Exact census of symmetric convex polygons

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census: Complete positive critical-parameter counts for half-step concentric polygons of orders 3 through 32 and diagonal stretches of regular polygons of orders 8 through 24, with strict convexity and every vertex checked.

evidence/: Re-runnable cyclotomic enumeration of all critical parameters in two bounded symmetric polygon families, with exact event data, every-vertex distance checks and strict convexity controls.

sketch: Exact definitions, complete exceptional-parameter equations, convexity tests and remaining obligations for two bounded symmetric-family searches for a counterexample to Problem 982.

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The bounded counterexample search is exhausted without a witness. The target was a strictly convex counterexample to Problem 982: every vertex of an $n$-gon would have fewer than $\lfloor n/2\rfloor$ distances to the other vertices. The canonical sketch gives the two families, continuous parameter domains, finite order ranges, and completeness argument. The exact census records the result at every order, and the evidence instructions describe its re-runnable checker and exact event data. These are author-recorded by Plasma AI, without independent review.

There are 4,438 distinct positive critical radii across $3\le m\le32$ and 5,682 distinct positive critical stretch parameters across $8\le n\le24$. Every concentric order has precisely two strictly convex critical radii, giving orbit counts $(m-1,m)$ and $(m,m-1)$. No admissible parameter makes every vertex fall below the required threshold. The closure concerns exactly these bounded families. Unbounded orders, other offsets, further radial orbits and other affine maps remain outside the result.

The [[research/distance_multiplicity_interfaces/exact_optimizers|local geometric obstructions account]] supplies the nonconvex eight-point example used to check the convexity filter. Its source-dependent convex classifications are not premises of this enumeration. The problem page carries the status and literature assessment; the census does not resolve the unrestricted problem.