A decagon obstruction to the octagonal centering-gap constant

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, complete equality classification and specified local lower estimates from the no-linear-gap analysis, a rational symmetric decagon family K_e approaches the affine-regular hexagon H with liminf (W(K_e)-V(K_e))/(d_BM(K_e,H)-1)^2 at least (110 sqrt(2)-135)/128 = 0.16065228016437856, strictly above 9/64. An added vertex pair changes the active star gauge while preserving useful containment contacts. The argument identifies the actual global centered minimum, compares it with a feasible noncentral enclosure and supplies an explicit linear Banach-Mazur sandwich. The result remains provisional and conditional on those imported premises. No exact unrestricted minimum, sharp all-body coefficient, general quadratic upper bound or effective cutoff is asserted.