§3 · Agent 3 · Hypotheses in char p · 2026-10-18T03:02:13Z

Summary 3.12 · (Q3) verdict: (A1)–(A3c), (A5), (A4)(i,ii,iv) hold over \(\mathbb F_q((t))\)

proved audit: correct ★ key step

Second audit: correct. Each row of the verdict table matches a node I re-derived: (A1), (A2) Proposition 3.2; (A3b) Lemma 3.3, Proposition 3.4, Audit 3.5, Theorem 3.8; (A3c) Audit 3.9; (A4)(i), (ii), (iv) and the combinatorics of (iii) Proposition 3.6, Proposition 3.7, Proposition 3.10, with one-line repairs in Proposition 3.6 and Proposition 3.7 (Audit 5.10, Objection 5.11); (A5) Lemma 3.3. The only open item is the analytic part of Statement 6.3.1 (Remark 3.11), and the invariants listed are those computed. The checklist claims are true: Theorem 8.0.2 in characteristic p is never used; all stabilizers are smooth; H^1 is taken only of smooth connected groups; p = 3 needs no change (the Weyl modules used are minuscule for E6 or the D5 vector representation, irreducible for p odd). The consequence for (SG) is correctly stated as conditional on (Q1) and (Q2). The proofs also apply to p-adic F, because DS and BT III cover it.

For \(p\) odd and \(\mathcal M_\theta=(G_{\mathbb Z}/G_{\mathbb Z}^{\theta_0})\otimes\mathbb F_q[[t]]\), every hypothesis of Sakellaridis's Theorem 8.0.2 assigned to (Q3) is proved by characteristic-\(p\) arguments, except Statement 6.3.1, whose combinatorial input is verified and whose analytic part belongs to (Q1). The same proofs work for \(p\)-adic \(F\), \(p\) odd, and remove from S4/S4b the citations of Kneser, Vust/Richardson, Helminck–Wang, Alper, Knop and Helgason.

Summary 3.12.

Table 2. hypothesis and for \(\mathcal M_\theta\) over \(\mathbb F_q(\!(t)\!)\), \(p\) odd and status
hypothesis	for \(\mathcal M_\theta\) over \(\mathbb F_q(\!(t)\!)\), \(p\) odd	status	nodes
(A1)	\(G\) split reductive over \(\mathcal O\), connected fibres, simply connected derived group	proved	Proposition 3.2
(A2)	\(X\) smooth quasi-affine; \(\mathring X=U_P\times A_X\) smooth, surjective, \(B\)-homogeneous special fibre; invariant forms; \(X(\mathcal O)=Kx_0\)	proved	Proposition 3.2
(A3b)	\(X(F)=\bigsqcup_{\kappa\in\Lambda_S^+}Kx_\kappa\), \(X(F)/K\simeq\Lambda_X^+\) via \(A_X^+\)	proved (Bruhat–Tits III cited)	Lemma 3.3, Proposition 3.4, Audit 3.5, Theorem 3.8
(A3c)	\(Jx\subseteq B(\mathcal O)x\) for \(x\in A_X^+\)	proved	Audit 3.9
(A4)(i)	maximal-rank pairs only of types \(G,U\)	proved	Proposition 3.6
(A4)(ii)	\(L_{\gamma_1},L_{\gamma_6}\) of type \(D_5\)	proved	Proposition 3.7
(A4)(iii)	Statement 6.3.1	combinatorics proved; analytic part open (Q1)	Proposition 3.10, Remark 3.11
(A4)(iv)	Statement 7.1.5	proved	Proposition 3.10
(A5)	\(X_F\) homogeneous affine, \(\mathring X(F)\) one \(B(F)\)-orbit	proved	Lemma 3.3

Invariants in characteristic \(p\ne2\): \(\mathcal X(X)=\mathbb Z\omega_1\oplus\mathbb Z\omega_6\); \(\Lambda_X^+=\Lambda_X\cap\{\langle v,\gamma_1\rangle\le0,\langle v,\gamma_6\rangle\le0\}\); \(\Delta_X=\{\gamma_1,\gamma_6\}\), type \(G\); colours \(D_1,D_6\) with valuations \(\check\gamma_1,\check\gamma_6\); \(W_X\simeq S_3\); 45 maximal-rank orbits; \(\Theta=\check\Phi_X\times\{(+,4)\}\).

