Negative answers to the printed totally mixed face questions

Abstract

A full-dimensional lattice-polytopal counterexample to both unrestricted assertions in Sam Payne's printed 2019 totally mixed face-poset question. For the standard four-simplex D and I=[0,1], the Minkowski sum of D×I and (-D)×I has a positive-dimensional, dimension-additive face poset whose order complex is two connected pure graphs, each with 50 vertices and 60 edges. Its rational Betti numbers are (2,22,0,...), so it is nonshellable and fails the printed sphere-wedge homology assertion. The shared interval factor deliberately gives a nongeneric example. Four-page research note with self-contained proof, TeX source and two independently authored standard-library exact verification programs. Classical simplex-difference and exact-decomposition geometry and the lower-dimensional six-point control are credited to prior work. The different mixed-face criteria in Adiprasito et al. and Adiprasito's stable-intersection work are distinguished explicitly. The revised July 2026 Hiatt–Simpson–Wang–Wu tropical-fan counterexample is credited as a different object, without assuming a two-summand exact-poset representation. No generic or stable tropical shellability problem is claimed resolved. A bounded current primary-source priority audit is included, with the uninspected paywalled 2024 article-body limitation disclosed; no exhaustive literature or first-ever priority certification is claimed. AI tools were used extensively in development, verification, literature review and writing. Unrefereed; no external human peer review or formal proof-assistant certification is claimed.