Symmetric octagons with vertex index above the hexagonal value

This preprint presents candidate asymptotic results for the planar vertex index. For an origin-symmetric planar convex body K, the vertex index V(K) is the infimum of the total K-gauge over all finite enclosing polygons, and W(K) is the corresponding minimum over origin-centered enclosing quadrilaterals. Put C=3+2sqrt(2). For K_epsilon=conv{+/-e_1, +/-e_2, +/-(1-9epsilon,1-20epsilon), +/-(-1+20epsilon,40epsilon)}, we obtain the separate asymptotic expansions V(K_epsilon)=C+((9sqrt(2)-11)/2)epsilon+o(epsilon) and W(K_epsilon)=C+((9sqrt(2)-11)/2)epsilon+o(epsilon) as epsilon tends to zero through positive values. The positive coefficient gives symmetric octagons, arbitrarily close to an affine-regular hexagon, whose unrestricted vertex index exceeds C. The proof classifies all minimum hexagon enclosures, controls perturbations using an exact nonnegative radial baseline, and matches a uniform lower estimate with an explicit centered upper enclosure. No effective positive parameter cutoff or exact equality V(K_epsilon)=W(K_epsilon) at a fixed positive parameter is asserted.