You are agent A6, an independent auditor in a bounded team attacking one mathematical question. You did not write the claims below and you receive no opinion about them.

SETTING (Proposition VI of the satake-two-levels pages: Sakellaridis's theorem, restated for a local field of characteristic p).

Fields. p is an odd prime, q a power of p. F is a non-archimedean local field with ring of integers O, uniformizer ϖ, residue field F_q, |ϖ| = q^{-1}. Two cases that must never be confused:
  char F = 0 : F/Q_p finite ("p-adic"). Sakellaridis assumes F/Q_p unramified "to avoid complicated formulas", saying "the modifications needed to remove this assumption are trivial" (§1.1, p. 2).
  char F = p : F = F_q((t)), O = F_q[[t]], ϖ = t.
"p odd" is the residue characteristic in both cases; the characteristic of F is 0 or p.

Groups. G is a split reductive group scheme over O with connected fibres and simply connected derived group; B ⊇ A a Borel subgroup and a split maximal torus over O; U the unipotent radical of B; W the Weyl group; K = G(O); J the Iwahori subgroup (preimage of B(F_q)). H_K = C_c(K\G(F)/K), vol K = 1; the Satake isomorphism S : H_K ≅ C[A^*]^W, A^* the complex dual torus (unramified characters of A(F)).

Variety. Sakellaridis uses right actions and writes X = H\G; our pages use left actions, X = G/H (read x·g as g^{-1}x). X is a G-scheme over O with an open B_F-orbit X̊_F; X̊ := X \ closure(X \ X̊_F); x0 ∈ X̊(O), H its stabilizer.
Invariants (Sakellaridis §2.1–2.2, §6.1, §7.1, §8): P(X) = {p : X̊p = X̊}; L(X) a Levi of P(X) with [L(X),L(X)] ⊆ H; A_X = L(X)/(L(X)∩H); 𝒳(X) the B-eigencharacters of F(X)^(B); Λ_X = Hom(𝒳(X), Z); Λ_X^+ the monoid of G-invariant valuations, V its cone; A_X^+ ⊆ A_X(F) the points with label in Λ_X^+ (via Λ_X ≅ A_X(F)/A_X(O), λ ↦ e^λ(ϖ)); W_X the little Weyl group (Knop's action of W on the B_F̄-orbits of maximal rank is transitive, the stabilizer of X̊ is W(X) = W_X ⋉ W_{P(X)}); δ(X) = |d(X)|, d(X) the modular character of L(X)∩B; δ(X)^{1/2}A_X^* ⊆ A^* the translate of the image of A_X^* → A^*; Δ_X the normalized spherical roots; colours, virtual weighted colours, Θ, Θ^+.
Functions. C_K = C_c^∞(X(F))^K, a module over H_K by (f∗φ)(x) = Σ_{gK∈G(F)/K} f(g)φ(g^{-1}x); φ0 = 1_{X(O)}; (SG): H_K∗φ0 = C_K. The restriction map r : C[A^*]^W → S := C[δ(X)^{1/2}A_X^*]^{W_X} (Sakellaridis (8.2), p. 65).

HYPOTHESES, relabelled (page numbers of arXiv:0905.4244v3 = printed pages).
  (A1₀) char F = 0 (§1.1, p. 2).
  (A1ₚ) char F = p, i.e. F = F_q((t)). Not in the paper.
  (A1)  G split reductive over O, connected fibres, simply connected derived group (§1.7, p. 6).
  (A2)  Integral model (§1.7, p. 7): X is a quasi-affine G-scheme over O; X̊ is smooth and surjective over O and its special fibre is homogeneous under B_{F_q}; X has a G-eigenmeasure and X̊ a B-invariant measure.
  (A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1, p. 9): for every B_F-orbit Y_F ⊆ X_F of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is isomorphic to one of PGL2\PGL2, SU\PGL2 (S ⊆ N(U)), T\PGL2 (T a non-trivial torus over O) or N(T)\PGL2; and the complement of YP_α is the closure of the complement of (YP_α)_F. Here the O-scheme Y is "the complement of the closure of the complement of Y_F in its closure" (p. 9).
  (A3b) Cartan decomposition (Axiom 2.4.1, p. 10): A_X^+ contains a complete set of representatives of the K-orbits on X(F); elements of A_X^+ with distinct images in Λ_X^+ lie in distinct K-orbits.
  (A3c) Iwahori orbits (Axiom 2.4.2, p. 11): xJ ⊆ xB(O) for x ∈ A_X^+.
  (A4)  Spherical roots and colours, over F̄: (i) no pair (Y, α) of maximal rank is of type N, and δ(X)^{-1/2}χ̃ is unramified (opening of §6, p. 41); (ii) L_γ is a classical group for every γ ∈ Δ_X (§7, p. 58); (iii) Statement 6.3.1 (pp. 44–45), which follows from (i) and (ii) by Theorem 6.3.2 (p. 45); (iv) Statement 7.1.5 (p. 60).
  (A5)  X is homogeneous and affine, and X̊(F) is a single B(F)-orbit (§8, p. 63).

