§14 · Agent X4 · Sakellaridis's theorem · 2026-10-17T21:18:09Z

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

open audit: error

Error. Fails at step 4, which leaves (E) open. (E) is settled for p odd. Delorme–Sécherre, Pacific J. Math. 251 (2011) 1–21, Theorem 1.1 (= Thm 4.13; arXiv math/0612545 Thm 0.4/4.8): for G k-split, p odd, H open in G^σ, S a σ-stable maximal k-split torus containing a maximal (σ,k)-split torus A, and K the stabilizer of a special point of the apartment of S, one has G_k = ∪{j∈J} H_k y_j S_k K, with J indexing the H_k-classes of maximal (σ,k)-split tori. For M_θ (S=T, A=our S, K=G(O)) J is a single class: any such A' is G(F)-conjugate to A (Helminck–Wang Prop 10.3), hence A' = hAh^{-1} with h∈H(F̄); splitness forces h^{-1}σ(h)∈Z_H(A)=L^θ=Spin8; H^1(F,Spin8)=1 (Kneser), so A' is H(F)-conjugate to A. Hence G(F)=H(F)T(F)K, i.e. G(F)=K T(F) H(F), and X(F)=K T(F)x0 since H^1(F,F4)=1. To reach A_X^+: ṅ=ṅ{α1}ṅ_{w_Lα1}∈N_G(T)(O) satisfies ṅθ(ṅ)^{-1}=e1(a)∈S(O), so ṅx0∈T(O)x0, and ṅ realizes s_{γ1} on labels; similarly for γ6. So Axiom 2.4.1 holds in full, and the step (in S4-6) that the three forms of (E) are equivalent also needed this W_X-in-K argument, which S4-6 omits. (I) is genuinely unverified. No other hypothesis is open beyond the cited inputs. The conclusion does not hold as stated.
Corrected by: Proposition X4.13 · $E_6/F_4$, $p$ odd: Axiom 2.4.1 holds; only Axiom 2.2.1 over $\mathcal O$ open.

After Proposition X4.3–Proposition X4.5, exactly two hypotheses of Theorem 8.0.2 remain unverified for the model $\mathcal M_\theta$: the existence half of Axiom 2.4.1 (every $K$-orbit on $X(F)$ meets $T(F)x_0$) and the integral form of Axiom 2.2.1 for the 45 maximal-rank Borel orbits. Both are stated exactly, with the routes that look closest.

Gap X4.6.

For $\mathcal M_\theta$ (model of Proposition X4.4), $p\ne2$, $F$ $p$-adic, the hypotheses (S1)–(S10) of Theorem (cited) X4.1 hold except possibly for:

(E) Existence half of Axiom 2.4.1 (p. 10). $$G(F)=K\cdot T(F)\cdot H(F),\qquad\text{equivalently}\qquad X(F)=\bigcup_{\kappa\in\Lambda_S^+}K\,x_\kappa,\qquad\text{equivalently}\qquad K\backslash X(F)\xrightarrow{\ \sim\ }\Lambda_S^+ .$$ (The three forms agree because $A_X(F)x_0=T(F)x_0$, $H^1(F,T^\theta)=1$, and by Proposition X4.5(1).)

(I) Integral half of Axiom 2.2.1 (p. 9). For each of the $45$ Borel orbits $Y_F$ of maximal rank, with $Y\subseteq X_{\mathcal O}$ the $\mathcal O$-scheme "complement of the closure of the complement of $Y_F$ in its closure", and each simple root $\alpha$: if $(Y_F,\alpha)$ has type $U$ then $YP_\alpha/R(P_\alpha)\simeq SU\backslash\mathrm{PGL}_2$ over $\mathcal O$ with $S\subseteq N(U)$; if it has type $G$ then $YP_\alpha/R(P_\alpha)\simeq\mathrm{PGL}_2\backslash\mathrm{PGL}_2$; 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).)

Nothing in this run or in the local sources proves (E) or (I) at a given place; Sakellaridis proves the axioms only at almost every place of a global field (pp. 9–11).

Why (E) is the remaining content of Axiom 2.4.1. Its second clause is Proposition X4.5(1). Its first clause says that $A_X^+x_0$ meets every $K$-orbit; since $A_X(F)x_0=T(F)x_0$ and points with the same label are $K$-conjugate, this is (E). Without (E), $C_K$ could contain indicators of $K$-orbits not of the form $Kx_\kappa$, and Theorem 8.0.2 would not apply.
What is proved towards (E). Iwasawa $G(F)=K\,P(F)$ and $L(F)\twoheadrightarrow A_X(F)$ ($H^1(F,\mathrm{Spin}_8)=1$) give $X(F)=K\cdot U_P(F)\cdot T(F)x_0$; the missing step is to remove $U_P(F)$. (Proved here; elementary.)
Route (i), Albert algebra (heuristic). With $G$ the norm-preserving group of the split Albert algebra $J$ and $H=\operatorname{Stab}(e)$, $X(F)\simeq\{x\in J(F):N(x)=1\}$ (one $G(F)$-orbit since $H^1(F,F_4)=1$) and $T(F)x_0$ is the set of diagonal elements of norm $1$. (E) would follow from an elementary-divisor theorem: every $x\in J(F)$ is $G(\mathcal O)$-equivalent to a diagonal element. Krutelevich (J. Algebra 314 (2007)) proves the analogue over $\mathbb Z$; the version over $\mathcal O$ with $p\ne2$, and the identification of $G(\mathcal O)$ with the structure group of $J(\mathcal O)$, are not checked here.
Route (ii), buildings (gap). Benoist–Oh (2007) and Delorme–Sécherre (2011), which Sakellaridis cites on p. 11 for "similar statements", give decompositions $G(F)=K\,A^+\,\mathcal F\,H(F)$ with a finite set $\mathcal F$; (E) needs $\mathcal F=\{1\}$ for $\mathcal M_\theta$. Their exact hypotheses and the form of $\mathcal F$ were not checked (not in the local sources).
(I). Every maximal-rank orbit is $w\cdot\mathring X$ for Knop's action, reached from $\mathring X$ by type-$U$ moves (p. 9). The open-orbit argument of Proposition X4.4 (steps 3–4: fibrewise open immersion plus EGA IV 21.12.7) would prove (I) once each $Y_F$ has a representative $y\in X(\mathcal O)$ with $B_y$ smooth over $\mathcal O$ and $Y=B\cdot y$ on both fibres. This is a finite verification, $45\times6$ pairs, not done here.
Blocked route recorded. Proving Axiom 2.4.2 at the uncovered labels with a single $P$-dominant cocharacter is impossible (Proposition X4.5); the wall-parabolic argument replaces it. No analogous trick is known here for (E).

Builds on

Theorem (cited) X4.1 · Sakellaridis's Theorem 8.0.2 and Corollary 8.0.4 with every hypothesis in his own terms
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 2.4.2, half of Axiom 2.4.1, types in 2.2.1

Built on by

Corollary X4.8 · Under Theorem 8.0.2, $C_K$ is not cyclic and needs exactly two generators
Summary X4.9 · $E_6/F_4$ function cell: two generators, modulo (E) and (I)
Re-check V4 · Re-check by V4

Sources

sakellaridis-v3.pdf, Axiom 2.2.1 p. 9; Axioms 2.4.1–2.4.2 and the remark on [Sa12], Benoist–Oh, Delorme–Sécherre, pp. 10–11

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

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

Builds on

Built on by

Sources

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