A local quadratic bound for the planar centering gap

Let V(K) be the vertex index of an origin-symmetric planar convex body and W(K) its restriction to enclosing origin-symmetric quadrilaterals with four actual vertices. Conditional on the earlier candidate results giving the affine-regular hexagon minimum, pentagon lower bound and complete minimizing-quadrilateral classification (H1 and H2), this preprint gives a provisional proof of a uniform local bound 0 <= W(K) - V(K) <= C(d_BM(K,H) - 1)^2. The distance is untranslated and linear. The constants are common to every such body sufficiently near the affine-regular hexagon H, without bounds on face count, boundary regularity or relative perturbation scales. The proof retains the same actual radial sum in the lower and centered upper comparisons, using bounded corner supports, convexity of the true gauge and a quadratic repair to full containment. No effective constant, radius or sharp all-body coefficient is asserted.