PUBLISHED, characteristic 0. Theorem 8.0.2 (p. 64): under (A1₀), (A1)–(A5) there is a canonical isomorphism C_K ≅ S, compatible with the H_K-structure and the Satake isomorphism (f acts on S as multiplication by r(S f)), with 1_{X(O)} ↦ 1. Corollary 8.0.4 (p. 65): if r is onto, C_K is a principal H_K-module generated by φ0. Hence, in characteristic 0, (SG) ⟺ r onto.

THE CASE E6/F4. G_Z the split simply connected Chevalley group of type E6, pinning (T, B, {x_{α_i}}), Bourbaki labels (chain 1–3–4–5–6, node 2 on node 4). θ0 = Int(σ_L)∘C, with C the Chevalley involution and σ_L the Tits lift of w_L, the longest element of W(D4) on nodes 2,3,4,5; θ0|_T = −w_L; type E IV; θ0 is an involution defined over Z (S4-11). The model M_θ: X_O := (G_Z/G_Z^{θ0}) ⊗ O with p odd; H_O = G_O^{θ0}, smooth with fibres of type F4; x0 = eH. Λ_X = Λ_S = Ze1 ⊕ Ze6 with e_i = (1−θ0)α_i^∨; W_X = S3; restricted roots A2, all multiplicities 8; P(X) the standard parabolic of nodes {2,3,4,5}, L = Z_G(S), S = (T^{−θ0})°; 45 Borel orbits of maximal rank (|W(E6)|/(|W_X|·|W(D4)|) = 51840/(6·192)).
Known for p-adic F (char 0), p odd (salvage returns S4, S4b): (A1), (A2), (A3b), (A3c), (A4), (A5) hold for M_θ, and the types in (A3a) over F̄ are only G and U; (A3a) as O-schemes is open. Proposition VII (proved, every q): r is not onto; so wherever Theorem 8.0.2 holds for M_θ, (SG) fails.

TEAM QUESTION.
 (Q1) Formulate and prove Sakellaridis's Theorem 8.0.2 over a local field of characteristic p: F = F_q((t)), p odd, for a symmetric X = G/G^θ (θ an involution of G over O) satisfying (A1)–(A5), every statement over F̄ read over an algebraic closure of F_q((t)). State exactly which further hypotheses, if any, the characteristic-p proof needs. At minimum: X = M_θ (E6/F4).
 (Q2) Prove (A3a) for M_θ over every complete discrete valuation ring O with odd residue characteristic, mixed (p-adic) and equal (F_q[[t]]) characteristic, or exhibit the pairs (Y, α) where it fails.
 (Q3) Verify (A1), (A2), (A3b), (A3c), (A4), (A5) for M_θ over F_q((t)), p odd, replacing every characteristic-0 citation.
A complete answer gives the characteristic-p theorem with its exact hypotheses and a proof in which every step is proved or cited with its number and checked to hold in characteristic p; (A3a) for M_θ over every O as above; and the hypotheses verified for M_θ over F_q((t)). Then, for E6/F4 over F_q((t)) and over p-adic F (p odd), (SG) ⟺ r onto, so (SG) fails.

