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We recast the persistence probability for the spin located at the origin of ahalf-space arbitrarily $m$-magnetized Glauber-Ising chain as a FredholmPfaffian gap probability generating function with a sech-kernel. This is thenspelled out as a tau-function for a certain Painlev\'e VI transcendent, thepersistence exponent $\theta(m)/2$ emerging as an asymptotic decay rate. Usinga known yet remarkable correspondence that relates Painlev\'e equations toBonnet surfaces, the persistence probability also acquires a geometric meaningin terms of the mean curvature of the latter, and even a topological one at themagnetization-symmetric point. Since the same sech-kernel with an underlyingPfaffian structure shows up in a variety of Gaussian first-passage problems,our Painlev\'e VI provides their universal first-passage probabilitydistribution, in a manner exactly analogous to the famous Painlev\'e IITracy-Widom laws. The tail behavior in the magnetization-symmetric case of ourfull scaling function allows to recover the exact persistence exponent$\theta(0)/2=3/16$ for the $2d$-diffusing random field or for random real Kac'spolynomials, a particular result found very recently by Poplavskyi and Schehr(Phys. Rev. Lett. {\bf 121}, 150601 (2018)). Our Painlev\'e VI tau-functioncharacterization of the persistence probability also bears a correspondencewith a $c=1$ conformal field theory, the monodromy parameters giving thedimensions of the associated primary fields. Thereby $\theta(0)/2=3 \beta/2$,with $\beta=1/8$ the Onsager-Yang magnetization exponent for the critical $2d$Ising model. This relates a nonequilibrium exponent to ordinary static criticalbehavior in one more space dimension, and suggests more generally that methodsof boundary conformal field theory should be helpful for determining thecritical properties of other unsolved nonequilibrium $1d$ processes.

Universal Painlev\'e VI Probability Distribution in Pfaffian Persistence and Gaussian First-Passage Problems with a sech-Kernel