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. The computation provides independently checkable certificates and agrees with external triangulation and Hamiltonicity counts. There are exactly 17 extremal twenty-vertex types: 11 admit the T₉ partner structure of Dillencourt's examples, while 6 have a different structure. The latter decompose along separating triangles into two octahedral blocks and eleven tetrahedral blocks. 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. The accompanying data provide the graph classifications, exact certificates and reproducible checking programs.