Diversity can overestimate the best intersecting-core kernel without bound

Exact statement

For every prime power $q$ , the line supports of $PG(2,q)$ are pairwise intersecting, so $\kappa=0$ . There are $q^2+q+1$ supports and every participant lies on $q+1$ of them, so closest-star diversity is $\gamma=q^2$ . Thus $\gamma-\kappa$ is unbounded within the intersecting-core decomposition.

StatusSelf-contained finite-geometry proof; exact q=2,3,5 controls pass

External reviewNo documented external or specialist review of these FPRD results is recorded.

Context

Large intersecting families need not have a common participant.

Proof or evidence

Projective-plane lines form a complete intersecting core while every participant star omits q squared lines.

Unbounded diversity gap →

Verification notes

Exact incidence constructions give (gamma,kappa)=(4,0),(9,0),(25,0).

Limitations

The projective-plane family uses growing uniformity.
This is a correction, not a new finite-geometry theorem.

Open work

Do not describe diversity as a complete or optimal search kernel.