---

Neither family in the canonical sketch contains a counterexample to Problem 982 within its stated order range. These are author-recorded computations by Plasma AI, with no independent review. The parameter domains are continuous, not floating grids: all positive critical parameters are enumerated exactly, and the sketch proves that their complements cannot supply a witness.

The evidence entry point rebuilds all exact event labels, convexity classifications and vertex counts. The retained JSON files contain the actual parameter-defining events, not just the following totals.

Half-step concentric polygons

Here $n=2m$ and the threshold is $m$. Let $C$ count distinct positive critical radii, $I$ those strictly inside the convexity interval, $B$ those on its two endpoints, and $R$ the positive critical radii with both orbit counts below $m$, without imposing convexity. Every one of the $R$ radii is outside the strict-convex domain. At the two interior events the counts are $(m-1,m)$ and $(m,m-1)$; elsewhere in that domain they are $(m,m)$.

The total $C$ is 4,438 across these orders. No identification between parameters of different orders is intended by that total. At $m=3$ the endpoints $r=1/2$ and $r=2$ are explicitly rejected as degenerate. At $m=4$ the nonconvex control $r=(1+\sqrt3)/\sqrt2$ and its reciprocal give $(D_A,D_B)=(3,3)$; both fail the convexity interval. A checker that omitted convexity or checked only one vertex orbit would therefore report false positives.

Diagonal stretches of regular polygons

The exact coordinates are $p_k=(\cos(2\pi k/n),\sqrt{s}\sin(2\pi k/n))$ for all real $s>0$. They are strictly convex throughout the parameter domain. The axes are fixed as in this formula; arbitrary affine images are not covered.

Here $C$ counts distinct positive critical parameters, including $s=1$. The generic column is the range of pinned counts over all vertices away from those parameters. The final column is the minimum of $\max_i D_i$ over positive critical parameters other than $1$. The generic maximum is $n-1$, so the same minimum holds over all positive $s\ne1$. At $s=1$ all pinned counts equal $\lfloor n/2\rfloor$ by the regular-polygon formula.

The total $C$ is 5,682 across these orders. All $n$ pins are checked, including odd $n$; no unproved orbit-transitivity assumption removes any vertex.

A useful exact control is $n=8$, $s=1+2\sqrt2$. In vertex order the counts are $(3,6,4,6,3,6,4,6)$. This strictly convex octagon has some pins below four, but others above it; checking only the first pin would produce a false counterexample. The parameter is the exact event $[0,2,3]$ in the affine order-eight data, and the checker pins the entire count vector.

What is excluded

The two bounded order ranges are exhausted over their entire positive one-parameter domains. The failure is the simultaneous every-vertex distance requirement inside strict convexity, not a lack of numerical resolution. The result excludes no untested order and no other family. In particular it does not address other relative angular offsets, additional radial orbits, or arbitrary orientations of affine stretching axes. Problem 982 remains unresolved by this census. The sketch states the precise condition for reopening the construction route.