Stability of the extra edge vertex in near-extremizing symmetric decagons

For origin-symmetric planar polygons with at most ten vertices sufficiently close to the affine-regular hexagon and with a sufficiently small shortfall from the limiting normalized centering-gap coefficient 1/4, the paper controls the tangential position and scaled normal lift of the extra edge vertex in a positively ordered anchored chart. The error is bounded by a constant times the sum of the normalized gap shortfall and the Banach–Mazur distance excess. The linear order in the shortfall is sharp for tangential position. The paper also gives a cubic upper remainder for the centering gap throughout the nearby vertex-count class. The results are provisional and conditional on the stated hexagon premises and local comparisons.