= Generator replacement proof of the small-support intersection lemma
Let $m$ be the smallest supporting prefix for a maximum <left-compressed set family> of $k$-sets that is <t-intersecting>, and set $N=n-m$. The <maximum-support generator fibre> gives exclusive count $\binom N{k-a}$ for a size-$a$ generator containing $m$. By <tight pairs of left-compressed generators>, only classes $a,b$ with $a+b=m+t$ can conflict after deleting $m$. For unequal such sizes, replacing both classes by the shortened generators of either class would imply
$$
\binom N{k-a+1}\binom N{k-b+1}\leq\binom N{k-a}\binom N{k-b}.
$$
When $n\geq2k-t+2$ and the classes have positive fibres, the reverse inequality is strict. For the remaining central class $a=t+j$, $m=t+2j$, some earlier coordinate is absent from at least $j/(m-1)$ of its shortened generators. Those generators are mutually t-intersecting. Replacing the central class by that subfamily has gain-to-loss ratio at least
$$
\frac{j}{m-1}\frac{N+1}{k-t-j+1},
$$
which exceeds one exactly when $n>(k-t+1)(2+(t-1)/j)$. This forces the support bound in the <small-support generating lemma for extremal intersecting families>.
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