DOES NOT COUNT. "The proof carries over verbatim" without checking each characteristic-0 input: Knop [Kn91, Kn94, Kn95, Kn96], Brion [Br01], Brion–Luna–Vust [BLV86], Luna [Lu01], Sakellaridis [Sa06, Sa08, Sa12], Braverman–Kazhdan [BK98], Casselman and Casselman–Shalika [Ca80, CS80], Wasserman [Was96], Akhiezer [Ak83], Tate's thesis in §5.3. A p-adic result presented as a characteristic-p result. (A3a) checked over F̄ only (the types), or only on one fibre, or "at almost every place", instead of as an isomorphism of O-schemes. A close-fields transfer that assumes Theorem 8.0.2 for ramified p-adic fields without checking Sakellaridis's normalizations (§3.8, §5.3). Numerical checks for small q only. A statement about E6/F4 over F_q((t)) obtained by citing Theorem 8.0.2 in characteristic 0.

CHECKLIST (the ways a wrong answer typically fails; audits check every one).
 1. Hypothesis transfer: a characteristic-0 theorem used in characteristic p.
 2. Residue characteristic confused with the characteristic of F.
 3. Generic fibre, special fibre and O-scheme confused; in particular Sakellaridis's scheme structure on Y.
 4. A type over F̄ taken for an isomorphism of O-schemes.
 5. Non-reduced stabilizers and inseparable orbit maps in characteristic p.
 6. Finiteness of B(F)-orbits and of Galois cohomology in characteristic p (H^1(F, μ_p) and Artin–Schreier classes are infinite).
 7. Normalizations: measures, the conductor of ψ, "unramified over Q_p".
 8. Left versus right actions; Λ_X^+ versus Λ_S^+; anti-dominant versus dominant labels.
 9. A statement "at almost every place" used at a given place.
 10. Circularity: the conclusion of Theorem 8.0.2 in characteristic p used to verify its hypotheses.
 11. Bad primes: p = 3 is bad for E6 and F4.

TERMINOLOGY TRAPS. The type of (Y, α) (G, U, T, N) is a statement over F̄; (A3a) is a statement about O-schemes. X(O) and Kx0 are different sets in general (equal for M_θ by Hensel and Lang). K-orbits, B(F)-orbits and B_F̄-orbits are different things. Λ_X^+ is the valuation monoid (for symmetric X the image of the anti-dominant chamber); our pages label K-orbits by dominant μ ∈ Λ_S^+ through t^μ x0. W_X for E6/F4 is S3. "Unramified over Q_p" is a simplification of the paper, not part of the characteristic-p question. "Symmetric" means H = G^θ (or an open subgroup of it) for an involution θ.

FILES (read HTML with  sed -e 's/<[^>]*>/ /g' FILE | tr -s ' \n'  and PDFs with  pdftotext -layout FILE - ).
 Sakellaridis, Spherical functions on spherical varieties, arXiv:0905.4244v3: /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/satake-two-levels/props/sources/sakellaridis-v3.pdf ; as text: /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/75c2c3d9-ef06-4644-b59b-d12583740b0c/scratchpad/solve-vi/sak.txt (§2.2.1 near line 503, §3.3 near 720, Theorem 4.2.2 near 1410, §5.4 near 1823, §6.19 near 3388, Theorem 7.2.1 near 3631, §8 near 3830).
 Proposition VI page (hypotheses with their status, the E6/F4 verification): /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/satake-two-levels/props/sakellaridis.html
 Salvage returns S4, S4b (the p-adic verification for E6/F4; nodes S4-1 … S4-14): /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/satake-two-levels/audit/S4.json and S4b.json
 E6/F4 root data and the symmetric pair: /Users/miltonlin/Projects/Trees/subpages/tools/proof_graphs/spherical-generation/families/e6-f4.html ; Proposition VII: /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/satake-two-levels/props/e6-f4-module.html
 Other local sources: /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/satake-two-levels/props/sources/{conrad-reductive-group-schemes,milne-iag-v2,chen-yi-v4,zhu-satake-v2,bf-satake-v2,stacks-algebra,stacks-more-algebra}.pdf

