§14 · Agent X4 · Sakellaridis's theorem · 2026-10-17T21:55:33Z

Proposition X4.13 · $E_6/F_4$, $p$ odd: Axiom 2.4.1 holds; only Axiom 2.2.1 over $\mathcal O$ open

proved correction ★ key step

Correction. Corrects Gap X4.6 · $E_6/F_4$: Cartan decomposition and Axiom 2.2.1 over $\mathcal O$ stay open.

Replaces Gap X4.6: the existence half of Axiom 2.4.1 holds for $p$ odd. Delorme–Sécherre's decomposition puts every $K$-orbit into $T(F)x_0$, and $W_X$ is realized inside $H(\mathcal O)\cap N_G(T)$ by products of two commuting Tits lifts, which moves every label into the dominant chamber. The only hypothesis of Sakellaridis's theorem still open for $E_6/F_4$ is Axiom 2.2.1 as $\mathcal O$-schemes.

Proposition X4.13.

Model $\mathcal M_\theta$ with $\theta=\theta_0$ (Proposition X4.11), $F$ $p$-adic, $p\ne2$.

$W_X\subseteq K$. For $i\in\{1,6\}$ let $\beta_i:=w_L\alpha_i=-\theta\alpha_i$ ($\beta_1=\alpha_1+\alpha_2+2\alpha_3+2\alpha_4+\alpha_5$) and

$$\dot n_i:=\sigma_{\alpha_i}\,\theta(\sigma_{\alpha_i})\in G(\mathbb Z).$$

Then $\dot n_i\in H(\mathcal O)\cap N_G(T)$, and on labels $\varphi(\mu)\in\Lambda_S$ it acts by the simple reflection $s_{\gamma_i}$ of $W_X$: $\dot n_i\,t^\mu x_0=t^{s_{\alpha_i}s_{\beta_i}\mu}x_0$ with $\varphi(s_{\alpha_i}s_{\beta_i}\mu)=s_{\gamma_i}\varphi(\mu)$.

Axiom 2.4.1 holds. $X(F)=\bigsqcup_{\kappa\in\Lambda_S^+}Kx_\kappa$. So $A_X^+x_0$ contains a complete set of representatives of $K\backslash X(F)$, distinct labels give distinct orbits, and $K\backslash X(F)\simeq\Lambda_S^+$.
What remains. With Proposition X4.3, Proposition X4.4, Proposition X4.5, Audit X4.10, Proposition X4.11 and Proposition X4.12, every hypothesis of Theorem 8.0.2 holds for $\mathcal M_\theta$, $p\ne2$, except possibly

(I) Axiom 2.2.1 over $\mathcal O$ (p. 9): for each of the $45$ Borel orbits $Y_F$ of maximal rank (with $Y$ its $\mathcal O$-scheme "complement of the closure of the complement of $Y_F$ in its closure") and each simple root $\alpha$, $YP_\alpha/R(P_\alpha)\simeq SU\backslash\mathrm{PGL}_2$ ($S\subseteq N(U)$) when $(Y_F,\alpha)$ has type $U$, and $\simeq\mathrm{PGL}_2\backslash\mathrm{PGL}_2$ when it has type $G$, as $\mathcal O$-schemes; and the complement of $YP_\alpha$ is the closure of the complement of $(YP_\alpha)_F$. (Types $T,N$ do not occur: Proposition X4.5(3).)

Proof of 1.

$\beta_i$ is a root, $(\alpha_i,\beta_i)=0$, and neither $\alpha_i+\beta_i=\gamma_i$ nor $\alpha_i-\beta_i$ is a root (c8_wx_in_K.py; for $i=1$, $\alpha_1-\beta_1=-(\alpha_2+2\alpha_3+2\alpha_4+\alpha_5)$ has coefficient $2$ at the outer node $3$ of the black $D_4$). So the root $\mathrm{SL}_2$'s of $\alpha_i$ and $\beta_i$ commute.
$\theta(\sigma_{\alpha_i})$ lies in $\theta(\mathrm{SL}_{2,\alpha_i})=\mathrm{SL}_{2,\theta\alpha_i}=\mathrm{SL}_{2,\beta_i}$, normalizes $T$, and lifts $s_{\beta_i}$. It commutes with $\sigma_{\alpha_i}$, so $\theta(\dot n_i)=\theta(\sigma_{\alpha_i})\,\sigma_{\alpha_i}=\dot n_i$ (using $\theta^2=\mathrm{id}$). Hence $\dot n_i\in H$; since $\theta=\theta_0$ is defined over $\mathbb Z$, $\dot n_i\in G(\mathbb Z)\subseteq K$.
$\dot n_i$ lifts $w:=s_{\alpha_i}s_{\beta_i}$. For $\lambda\in\Lambda_S$: $\langle\beta_i,\lambda\rangle=\langle-\theta\alpha_i,\lambda\rangle=-\langle\alpha_i,\theta\lambda\rangle=\langle\alpha_i,\lambda\rangle$ and $\beta_i^\vee=-\theta\alpha_i^\vee$, so $w\lambda=\lambda-\langle\alpha_i,\lambda\rangle(\alpha_i^\vee-\theta\alpha_i^\vee)=\lambda-\langle\alpha_i,\lambda\rangle e_i$: the reflection $s_{\gamma_i}$, i.e. $(a,b)\mapsto(-a,a+b)$ for $i=1$ and $(a,b)\mapsto(a+b,-b)$ for $i=6$ (Computation 6.4(3); checked on sample vectors in c8_wx_in_K.py). $w$ commutes with $\theta$ on $\Lambda_T$ (it is a product of reflections in a $\theta$-stable pair), hence with $\varphi=1-\theta$.
$\dot n_it^\mu x_0=(\dot n_it^\mu\dot n_i^{-1})\,\dot n_ix_0=t^{w\mu}x_0$, since $\dot n_i\in H$.

