Weil positivity in compact windows: certified two-sided bounds and a Landau--Widom decay law

Weil’s criterion equates the Riemann Hypothesis with the positivity of an explicit quadratic form $Q(f)$. For test functions supported in a window $[-L,L]$ we study the profile ${\lambda }^{\ast }(L)=\inf Q(f)\mathrm{/}\mathrm{\parallel }f{\mathrm{\parallel }}_2^2$ from both sides. A one-stroke reduction converts window positivity into positive semidefiniteness of a finite matrix; executing it at $L=0.8$ yields the unconditional certificate $Q(f)\ge 8.9\times {10}^{-18}\mathrm{\parallel }f{\mathrm{\parallel }}_2^2$ for all (complex) $f$ of support $1.6$, extending the classical range $\log 2$ of Yoshida and Connes–Consani by a factor $2.3$. The same computation shows that the window ground state is simple and even. On the other side, interval-arithmetic Rayleigh quotients, evaluated on the geometric side of the explicit formula with no appeal to zeros or to RH, give unconditional upper bounds down to $3.2\times {10}^{-283}$ at $L=2$. They follow $-\ln {\lambda }^{\ast }(L)\simeq 2{\pi }^{2}N(T^{\ast })\mathrm{/}\ln N(T^{\ast })$ with $T^{\ast }=2\pi e^{2L}$, the Landau–Widom plunge rate for time–band limiting. At $L=0.8$ the two halves enclose the profile, $8.9\times {10}^{-18}\le {\lambda }^{\ast }(0.8)\le 2.27\times {10}^{-17}$. Any pointwise-envelope certificate must resolve frequencies up to $T_1=2\pi e^{A_L}$ with $A_L\sim 4e^L$, a doubly exponential barrier. Under RH we prove ${\lambda }^{\ast }(L)\le \exp (-L{e}^L)$ for all large $L$.