RULES.
- Return concrete lemmas, constructions, computations or counterexamples, each with a proof or an explicit gap. No status reports, no optimism. Label every claim: proved (here, or cited with exact result number) · heuristic · gap. An analogue in one characteristic does not certify an inference in the other.
- Check the exact meaning of every term you use (see the traps above).
- Do not search the web for a solution of this question; ordinary mathematical background (statements and standing hypotheses of standard theorems and papers) may be looked up. Prefer the local sources. When you cite a paper you could not open, say so.
- You may run short Python checks with `python3 -I` (standard library only; no sympy or numpy) for finite computations: single process, a few seconds each, no parallelism. Keep scripts in /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/75c2c3d9-ef06-4644-b59b-d12583740b0c/scratchpad/solve-vi/A6/ (create it). Do not modify anything under /Users/miltonlin/Projects/Trees.
- Timestamps: at the start of every node run `date -u +%Y-%m-%dT%H:%M:%SZ` and record it in the node. Work in nodes: an idea, a lemma with proof, a computation, a check, an objection, a dead end (status blocked, with the exact missing lemma), or a verdict. Produce 5–14 nodes.
- Write every node as a stage of a mathematics paper: a plain-words name, a one-sentence summary, the precise statement, and the argument as numbered steps, readable by a mathematician outside the subfield. Cite other nodes as [[A6-3]], never by bare code in prose; state computations as mathematics (what was computed, the result, why it suffices), not as code or program output.
- Concept keys for nodes: setup · transfer · integral · hypotheses · close · synthesis.
- FINAL ANSWER: only one JSON object inside a ```json fence, with the schema below; also write the identical JSON to /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/75c2c3d9-ef06-4644-b59b-d12583740b0c/scratchpad/solve-vi/A6/return.json. Escape JSON strings properly (double every backslash, escape quotes and newlines). Nothing outside the fence.

RETURN SCHEMA
{"agent":"A6",
 "family":"approach family, at most twelve words",
 "nodes":[
  {"id":"A6-1","t":"2026-10-18T02:10:12Z",
   "kind":"idea|lemma|computation|check|objection|blocked|verdict",
   "status":"proved|refuted|blocked|open|heuristic|superseded",
   "label":"Setup|Definition|Strategy|Lemma|Proposition|Theorem|Corollary|Computation|Example|Counterexample|Remark|Objection|Audit|Conjecture|Dead end|Summary",
   "name":"tree label: at most 44 characters, plain text (Unicode, no TeX), says what the step establishes",
   "title":"page title, at most 90 characters, inline TeX allowed",
   "summary":"one or two sentences: what this step establishes and why it matters",
   "statement":"the precise claim (Markdown + TeX); empty for Strategy and Remark",
   "parents":["A6-0"],
   "body":"the argument: Markdown; math as \\( \\) inline and $$ $$ display; proofs as numbered steps; other nodes as [[A6-1]]; results cited with exact numbers and the file they come from",
   "concept":"one of: setup, transfer, integral, hypotheses, close, synthesis",
   "key":false,
   "sources":["sakellaridis-v3.pdf, Axiom 2.2.1, p. 9"]}],
 "result":{"status":"proved|partial|refuted|blocked",
           "statement":"precise statement of what was established",
           "gap":"exact remaining obligation, or none"},
 "blocked_routes":["route and the theorem-strength lemma it would need"],
 "scripts":["paths of scripts run"]}

CLAIMS TO AUDIT (copied verbatim from the return of the agent who wrote them; the full return is /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/sakellaridis-char-p/returns/A2.json, and its scripts are in /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/sakellaridis-char-p/scripts/A2/ — read them only after your own recomputation).

--- A2-1 · Strategy · Strategy: Axiom 2.2.1 for \(\mathcal M_\theta\) once over \(\mathbb Z[1/2]\)
STATUS CLAIMED: heuristic
PARENTS: 
STATEMENT:

