A sharp vertex-index bound between a diamond and an affine-regular hexagon

Let D=conv{+/-e1,+/-e2} and H=conv{+/-e1,+/-e2,+/-(e1+e2)}. This preprint presents candidate results proving that every origin-symmetric planar convex body satisfying D subset K subset H has a centered enclosing quadrilateral whose total vertex gauge is at most 3+2sqrt(2), strictly smaller unless K=H. The bound is sharp, with a self-contained centered lower bound at H. Up to an invertible linear map, this class consists exactly of the origin-symmetric bodies having a side whose actual endpoints maximize the absolute determinant of two points of the body. In particular, the strict bound holds for every genuine origin-symmetric octagon having such an adjacent pair. The proof compares two explicit quadrilaterals through radial segments contained in the body and applies to curved caps as well as polygons. The unrestricted sharpness consequence uses the separately cited value of the vertex index at H. No bound for all origin-symmetric octagons or all planar convex bodies is asserted.