The agent was asked to decide the categorical statement for \(X=\mathrm{SL}_2/\mathrm{SO}_2\), \(G=\mathrm{SL}_2\), a row the pages left open because both conditions of their Proposition III fail. It first computed the arc-group stabilizers: away from the base orbit each is \(\{\pm I\}\) times a connected pro-unipotent group, so the component group comes from the central element \(-I\) (Computation 2.1).
Because \(-I\) is central and acts trivially on the loop space, it acts on every equivariant sheaf by a natural involution \(\varepsilon\), which splits the category into an even and an odd part. The base sheaf and all Hecke kernels are even, and convolution multiplies \(\varepsilon\), so every Satake object is even (Lemma 2.3). The sign local system on the first orbit is odd, hence not generated (Theorem 2.4). The argument is topological, over \(\mathbb C\): no weights, no decomposition theorem, no transfer from finite fields. Since the convolution maps are finite and birational, each Satake object is a single IC sheaf, which gives the exact Satake span: the IC sheaves of the even orbits with constant coefficients (Proposition 2.7). At function level the \(\mathrm{SL}_2\) Hecke algebra is generated by the distance-two operator on the tree, and a dimension count shows that spherical generation fails for every \(q\), not only for \(q\) odd (Computation 2.8).
The audit found everything correct, with two small gaps: closure of the even part under cofibres needs one extra line, and a closing trace identification used a characteristic-0 lemma without comment (it still holds). The result is Theorem 1 of the verdict, and the only sheaf-level decision proved directly in the Betti category.