A Finite-to-Infinite Certification Program for Zero-Free Disks of the Dirichlet η-Function via Log-Moment Coefficients, Padé Approximants, and Rouché's Theorem

We present a concrete finite-to-infinite certification pipeline for proving local nonvanishing of the Dirichlet eta function $\eta(s)=\sum_{n\ge1}(-1)^{n-1}n^{-s}$ on disks contained in the half-plane $\Re(s)>\tfrac12$. The method has three components: (i) an explicit log-moment structure for the Taylor coefficients of $\eta(s_0+z)$ about a general center $s_0=\sigma+it$, (ii) diagonal Pad\'e approximation built from finitely many Taylor coefficients recovered from boundary values via a discrete Cauchy/Fourier formula, and (iii) a Rouch\'e/argument-principle certificate that transfers a zero count from the Pad\'e approximant to $\eta$. This yields a notion of certificate complexity---the minimal Pad\'e order required to certify a given disk---and motivates large-scale "spike hunts" in $t$ as a new quantitative window into the local geometry of $\eta$ off the critical line. As a pilot demonstration in a near-critical stress geometry, we implement an order-1 (type $[1/1]$) certificate for disks of radius $r=0.049$ centered on the vertical line $\sigma=0.55$ (left edge at $\Re(s)=0.501$). For eight sample heights in $[10,5000]$ (including the first three critical-line ordinates of $\zeta$ and five pseudo-random heights), the sampled certificate succeeds with a wide range of safety margins, including one instance close to threshold that illustrates where higher Pad\'e order can become relevant.