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.
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- Stable ID
math-truncated-octahedral-conjecture-proof-2026- Revision
math-truncated-octahedral-conjecture-proof-2026.v2- 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
- Current v2 revised September 11, 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 simplified 23-page v2 remains unrefereed. Its exact Bernstein certificate was introduced earlier in the argument, but this maintenance cycle did not independently replay the full computation; the theorem concerns translative polyhedral tilings, not unrestricted Kelvin foams.
Sources
Correction and revision history
- 2026-09-04 — Added after primary-source, scope, status, AI-role and limitation review.
- 2026-10-02 — Reviewed the shortened v2, which introduces the exact Bernstein certificate earlier; no independent replay or status upgrade located.
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