Frankl-complete configurations and Morris's asymptotic conjecture

It is proved that \(\mathrm{FC}(k,n)=\Theta_k(n^{k-2})\) for every fixed \(k\ge2\), resolving Morris’s asymptotic conjecture. The new upper bound combines robust pair lifting of weighted certificates with extremal forcing, matching Morris’s lower bound. Sharp four-uniform sunflower thresholds are obtained, while exact finite certificates establish \(\mathrm{FC}(4,9)=16\) and refute a lexicographic extremality conjecture of Pulaj and Wood. The general asymptotic proof is independent of the finite classification. V2 extends the four-uniform result of V1 to all fixed uniformities, retaining its sharp and finite results. Code and exact certificates are available in the companion repository: https://github.com/michaeliu4/frankl-complete-four-sets/releases/tag/V1