Any intersecting support core factorizes exact overhead

Exact statement

Let $\mathcal C\subseteq\mathcal F$ be any pairwise-intersecting support family and $\mathcal B=\mathcal F\setminus\mathcal C$ , with $|\mathcal B|=k$ . Every compatible family is an exceptional matching in $\mathcal B$ , with at most one compatible member of $\mathcal C$ . Hence the target-oracle and one-shot factorizations hold with $k$ in place of diversity. Among decompositions obtained from one intersecting core, the smallest possible exception count is $\kappa(\mathcal F)=\tau(D_{\mathcal F})$ , and $\kappa\le\gamma$ .

StatusSelf-contained proof; exact minimum-cover and target audits pass

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

Context

A closest star is only one pairwise-intersecting core; its complement need not be minimum.

Proof or evidence

Every support matching uses at most one core support, so the previous inclusion-exclusion and zeta proofs apply verbatim.

Intersecting-core factorization →

Verification notes

The preserved verifier covers 1,088 complete families, 2,600 larger policies, and 1,200 valid target systems. A separate checker exhaustively confirms 16,384 support families and 181,196 matching states.

Limitations

Kappa is optimal only within the displayed intersecting-core factorization.
Finding a minimum cover has its own FPT cost; the factorization is cleanest for a supplied cover.
Arbitrary targets still require the expression-dependent oracle.
The graph machinery is classical.

Open work

Use kappa only as the optimum of this supplied-core decomposition; do not infer a general arithmetic lower bound.