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arxiv: 2606.31640 · v2 · pith:ZNCEFJXQnew · submitted 2026-06-30 · 🧮 math.CO · math.OC

A Counterexample to Ziegler's Cross-Polytope Conjecture for Simplicial 0/1-Polytopes

Volker Kaibel , Sebastian Pokutta This is my paper

Pith reviewed 2026-07-03 22:12 UTC · model grok-4.3

classification 🧮 math.CO math.OC
keywords 0/1-polytopessimplicial polytopescentral symmetrycross-polytopecounterexampleenumerationcube symmetries
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The pith

There exists a simplicial 7-polytope with 14 vertices in the 0/1-cube that is not centrally symmetric.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Ziegler showed every simplicial d-dimensional 0/1-polytope has at most 2d vertices and conjectured that reaching this bound forces the polytope to be centrally symmetric, hence a 0/1-realization of the cross-polytope. The paper constructs an explicit set of 14 points in {0,1}^7 whose convex hull is simplicial yet lacks central symmetry. Exhaustive search up to cube symmetries finds exactly five such polytopes in dimension 7, belonging to two combinatorial types.

Core claim

The convex hull of a specific collection of 14 vertices chosen from {0,1}^7 is a simplicial 7-polytope whose vertex set is not invariant under central inversion.

What carries the argument

An explicit 14-vertex subset of the 7-cube whose convex hull is simplicial and asymmetric.

Load-bearing premise

The 14 chosen points form a simplicial polytope whose vertex set is not centrally symmetric.

What would settle it

A direct computational check confirming whether the listed 14 points in {0,1}^7 produce a simplicial convex hull without central symmetry.

Figures

Figures reproduced from arXiv: 2606.31640 by Sebastian Pokutta, Volker Kaibel.

Figure 1
Figure 1. Figure 1: A two-dimensional projection, chosen to separate all points, of the centered vertex set V − 1 2 1. Blue points form the five present cube-antipodal pairs. Red points are the four unmatched vertices; hollow red markers indicate where their missing cube antipodes would project. A single coordinate swap matches the remaining red vertices to obtain a cross polytope. 4 The complete classification in dimension s… view at source ↗
Figure 2
Figure 2. Figure 2: places the principal milestones of the agentic research process on a wall-clock timeline; each bar spans the work turn in which the corresponding milestone was reached as well as the lead-up time. 06-23 12:00 06-24 00:00 06-24 12:00 06-25 00:00 06-25 12:00 06-26 00:00 06-26 12:00 06-27 00:00 06-27 12:00 06-28 00:00 Milestone 7 Milestone 6 Milestone 5 Milestone 4 Milestone 3 Milestone 2 Milestone 1 lead-up … view at source ↗
read the original abstract

Ziegler proved that every simplicial $d$-dimensional $0/1$-polytope has at most $2d$ vertices, and asked whether equality forces the polytope to be centrally symmetric and hence, equivalently, a $0/1$-realization of the $d$-dimensional cross polytope. In this note, we give a negative answer, exhibiting an explicit set of $14$ vertices in $\{0,1\}^7$ whose convex hull is a simplicial $7$-polytope and is not centrally symmetric. Moreover, via exhaustive enumeration we show that up to the symmetries of the cube, there are precisely five such polytopes in dimension $7$ (of two combinatorial types) that are not centrally symmetric.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript disproves Ziegler's conjecture that every simplicial d-dimensional 0/1-polytope with exactly 2d vertices must be centrally symmetric (hence combinatorially a cross-polytope). It exhibits an explicit 14-vertex subset of {0,1}^7 whose convex hull is a simplicial 7-polytope whose vertex set is not closed under the map x ↦ 1−x. Exhaustive enumeration up to the action of the hypercube symmetry group shows there are precisely five such polytopes in dimension 7, falling into two combinatorial types.

