The agent was asked to remove Sakellaridis's hypotheses from the function-level column: classify the \(K\)-orbits over a general local field, bound the supports of Hecke translates, and apply the pages' triangular criterion.
For the spheres it showed that the \(K\)-orbits are the shells of fixed pole order, in every residue characteristic, and concluded spherical generation for all \(n\ge2\) over every local field (Theorem 5.3). For \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) it used the model of pairs of complementary subspaces, a lattice normal form and a componentwise support bound. For \(\mathrm{GL}_4/\mathrm{Sp}_4\), counts of self-dual lattices give a combination of two translates equal to a multiple of the indicator of the uncovered orbit (Computation 5.5). In general rank it used the Hecke row \((b_1,\dots,b_m,0,\dots,0)\), which is not a lift of \(b\), and proved by a submodule inequality that its translate vanishes on every orbit not dominating \(b\) (Lemma 5.6). A downward induction inside finite sets of labels then gives generation for \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) in all ranks, over every local field (Theorem 5.7). \(E_6/F_4\) stayed blocked: no integral orbit classification for its Albert-algebra model is available.
The audit confirmed the theorems. It found two wrong entries in a hand-made check table (not used), an appeal to the triangular criterion that is really a direct induction, two stale remarks, and an input of a support-bound transfer that fails in residue characteristic 2 for the pairs model (Proposition 5.8); a correction proves that bound directly. These results are Theorem 4 of the verdict; through the transfer theorem they give sheaf-level equality for the symplectic spaces, which left \(E_6/F_4\) as the only open row.
Result as the agent reported it partial
Unconditionally, for every non-archimedean local field F (any characteristic, any residue field): (SG) holds for Spin_{2n}/Spin_{2n-1} (all n>=2), GL_n/(GL_p x GL_q) (pair model; all p<=q), GL_{2m}/Sp_{2m} (all m) and SL_{2n}/Sp_{2n} (all n), proved via Proposition B with explicit orbit classifications, lattice support bounds, and for the uncovered symplectic labels the rows T_{(b,0,...,0)} ordered by reverse dominance (vanishing from a submodule-type inequality). GL_4/Sp_4, label (1,1): f = T_{(1,1,0,0)} - T_{(2,0,0,0)}/(q(q+1)) gives f*phi_0 = (q+1)(q^2+1) 1_{O_{(1,1)}} (counts checked over F_2[[t]], F_3[[t]]). Function-level H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ.(a) transfers to every F. No row disagrees between the levels; the sheaf side stays undecided where coverage fails.
Gap
E_6/F_4 at function level: orbit classification of E_6(O) on the norm-one hypersurface of the Albert algebra over O and a triangular Hecke system around 3*omega_1^vee (node 9). The m=3 symplectic counts in node 6 are hand computations (not machine-checked); the general theorem does not depend on them. Equality of K t^lambda X(O) with the F_q-points of L^+G t^lambda L^+X (as opposed to inclusion) is not re-proved.
Blocked routes
- E_6/F_4 by the method of nodes 3-7: needs K-orbit classification of the norm-one hypersurface Q_N(F) of the split Albert algebra by Lambda_S^+ and a two-sided Hecke support sandwich replacing the symplectic self-dual-lattice inequality (node 6, Lemma 3).
- Deciding C^Sat = C for GL_{2m}/Sp_{2m}, m>=2, from the function level: would need IC(closure O_b) to be a retract of V_{lambda*(b)} * IC_0 for uncovered b; the function-level nonvanishing only shows the restriction to O_b is nonzero.