---
id: "2026-11-03-kronheimer-mrowka-dodecahedron-web-instanton-homology-64-chatgpt"
url: "https://postcutoff.com/e/2026-11-03-kronheimer-mrowka-dodecahedron-web-instanton-homology-64-chatgpt/"
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: Unverified claim
sources: 2
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-kronheimer-mrowka-dodecahedron-web-instanton-homology-64-chatgpt/

# Kronheimer–Mrowka: dodecahedron instanton homology is 64 (ChatGPT Astra 6)

Full title: Kronheimer and Mrowka post a primarily AI-generated proposed proof that the dodecahedron web's instanton homology has dimension 64, not its 60 Tait colourings

Peter Kronheimer (Harvard) and Tomasz Mrowka (MIT) deposited on Hexagon (2611.00021, published 3 Nov 2026) a "proposed proof" that the SO(3) instanton homology J♯ of the planar dodecahedron web has dimension 64. Earlier work had narrowed it to 60 or 64; 60 is the number of Tait colourings of the dodecahedron, the value their 2015 programme towards the four-colour theorem conjectured for planar webs. The deposit is declared primarily AI-generated (ChatGPT Astra 6, building on earlier ChatGPT and Claude calculations), with only some parts checked by a human.

## Key facts
- Claim (abstract): the octahedral diagram reduces dim J♯(D) to 60 or 64; 'We give a proposed proof that all three [secondary-map coefficients] vanish and that the dimension is 64'
- Context: Kronheimer–Mrowka, 'Tait colorings, and an instanton homology for webs and foams' (JEMS 2019, arXiv 1508.07205) defined J♯ and asked whether its dimension equals the number of Tait colourings for planar webs; 64 ≠ 60 would answer that in the negative for the dodecahedron
- Provenance (verbatim): 'Primarily AI-generated'; 'Some parts of this contribution, but not all, have been checked and understood by a human'; 'ChatGPT Astra 6 was prompted to consider the question of whether the instanton homology of the dodecahedron had dimension 60 or 64, having been provided with previous calculations from ChatGPT and Claude.'
- Authorship mode on Hexagon: no-human-author-claimed; sponsor Peter Kronheimer; exact matrices and foam calculations 'provided separately'
- Verification status: unrefereed Hexagon deposit, described by the authors as a proposed proof; no Lean formalization; no HN discussion found on 3 Nov

## What happened

On 2 Nov 2026 Peter Kronheimer deposited, and on 3 Nov Hexagon published, a paper by him and Tomasz Mrowka on the instanton homology of the planar dodecahedron web. Known calculations left two values, 60 and 64; the paper gives a proposed proof that three coefficients of a secondary map vanish, which gives 64. The provenance record says the text is primarily AI-generated by ChatGPT Astra 6, prompted with earlier calculations from ChatGPT and Claude, and that only some parts have been checked by a human.

## Why it matters

Kronheimer and Mrowka built J♯ in 2015 as a possible route to the four-colour theorem, asking whether its dimension counts Tait colourings for planar webs. The dodecahedron has 60 Tait colourings, so a dimension of 64 would answer that question in the negative for this web. The result is a proposed proof, not checked in full by its sponsors.

## What is disputed or not yet verified

- Verification: Unrefereed Hexagon deposit; partly human-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. [Hexagon 2611.00021: The dodecahedron web and its instanton homology (Kronheimer, Mrowka)](https://hexagonmath.org/abs/2611.00021) (hexagonmath.org, paper)
2. [Kronheimer & Mrowka, Tait colorings, and an instanton homology for webs and foams (arXiv 1508.07205)](https://arxiv.org/abs/1508.07205) (arxiv.org, paper)

## Changes

- 2026-11-03 (filed): Created (Hexagon provenance record read through the Hexagon API)

## 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)
- People: [Peter Kronheimer](https://postcutoff.com/person/peter-kronheimer/), [Tomasz Mrowka](https://postcutoff.com/person/tomasz-mrowka/)