The agent was asked to audit the pages' status table: check each row against the page where it is proved, list the cells where a divergence could still hide, and check that the two statements are comparable at all. It rebuilt the table row by row (Setup 1.1) and pointed out a basic problem. The sheaf statement is about Betti sheaves over \(\mathbb C\) but is proved with \(\ell\)-adic inputs over \(\overline{\mathbb F}_p\); Sakellaridis's theorem needs a \(p\)-adic field; and the only setting where a functor (the trace of Frobenius) links the two levels is \(\mathbb F_q(\!(t)\!)\) against sheaves over \(\overline{\mathbb F}_q\) (Objection 1.8).
It then settled several cells. The spheres satisfy spherical generation for every \(n\ge2\) over any local field with odd residue characteristic. For \(\mathrm{GL}_4/\mathrm{Sp}_4\) the orbit \((1,1,1,1)\), which has no dominant lift, is reached by cancelling two Hecke translates, so generation holds over every local field (Proposition 1.4). Over \(\mathbb F_q(\!(t)\!)\), \(q\) odd, with connected stabilizers and the decomposition theorem for Satake objects, function-level generation forces sheaf-level generation (Theorem 1.6). For \(\mathrm{SL}_2/\mathrm{SO}_2\), functions fail for every \(q\) by a parity argument, and sheaves fail \(\ell\)-adically because sign local systems never occur in Satake objects (Proposition 1.7).
The audit found a false side remark (\(\mathrm{IC}_0\) is not a summand of \(V_{(1,0)}\ast\mathrm{IC}_0\); that object is simple), repeated in the final classification along with three stale cells, and a wrong list of the Hecke translates reaching \((1,1,1,1)\). None was used for a conclusion, and all conclusions survived. The transfer theorem is one of three proofs of Theorem 3 of the verdict; the three-field analysis became its §4.
Result as the agent reported it partial
No decided cell diverges. New decisions: (1) SL2/SO2 is both-fail: (SG) fails for every q (Hecke parity plus square classes), and C^Sat != C in Chen-Yi's l-adic category over F_q-bar (q odd) because the sign local systems on O_m never occur in any Satake object (trace of Frobenius + pointwise purity); Betti C over C follows by the pages' standard comparison (open item 1). (2) GL4/Sp4 is both-hold: (SG) holds over every non-archimedean F by a Lagrangian count (f = T_(1,1,0,0) - (c'/c) T_(2,0,0,0) reaches the uncovered label (1,1,1,1)), and C^Sat = C l-adically via the transfer theorem, although 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. fails; so 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. is not necessary. (3) Spheres Spin_2n/Spin_2n-1: (SG) holds for all n >= 2 over any F with odd residue characteristic (Corollary C with the pole-order orbit classification), closing the n >= 4 cell. (4) Transfer theorem: over F = F_q((t)), q odd, with G^theta and Z_{G^theta}(S) connected, (SG) implies C^Sat = C (l-adic), via unitriangularity of IC traces and semisimplicity of Satake objects; and C^Sat = C iff every simple IC(O_mu, L) is a shifted summand of a Satake object. The reverse implication (sheaf => function) is not established.
Gap
Undecided cells: GL_{2m}/Sp_{2m} (m>=3) and SL_{2n}/Sp_{2n} (n>=3): (SG) over F_q((t)) for one odd q (or the summand condition for the uncovered IC_mu) would decide the sheaf level; E6/F4: K-orbit classification over F and the labels 3k omega_1^vee (k odd) at either level; GL_n/(GL_p x GL_q), (p,q) != (1,1): K-orbit classification over F to make (SG) unconditional; all sheaf verdicts are l-adic over F_q-bar, p odd, and reach the Betti C of the pages only through the comparison of decomposition multiplicities (the pages' open item 1).
Blocked routes