Minimal uniformity of the non-Archimedean vertex types in unit-edge tilings

Of the 21 cyclic sequences of regular polygons that can surround a vertex of the Euclidean plane, eleven — the Archimedean types — occur in vertex-transitive tilings by regular polygons; the remaining ten, here called the non-Archimedean vertex types, do not. For each such species z we ask how uniformly the plane can be tiled around it when irregular tiles are admitted as sparingly as possible: in the class U(z), all tiles are simple polygons with unit edges, some vertex shows exactly z in regular tiles, every vertex carries at most one irregular tile with its regular tiles reading a contiguous arc of z — so the irregular tiles are pairwise disjoint and every vertex still exhibits z — and no regular polygon can be split off an irregular tile. The minimal uniformity of z is the least number of vertex orbits of a tiling in U(z). We prove all ten values: 3 for (3.8.24), (3.4.3.12) and (3.4².6); 4 for (3.9.18), (5².10) and (3².4.12); 5 for (3.10.15) and (3².6²); 7 for (4.5.20); and 10 for (3.7.42). Lower bounds are exhaustive scans of Delaney–Dress symbol catalogues — certified complete by DRAT-checked UNSAT proofs at up to two vertex orbits and, chamber count by chamber count, for the three-orbit catalogue, by a cross-validated enumeration beyond — sharpened by a chamber bound, stabiliser counting and a lattice argument proved from the definition, each surviving candidate refuted by the same exact arithmetic that realizes the witnesses; upper bounds are rigid explicit tilings, realized in exact cyclotomic arithmetic through a torus-covering criterion. Two questions remain open: whether (3².4.12) has a convex witness, and whether it has a member whose irregular tile is no edge-to-edge union of regular polygons. Keywords: minimal uniformity; k-uniform tiling; vertex configuration; unit-edge polygon; Delaney–Dress symbol; computer-assisted proof. MSC 2020: 52C20 (primary); 05B45, 52C25, 68V05.