The sharp intrinsic quadratic centering gap for symmetric octagons

Let V(K) be the vertex index of an origin-symmetric planar convex body and let W(K) restrict the enclosing polygons to origin-centered quadrilaterals. Conditional on the earlier hexagon minimum, pentagon bound and complete quadrilateral equality classification, this preprint determines the sharp local coefficient of W(K)-V(K) in linear Banach-Mazur distance from an affine-regular hexagon. Over genuine origin-symmetric octagons the coefficient is 9/64. The upper comparison covers arbitrary approaching sequences, including multiscale endpoint degenerations. It compares centered enclosures at the same radial sum, retains the exact contact switches and combines a metric dual inequality with a sharp square identity. An explicit polynomial octagon family attains the coefficient asymptotically. The main result remains provisional and conditional on the stated earlier candidate results. No effective radius or exact finite-neighborhood bound with coefficient 9/64 is asserted.