Equality and symmetry breaking in a planar octagon family

For the origin-symmetric octagons Lε,t = conv{±(1,0), ±(0,1), ±(1−tε,1−ε), ±(−1+ε,2ε)}, we compare the vertex index V, which permits arbitrary finite enclosing polygons, with its origin-centered quadrilateral restriction W. For each fixed real t and all sufficiently small positive ε, exact equality V = W holds when t < √2−1 or t ≥ √2/3. For √2−1 < t < √2/3, the gap is W(Lε,t) − V(Lε,t) = (1−t)(t−√2+1)ε2/(4√2) + O(ε3) > 0. At t = √2−1, the first nonzero term is instead (23√2−32)ε3/16, with remainder O(ε4). A moving-facet gauge plateau controls arbitrary nearby enclosures; exact product monotonicity centers the competing orientation. Matching feasible polygons yield scalar envelopes of the actual indices. This provisional candidate classification is conditional on the previously obtained minimum, genuine-pentagon bound and complete equality classification for the limiting affine-regular hexagon. No effective parameter cutoff, uniform classification across the transitions or classification of all optimizing polygons is claimed.