§14 · Agent X4 · Sakellaridis's theorem · 2026-10-17T21:19:54Z
Summary X4.9 · \(E_6/F_4\) function cell: two generators, modulo (E) and (I)
proved correction ★ key step
Correction. Corrects Summary 7.32 · Final audit verdict: the five conclusions survive; \(E_6/F_4\) is the one divergence, under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies., Summary 9.4 · Synthesis: \(E_6/F_4\) is the one divergence (categorical generation holds, functions fail under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies.); no reverse case.
Re-derives the verdict's \(E_6/F_4\) function cell: what is unconditional, what holds under Sakellaridis's Theorem 8.0.2, which of its hypotheses are now verified for an explicit \(\mathcal O\)-model with \(p\) odd, and which two remain; the cell stays a \(p\)-adic statement.
Summary X4.9.
\(E_6/F_4\), functions. \(F\) a \(p\)-adic field (characteristic \(0\)), residue characteristic \(p\ne2\), \(X\) the model \(\mathcal M_\theta=G_{\mathcal O}/G_{\mathcal O}^\theta\) (split simply connected \(E_6\) over \(\mathcal O\), \(\theta\) an \(\mathcal O\)-involution of type E IV, \(\theta(K)=K\)), \(\varphi_0=\mathbf 1_{X(\mathcal O)}=\mathbf 1_{Kx_0}\).
| claim | status | where |
|---|---|---|
| \(\operatorname{im}r\) is the codimension-one subring \(\{g:g(y_1)=g(\sigma y_1)\}\) | proved, unconditional algebra, every \(q\) | Proposition X4.7 |
| (SG) fails; \(C_K\) is not cyclic for any generator; \(C_K=\mathcal H_K\varphi_0\oplus\mathbb C\mathbf 1_{O_{3\omega_1^\vee}}\); \(C_K/\mathcal H_K\varphi_0\) is the character \(q^{\rho-\omega_4}\) | proved under Theorem 8.0.2 for \(\mathcal M_\theta\) | Corollary X4.8 |
| §1.1/§1.7 conditions on \(F\) and \(G\) (characteristic \(0\); split, simply connected derived group) | verified, every \(p\) (with the paper's "unramified over \(\mathbb Q_p\)" simplification) | Proposition X4.4 |
| §1.7 conditions on \(X\) over \(\mathcal O\) (quasi-affine, \(\mathring X\) smooth surjective and \(B\)-homogeneous, eigenmeasures) | verified for \(p\ne2\), modulo cited integral-model facts | Proposition X4.4 |
| §7: no type \(N\), \(L_\gamma\) classical; Statements 6.3.1 and 7.1.5 | verified (statements over \(\bar F\)) | Proposition X4.3 |
| §8: homogeneous, affine, one open \(B(F)\)-orbit | verified, every \(p\) | Proposition X4.4 |
| Axiom 2.4.2Sakellaridis's Axiom 2.4.2, relabelled (A3c): Iwahori orbits at the antidominant torus points. (\(xJ\subseteq xB(\mathcal O)\) on \(A_X^+\)) | verified, \(p\ne2\) | Proposition X4.5 |
| Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits., distinct labels give distinct \(K\)-orbits | verified | Proposition X4.5 |
| Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits., every \(K\)-orbit meets \(T(F)x_0\) | open | Gap X4.6 (E) |
| Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. For E6/F4 it holds over every O with 2 invertible (VI run).: only types \(G,U\) | verified over \(\bar F\) | Proposition X4.5 |
| Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. For E6/F4 it holds over every O with 2 invertible (VI run). as \(\mathcal O\)-schemes for the 45 maximal-rank orbits | open | Gap X4.6 (I) |
Cell. "Fails, and \(C_K\) needs exactly two generators (\(\varphi_0\) and \(\mathbf 1_{O_{3\omega_1^\vee}}\)); \(p\)-adic \(F\), \(p\) odd, model \(\mathcal M_\theta\); conditional only on the Cartan decomposition \(X(F)=\bigcup_\kappa Kx_\kappa\) and Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. For E6/F4 it holds over every O with 2 invertible (VI run). over \(\mathcal O\)." Over \(\mathbb F_q(\!(t)\!)\) the cell is undecided: Sakellaridis assumes characteristic \(0\), and the algebra of Proposition X4.7 does not by itself compute \(C_K\) there.
