A symmetric octagon above the hexagonal quadrilateral threshold

For an origin-symmetric planar convex body K, let W(K) be the least sum of its four vertex gauges among origin-centered quadrilaterals containing K. We exhibit the rational octagon K = conv{±(500,0), ±(491,480), ±(0,500), ±(−480,40)} and give a written proof and exact rational certificates for 11657/2000 ≤ W(K) ≤ 12859729099/2206320000. The lower endpoint exceeds 3+2√2, the affine-regular hexagon value, so that value is not a universal upper bound for centered quadrilateral enclosures, even for symmetric octagons. The proof reduces all supporting-strip directions to finitely many facet pairs and four one-variable families, then verifies explicit rational quadratics. A second certificate covers the complete two-normal parameter space. This provisional candidate result does not determine the exact W(K) or settle the corresponding bound for the unrestricted vertex index.