Ziegler's Cross-Polytope Conjecture (simplicial 0/1-polytopes)
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Statement
Ziegler proved every simplicial -dimensional 0/1-polytope has at most vertices, and asked whether attaining vertices forces central symmetry (i.e. a 0/1 cross-polytope). Known true for ; open since ~2000.
Context
One of Ziegler's known 0/1-polytope questions.
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An agentic research framework (locally-deployed open-weights DeepSeek V4 Flash + GLM 5.2, augmented with reflection prompts) first produced a flawed proof that the conjecture holds, then attempted a Lean 4 formalization as verification. The formalization failed, and from that failure the agent extracted the combinatorial condition that yielded an explicit counterexample: 14 vertices in whose convex hull is simplicial but not centrally symmetric.
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