§2 · Agent 2 · Integral models · 2026-10-18T03:21:51Z

Theorem 2.8 · Theorem: Axiom 2.2.1 holds for $\mathcal M_\theta$ over every DVR with $2$ invertible

proved audit: correct ★ key step

Second audit: correct. Re-derived: item 1 from Proposition 2.3 over F̄ (each of the 45 orbits has a Z[1/2]-point, so it is defined over F, of rank 2); Y_x is flat and locally closed, so its closure is Z_x and Y_x is open in Z_x; the boundary of Y_{x,F} has closure ⋃{v∈Σ_x∖{x}}Z_v, whose special fibre is ⋃{v∈Σ_x∖{x}}Y_{v,κ} by Lemma 2.7 item 3 and transitivity (Σ_v⊆Σ_x∖{x}), so Y^can and Y_x are open reduced subschemes of Z_x with the same points, hence equal. For α∉Φ_x, E=Σ_u∖{u,u′} is closed under v↦Σ_v (an orbit of dimension dim Y_u−1 other than Y_{u′} cannot contain Y_{u′} in its closure), so 𝒫=𝒫^can: the complement clause, read inside closures as the text forces; for α∈Φ_x the clause is Y=Y^can. Quotients from Proposition 2.5 (its two supplied lines change nothing here) and counts 90/180 from Computation 2.6; the side remark on P_{α₁}X̊, P_{α₆}X̊ with complements in X (E closed, Z_{w_L}=X_O) is also right.

Sakellaridis's $\mathcal O$-scheme of each of the $45$ maximal-rank orbits is the smooth orbit scheme $Y_x\simeq B/B_x$; its $P_\alpha$-saturation is the canonical structure on the $P_\alpha$-orbit (complement clause); the rank-one quotient is $\mathrm{PGL}_2\backslash\mathrm{PGL}_2$ (90 pairs) or $B_2\backslash\mathrm{PGL}_2$ (180), as $\mathcal O$-schemes, $p=3$ included.

Theorem 2.8.

Let $\mathcal O$ be a DVR with $2\in\mathcal O^\times$ (e.g. any complete DVR of odd residue characteristic $p$, $\operatorname{char}F\in\{0,p\}$, $p=3$ included), $X_{\mathcal O}=(G_{\mathbb Z}/G_{\mathbb Z}^{\theta_0})\otimes\mathcal O$.

The maximal-rank $B_F$-orbits are the $45$ orbits $Y_{x,F}$, $x\in\mathcal C$; they are all the $B_F$-orbits and are defined over $\mathbb Z[1/2]$.
Sakellaridis's $Y^{\rm can}$ of $Y_{x,F}$ is $Y_x\simeq B/B_x$: smooth, $B$-homogeneous fibres, $\mathcal O$-point $y_x$.
For every $\alpha$: $P_\alpha Y^{\rm can}=(P_\alpha Y_{x,F})^{\rm can}$ (complement clause), and $P_\alpha$-equivariantly

$$Y^{\rm can}P_\alpha/R(P_\alpha)\simeq\begin{cases}\mathrm{PGL}_2\backslash\mathrm{PGL}_2&\alpha\in\Phi_x\ (90\text{ pairs}),\\ B_2\backslash\mathrm{PGL}_2\simeq\mathbb P^1_{\mathcal O}\ (S=T_2)&\alpha\notin\Phi_x\ (180\text{ pairs}).\end{cases}$$

Types $T,N$ never occur. Hence (A3a) holds for $\mathcal M_\theta$ over $\mathcal O$.