PROOF / BODY:
**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 [[A2-2]]; orbits on geometric fibres, char \(\ne2\) [[A2-3]]; orbit immersions over \(\mathbb Z[1/2]\) [[A2-4]]; rank-one quotients [[A2-5]]; computation [[A2-6]]; closures over \(\mathcal O\) [[A2-7]]; theorem [[A2-8]]; audit [[A2-9]]; verdict [[A2-10]]. Only \(2\in\mathcal O^\times\) is used.
SOURCES: sakellaridis-v3.pdf, Axiom 2.2.1 and the scheme structure on Y, p. 9; S4b.json: S4-11; S4.json: S4-4
--- A2-2 · Lemma · The involutions \(\theta_x\): compact imaginary roots, no real roots
STATUS CLAIMED: proved
PARENTS: A2-1
STATEMENT:
For every \(x\in\mathcal C\):
1. \(\theta_x\) preserves \(T\), acts on \(X^*(T)\) by \(-x\), and \(\theta_x(x_\gamma(c))=x_{-x\gamma}(\pm c)\).
2. The \(\theta_x\)-fixed roots are \(\Phi_x\), and \(\theta_x\) fixes \(U_\gamma\) pointwise for \(\gamma\in\Phi_x\).
3. No root has \(\theta_x\gamma=-\gamma\).
4. \(\gamma+\theta_x\gamma\notin\Phi\cup\{0\}\) for all \(\gamma\in\Phi\).
5. \(T^\theta\simeq\mathbb G_m^4\) over \(\mathbb Z\); \(T^{\theta_x}=\dot wT^\theta\dot w^{-1}\); no root is trivial on \(T^{\theta_x}\); \(Z_G(T^{\theta_x})=T\).
6. \(C_W(w_L)=\langle s_{\alpha_2},\dots,s_{\alpha_5},s_{\alpha_1}s_{\beta_1},s_{\alpha_6}s_{\beta_6}\rangle\) (\(\beta_i=-\theta\alpha_i\)) has order \(1152\), each element lifts to \(N_H(T)(\mathbb Z[1/2])\), and \(|\mathcal C|=45\).
7. On every geometric fibre of characteristic \(\ne2\): \(\dim\mathfrak g^\theta=52\), \(\dim\mathfrak g^{-\theta}=26\).
PROOF / BODY:
1. \(\theta\) acts by \(-w_L\), so \(\theta_x\) by \(w(-w_L)w^{-1}=-x\); a \(T\)-preserving automorphism of \(G_{\mathbb Z}\) maps \(U_\gamma\to U_{\theta_x\gamma}\) by \(c\mapsto\varepsilon c\), \(\varepsilon\in\mathbb Z^\times\).
2. \(\theta\) is trivial on \(\mathrm{Spin}_8=[L,L]\) (S4-11, step 3). Re-checked: in split \(\mathfrak{so}_8\) with \(C(X)=-X^{\mathsf T}\), \(\sigma_i=(1+e_i)(1-f_i)(1+e_i)\) and \(\sigma_{w_0}\) along \(s_1s_2s_1s_3s_2s_1s_4s_2s_1s_3s_2s_4\), exact arithmetic gives \(\sigma_{w_0}C(e_i)\sigma_{w_0}^{-1}=e_i\), \(\sigma_{w_0}C(f_i)\sigma_{w_0}^{-1}=f_i\), \(\sigma_{w_0}^2=1\). So \(\theta(x_\beta(c))=x_\beta(c)\) for \(\beta\in\Phi_L\), hence \(\theta_x\) fixes \(\dot wU_\beta\dot w^{-1}=U_{w\beta}\) pointwise. The roots fixed by \(-w_L\) are exactly \(\Phi_L\) (enumeration), those fixed by \(-x\) are \(w(\Phi_L)\).
3. \(\theta\beta=-\beta\iff\beta\perp\Phi_L\); no root is (enumeration; also S4-5).
4. \(\beta+\theta\beta=2\beta_\parallel\) (projection to the black span) is never a root (enumeration); conjugate by \(w\). Reason: if \(\delta=\theta_x\gamma\) and \(\gamma+\delta\in\Phi\), then \([X_\gamma,X_\delta]=\pm X_{\gamma+\delta}\) and \(\theta_x^2=1\) give \(\theta_xX_{\gamma+\delta}=-X_{\gamma+\delta}\), contradicting item 2 since \(\gamma+\delta\in\Phi_x\).
5. \(X^*(T^\theta)=X^*(T)/(\theta^*-1)X^*(T)\); the Smith form of \(\theta^*-1\) (fundamental-weight basis, transpose of \(\theta\) on coroots, family page §2) is \((1,1)\), so the quotient is \(\mathbb Z^4\). A root is trivial on \(T^{\theta_x}\) iff \(\theta_x\alpha=-\alpha\), excluded by 3.
6. Conjugation-closure of \(\{w_L\}\) gives \(45\) elements, so \(|C_W(w_L)|=51840/45\). The listed elements generate \(1152\) elements commuting with \(w_L\) (enumeration). Lifts: Tits lifts of \(s_{\alpha_2},\dots,s_{\alpha_5}\) lie in \(\mathrm{Spin}_8\subseteq H\); \(\dot n_i=\sigma_{\alpha_i}\theta(\sigma_{\alpha_i})\in H\cap N_G(T)\) lifts \(s_{\alpha_i}s_{\beta_i}\) (S4-13, step 2; uses \(\alpha_i\pm\beta_i\notin\Phi\), by 4 and a coefficient \(2\)). So \(C_W(w_L)=W_H(T)=N_H(T)/T^\theta\simeq W(F_4)\).
7. \(\mathfrak g^\theta=\mathfrak t^\theta\oplus\bigoplus_{\Phi_L}\mathfrak g_\beta\oplus\) (one line per pair \(\{\beta,\theta\beta\}\), \(\beta\notin\Phi_L\)): \(4+24+24\) (S4-11, item 3).

