An improved cubic centering-gap lower bound for symmetric decagons

For a perturbed family of origin-symmetric decagons, the linear Banach–Mazur distance from the affine-regular hexagon is determined exactly and its optimizing maps are classified. Conditional on the specified hexagon and orientation results, the unrestricted and centered vertex-index minima reduce to scalar branches. Their comparison along a threshold path gives the intrinsic cubic lower bound (21√2+2)/32. The results are provisional.