A quadratic symmetry-breaking gap for symmetric octagons

For an origin-symmetric planar convex body K, the vertex index V(K) minimizes the sum of vertex gauges over all finite polygons enclosing K. Let W(K) be the same minimum restricted to origin-centered quadrilaterals. This preprint studies the octagons Kε = conv{±e₁, ±e₂, ±(1−9ε, 1−20ε), ±(−1+20ε, 40ε)}. For every sufficiently small positive ε, the proposed theorem gives W(Kε)−V(Kε) = (319√2/8−55)ε² + O(ε³) > 0. The constant and first-order terms agree. A uniform four-contact lower relaxation, transverse gauge cusp and endpoint clamp identify the two minima with analytic scalar problems. The unrestricted minimum selects noncentral endpoints of the limiting hexagon family, while the centered problem remains at its central member. The results are provisional candidate theorems conditional on the cited earlier hexagon classification and localization results. No effective positive parameter cutoff is obtained. The supporting archive contains proof sources, a verification and reproducibility report, finite checks and their evidence, and the exact imported articles.