- Inputs from the run. Theorem 6.6 proves that \(q^{\rho_L}y_1\) and \(q^{\rho_L}\sigma y_1\) are \(W(E_6)\)-conjugate while \(y_1,\sigma y_1\) are not \(W_X\)-conjugate; this is pure algebra about \(r\) and holds as stated. Summary 7.32 records the \(E_6/F_4\) function cell as "(SG) fails under (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. (Sakellaridis's field/group, \(\mathcal O\)-model, axioms, spherical-root/colour and affine-one-orbit hypotheses), 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. (the \(\mathcal O\)-model), (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). (the axioms) 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. (spherical roots and colours) beyond '\(L_\gamma\) classical' unverified".
- Sharpening of the conclusion. Proposition X4.7 identifies \(\operatorname{im}r\) exactly; with Theorem 8.0.2, Corollary X4.8 turns "\(\varphi_0\) does not generate" into "nothing generates", gives the second generator, and the cokernel.
- Sharpening of the hypotheses. Audit X4.2 fixes what Theorem 8.0.2 needs and that a model must be named. For \(\mathcal M_\theta\): the spherical-root and colour hypotheses hold in full (Proposition X4.3), contrary to "unverified" in Lemma 6.9 and Summary 7.32; the \(\mathcal O\)-model conditions hold for \(p\ne2\) (Proposition X4.4); of the three axioms, 2.4.2 and half of 2.4.1 hold (Proposition X4.5). What is left is (E) and (I) of Gap X4.6.
- Fields. Everything uses Sakellaridis's characteristic-\(0\) theorem, so the cell is \(p\)-adic. A comparison with the sheaf side over \(\overline{\mathbb F}_q\) remains a comparison across fields, exactly as Summary 7.32 says.
- Effect on the team verdict. The divergence "categorical generation holds, function-level generation fails" at \(E_6/F_4\) is unchanged in direction and strengthened in content; its function half now rests on two named statements, (E) and (I), instead of three families of unverified hypotheses.
Builds on
- Audit X4.2 · How Theorem 8.0.2 is proved, where each hypothesis enters, what the pages drop
- Proposition X4.3 · \(E_6/F_4\): type-\(G\) spherical roots, two colours; Statements 6.3.1, 7.1.5 hold
- Proposition X4.4 · \(E_6/F_4\), \(p\) odd: the \(\mathcal O\)-model meets §1.7 and §8
- Proposition X4.5 · \(E_6/F_4\), \(p\) odd: Axiom 2.4.2Sakellaridis's Axiom 2.4.2, relabelled (A3c): Iwahori orbits at the antidominant torus points., half of Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits., types in 2.2.1
- Gap X4.6 · \(E_6/F_4\): Cartan decomposition and Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. For E6/F4 it holds over every O with 2 invertible (VI run). over \(\mathcal O\) stay open
- Proposition X4.7 · \(\operatorname{im}r=\{g:g(y_1)=g(\sigma y_1)\}\), of codimension one, every \(q\)
- Corollary X4.8 · Under Theorem 8.0.2, \(C_K\) is not cyclic and needs exactly two generators
- Theorem 6.6 · \(E_6/F_4\): two points of the shifted torus that are not \(W_X\)-conjugate are \(W(E_6)\)-conjugate, so \(r\) is not onto and (SG) fails under (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.
- Summary 7.32 · Final audit verdict: the five conclusions survive; \(E_6/F_4\) is the one divergence, under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies.
Built on by
- Re-check V5 · Re-check by V5
Sources
- data.json: Theorem 6.6, Lemma 6.9, Summary 7.32, Summary 9.4
- sakellaridis-v3.pdf, Theorem 8.0.2 p. 64; Corollary 8.0.4 p. 65