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Claim. Let $3\le m\le32$, put $n=2m$, and let $r$ be real with $\cos(\pi/m)<r<\sec(\pi/m)$. The $n$ points of the plane, written as complex numbers,

$$A_k=e^{2\pi ik/m},\qquad B_k=re^{(2k+1)\pi i/m},\qquad 0\le k<m,$$

form a strictly convex polygon, and some vertex has at least $m=\lfloor n/2\rfloor$ distinct distances to the other vertices. Let $8\le n\le24$ and let $s>0$ be real. The $n$ points

$$p_k=\bigl(\cos(2\pi k/n),\sqrt s\,\sin(2\pi k/n)\bigr),\qquad 0\le k<n,$$

form a strictly convex polygon, and some vertex has at least $\lfloor n/2\rfloor$ distinct distances to the other vertices. Thus neither family contains a counterexample in these ranges.

Covers. Two one-parameter families of convex polygons on Problem 982: the half-step concentric pairs of regular $m$-gons, of even orders $6\le n\le64$, and the regular $n$-gons stretched along the fixed coordinate axes, of orders $8\le n\le24$. It settles no order $n$: other convex $n$-gons, other angular offsets between the two orbits, further radial orbits, stretches along other axes and all orders outside these ranges are not covered, and the problem remains open.

Standing. Tier 0. The reduction to finitely many critical parameters, the strict-convexity tests and the census are author-recorded by Plasma AI, without independent review. The lead's evidence re-runs the exact enumeration in cyclotomic arithmetic, with certified real enclosures deciding only strict signs; it is author evidence and confers no tier. No Lean proof is asserted. Catalog acceptance is pending.

Depends on. the symmetric-family census lead (tier 0), with its critical-parameter reduction and its exact census.