Significance. The result is significant: it supplies a concrete, finitary counterexample to a natural conjecture about the extremal structure of simplicial 0/1-polytopes. The explicit vertex list together with the claim of machine-verifiable simpliciality and asymmetry constitute a strong form of evidence. The additional enumeration of all examples up to symmetry provides a complete classification in the first dimension where counterexamples appear, which is useful for future work on 0/1-polytopes.

minor comments (2)
  1. The manuscript states that simpliciality and non-symmetry were confirmed by exhaustive enumeration, but does not indicate the software or exact enumeration strategy used; a brief description of the verification pipeline would increase reproducibility.
  2. Table or listing of the five polytopes (or at least representatives of the two types) would benefit from an explicit statement of the vertex coordinates in a machine-readable format.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of its significance and the recommendation to accept.

Circularity Check

0 steps flagged

Explicit construction with machine-checkable verification; no circularity

full rationale

The paper's central claim is established by exhibiting an explicit list of 14 points in {0,1}^7, followed by direct verification that their convex hull is simplicial and not centrally symmetric. Both properties are finitary and confirmed by exhaustive enumeration up to cube symmetry, with no fitted parameters, self-referential definitions, or load-bearing self-citations. The argument is self-contained against external benchmarks and does not reduce any prediction or uniqueness claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests only on standard definitions from convex geometry with no free parameters, ad-hoc axioms, or invented entities.

axioms (2)
  • standard math Convex hull of a finite point set in R^d is a convex polytope whose faces are defined by supporting hyperplanes.
    Invoked to assert that the 14 points form a polytope with the claimed facial structure.
  • domain assumption A polytope is centrally symmetric if its vertex set V satisfies V = c - V for some center c.
    Standard definition used to check the absence of central symmetry in the constructed example.

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

  1. [1]

    GEPA: Reflective Prompt Evolution Can Outperform Reinforcement Learning

    URL https://arxiv.org/ abs/2507.19457. Oswin Aichholzer . Extremal properties of0/1-polytopes of dimension

    work page internal anchor Pith review Pith/arXiv arXiv

  2. [2]

    Catherine Babecki, Tycho Elling, and Asaf Ferber

    doi: 10.1007/978-3-0348-8438-9_5. Catherine Babecki, Tycho Elling, and Asaf Ferber . Sharp threshold for cliques in random 0/1 polytope graphs,

    work page doi:10.1007/978-3-0348-8438-9_5

  3. [3]

    URL https://arxiv.org/abs/2507.03212. Louis J. Billera and Carl W. Lee. Sufficiency of McMullen’s conditions for f -vectors of simplicial polytopes. Bulletin of the American Mathematical Society , 2(1):181–185,

    work page arXiv

  4. [4]

    Imre Bárány and Attila Pór

    doi: 10.1090/S0 273-0979-1980-14712-6. Imre Bárány and Attila Pór . On 0-1 polytopes with many facets. Advances in Mathematics , 161 (2):209–228,

    work page doi:10.1090/s0 1980

  5. [5]

    doi: https://doi.org/10.1006/aima.2001.1991

    ISSN 0001-8708. doi: https://doi.org/10.1006/aima.2001.1991. URL https://www.sciencedirect.com/science/article/pii/S0001870801919917. Asaf Ferber , Michael Krivelevich, Marcelo Sales, and Wojciech Samotij. On the edge expansion of random polytopes, pages 3022–3035

    work page doi:10.1006/aima.2001.1991 2001

  6. [6]

    URL https: //epubs.siam.org/doi/abs/10.1137/1.9781611978971.111

    doi: 10.1137/1.9781611978971.111. URL https: //epubs.siam.org/doi/abs/10.1137/1.9781611978971.111. Tamás Fleiner , Volker Kaibel, and Günter Rote. Upper bounds on the maximal number of facets of 0/1-polytopes. European Journal of Combinatorics, 21(1):121–130,

    work page doi:10.1137/1.9781611978971.111

  7. [7]

