A Counterexample to a Corner-Peeling Conjecture for Subcomplexes of Z^3

Let G be a finite connected subgraph of the grid Z^3, not necessarily induced, and let X(G) be the cube complex whose cells are the squares and 3-cubes of Z^3 all of whose vertices and edges lie in G. A corner is a vertex that lies in exactly one maximal cell, and a corner peeling is an ordering v_1,...,v_n of the vertices in which each v_i is a corner of the subgraph induced by v_1,...,v_i. In the problem session of the 2026 Oberwolfach workshop Median Geometry and Applications, V. Chepoi, from discussions with J. Chalopin and M. Kokkou, conjectured that G admits a corner peeling whenever X(G) is simply connected and every connected component of every section of X(G) by a coordinate plane is simply connected. We show that the conjecture is false. We give a subgraph G_62 of Z^3 with 62 vertices in [0,3]^3 and an induced subgraph H_73 with 73 vertices in [0,4]^3 such that the cube complex and all its sections are collapsible, but there is no corner at all. Both examples have a symmetry group of order 6. The proofs can be checked by hand; explicit sequences of elementary collapses are supplied as machine-checkable certificates. A subdivision construction turns every finite subgraph into an induced subgraph whose pieces and corners are dilates of the original ones, so the partial- and induced-subgraph formulations are equivalent. Uncertified solver computations indicate that no counterexample fits into a 3 x 3 x 3 box of lattice points. Scope: This paper supplies exact counterexamples to the partial- and induced-grid-graph versions of the corner-peeling conjecture corresponding to corpus record OWR-14299911-029. The 62- and 73-vertex examples and their collapse certificates are verified. Small-box solver results are explicitly uncertified; no smallest-size counterexample is proved, and the least size remains unknown. A formulation with the additional isometry requirement (3) is not refuted here. No absolute priority claim is made. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-14299911-029, Oberwolfach Reports, Report 8/2026, Problem 15. Public paper page: https://eulersolve.org/papers/owr-14299911-029/