Microlocal Labels for the Non-Tempered θ10 Packet: An Explicit Four-Object Classification Across Pure Inner Forms

Let $F$ be a non-archimedean local field of characteristic zero and let $\Psi_{E,\eta}$ be the non-tempered $\theta_{10}$ Arthur parameter of Gurevich–Szpruch in the quadratic-field symmetric branch $E/F$ with $\eta=\eta^\sigma$. We give a self-contained explicit calculation of the associated rank-two Vogan carrier and of the four packet objects across the split and nonsplit five-dimensional orthogonal pure inner forms. The carrier is the toric model $((\mathbb{G}_m)^2, \mathbb{A}^2)$ with square weights. Its strongly regular closed conormal has stabilizer $\mu_2^2$, and the ambiguity in choosing a norm extension of $\eta$ is exactly an axis-exchange torsor acting on component-group coordinates by $(a,b)\mapsto(a+b,b)$. We then identify the four representation-to-orbit/local-system attachments, transport the geometrically normalized two-factor microlocal calculation of Cunningham–Fiori–Moussaoui–Mracek–Xu to this carrier, and isolate the Whittaker dependence in the enhanced-LLC map. The resulting four microlocal characters agree, object by object, with the explicit Gurevich–Szpruch component-group characters after the basis conversion $(a,b)=(p,p+q)$. The statement is deliberately scoped to this explicit symmetric $\theta_{10}$ model; no general non-tempered ABV–Arthur theorem or abstract pure-inner packet-equality claim is made.