Breakthrough Tracker record
A preprint claims the truncated octahedron is the unique least-area parallelohedron
Thomas Hales and Lark Song claim that, among three-dimensional parallelohedra of fixed volume, the Archimedean truncated octahedron uniquely minimizes surface area. The final scalar inequality is supported by an exact computer-assisted certificate.
← Back to the filtered Breakthrough Tracker
- Stable ID
math-truncated-octahedral-conjecture-proof-2026- Revision
math-truncated-octahedral-conjecture-proof-2026.v1- Field
- Mathematics · Convex and metric geometry, tilings and computer-assisted proof
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional computer-assisted proof
- Last checked
AI role
The authors state that GPT-5.6 Sol in ChatGPT was used substantially at every stage. They take responsibility for the proof, code and certificate; Song’s local-minimality result was obtained without machine assistance.
Record details
- Problem or result
- Károly Bezdek’s 2006 fixed-volume surface-area conjecture for three-dimensional parallelohedra
- Authors
- Thomas Hales and Lark Song
- Institutions
- University of Pittsburgh; University of Wisconsin–Madison
- Result date
- First-version preprint submitted September 1, 2026
Why it matters
The result would give a sharp global isoperimetric classification across all five three-dimensional parallelohedron types and settle a geometric problem tied to space-filling cells.
Limits
The manuscript is a 45-page first-version preprint and has not been peer reviewed. The certificate is described as exact, but this maintenance cycle did not independently reproduce its full computation; the theorem concerns translative polyhedral tilings, not the unrestricted Kelvin foam problem.
Sources
Correction and revision history
- 2026-09-04 — Added after primary-source, scope, status, AI-role and limitation review.
Machine-readable: JSON v1 · CSV v1 · Schema v1
This individual record remains noindex until a story-specific featured image passes Kingy’s rendered-pixel visual review. The source-linked tracker hub remains the public index.