Proof of 2.

By Proposition X4.12, every $x\in X(F)$ is $K$-conjugate to some $t\,t^\mu x_0$ with $t\in T(\mathcal O)$, $\mu\in\Lambda_T$; $T(\mathcal O)\subseteq K$ removes $t$.
$W_X=\langle s_{\gamma_1},s_{\gamma_6}\rangle$ and every $\kappa\in\Lambda_S$ is $W_X$-conjugate to a dominant one. Applying the $\dot n_i$ (step 4) moves $t^\mu x_0$ inside its $K$-orbit to a point with dominant label, i.e. into $A_X^+x_0$.
Distinctness and the identification with $\Lambda_S^+$ are Proposition X4.5(1).

Where Gap X4.6 stood. It listed (E) as open. (E) is the content of steps 5–6, so Gap X4.6 is replaced by this node; its route (ii), "need $\mathcal F=\{1\}$", is what $|J|=1$ in Proposition X4.12 provides. (I) is unchanged: it is a finite check ($45\times6$ pairs), not done here.

Cited inputs used: Delorme–Sécherre Theorem 4.8 and Helminck–Wang Prop. 10.3 (through Proposition X4.12); conjugacy of maximal $\theta$-split tori over $\bar F$ (Vust, Richardson); Kneser's $H^1(F,\mathrm{Spin}_8)=H^1(F,F_4)=1$; Bruhat–Tits for $K=\operatorname{Stab}(x)$.

Builds on

Proposition X4.12 · $E_6/F_4$, $p$ odd: $G(F)=K\,T(F)\,H(F)$ by Delorme–Sécherre
Proposition X4.11 · An involution of type E IV of $E_6$ defined over $\mathbb Z$
Proposition X4.5 · $E_6/F_4$, $p$ odd: Axiom 2.4.2, half of Axiom 2.4.1, types in 2.2.1
Proposition X4.4 · $E_6/F_4$, $p$ odd: the $\mathcal O$-model meets §1.7 and §8
Computation 6.4 · $E_6/F_4$: root data, the shift $q^{\rho_L}$, the little Weyl group $S_3$, and the three generators of the target

Built on by

Summary X4.14 · $E_6/F_4$ function cell: two generators, modulo Axiom 2.2.1 over $\mathcal O$
Re-check V4 · Re-check by V4
Summary 9.6 · Verdict after the repairs: every input held or repaired; $E_6/F_4$ still the one divergence

Sources

Delorme–Sécherre, arXiv:math/0612545v1, Theorem 4.8 (via S4-12)
sakellaridis-v3.pdf, Axiom 2.2.1 p. 9; Axiom 2.4.1 p. 10
families/e6-f4.html §4

Table 1. Page production by session
	tokens	US $
This conversation: the search, the audit, the pages and propositions, and the VI characteristic-p run01:44–05:19 UTC	802 M	$490
Setup pages16:17–19:42 UTC	37 M	$19
Shareable animation17:16–17:22 UTC	1 M	$1
Proposition IIb: figures and the linear-algebra lemma19:48–20:05 UTC	14 M	$7
Audit pass: the repairs by Agents X1–X4, their re-checks and Propositions VI–VII (with the audit skill)19:35–02:12 UTC	275 M	$113
Proposition IX: converse search, verdict and trace formalism19:56–02:12 UTC	199 M	$106
Audit pass on the VI characteristic-p run; the simplified VIb page and the VI corrections05:21–07:33 UTC	268 M	$107
Removing the detail switch; Proposition IX restated; the hub and verdict reviewed and simplified05:34–15:56 UTC	38 M	$20
C_K for the categorical statements, in the Iwahori run's conversation18:27–18:32 UTC	9 M	$3
Total	1,642 M	$866