Slip Certificates for the Riemann ξ–Function via Poisson Forcing and Carleson Tents a Stagewise One-Dimensional Criterion for Zero-Free Rectangles

Let $\xi(s)$ be the completed Riemann zeta function and $\Xi(z)=\xi(\tfrac12+\ii z)$. We study the logarithmic derivative field \[ m(z):=-\frac{\Xi'(z)}{\Xi(z)} \] and introduce a one-dimensional functional along horizontal scan lines $z=t+\ii\eta$: for a bounded interval $I\subset\RR$ and $\eta>0$, \[ \Slip^+_{\eta}(I):=\int_I \pos{-\Ima m(t+\ii\eta)}\,\dd t, \] with $\Slip^+_{\eta}(I)=+\infty$ if $\Xi(t+\ii\eta)=0$ for some $t\in I$. Writing $s=\tfrac12+\ii z=(\tfrac12-\eta)+\ii t$, one has \[ -\Ima m(t+\ii\eta)=\Rea\!\left(\frac{\xi'}{\xi}(s)\right) =\frac{\partial}{\partial\sigma}\log|\xi(\sigma+\ii t)| \qquad(\sigma=\tfrac12-\eta), \] so $-\Ima m$ is a vertical derivative of $\log|\xi|$.Our first main result is a local coercivity principle with a Poisson/harmonic-measure interpretation: if $\Xi$ has a zero $z_0=t_0+\ii\eta_0$ of multiplicity $k$, then on every scan line just below $z_0$ the positive-part argument-variation on the symmetric window $[t_0-d,t_0+d]$ is bounded below by $k\pi/4$. Geometrically, a zero forces a quantized defect on the Carleson tent (cone) directly beneath it.We then prove a stagewise transducer from one-dimensional slip control to two-dimensional zero-free rectangles: if $\Slip^+_{\eta}(I)