Uniform quadratic control of the centering gap near the affine-regular hexagon

A provisional proof is given, conditional on the earlier hexagon minimum, pentagon lower bound and complete quadrilateral equality classification, of a uniform estimate 0 ≤ W(K) − V(K) ≤ C(d_BM(K,H) − 1)² for every genuine origin-symmetric planar octagon sufficiently near the affine-regular hexagon H. Here V is the vertex index and W restricts enclosing polygons to quadrilaterals centered at the origin. The constants are common to the whole neighborhood, including multiscale approaches in which an inserted vertex approaches an edge endpoint at a different rate from the geometric error. The proof adapts first-order lower-bound devices, controls degenerating contacts before expansion, and matches actual-minimizer lower bounds with feasible centered upper enclosures. The exponent 2 is optimal conditional additionally on the earlier positive quadratic-gap family. No effective numerical constant, radius, best constant, exact equality of the indices, or centeredness of unrestricted optimizers is asserted.