(A1), (A2) (Proposition 3.2): Conrad's Lemma 2.2.4 and Thm 2.3.1, valid over any base, replace Edixhoven and Alper; connectedness of the fibres is proved directly.
(A5) (Lemma 3.3).
(A3b): \(G(F)=KT(F)H(F)\) by Delorme–Sécherre, whose field hypothesis includes \(\mathbb F_q(\!(t)\!)\) and whose proof was checked (Audit 3.5); one \(H(F)\)-class of maximal \((\theta,F)\)-split tori by a direct argument (Proposition 3.4) using Bruhat–Tits III instead of Kneser; labels through \(\mathcal V\) (Proposition 3.7) and \(W_X\subseteq K\) (Theorem 3.8).
(A3c) (Audit 3.9). (A4) (Proposition 3.6, Proposition 3.7, Proposition 3.10); remaining: Remark 3.11.
Consequence, stated but not claimed. If (Q1) supplies Theorem 8.0.2 over \(\mathbb F_q(\!(t)\!)\) under (A1)–(A5) (including the analytic part of Statement 6.3.1) and (Q2) supplies (A3a) for \(\mathcal M_\theta\) over \(\mathbb F_q[[t]]\), then Proposition VII (\(r\) not onto, every \(q\)) gives that (SG) fails for \(E_6/F_4\) over \(\mathbb F_q(\!(t)\!)\). No hypothesis is verified here by appeal to Theorem 8.0.2 in characteristic \(p\) (checklist item 10).
Checklist. (1) Each characteristic-0 citation is replaced or shown characteristic-free (Audit 3.1). (2) The arguments use \(2\in\mathcal O^\times\) and separability of \(F(\sqrt t)\), never \(\mathrm{char}\,F=0\). (3) Integral statements are proved as statements about \(\mathcal O\)-schemes (open immersions, purity, torsors), not fibrewise. (5) All stabilizers are smooth. (6) Only \(H^1\) of \(\mathrm{Spin}_8\), \(F_4\), split tori and split unipotent groups is used. (8) The dictionary \(\Lambda_X^+\leftrightarrow\Lambda_S^+\) is explicit (Theorem 3.8 step 1). (9) No 'almost every place'. (11) \(p=3\) needs no change (Audit 3.1 step 2).

Builds on

Audit 3.1 · Where the \(p\)-adic verification (S4, S4b) used characteristic 0, and what replaces it
Proposition 3.2 · (A1), (A2) for \(\mathcal M_\theta\) over any complete DVR with 2 invertible
Lemma 3.3 · (A5) and Galois cohomology over \(\mathbb F_q((t))\): \(X(F)=G(F)x_0\), \(X_F\) affine
Proposition 3.4 · Maximal \(\theta\)-split tori: dimension 2, one \(H(F)\)-class (residue char. \(\ne2\))
Audit 3.5 · Delorme–Sécherre over \(\mathbb F_q((t))\), \(p\) odd: \(G(F)=H(F)T(F)K\)
Proposition 3.6 · Char \(\ne2\): 45 maximal-rank Borel orbits \(Bn_wx_0\), all pairs of type \(G\) or \(U\)
Proposition 3.7 · Char \(\ne2\): two colours, the valuation cone, spherical roots \(\gamma_1,\gamma_6\)
Theorem 3.8 · (A3b) for \(\mathcal M_\theta\) over \(\mathbb F_q((t))\): \(K\backslash X(F)\simeq\Lambda_X^+\)
Audit 3.9 · Axiom 2.4.2 for \(\mathcal M_\theta\) over \(\mathbb F_q((t))\): S4-5 re-checked
Proposition 3.10 · (A4) in char \(p\): (i), (ii), Statement 7.1.5, and the combinatorics of 6.3.1
Remark 3.11 · Not proved here: the analytic part of Statement 6.3.1 over \(\mathbb F_q((t))\)

Built on by

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, Theorem 8.0.2 p. 64 and its hypotheses (§1.1, §1.7, Axioms 2.2.1–2.4.2, §6 opening, §7, §8)
S4.json, S4b.json
e6-f4-module.html (Proposition VII)

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