AI 中文总结
通过计算机辅助证明，结合独立与条件独立采样及全局耦合，证明了有限并闭族中某元素出现频率至少为0.38288525，并给出依赖族大小的额外频率界。
AI 中文摘要
我们给出了一个计算机辅助证明：每个包含非空集合的有限并闭族中，至少有一个元素出现在其0.38288525比例的成员中。该论证将独立采样与条件独立采样相结合，并利用一个二元熵核的显式乘积下界。我们通过一个单变量平稳方程确定了该逐点方法的极限常数。我们还通过平衡逆并重数构造了一个全局耦合，并证明了相对于独立采样的严格正定量熵增益。将这两个论证相结合，可得到一个依赖于族大小的额外频率界。
英文摘要
We give a computer-assisted proof that every finite union-closed family containing a nonempty set has an element in at least 0.38288525 of its members. The argument combines independent sampling with conditionally independent sampling and an explicit product lower bound for a binary-entropy kernel. We identify the limiting constant of this pointwise method through a one-variable stationary equation. We also construct a global coupling by balancing inverse union multiplicities and prove a strictly positive, quantitative entropy gain over independent sampling. Combining the two arguments gives an additional frequency bound depending on the size of the family.
