= Entropy bound for a union-closed family
Let $\mathcal F$ be a finite <union-closed family> whose <union> is nonempty. Some ground element belongs to at least a proportion
$$
c=\frac{3-\sqrt5}{2}
$$
of its members. Requiring a nonempty <union> excludes both the empty <set family> and $\{\varnothing\}$, for which no element can be chosen.
Here is the <entropy> mechanism. Choose independent uniform random members $A,B$ of $\mathcal F$. Then $H(A)=\log|\mathcal F|$ and $H(A\cup B)\leq\log|\mathcal F|$, because the <union> still belongs to $\mathcal F$. If every element had <probability> at most $u<c$ of belonging to $A$, <Will Sawin>'s <binary entropy> estimate would give
$$
H(A\cup B)\geq\frac{h_2(2u-u^2)}{h_2(u)}H(A)>H(A),
$$
a contradiction when $|\mathcal F|>1$. A singleton <set family> with nonempty <union> satisfies the conclusion directly.<An improved lower bound for the union-closed sets conjecture by Will Sawin>[Theorems1–2 and proof of Theorem1]{ref}
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