Uniform zero-overhead policies obey the Kneser stability trichotomy

Exact statement

For a one-support-per-letter policy with support family $\mathcal F\subseteq\binom{[n]}r$ , universal zero overhead is equivalent to $\mathcal F$ being independent in $KG_{n,r}$ . If $n<2r$ , every support family has zero overhead. If $n\geq2r$ , then $|\mathcal F|\leq\binom{n-1}{r-1}$ ; for $n>2r$ , equality requires a full star. When $r\geq2$ , $n>2r$ , and the core is empty, the sharp Hilton–Milner ceiling is $\binom{n-1}{r-1}-\binom{n-r-1}{r-1}+1$ . At $n=2r$ , maximum families choose one support from each complementary pair, and empty-core examples exist for $r\geq2$ .

StatusExact automata reduction with classical EKR and Hilton–Milner inputs; equality cases audited

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

Context

The universal zero-overhead policy criterion becomes an independence problem in the Kneser graph when every letter has one support of the same size.

Proof or evidence

FPRD-SH-T14 supplies the exact automata reduction. The case n<2r is immediate because no two r-sets are disjoint; EKR gives the ceiling from n=2r onward, and Hilton–Milner gives the exact empty-core ceiling for r>=2 and n>2r.

Uniform-support trichotomy →

Verification notes

Complete small domains cover 1,082,432 support families. At (n,r)=(7,3), all 175 maximum empty-core families are exactly the classical 140 Hilton–Milner and 35 three-set exceptional families.

Limitations

The extremal set theorems are classical; only their automata interpretation is FPRD work.
The theorem assumes one support per letter and a uniform support size; the Hilton–Milner clause requires r>=2 and n>2r.
Universal zero overhead does not imply that every disjoint-support policy has overhead for every expression tuple; it supplies an exact one-shot witness.

Open work

Use FPRD-SH-T23 for the quantitative receipt once the EKR ceiling is crossed.