT-6 Astra

Full title: First FPRAS for mixed volumes of convex bodies, disproving a 2009 conjecture of Gurvits; main ideas 'generated entirely by GPT 6 Astra' (Chen, Liu)

Xiaoyu Chen and Kuikui Liu (arXiv 2611.01368, 3 Nov 2026) give the first fully polynomial-time randomized approximation scheme for the mixed volume of n compact convex sets in R^n given by weak membership oracles, which they say disproves a long-standing conjecture of Gurvits (2009). Their disclosure: "The main ideas were generated entirely by GPT 6 Astra"; the paper is a human-written exposition.

## Key facts
- Claim: first FPRAS for mixed volumes of arbitrary compact convex sets given by weak membership oracles; 'This disproves a longstanding conjecture of Gurvits [Gur09a, Section 4, item 2]'
- AI disclosure (verbatim): 'The main ideas were generated entirely by GPT 6 Astra. The purpose of this paper is to provide a human-written exposition of the algorithm.'
- Running time: GPT-6 Astra estimated about n^900 ε^-220 · poly(L); the authors say they have not verified it and 'do not claim it as a proved bound'
- Verification status: unrefereed preprint, rewritten and checked by the authors; no formalization

## What happened

Chen and Liu posted an algorithm that approximates mixed volumes to any relative accuracy in randomized polynomial time. They credit the main ideas entirely to GPT-6 Astra and present the paper as their human-written exposition. The polynomial is very large (an unverified model estimate puts it near n^900).

## Why it matters

Mixed volumes generalize permanents and volumes; Gurvits had conjectured that no such scheme exists. The result is theoretical, and refereeing has not started.

## What is disputed or not yet verified

- Verification: Preprint, author-checked

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 187 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 126 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 217 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 156 days after its cutoff

## Sources

1. [arXiv 2611.01368: An FPRAS for Mixed Volumes (Chen, Liu)](https://arxiv.org/abs/2611.01368) (arxiv.org, paper)

## Changes

- 2026-11-03 (filed): Created

## Related

- 2026-11-03: [AI-assisted results, 3 Nov: arXiv of 3 Nov and Hexagon of 2–3 Nov](https://postcutoff.com/e/2026-11-03-ai-assisted-results-november-3-listing/index.md)