A Genus-Two Algebraic Family with Vanishing Real Toledo Numbers and Nonzero Generic Witt Class

We give four explicit matrices defining a genus-two surface-group representation over \(\mathbb{R}[s,t,1/(st(1+t^2))]\). Every real specialization has zero Witt class and zero Toledo number. Over \(\mathbb{R}(s,t)\), however, the Witt class is the anisotropic Pfister form \(\langle 1,-(1+t^2)\rangle\otimes\langle 1,-s\rangle\), of additive order two. A nonzero second residue and a trace pole detect bad reduction at the nonreal divisor \(1+t^2=0\). The calculation shows concretely that the real Toledo function does not determine the generic Witt-valued invariant. General realization results of Dymara and Januszkiewicz already imply existence of such examples; this note provides a compact matrix presentation and its direct verification. It is not a solution of the full open-ended AIM programme, and no priority claim is made.