No torus with ten convex quadrilateral faces

No acoptic polyhedral torus in R3 has exactly 10 strictly convex quadrilateral faces. This is the F = 10 case of Grünbaum–Szilassi Conjecture 3.2; with Seamons's 11-face torus it settles which face counts occur. The proof reduces to the map {4,4}(3,1) and certifies, by an LRAT proof checked with the formally verified checker cake_lpr, that it has no planar strictly convex realisation, even allowing self-intersections. Code, certificates and a one-command checker: https://github.com/LeeMarshall1113/ai-math-results (results/torus-quad10).

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-10-31
DOI
https://doi.org/10.5281/zenodo.23842783
Primary Topic
Advanced Combinatorial Mathematics
Type
preprint
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preprint

No torus with ten convex quadrilateral faces

Lee Marshall
Zenodo (CERN European Organization for Nuclear Research)
Advanced Combinatorial Mathematics
preprint

No torus with ten convex quadrilateral faces

Lee Marshall
preprint en

Abstract

No acoptic polyhedral torus in R3 has exactly 10 strictly convex quadrilateral faces. This is the F = 10 case of Grünbaum–Szilassi Conjecture 3.2; with Seamons's 11-face torus it settles which face counts occur. The proof reduces to the map {4,4}(3,1) and certifies, by an LRAT proof checked with the formally verified checker cake_lpr, that it has no planar strictly convex realisation, even allowing self-intersections. Code, certificates and a one-command checker: https://github.com/LeeMarshall1113/ai-math-results (results/torus-quad10).

Zenodo (CERN European Organization for Nuclear Research)
Advanced Combinatorial Mathematics
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No torus with ten convex quadrilateral faces — Lee Marshall · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS