The Kneser spectrum gives an above-threshold overhead receipt

Exact statement

Assume $1\leq r\leq n/2$ . Let $N=\binom nr$ , $d=\binom{n-r}r$ , $\tau=\binom{n-r-1}{r-1}$ , and $\alpha=\binom{n-1}{r-1}$ . If an $r$ -uniform policy has $m>\alpha$ supports, then its number of disjoint pairs is at least $\left\lceil\frac{d+\tau}{2N}m(m-\alpha)\right\rceil$ . The depth-one one-shot overhead dominates this count.

StatusSelf-contained spectral derivation from the known Kneser spectrum; exact finite audit passes

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

Context

EKR says a collision must occur above the independent-set ceiling; the spectral calculation counts a forced first layer of collision witnesses.

Proof or evidence

The least Kneser eigenvalue bounds the induced edge count. Every induced edge is a size-two support matching, and FPRD-SH-T20 makes that a unit contribution to one-shot overhead.

Spectral collision receipt →

Verification notes

The rational inequality passes on every complete small-domain family and on 640 deterministic larger policies covering 10,466 matching states.

Limitations

The bound is not claimed sharp; stronger supersaturation theorems are known in restricted parameter ranges.
It requires 1<=r<=n/2 and lower-bounds the one-shot witness family, not the overhead of every expression tuple.
The spectral argument is classical framework synthesis, not a new Kneser-graph theorem.

Open work

Replace the general spectral receipt with the exact diversity and supersaturation hierarchy in its valid parameter ranges.