AI 中文总结
针对Lukic猜想，通过GPT-5.6构造了重数为3的两临界点混合共振反例，该序列满足分解条件但加权熵为-∞，推翻了原猜想。
AI 中文摘要
设μ为单位圆上的概率测度，其Verblunsky系数为α。Lukic猜想，具有有限个临界点的加权熵条件等价于α在这些点处的局部分量分解。我们给出了由GPT-5.6找到的、针对重数为3的两个临界点的反例，该构造使用了两个具有相同幂衰减指数3/20的相位模式。该序列满足Lukic的分解条件，但对应两点权重的加权熵等于-∞。
英文摘要
Let $μ$ be a probability measure on the unit circle with Verblunsky coefficients $α$. Lukic conjectured that a weighted entropy condition with finitely many critical points is equivalent to a decomposition of $α$ into components localized at those points. We give a counterexample for two critical points of multiplicity three, found by GPT-5.6. The construction uses two phase modes with common power-decay exponent $3/20$. The sequence admits the Lukic's decomposition conditions, but its weighted entropy for the corresponding two-point weight equals $-\infty$.
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