Necessary Conditions for Single-Critical-Point Higher-Order Szegő Sum Rules in OPUC

We prove the necessity part of the higher-order Szegő theorem on the unit circle for the single-critical-point weights $H_m(e^{i\theta })=(1-\cos \theta {)}^m$, $m\ge 1$. If $\{{\alpha }_n{\}}_{n\ge 0}$ are the Verblunsky coefficients of a nontrivial probability measure $d\mu =w(\theta )d\theta \mathrm{/}(2\pi )+d{\mu }_{\mathrm{s}}$, then the weighted Szegő condition ${\int }_0^{2\pi }(1-\cos \theta {)}^m\log w(\theta )\frac{d\theta }{2\pi }>-\mathrm{\infty }$ implies ${\mathrm{\Delta }}^m\alpha \in {\mathrm{\ell }}^2,\alpha \in {\mathrm{\ell }}^{2m+2}\mathrm{.}$ The proof uses a finite-volume version of Yan’s higher-order sum rule. The quadratic part yields the $m$-th difference energy, and the logarithmic tail yields the ${\mathrm{\ell }}^{2m+2}$-control. The non-sign-definite critical terms are treated in two steps. First, the quartic principal critical block is isolated using the Yan quotient-algebra normal representative and shown to have a positive semidefinite Gram representation. Second, the remaining non-principal critical terms are controlled by the diagonal-vanishing property ${\mathcal{Y}}_{k,\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{(m)}\in {\mathfrak{I}}_k^{m+1-k},2\le k\le m,$ together with the Breuer–Simon–Zeitouni normal form, discrete interpolation, and Young’s inequality. These estimates yield a uniform finite-volume coercive bound, from which the necessity theorem follows for all $m\ge 1$.