Title:Uniform sine-kernel determinant asymptotics, tail-side quantiles, and prolate eigenvalue bounds

Abstract:Let $S_c=P_{(0,c)}QP_{(0,c)}$ be the one-dimensional sinc-kernel concentration operator, let $N_a(c)=\#{n:\lambda_n(c)>a}$, and set $\bar L=\log((1-\delta)/\delta)$. We prove, uniformly for each fixed $A>0$, the tail-side quantile formula $N_\delta(c)=c+\pi^{-2}\bar L\log(4\pi^2c/\bar L)+O_A(\log c+\bar L)$ for $6\le\bar L\le A\log c$. It yields corresponding additive formulas for the lower half and full plunge, with main terms respectively $\pi^{-2}\bar L\log(4\pi^2c/\bar L)$ and twice this quantity. An exact one-tail-coordinate selection gives, for fixed $A>0$, $d\ge1$, and $q\in(1/2,1)$, the one-sided tensor-product bound $\Lambda_\delta(c;d)\ge\pi^{-2}d c^{d-1}\bar L\log(4\pi^2c/\bar L)-O_{A,d,q}(c^{d-1}(\log c+\bar L))$ for $L_{d,q}\le\bar L\le A\log c$, where $L_{d,q}=\log(q^{-(d-1)}(e^6+1)-1)$; the tensor content is nontrivial for $d\ge2$. The analytic input is a signed growing-parameter sine-kernel determinant asymptotic: uniformly for $0\le\omega\le A\log s$, $\log\det(I+(e^{2\omega}-1)K_s)=4\omega s/\pi+2\pi^{-2}\omega^2\log(4s)+2\log|G(1+i\omega/\pi)|^2+O_A((1+\omega)^4\log^2s/s)$, where $G$ is the Barnes $G$-function. We prove this negative-coupling counterpart of the Bothner--Deift--Its--Krasovsky theorem by direct IIKS steepest descent. We also retain the uniform head-side results and use a two-way determinant reduction to obtain the moving-depth lower-half bridge bound with constant $1/(32\pi^2)$; extending it to the deeper range uses Kulikov--Dam Larsen and may require a smaller constant. These counting formulas are additive. Their errors become uniformly relative when $\bar L$ tends uniformly to infinity; fixed thresholds are covered separately by Landau--Widom. A Lambert-$W_{-1}$ formula is recorded only for the continuous main term, not for individual eigenvalues.