Entropy-type traces and moment expansions for the sine-kernel time-band limiting operator

We study the sine-kernel time–band limiting operator $S_c$ on $L^2(0,c)$ through the entropy-type trace $\mathrm{Tr}{\phi }_u(S_c)$, where ${\phi }_u(x)=\log (1+(e^u-1)x)-ux$. A Fourier factorization gives $S_c=A_c^{\ast }{A}_c$ and $\mathrm{Tr}{S}_c=c$. We derive an exact identity for $\mathrm{Tr}(S_c-S_c^2)$ and the lower bound $\mathrm{Tr}(S_c-S_c^2)\ge {\pi }^{-2}\log c-{\pi }^{-3}$, which in turn yields $\mathrm{Tr}{\phi }_u(S_c)\ge u\log c\mathrm{/}(4{\pi }^2)$. We also obtain a positive moment expansion $\mathrm{Tr}{\phi }_u(S_c)={\sum }_{n\ge 1}{A}_n(u)T_n(c)$ and, for each fixed $n$, the Landau–Widom asymptotic $T_n(c)=\log c\mathrm{/}({\pi }^{2}n)+o_n(\log c)$. Three natural uniformity hypotheses, denoted (LC), (MT), and (QM), lead to a quadratic lower bound of order $u^2\log c$ on logarithmically growing windows. We prove the implications among these hypotheses and the resulting bounds, with explicit constants, and explain why the fixed-index asymptotic alone does not provide the required uniformity. The hypotheses themselves remain open. Finally, a signed growing-parameter sine-kernel determinant theorem from a companion paper gives the quadratic lower bound independently of (LC), (MT), and (QM). This separates the conclusions available from elementary operator methods from those that use Riemann–Hilbert asymptotics.