A Mixed-Resonance Counterexample to the Lukic Conjecture

Abstract

Let $\hat{I}¼$ be a probability measure on the unit circle with Verblunsky coefficients $\hat{I}\pm$. Lukic conjectured that a weighted entropy condition with finitely many critical points is equivalent to a decomposition of $\hat{I}\pm$ 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 $-\infty$.

12 pages

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