    Ewgenij Gawrilow and Michael Joswig

    doi: 10.1006/eujc.199 9.0326. Ewgenij Gawrilow and Michael Joswig. polymake: A framework for analyzing convex polytopes. In Polytopes—Combinatorics and Computation, volume 29 of DMV Seminar, pages 43–73. Birkhäuser Basel, Basel,

    work page doi:10.1006/eujc.199

  8. [8]

    URL https://polymake.org

    doi: 10.1007/978-3-0348-8438-9_2. URL https://polymake.org. Volker Kaibel. Low-dimensional faces of random 0/1-polytopes. In Daniel Bienstock and George Nemhauser , editors,Integer Programming and Combinatorial Optimization. Proceedings of IPCO X, New York, NY, volume 3064 of Lecture Notes in Computer Science , pages 401–415. Springer ,

    work page doi:10.1007/978-3-0348-8438-9_2

  9. [9]

    URL https://cloud.ovgu.de/s/2pabi8c4Lsz46gp

    doi: 10.1007/978-3-540-25960-2_30. URL https://cloud.ovgu.de/s/2pabi8c4Lsz46gp. Volker Kaibel and Martin Wolff. Simple 0/1-polytopes. European Journal of Combinatorics , 21(1): 139–144,

    work page doi:10.1007/978-3-540-25960-2_30

  10. [10]

    7 Ulrich Kortenkamp, Jürgen Richter-Gebert, Ambatipudi Sarangarajan, and Günter M

    doi: 10.1006/eujc.1999.0328. 7 Ulrich Kortenkamp, Jürgen Richter-Gebert, Ambatipudi Sarangarajan, and Günter M. Ziegler . Ex- tremal properties of 0/1-polytopes. Discrete & Computational Geometry , 17(4):439–448,

    work page doi:10.1006/eujc.1999.0328 1999

  11. [11]

    Peter McMullen

    doi: 10.1007/PL00009303. Peter McMullen. The numbers of faces of simplicial polytopes. Israel Journal of Mathematics , 9: 559–570,

    work page doi:10.1007/pl00009303

  12. [12]

    Richard P

    doi: 10.1007/BF02771471. Richard P. Stanley . Cohen–Macaulay rings and constructible polytopes. Bulletin of the American Mathematical Society, 81(1):133–135,

    work page doi:10.1007/bf02771471

  13. [13]

    Günter M

    doi: 10.1090/S0002-9904-1975-13670-6. Günter M. Ziegler . Lectures on 0/1-polytopes. In Polytopes—Combinatorics and Computation , vol- ume 29 of DMV Seminar , pages 1–41. Birkhäuser Basel, Basel,

    work page doi:10.1090/s0002-9904-1975-13670-6 1975

  14. [14]

    Günter M

    doi: 10.1007/978-3-034 8-8438-9_1. Günter M. Ziegler . Lectures on Polytopes , volume 152 of Graduate Texts in Mathematics . Springer , New Y ork, updated seventh printing of the first edition edition,

    work page doi:10.1007/978-3-034

  15. [15]

    doi: 10.1007/978-1-4613-8431-1

    ISBN 978-0-387-94365-7. doi: 10.1007/978-1-4613-8431-1. Corrected and updated printing. Max Zimmer , Nico Pelleriti, Christophe Roux, and Sebastian Pokutta. The Agentic Researcher: A Practical Guide to AI-Assisted Research in Mathematics and Machine Learning. Proceedings of the ICML 2026 Workshop on AI as a Tool for Mathematics, Computer Science, and Mach...

    work page doi:10.1007/978-1-4613-8431-1 2026

  16. [16]

    established

    doi: 10.48550/arXiv.2603.15914. 8 A The five not centrally symmetric examples in dimension seven Table 3gives explicit generating vertex sets for the five simplicial, not centrally symmetric7-dimensional 0/1-polytopes with 14 vertices of Theorem 4.3, one representative per orbit under the cube symme- try group B7. V1 V2 V3 V † 4 V5 0000000 0000000 0000000...

    work page doi:10.48550/arxiv.2603.15914