A Complete Proof of the Simon--Lukic Conjecture for Higher-Order Szeg\H{o} Theorems

This paper provides a complete proof of Simon-Lukic conjecture for orthogonal polynomials on the unit circle. For a probability measure $d\mu = w(\theta) \frac{d\theta}{2\pi} + d\mu_s$ with Verblunsky coefficients $\alpha=\{\alpha_n\}_{n=0}^\infty$, distinct singular points $(\theta_k)_{k=1}^{\ell}$, and multiplicities $(m_k)_{k=1}^{\ell}$, we establish the equivalence between the entropy condition $$\int_0^{2\pi} \prod_{k=1}^{\ell} [1 - \cos(\theta - \theta_k)]^{m_k} \log w(\theta) \frac{d\theta}{2\pi} > -\infty$$ and the decomposition condition $$\exists \beta^{(1)}, \ldots, \beta^{(\ell)} : \alpha = \sum_{k=1}^\ell \beta^{(k)} \,\, \text{with} \,\, (S - e^{-i\theta_k})^{m_k} \beta^{(k)} \in \ell^2, \,\, \beta^{(k)} \in \ell^{2m_k + 2}.$$ The proof synthesizes unitary transformations, discrete Sobolev-type inequalities, higher-order Szeg\H{o} expansions, and a novel algebraic decomposition technique. Our resolution affirms that spectral theory is fundamentally local-global behavior emerges from the superposition of local resonances, each governed by its intrinsic scale.