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 $z=t+iieta$: for a bounded interval $IsubsetRR$ and $eta>0$, [ Slip^+_{eta}(I):=int_I pos{-Ima m(t+iieta)},dd t, ] with $Slip^+_{eta}(I)=+infty$ if $Xi(t+iieta)=0$ for some $tin I$. Writing $s=tfrac12+ii z=(tfrac12-eta)+ii t$, one has [ -Ima m(t+iieta)=Rea!left(frac{xi’}{xi}(s)right) =frac{partial}{partialsigma}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 $z_0=t_0+iieta_0$ 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 $kpi/4$. 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 $Slip^+_{eta}(I)<pi/4$ holds for all scan heights $etain[eta_star,tfrac12]$ and for all $I$ in a fixed two-shift unit-interval cover of $[T,2T]$, then $Xi$ is zero-free in the corresponding upper-half-plane window, and hence every $xi$-zero $beta+iigamma$ with $T<gamma<2T$ satisfies $|beta-tfrac12|<eta_star$. A finite-mesh reduction replaces the continuum of scan heights by finitely many scan lines using height-Lipschitz control; we also record perturbation stability of slip.To support formally checkable certificates, we record an explicit validated-numerics layer: disk enclosure arithmetic, Euler–Maclaurin remainder bounds for $zeta,zeta’,zeta”$, and Stirling-type remainder bounds for $psi$ and $psi’$. Appendices include certificate-format completeness and optional adaptive scan-height propagation.