 Small-support generating lemma for extremal intersecting families

Assume $1\le t\le r\le n$, $n>2r-t$, and $j\ge 0$. A maximum-cardinality left-compressed set family of $r$-sets that is t-intersecting has a generating family for a uniform set family supported on $[t+2j]$ under the displayed strict inequality. Reflection gives a right-end supporting interval for a right-compressed optimum. The strict threshold is important: at a tie between consecutive candidate families, use the next larger support bound. This auxiliary generating result, combined with the compatibility bound for complementary generating families, supplies the upper bound in the Ahlswede-Khachatrian theorem.