Item 1: Proposition 2.3 over $\overline F$ and Lemma 2.4.
$Y_x\subseteq Y^{\rm can}$. $Y_x$ is locally closed and flat, so open in $Z_x$; $\partial_F=Z_{x,F}\smallsetminus Y_{x,F}\subseteq Z_x\smallsetminus Y_x$ (closed), so $\overline{\partial_F}\cap Y_x=\emptyset$.
Same points. $\partial_F=\bigcup_{v\in\Sigma_x\smallsetminus\{x\}}Y_{v,F}$, $\overline{\partial_F}=\bigcup Z_v$, and by Lemma 2.7 $Z_v\cap X_\kappa=\bigcup_{\Sigma_v}Y_{u,\kappa}$ with $\Sigma_v\subseteq\Sigma_x\smallsetminus\{x\}$. So $Y^{\rm can}\cap X_\kappa=\overline{Y_{x,\kappa}}\smallsetminus\bigcup_{\Sigma_x\smallsetminus\{x\}}Y_{v,\kappa}=Y_{x,\kappa}$.
Schemes. $Y_x$ (smooth) and $Y^{\rm can}$ (open in reduced $Z_x$) are reduced locally closed with the same points, hence equal.
$\alpha\in\Phi_x$: $P_\alpha Y_x=Y_x$, quotient $\operatorname{Spec}\mathcal O$ (Proposition 2.5, 1).
$\alpha\notin\Phi_x$: with $u\in\{x,s_\alpha xs_\alpha\}$ the open one, $u'=s_\alpha us_\alpha$, $\mathcal P=P_\alpha Y_x$ (Proposition 2.5) has closure $Z_u$; $E=\Sigma_u\smallsetminus\{u,u'\}$ is closed under $v\mapsto\Sigma_v$ (dimensions), so as in 2–4, $\mathcal P^{\rm can}\cap X_\kappa=Y_{u,\kappa}\sqcup Y_{u',\kappa}$ and $\mathcal P=\mathcal P^{\rm can}$. Quotient: Proposition 2.5, 2(c).
Counts from Computation 2.6, 4. For the open saturations $P_{\alpha_1}X^\circ,P_{\alpha_6}X^\circ$ the clause also holds with complements in $X$ ($E=\mathcal C\smallsetminus\{w_L,s_\alpha w_Ls_\alpha\}$). Orbits over $F$ and $\kappa$ correspond via $\mathcal C$, each with an $\mathcal O$-point, as Sakellaridis's remark (p. 9) predicts.

Builds on

Proposition 2.3 · Borel orbits of $E_6/F_4$ over an algebraically closed field of characteristic $\ne2$
Lemma 2.4 · Smooth stabilizers; orbit immersions $B/B_x\hookrightarrow X$ over $\mathbb Z[1/2]$
Proposition 2.5 · Rank-one quotients: a point (type $G$) or $\mathbb P^1_{\mathcal O}$ (type $U$)
Computation 2.6 · The $45\times6$ pairs: 90 of type $G$, 180 of type $U$
Lemma 2.7 · Orbit closures over $\mathcal O$ have the expected special fibres

Built on by

Audit 2.9 · Audit of the proof of (A3a): characteristic, $p=3$, scheme structure, citations
Summary 2.10 · Verdict on (Q2): (A3a) holds for $\mathcal M_\theta$ over every such $\mathcal O$
Audit 6.7 · Audit of Sakellaridis's scheme $Y$ and the complement clause (claims A2-1, A2-8)
Remark 7.2 · Registry after the integral-model return: (A3a) claimed over $\mathbb Z[1/2]$; launch its audit
Theorem 7.4 · Verdict: Sakellaridis's theorem over $\mathbb F_q(\!(t)\!)$; for $E_6/F_4$ in both characteristics, (A3a) included

Sources

sakellaridis-v3.pdf, Axiom 2.2.1 and remark, p. 9
S4.json: S4-4 step 4 (open orbit, x = w_L)

Table 1. Page production by session
	tokens	US $
The run, its in-run audits and the pages (from 01:50 UTC)01:50–05:20 UTC	349 M	$144
This conversation: the audit pass (eight reviewers, one salvage agent, re-checks) and the simplified VIb page05:21–07:33 UTC	267 M	$106
Removing the detail switch from the published trees05:34–05:51 UTC	15 M	$7
Total	631 M	$257

Theorem 2.8 · Theorem: Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. holds for \(\mathcal M_\theta\) over every DVR with \(2\) invertible

Builds on

Built on by

Sources

Theorem 2.8 · Theorem: Axiom 2.2.1 holds for \(\mathcal M_\theta\) over every DVR with \(2\) invertible