Objection 7.31 · Objection to the mirror step and the verdict: generation holds, \(\sigma\) needs normalizing, and the divergence needs its hypotheses stated

1. Mirror step (cosmetic). \(\sigma\theta\sigma^{-1}\) agrees with \(\theta\) on \(T\), but in general it equals \(\mathrm{Int}(t)\circ\theta\) for some \(t\in T\), so \(\sigma\) must be normalized to preserve \(G^\theta\). Alternatively, use the rerun with nodes \(1\leftrightarrow6\); an independent script confirms the mirror numbers. For \(k=1\) the same \(\lambda_1=\omega_1^\vee+\omega_6^\vee\) serves both \(3\omega_1^\vee\) and \(3\omega_6^\vee\), since \(\varphi(\lambda_1)-(0,3)=e_1\).
2. Generation holds. Lemma 8.9 (2), (3) and Summary 8.10, via the summand criterion, hold under H1–H3Hypotheses of the pages' Proposition I (H1–H5 imply categorical generation): H1 finite orbit filtration; H2 equivariant recollement; H3 constant coefficients on orbits; H4 the Hecke action preserves the category; H5 every orbit carries a Satake object with nonzero top splitting. and (A2)(A2) Integral model: X is a quasi-affine G-scheme over the integers whose open Borel orbit is smooth and surjective over them, with a G-eigenmeasure on X and a B-invariant measure on the open orbit.. So \(\Csat=\cC_K\) for \(E_6/F_4\) over \(\overline{\F}_q\), \(p\ne2\). 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 \(3k\omega_1^\vee\), so this is a second example, after \(\mathrm{GL}_4/\mathrm{Sp}_4\), showing that 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. Assumptions. Remark 8.12 is complete, with three remarks: (i) purity of \(V_\lambda\) is contained in (A2)(A2) Integral model: X is a quasi-affine G-scheme over the integers whose open Borel orbit is smooth and surjective over them, with a G-eigenmeasure on X and a B-invariant measure on the open orbit., and no IC stalk of \(\Gr^{\le\lambda_k}\) is used, as confirmed; (ii) Chen–Yi, Prop. 64 holds for all \(p\ne2\); (iii) the dimension bookkeeping in (A3)(A3) Sakellaridis's three axioms over O, relabelled: (A3a) rank-one models over O (Axiom 2.2.1); (A3b) the Cartan decomposition (Axiom 2.4.1); (A3c) Iwahori orbits (Axiom 2.4.2). For E6/F4 all three hold in both characteristics (VI run). is replaced by Objection 7.28.
4. Divergence: hypotheses. On the sheaf side the result holds over \(\overline{\F}_q\) under the workspace's 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. (Chen–Yi's geometry, proper base change, the decomposition theorem); the Betti statement over \(\C\) follows label by label by spreading out as in BBD §6 (the input 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).). On the function side (SG) fails only for \(p\)-adic \(F\), and only under Sakellaridis's (A1)–(A5)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., with (A2)(A2) Integral model: X is a quasi-affine G-scheme over the integers whose open Borel orbit is smooth and surjective over them, with a G-eigenmeasure on X and a B-invariant measure on the open orbit., (A3)(A3) Sakellaridis's three axioms over O, relabelled: (A3a) rank-one models over O (Axiom 2.2.1); (A3b) the Cartan decomposition (Axiom 2.4.1); (A3c) Iwahori orbits (Axiom 2.4.2). For E6/F4 all three hold in both characteristics (VI run). at the place and (A4)(A4) Spherical roots and colours: no spherical root of type N, every Levi L_γ of a simple spherical root classical, and his Statements 6.3.1 and 7.1.5 on the spherical roots and the virtual coloured weights. beyond '\(L_\gamma\) classical' unverified (Lemma 6.9). (SG) over \(\F_q(\!(t)\!)\) for \(E_6/F_4\) is undecided, so the two levels have not been compared inside the trace dictionary (Objection 1.8, Audit 7.16).
5. Heuristic. Conjecture 8.11 is correctly labelled heuristic; it is consistent with \(\chi(F_k)=\#F_k^{T'}=8=\chi(Q^7)\).

Verified by computation: e6_fibre_A7.py