The minimum order of a non-Hamiltonian inscribable simplicial polyhedron

We prove, without assuming the auxiliary conjectures in Dillencourt's 1996 study, that the minimum order of a non-Hamiltonian inscribable simplicial polyhedron is twenty. Complete enumeration gives the same numbers of eighteen- and nineteen-vertex 1-supertough candidates as his constructions: 698 and 9,232. Every candidate has an exact rational obstruction to inscribability. The eighteen-vertex exclusion already follows from his published count and earlier total enumerations; the substantive new exclusion is the complete nineteen-vertex case. There are exactly 17 extremal twenty-vertex types: 11 admit the T9 partner structure of Dillencourt's examples, while 6 have a different structure. A second computer-assisted proof of the bound and of this count does not enumerate the twenty-vertex triangulations. It rests on the fact that collapsing one side of a separating triangle of an inscribed simplicial polytope to a single vertex preserves inscribability, which reduces the search to gluings, along separating triangles, of pieces with at most fourteen vertices and of triangulations without separating triangles. The same computation shows that there are exactly 717 combinatorial types of non-Hamiltonian inscribable simplicial polyhedra with twenty-one vertices; 390 of them arise from the twenty-vertex types by adding a vertex on a face at a vertex of degree three. Every exclusion of a non-Hamiltonian candidate and every non-Hamiltonicity claim is backed by an independently checkable archived certificate. As a geometric consequence, any set of at most nineteen distinct points on a sphere with three-dimensional convex hull has a simple polygonal rim with exactly those vertices that bounds two disks on the hull boundary. This statement includes nonsimplicial hulls and is sharp at twenty points. MSC 2020: 05C45, 05C10, 52B10. Files: the paper (PDF) and its LaTeX source. Enumeration data: doi:10.5281/zenodo.22876289. Assembly certificates and programs: doi:10.5281/zenodo.22937208.