The sharp local planar centering gap

For an origin-symmetric planar convex body K, let V(K) be its vertex index and let W(K) restrict the enclosing polygons to genuine origin-centered quadrilaterals. Conditional on the stated earlier hexagon minimum, pentagon lower bound and complete minimizing-quadrilateral classification, this preprint presents a provisional proof that the sharp local supremal coefficient of [W(K)−V(K)]/[d_BM(K,H)−1]^2 near the affine-regular hexagon H is (4+3√2)/32. The distance is untranslated and linear. The upper bound covers arbitrary convex bodies, including curved boundaries and multiscale approaches. An exact convex-duality certificate produces a containing centered quadrilateral; a quadratic cancellation and retained nonnegative excesses turn its comparison with an actual unrestricted minimum into sharp metric inequalities. Genuine twelve-gons approach the coefficient in an iterated limit. No effective radius, exact unrestricted minimum for this family or complete extremizer classification is asserted.