Tagewerk XXII

Kernel-attested counterexample to Ziegler's cross-polytope conjecture for simplicial 0/1-polytopes (Kaibel–Pokutta 2026)

A fresh 2026 result in a domain new to the ledger — polytope theory / discrete geometry — independently confirmed and re-decided by the Lean kernel. Ziegler proved every simplicial d-dimensional 0/1-polytope has at most 2d vertices and asked (Question 1.1) whether equality forces central symmetry, i.e. a 0/1-realization of the d-dimensional cross polytope. Kaibel & Pokutta (arXiv:2606.31640) answer NO with an explicit 14 = 2·7 vertices in {0,1}^7 whose convex hull is a simplicial 7-polytope that is not centrally symmetric (d=7 is the first dimension where this can occur). Leibniz re-decides the counterexample from the 14 vertices by exact rational linear algebra — the paper's own method ("carried out exactly over ℚ"): dim P = 7 (rank[1|V]=8); exact facet enumeration finds exactly 136 supporting facets, each an affinely-independent 6-simplex on 7 of the 14 vertices; the 136 facets form a CLOSED pseudomanifold — every one of the 476 ridges lies in exactly 2 facets, so (since ∂P is a connected pseudomanifold) the enumeration is complete and P is simplicial; and P is not centrally symmetric — V is balanced (each coordinate sums to 7, so the only possible centre is (½,…,½)), yet four vertices lack their cube antipode 1−v. The Lean 4.31 kernel then independently re-decides (plain decide, report-only) the dimension, the supporting-hyperplane structure (each facet cut out by a hyperplane touching exactly 7 vertices), the closed-pseudomanifold completeness, and the non-central-symmetry; affine-independence of the 136 facets (136 nonzero determinants) is carried by the exact-rational leg, exceeding the kernel's decide budget. Report-only, audit tier — the kernel observes; nothing sets kernel_verified. LLMs propose nothing; exact linear algebra and the kernel decide.