The agent was asked to decide the categorical statement for \(E_6/F_4\), the one row where a divergence was still possible. The difficulty is the labels \(\mu_k=3k\omega_1^\vee\), \(k\) odd (and their mirror images), which have no dominant lift. It took \(\lambda_k=\tfrac{3k-1}{2}\omega_1^\vee+\omega_6^\vee\), whose image \(\mu_k'\) lies just above \(\mu_k\): nothing in between, codimension 8 (Lemma 8.2).
It showed that both convolution fibres, over \(\mu_k\) and over \(\mu_k'\), lie in the open Schubert cell, so the stalks of \(V_{\lambda_k}\ast\mathrm{IC}_0\) are shifted fibre cohomology and the singular Schubert variety never enters (Lemma 8.4). The fibre over \(\mu_k'\) is a single point. The fibre over \(\mu_k\) has eight torus-fixed points, and through each passes a 7-dimensional orbit of \(\mathrm{Spin}_9\) (Computation 8.6). A detection lemma turns its top cohomology class into a summand: \(\mathrm{IC}_{\mu_k}\) is a direct summand of \(V_{\lambda_k}\ast\mathrm{IC}_0\) (Theorem 8.8). With the mirror labels and the labels that do have lifts, every IC sheaf is a summand, so \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) over \(\overline{\mathbb F}_q\), \(p\ne2\), assuming Chen–Yi's theory and the decomposition theorem for Satake objects; the Betti case needs the comparison 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). (Summary 8.10).
The audit found the proof that the fibre over \(\mu_k'\) is a single point wrong as written (a dimension count in congruence quotients); the statement is true by the auditor's fixed-point argument. The mirror step needed a normalized diagram automorphism, or a rerun. The final summary called the row a genuine divergence while naming only sheaf-side assumptions; the function side is \(p\)-adic and rests on Sakellaridis's partly unverified hypotheses, and the restated divergence carries both. The sheaf theorem survived and is the sheaf half of the divergence.
Result as the agent reported it proved
For X = E_6/F_4 (G simply connected, G^theta = F_4, Z_{G^theta}(S) = Spin_8 connected), C^Sat = C at the sheaf level, modulo (E1) base field F_q-bar with Chen-Yi's l-adic theory (over C: criterion.html open item 1), (E2) the decomposition of V_lambda * IC_0 into shifted IC's with constant coefficients (Chen-Yi Prop. 70 / Thm 93 proof + H3H3 · Constant coefficients: on each orbit the heart consists of finite sums of the constant sheaf.), (E3) the finite-type equivariant model for stalks and dimensions. Proof: for the uncovered labels mu_k = 3k omega_1^vee (k odd) take lambda_k = ((3k-1)/2) omega_1^vee + omega_6^vee, phi(lambda_k) = mu_k' = mu_k + phi(alpha_6^vee), codim 8, and the interval [mu_k, mu_k'] has two elements. Both convolution fibres over x_{mu_k} and x_{mu_k'} lie in the open cell Gr^{lambda_k} (support bound + no dominant lift of mu_k + no dominant lambda'' < lambda_k with the same phi), so the stalks are H^*(fibre)[dim Gr^{lambda_k}] with dim Gr^{lambda_k} = delta(mu_k'). The fibre over the covering orbit is finite (dimension count), so IC_{mu_k'} occurs unshifted only; the fibre over x_{mu_k} contains a 7-dimensional orbit of Z_{G^theta}(mu_k) = Spin_9 through each of its 8 torus-fixed points t^nu (nu in W lambda_k, phi(nu) = mu_k), so its top cohomology sits in degree >= -delta(mu_k) and the detection lemma makes IC_{mu_k} a direct summand (indeed E_{mu_k, 2 dim F_k - 8} != 0). 3k omega_6^vee follows by the diagram automorphism sigma, covered labels by Prop. III step 6, and (F1) = Lemma 4.5 gives C^Sat = C. Combined with Theorem 6.6 (SG fails at function level under Sakellaridis's hypotheses), the E_6/F_4 row is a genuine divergence between the two levels; 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 at 3k omega_1^vee yet generation holds, so Prop. I's hypotheses are sufficient, not necessary.
Gap
(E1)-(E3) as listed in Remark 8.12 (all already assumed elsewhere in the workspace: Chen-Yi's theory over F_q-bar; comparison to C open; decomposition theorem and finite-type model as in criterion.html Step 6 and Lemma 4.4.2). The exact fibre F_k = Q^7 and the full multiplicity space k^7 in degrees -6..6 are heuristic (Conjecture 8.11); only dim F_k >= 7 and dim F_k' = 0 are proved. The dual-group reading (V_lambda * IC_0 = restriction to G_X-dual with the 2<rho_L,.>-grading) is not proved here and was not used.
Blocked routes
- Pure degree-versus-Frobenius-weight detection of IC_mu among shifted summands: all summands IC_nu[-n] of a pure weight-0 complex are pointwise of weight <= degree, so weights do not separate different shifts n; one needs the finiteness of the fibre over the covering orbit (Lemma 8.5) instead.
- Deriving C^Sat = C from the function level via Theorem 4.6: impossible here, since (SG) fails for E_6/F_4 (Theorem 6.6); the implication only runs (SG) => C^Sat = C.
- Using the real-symmetric equivalence [CN24] + Nadler Thm 1.2.2 to read V_lambda * IC_0 as a graded restriction to the dual subgroup sl_3 in E_6: the identification of the Hecke action with Nadler's specialization is not in the sources (recorded as a gap in Conjecture 8.11.2).
- Minuscule lambda: E_6 simply connected has Lambda_T = coroot lattice, so no minuscule Hecke label exists and Gr^{<= lambda} is singular for every lambda != 0; the route around this is Lemma 8.4 (fibres avoid the singular locus), not IC stalk computations.
Brief as sent
Target: decide \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) for \(E_6/F_4\) at sheaf level, the one row where a divergence is now live (A6: (SG) fails under (A)). Given: A4-5 (thick generation iff every IC is a summand of a Satake object, under H1–H3 and the decomposition theorem), A4-9 (GL\(_4/\)Sp\(_4\): fibre LG(2,4) over the uncovered orbit, codimension 4, so IC is a summand), the E\(_6/F_4\) root data of families/e6-f4.html, and the SL\(_6/\)Sp\(_6\) analogy. Route: model \(E_6/F_4\) as norm-one elements of the Albert algebra \(H_3(\mathbb O)\), with the uniform family \(H_3(K_d)\), \(d=1,2,4,8\); dimension formula \(\dim\mathcal O_\mu-\dim\mathcal O_0=\langle\rho,\mu\rangle\) giving codimension \(d\) of \(\mathcal O_{(2,-1,-1)}\) in the closure of \(\mathcal O_{(2,0,-2)}\); compute the convolution fibre over \(x_{(2,-1,-1)}\) through the rank-two spin-factor block; prove the stalk-degree criterion for 'IC\(_\mu\) is a summand'; check \(2\dim(\text{fibre})\ge d\) or the IC-stalk version when the Schubert variety is singular; propagate to \(3k\omega_1^\vee\), \(k\) odd, and to \(3k\omega_6^\vee\) by \(\sigma\). State all assumptions; if the route fails, record the exact missing computation and the logical status.