A3 re-checked over \(F=\mathbb F_q(\!(t)\!)\) every step of the earlier \(p\)-adic verification of Sakellaridis's hypotheses for \(E_6/F_4\), replacing each characteristic-0 citation. It first listed where the old check leaned on characteristic 0 ( (Audit 3.1)): Kneser's vanishing of Galois cohomology, Vust and Richardson on \(\theta\)-split tori, Knop's transport of weight lattices.
The integral model, its open orbit and the integral points need only that 2 is invertible ( (Proposition 3.2)); Bruhat–Tits give \(H^1(F,\mathrm{Spin}_8)=H^1(F,F_4)=1\) in characteristic \(p\) ( (Lemma 3.3)). A3 proved directly that the maximal \(\theta\)-split tori form one \(H(F)\)-class ( (Proposition 3.4)), checked that Delorme–Sécherre's proof uses only residue characteristic \(\ne2\), and so obtained the Cartan decomposition over \(\mathbb F_q(\!(t)\!)\) independently of A4 ( (Theorem 3.8)). Over an algebraic closure of characteristic \(p\) it re-derived the orbits, colours, valuation cone, spherical roots and \(\Theta\) ( (Proposition 3.7), (Proposition 3.10)).
One input stayed out of reach: Statement 6.3.1 in its analytic form, an identity between intertwining integrals that belongs to the proof itself ( (Remark 3.11)). The close-field route does not need it. The audit A5 found no failing step; it supplied a converse needed for the orbit action and a correct reason for the colour weights, the diagram automorphism of \(E_6\).
A fresh review re-derived every step and found no error.
Result as the agent reported it partial
For \(p\) odd and the model \(\mathcal M_\theta=(G_{\mathbb Z}/G_{\mathbb Z}^{\theta_0})\otimes\mathbb F_q[[t]]\) of \(E_6/F_4\) over \(F=\mathbb F_q(\!(t)\!)\) (including \(p=3\)): (A1)(A1) Field and group: F a non-archimedean local field of characteristic 0 (Sakellaridis §1.1), G split reductive over O with connected fibres and simply connected derived group (§1.7)., (A2)(A2) Integral model: X quasi-affine over O, its open Borel orbit smooth and surjective over O with B-homogeneous special fibre; a G-eigenmeasure on X and a B-invariant measure on the open orbit (§1.7)., (A3b)(A3b) Cartan decomposition (Sakellaridis's Axiom 2.4.1): the X-antidominant torus points represent every K-orbit on X(F), and points with distinct labels lie in distinct K-orbits., (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x., (A5)(A5) X homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points (§8). hold, and (A4)(A4) Spherical roots and colours: no pair of maximal rank of type N; classical Levis L_γ; Statements 6.3.1 and 7.1.5.(i), (ii), (iv) hold, with characteristic-\(p\) proofs. Explicitly: \(H=G^{\theta}\) is smooth with connected \(F_4\) fibres, \(X\) is smooth quasi-affine over \(\mathcal O\), \(\mathring X=U_P\times A_X\) over \(\mathcal O\), \(X(\mathcal O)=Kx_0\), \(X(F)=G(F)x_0\), \(X_F\) affine, \(\mathring X(F)=B(F)x_0\); \(G(F)=KT(F)H(F)\) (Delorme–Sécherre, proof checked for \(\mathbb F_q(\!(t)\!)\), with a direct proof that the maximal \((\theta,F)\)-split tori form one \(H(F)\)-class) and \(X(F)/K\simeq\Lambda_S^+\simeq\Lambda_X^+\); \(Jx\subseteq B(\mathcal O)x\) on \(A_X^+\); over \(\bar F\) the 45 maximal-rank Borel orbits are \(Bn_wx_0\), every pair has type \(G\) or \(U\), \(\mathcal X(X)=\mathbb Z\omega_1\oplus\mathbb Z\omega_6\), two colours with valuations \(\check\gamma_1,\check\gamma_6\), \(\mathcal V=\{\langle v,\gamma_1\rangle\le0,\langle v,\gamma_6\rangle\le0\}\), \(\Lambda_X^+=\Lambda_X\cap\mathcal V\), \(\Delta_X=\{\gamma_1,\gamma_6\}\) of type \(G\) with \(D_5\) Levis, \(W_X\simeq S_3\), \(\Theta=\check\Phi_X\times\{(+,4)\}\), Statement 7.1.5 holds, and the Brion paths satisfy the combinatorial hypothesis of Sakellaridis's Prop. 6.4.1. The same proofs hold for \(p\)-adic \(F\), \(p\) odd.
Gap
(A4)(A4) Spherical roots and colours: no pair of maximal rank of type N; classical Levis L_γ; Statements 6.3.1 and 7.1.5.(iii), Statement 6.3.1 for \(\mathcal M_\theta\) over \(\mathbb F_q(\!(t)\!)\): its analytic part — the type-\(U\) functional equation of Sakellaridis §5, Example 5.4.1 and identity (6.10) for \(P_\alpha\backslash\mathrm{SL}_3\), with the measure and \(\psi\) normalizations of §§3.3, 3.8, over \(\mathbb F_q(\!(t)\!)\) — which belongs to (Q1). Cited without being opened (all characteristic-free in their standing hypotheses): Bruhat–Tits III Thm 4.7 (\(H^1(F,\mathrm{Spin}_8)=H^1(F,F_4)=1\)), Steinberg 1968 Thm 7.5, Richardson 1977 / Haboush, Prasad–Yu 2002 Thm 2.1, Bruhat–Tits I–II and Landvogt inside Delorme–Sécherre. (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary. is not addressed (Q2).
Blocked routes
- Deriving Statement 6.3.1 from the geometry of X alone: impossible, it is an identity of intertwining integrals; it needs the characteristic-p versions of Sakellaridis's §5 type-U functional equation, Example 5.4.1 and identity (6.10) (part of Q1).
- Cartan decomposition from Chen–Yi, Prop. 17 (loop spaces over k = F̄_p, p ≠ 2): its proof cites Gaitsgory–Nadler [GN10, Thm 8.2.9] and Nadler [Nad04, Thm 4.2], both over C, so it is a characteristic-0 citation in characteristic p; descent from F̄_q((t)) to F_q((t)) would also need connected L^+G-stabilizers. Not used; replaced by Delorme–Sécherre plus the direct one-class lemma (Proposition 3.4).
- Helminck–Wang, Prop. 10.3 (G(F)-conjugacy of maximal (θ,F)-split tori): field hypotheses not checked; not needed, since Proposition 3.4 proves dim = 2 and H(F)-conjugacy directly.
- Upper bound for the valuation cone through the E6-invariant cubic form on V(ω1) (f_{ω1} in the square of the G-module of f_{ω6}): needs an integral cubic with unit coefficient on the triad {ω1, ω6−ω1, −ω6} (all 45 triads form one W-orbit, computed) and the H-fixed vector mod p; feasible, but replaced by the simpler localization to the D5-Levis with the Spin10 quadratic form (Proposition 3.7 step 5).
Brief as sent
Target (Q3): (A1), (A2), (A3b), (A3c), (A4), (A5) for the \(E_6/F_4\) model over \(\mathbb F_q(\!(t)\!)\). Approach: re-verify every step of the \(p\)-adic verification (S4, S4b), replacing each characteristic-0 citation (Kneser, Delorme–Sécherre's inputs, Vust–Richardson, Knop's transport) by one valid in characteristic \(p\) or by a direct proof.