The root agent started six agents on different approaches, kept a running table of which rows were settled, and redirected as results came in. After the first three returns it declared \(E_6/F_4\) the only row that could still diverge and launched an adversarial auditor and a direct attack on the \(E_6/F_4\) sheaf level (Strategy 9.1). Once the lattice-count results were audited it recorded the symplectic rows as decided at both levels and re-checked the \(E_6\) Weyl-group conjugacy (Audit 9.3).
Its synthesis gives the answer: every symmetric space on the pages agrees at both levels except \(E_6/F_4\), where \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) but the Hecke translates of the basic function do not span (Summary 9.4). The status is partial. The sheaf side is proved over \(\overline{\mathbb F}_q\) under the standing inputs D0–D2The standing inputs of the sheaf–function dictionary over F̄_q, p ≠ 2: D0 Chen–Yi's orbit geometry; D1 traces of Frobenius and the perverse t-structure; D2 decomposition of Satake objects into shifted IC sheaves; D3 Lang's theorem; D4 unitriangularity of Satake traces; D5 comparison of F̄_q with C. (Betti through D5D5 · Comparison with C: “IC_μ is a summand of V_λ ∗ IC_0” has the same answer over F̄_q and over C (BBD §6.1, for large p).); the function side over \(p\)-adic fields under Sakellaridis's hypotheses, of which (A2)–(A4)Sakellaridis's standing assumptions for his Theorem 8.0.2 (the pages' Theorem E): (A1) a characteristic-0 field and a split group with simply connected derived group; (A2) an integral model of X with a smooth open Borel orbit and invariant measures; (A3a) rank-one models over O, (A3b) the Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) conditions on the spherical roots and colours; (A5) X homogeneous and affine with one open Borel orbit. are unverified; and the two sides live over different fields. No case of the reverse divergence exists among the rows. After the second audit it wrote the final verdict, with a table of every error and its correction (Summary 9.5).
The second audit found two problems in the root's own steps. The first 'only possible divergence' claim was premature: the higher-rank symplectic rows were still undecided then, and the claim became true only after the lattice-count theorem. The synthesis's mechanism paragraph asserted that the multiplicity vectors fail to span at the given \(q\); that is spherical generation over \(\mathbb F_q(\!(t)\!)\), which is open, and a correction restates the mechanism field by field. The answer did not change.
After Agents X1–X4 repaired the steps the run had left broken, it wrote the final verdict (Summary 9.6): of the twelve statements the answer rests on, six held and six were repaired, none lost, and the answer did not change. The \(E_6/F_4\) function cell became sharper: under Sakellaridis's theorem \(C_K\) needs two generators, and of that theorem's hypotheses only one remains open.