\section{Windows and Weil positivity: a summary}\label{ce:sec:windows}
The margin is Weil's form $Q$ for one test function. The companion papers \cite{P20,P21,P22} study $Q$ on whole spaces of test functions. We summarize the results, with their status, and refer to those papers for proofs and data.

\emph{Dirichlet polynomials} \cite{P20}. Because $\Lambda\ge0$, the prime part of $Q$ on Gaussian-smoothed Dirichlet polynomials is unbounded below. So positivity is a cancellation between the archimedean place and the coherent sum of all primes, not a diagonal dominance (proved, elementary). With ball arithmetic, $Q$ is positive definite on these polynomials for every length $N\le179$ (computer-assisted, for smoothing width $0.005$). No translation-invariant square $\|Xk\|^2$ with $X$ a Fourier multiplier fits below $Q$ on the whole line; in particular, no Euler-product or mollifier multiplier does (proved).

\emph{The margin as $W-E$} \cite{P21}. Following Weil's proof for curves, write $Q=W-E$ with the pole $W=2\widehat g(\frac i2)\widehat g(-\frac i2)$. Then $W$ is a hyperbolic plane, like the two fibres of $C\times C$, and $Q\ge0$ on a space implies that $E$ has at most one positive direction there (proved; not conversely). The archimedean place alone has $(t_0/\pi+o(1))\log N$ positive directions, $t_0=6.2898$ (proved, of Szeg\H{o} type). Numerically, the primes remove them only all together and only with their powers.

\emph{Windows} \cite{P22}. In a window of length $L=\log N$ the form is one symbol. A positive completion of that symbol gives an exact sum of squares, and a coercive form has the completion $\pi/|E_L|^2$ with $E_L$ the de Branges function of the window (proved, classical in substance). For even $k$ of bounded variation supported in $[-\frac L2,\frac L2]$, with $K=\widehat k$, the explicit formula applied to $k*\tilde k$ gives
\[
Q(k)=\sum_\rho K(\gamma_\rho)^2 ,
\]
so under RH the least eigenvalue of a window is the minimum of $2\sum_{\gamma>0}K(\gamma)^2/\|k\|^2$ (proved). Connes observed that the minimizer reproduces the zeros \cite{Co26}. In high precision, it does so to $10^{-2}$ up to $1.3$ to $1.9$ times $T^*=2\pi N$, the height where the density $\frac1{2\pi}\log\frac t{2\pi}$ of the zeros equals the Nyquist density $L/2\pi$ of the window (numerical; Figure~\ref{ce:fig:edge}). For the Davenport--Heilbronn function, the agreement stops at the corresponding scale $2\pi N/5$. The zeros below $T^*$ carry a fraction between $10^{-25}$ and $10^{-8}$ of the least eigenvalue.

\begin{figure}[htbp]\centering\includegraphics[width=\textwidth]{fig/CE22_borde.png}
\caption{The minimizer of Weil's form in the windows $N=4.5,\,7.4,\,13,\,20.1$ \cite{P22}. Left: $\log_{10}$ of the distance from each zero of $\zeta$ to the nearest zero of the minimizer's transform. Right: the same data with the height in units of $T^*=2\pi N$ and the error in units of $\lambda_{\min}$; dashed $T^*$, dotted $eT^*$.}\label{ce:fig:edge}
\end{figure}