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The truncated octahedral conjecture

A mix of human-written and AI-generated textHuman understanding: all partsmath.MG — Metric Geometry

Contributed by Thomas Hales, Lark Song ↗

Version 1 / Oct 12, 2026/ arXiv: 2609.00997 / CC BY 4.0

Abstract

Among three-dimensional parallelohedra of fixed volume, the Archimedean truncated octahedron uniquely minimizes surface area, affirming a conjecture of K. Bezdek from 2006. One inequality in the proof is verified by an exact computer-assisted certificate. The simplex, frame, and graphical-zonotope inequalities used in the proof hold in arbitrary dimension.

Provenance statement

The GPT-5.6 Sol model in ChatGPT was used in a substantial way at all stages, including proof discovery and manuscript drafts. It wrote the Python code, test suites, and code documentation. We substantially reworked and simplified the proofs. ChatGPT also provided editorial suggestions and feedback on the manuscript.

Tools used

OpenAI
ChatGPTVersion 5.6 Sol

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Version history

  1. v1Submitted by Lark SongInitial depositCurrentOct 12, 2026