Policy diversity gives a quantitative obstruction receipt

Exact statement

For an $r$ -uniform support family $\mathcal F$ , let $m=|\mathcal F|$ , let $d$ be its maximum participant degree, let $\gamma=m-d$ , and let $q(\mathcal F)$ count disjoint support pairs. Then $q(\mathcal F)\geq\gamma\max\{0,d-[{n-1\choose r-1}-{n-r-1\choose r-1}]\}$ . Every counted pair is a realizable two-signature prefix-splitting receipt.

StatusSelf-contained all-length counting proof; complete small-family audit passes

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

Context

Hilton–Milner says when a non-star collision must exist; the diversity receipt counts a forced block of collisions around a maximum-degree participant.

Proof or evidence

A support avoiding the maximum-degree participant can meet at most the Hilton–Milner star slice. Every remaining hub support is disjoint from it, and crossing pairs are counted once.

Diversity obstruction receipt →

Verification notes

The inequality passes on all 1,082,368 completely enumerated small uniform families.

Limitations

The set-system inequality is elementary and may be classical in substance; no priority claim is made.
Disjoint-pair receipts need not equal total reachable automaton-state overhead because several receipts may meet at one product state.

Open work

Test sharpness at the first positive-overhead layer and determine how pair receipts aggregate into total prefix-state overhead.