Uniform sine-kernel determinant asymptotics and prolate eigenvalue bounds

Let $S_c=P_{(0,c)}Q{P}_{(0,c)}$ be the one-dimensional sinc-kernel concentration operator, with eigenvalues $1>{\lambda }_0(c)\ge {\lambda }_1(c)\ge \cdots >0$. We study the lower-half count $D(\delta ,c)=\mathrm{\#}\{n:\delta <{\lambda }_n(c)\le 1\mathrm{/}2\}$ and prove, for a fixed $\alpha \ge 4$ and all sufficiently large $c$, that $D(\delta ,c)\ge (1\mathrm{/}32{\pi }^2)L\ln (\alpha c\mathrm{/}L)$ with $L=\ln (1\mathrm{/}\delta )$, uniformly on $c^{-\alpha }\le \delta \le {\delta }_0$. The complementary deep range follows from the theorem of Kulikov and Larsen. The main analytic input is a signed, growing-parameter sine-kernel determinant asymptotic: for each fixed $A>0$, uniformly for $0\le \omega \le A\ln s$, $\ln \det (I+(e^{2\omega }-1)K_s)=4\omega s\mathrm{/}\pi +(2{\omega }^2\mathrm{/}{\pi }^2)\ln (4s)+2\ln \mathrm{\mid }G(1+i\omega \mathrm{/}\pi ){\mathrm{\mid }}^2+O_A((1+\omega {)}^4{\ln }^{2}s\mathrm{/}s)$, where $G$ is the Barnes $G$-function. This is the $\gamma <0$ counterpart of the theorem of Bothner, Deift, Its and Krasovsky. It is proved by a direct IIKS steepest-descent analysis rather than by continuation in the parameter: positivity gives solvability, the confluent-hypergeometric model is derived from its defining Kummer functions, and the endpoint matching is estimated after the Stokes-sector cancellations. The unitary equivalence $S_c\simeq K_{\pi c\mathrm{/}2}$, a Barnes-$G$ lower bound, and a two-way determinant reduction then yield the bridge estimate. We also prove the uniform head quantile formula $N_{1-\epsilon }(c)=c-{\pi }^{-2}\bar{L}\ln (4{\pi }^{2}c\mathrm{/}\bar{L})+O(\ln c+\bar{L})$ for $6\le \bar{L}=\ln ((1-\epsilon )\mathrm{/}\epsilon )\le c^{1\mathrm{/}3}$, together with its upper-half, tensor-product, and joint-range consequences.