GPT 6 Astra Ultra attempt on Math-1041

Cube-Simplex conjecture

Definitions and mathematical statement

A convex polytope is the convex hull of finitely many points in a finite-dimensional real affine space. Its dimension is the dimension of its affine span. A face is the set on which an affine linear functional attains its maximum on the polytope; the whole polytope is also allowed as an improper face.

A $k$-simplex is the convex hull of $k+1$ affinely independent points. A $k$-dimensional face is combinatorially a cube when its face lattice, ordered by inclusion, is isomorphic to the face lattice of $[0,1]^k$. This condition fixes incidences, not Euclidean lengths, angles, or a particular affine realization.

The cube--simplex conjecture is the quantified statement For every integer $k\ge1$, there is an integer $d\ge k$ such that every convex polytope $P$ of dimension at least $d$ has a $k$-face $F$ satisfying \[ F\text{ is a simplex or a combinatorial cube}. \] The dimension threshold may depend on $k$ but must be uniform over all polytopes and all their numbers of vertices. Either alternative may occur; it need not be the same alternative for all polytopes.

The target is a face of the original polytope. A simplex or cube found as an arbitrary subset, a section, a projection or a subdivision cell does not meet the requirement. There is no simplicity or simpliciality assumption on $P$. The assertion can also be expressed by asking whether the least such threshold $d$ is finite for every prescribed $k$.

Short English statement

Must every sufficiently high-dimensional convex polytope have a face of a prescribed dimension that is a simplex or combinatorially a cube?

Research attempt

Literature check and scope.

The catalogue entry [S2] predates the recent preprint by De Loera, Fang, Guo, Lu and Zheng [A1], whose abstract reports finiteness of the corresponding threshold for simple polytopes. That is a reported theorem of those authors; it is not independently proved here and does not establish the unrestricted conjecture. This attempt finds the exact threshold within the smaller class of Cartesian products of simplices, illustrating both a positive mechanism and a sharp obstruction within that class. Sources were checked 1 October 2026.

Faces of a product.

Let $P=\prod_{i=1}^r\Delta_{d_i}$ with all $d_i\ge1$. A linear functional on the product is a sum of factor functionals. Its maximum is attained exactly on the product of their maximizer faces. Thus every nonempty face is $\prod_i\Delta_{a_i}$, where $0\le a_i\le d_i$, and its dimension is $\sum_i a_i$. Conversely every such product is exposed by choosing exposing functionals independently in the factors. Zero-dimensional factors are points and do not change its combinatorial type.

A sharp restricted threshold.

Fix $k\ge2$ and suppose \[ \dim P\ge(k-1)^2+1. \] If some $d_i\ge k$, take a $k$-face in that simplex and vertices in all other factors. This gives a simplex $k$-face of $P$. Otherwise every $d_i\le k-1$. The dimension inequality then forces $r\ge k$. Choose an edge in each of $k$ factors and vertices in all the rest. The resulting face is a product of $k$ segments, hence a combinatorial $k$-cube. This proves the conjectural conclusion for this class with threshold $(k-1)^2+1$.

To see sharpness, take $P_0=(\Delta_{k-1})^{k-1}$, of dimension $(k-1)^2$. Any $k$-face would have factor dimensions $a_i$ summing to $k$, each at most $k-1$. At least two are positive. Its number of vertices is $\prod_i(a_i+1)>1+\sum_i a_i=k+1$, because the product expansion has at least one positive cross term. Hence it is not a simplex. It is not a cube either. If every positive $a_i$ were one, their sum would be at most $k-1$. Thus some $a_i\ge2$, and a triangular $2$-face in that factor, with vertices in all others, gives a triangular $2$-face of the candidate. Every $2$-face of a cube is a quadrilateral, a combinatorial contradiction. For $k=2$ the example simply has dimension one and no $2$-face; for $k\ge3$ the argument excludes every possible $k$-face explicitly.

Concrete checks.

For $k=3$, the product of two triangles has dimension four. Its $3$-faces are triangular prisms, with six vertices and triangular faces, so none is a tetrahedron or a cube. In dimension five any product of simplices either has a factor of dimension at least three or has at least three nontrivial factors, giving the desired face. The accompanying check enumerates the factor-dimension possibilities in the obstruction products for $3\le k\le6$; it verifies the vertex-count and triangular-face obstructions separately. The general proof is the dimension and product-face argument, not this finite enumeration.

First remaining gap.

An arbitrary convex polytope need not have any decomposition into Cartesian factors, and high dimension does not itself supply independent edge choices whose product is a face. Nor is an embedded cube subgraph sufficient: the requirement concerns an exposed face and its entire face lattice. The proof therefore cannot replace the product hypothesis by arbitrary convexity. The stronger simple-polytope result reported in [A1] also cannot be extended to nonsimple polytopes merely by perturbing vertices, because perturbation can destroy the particular face incidences being sought.

Reproducibility.

The finite calculations above are implemented in scripts/check_attempt_batch_2026_10_01.py; run it with Python 3 from the repository. It uses exact integer arithmetic and no external packages.

Final status.

UNRESOLVED. The exact threshold for products of simplices is proved; the full all-polytopes assertion is not. The restricted threshold is not claimed to be historically new. Completion estimate: 5%.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3099 (OPG-778), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Cube-Simplex conjecture. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/cube_simplex_conjecture.
• A1 J. A. De Loera, E. X. Fang, S. Guo, J. Lu and H. Zheng, On Unavoidable Faces of High-Dimensional Polytopes, arXiv:2609.00397v1, submitted 31 August 2026. Abstract and manuscript consulted for current context; its general theorem is not reproved here. https://arxiv.org/abs/2609.00397.