§2 · Agent 2 · Integral models · 2026-10-18T03:16:48Z

Strategy 2.1 · Strategy: Axiom 2.2.1 for $\mathcal M_\theta$ once over $\mathbb Z[1/2]$

heuristic audit: correct

Second audit: correct. Re-derived each claim: τ(gH)=gθ(g)⁻¹ is a monomorphism (τ(g)=τ(g′) iff g′⁻¹g ∈ G^θ=H), θ_x acts on X*(T) by w(−w_L)w⁻¹=−x, Stab_B(ẇx₀)=B∩G^{θ_x}, and R(P_α)=ker α⋉U_{P_α} is Sakellaridis's radical (p. 7, 'R(M) its radical'; ker α is connected because α_i is primitive in the weight lattice). Against the text of p. 9, the reading Y^can = Ȳ∖closure(Ȳ_F∖Y_F) with closures in X_O is the only one under which the O-scheme Y can be smooth and surjective as the remark after the axiom says, and the complement clause read with complements in X fails for every non-open saturation, so 'complements inside closures' is forced. The plan uses only 2∈O^×. Not mentioned, no claim affected: Sakellaridis's V/Γ is a geometric quotient (p. 7), see Proposition 2.5.

Fixes notation and reduces (A3a) for $\mathcal M_\theta$ over every DVR with $2$ invertible to $45$ orbit schemes $B\dot wx_0$ defined over $\mathbb Z[1/2]$, without Knop's characteristic-0 theory.

Notation. $G=G_{\mathbb Z}$ split simply connected $E_6$, pinning $(T,B,\{x_{\alpha_i}\})$, $\Phi\supset\Phi^+\supset\Delta$ (Bourbaki), $W$, $\ell$, $w_0$; $\dot w$ Tits lifts. $\Phi_L$: the $24$ roots of the black $D_4$ (nodes 2–5), $w_L$ its longest element. $\theta=\theta_0=\operatorname{Int}(\sigma_L)\circ C$, an involution over $\mathbb Z$ with $\theta|_T=-w_L$ (S4-11). Over $\mathbb Z[1/2]$: $H=G^\theta$ (smooth, fibres $F_4$), $X=G/H$ ($\mathcal M_\theta=X\otimes\mathcal O$, smooth separated quasi-affine by S4-4), $x_0=eH$, $\tau:X\to G$, $gH\mapsto g\theta(g)^{-1}$ (a monomorphism). $\mathcal C=\{ww_Lw^{-1}\}\subseteq W$; for $x=ww_Lw^{-1}\in\mathcal C$: $$y_x=\dot wx_0,\qquad\theta_x=\operatorname{Int}(\dot w)\theta\operatorname{Int}(\dot w)^{-1},\qquad\Phi_x=w(\Phi_L),\qquad B_x=\operatorname{Stab}_B(y_x)=B\cap G^{\theta_x};$$ $\theta_x$ acts on roots by $-x$. $\mathcal O$: any DVR with $2\in\mathcal O^\times$ (every complete DVR of odd residue characteristic, $\operatorname{char}F=0$ or $p$, $p=3$ allowed). Sakellaridis's $YP_\alpha$ is our $P_\alpha Y$; $R(P_\alpha)=\ker\alpha\ltimes U_{P_\alpha}$.

Sakellaridis's structure (p. 9). For locally closed $Y_F\subseteq X_F$: $\overline Y$ its (reduced, flat) closure in $X_{\mathcal O}$, $Y^{\rm can}:=\overline Y\smallsetminus\overline{(\overline Y_F\smallsetminus Y_F)}$. (A3a) asks, for each maximal-rank $Y_F$ and $\alpha\in\Delta$: (i) $P_\alpha Y^{\rm can}/R(P_\alpha)$ is a standard model; (ii) complements taken in closures, $P_\alpha Y^{\rm can}=(P_\alpha Y_F)^{\rm can}$. (Complements in $X$ would force $P_\alpha Y$ open.)

Plan. Root/sign data Lemma 2.2; orbits on geometric fibres, char $\ne2$ Proposition 2.3; orbit immersions over $\mathbb Z[1/2]$ Lemma 2.4; rank-one quotients Proposition 2.5; computation Computation 2.6; closures over $\mathcal O$ Lemma 2.7; theorem Theorem 2.8; audit Audit 2.9; verdict Summary 2.10. Only $2\in\mathcal O^\times$ is used.

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Lemma 2.2 · The involutions $\theta_x$: compact imaginary roots, no real roots
Audit 6.7 · Audit of Sakellaridis's scheme $Y$ and the complement clause (claims A2-1, A2-8)

Sources

sakellaridis-v3.pdf, Axiom 2.2.1 and the scheme structure on Y, p. 9
S4b.json: S4-11; S4.json: S4-4

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

Strategy 2.1 · Strategy: Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. for \(\mathcal M_\theta\) once over \(\mathbb Z[1/2]\)

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Sources

Strategy 2.1 · Strategy: Axiom 2.2.1 for \(\mathcal M_\theta\) once over \(\mathbb Z[1/2]\)