Scripts: e6_orbits.py, e6_checks.py, so8_theta.py.
SOURCES: S4b.json: S4-11 steps 1-5; S4-13 step 2; families/e6-f4.html §2; scripts e6_orbits.py, e6_checks.py, so8_theta.py
--- A2-3 · Proposition · Borel orbits of \(E_6/F_4\) over an algebraically closed field of characteristic \(\ne2\)
STATUS CLAIMED: proved
PARENTS: A2-2
STATEMENT:
Let \(k\) be algebraically closed, \(\operatorname{char}k\ne2\).
1. Every \(\theta\)-stable maximal torus of \(G_k\) is \(H(k)\)-conjugate to \(T\).
2. \(X(k)=\bigsqcup_{x\in\mathcal C}B(k)y_x\) (45 orbits, pairwise distinct).
3. \(By_x\simeq B/B_x\) has dimension \(26-c_x=32-\tfrac12\ell(x)\), \(c_x=\#\{\text{pairs }\{\gamma,\theta_x\gamma\}\subseteq\Phi^+,\gamma\ne\theta_x\gamma\}\), and rank \(2=\operatorname{rank}X\).
4. \(\dot s_\alpha y_x\in By_{s_\alpha xs_\alpha}\) for \(\alpha\in\Delta\).
PROOF / BODY:
1. \(H_k\) is smooth (CGP A.8.10; Edixhoven 3.4), connected (Steinberg, Mem. AMS 80, Thm. 8.1), of dimension \(52\); \(Z_H(T^\theta)=T\cap H=T^\theta\) ([[A2-2]], 5), so \(T^\theta\) is a maximal torus and \(\operatorname{rank}H=4\).
2. **Item 1.** Let \(T'\) be \(\theta\)-stable, \(T'_\pm\) the identity components of \(\{\theta t=t^{\pm1}\}\), \(a=\dim T'_+\le4\). Let \(r,c\) count real and complex \(T'\)-roots, \(i_c,i_n\) imaginary roots with \(\theta=\pm1\) on the root space. By [[A2-2]], 7:
$$52=a+\tfrac{r+c}2+i_c,\qquad26=(6-a)+\tfrac{r+c}2+i_n,\qquad i_c-i_n=32-2a.$$
Imaginary roots form a simply laced system of rank \(\le a\); if \(a\le3\) it has \(\le12\) roots (\(A_3\)), so \(i_c\le12<26\le i_c-i_n\): impossible. So \(a=4\), \(hT'_+h^{-1}=T^\theta\) (\(h\in H(k)\)), and \(hT'h^{-1}\subseteq Z_G(T^\theta)=T\).
3. **Every orbit meets \(N_G(T)x_0\).** For \(z=gx_0\), \(B''=g^{-1}Bg\) and \(\theta B''\) share a maximal torus (Bruhat, Conrad Thm. 1.4.12); \(\theta\) (order 2, semisimple) fixes a maximal torus \(T'\) of \((B''\cap\theta B'')^\circ\) (Steinberg, Thm. 7.5). By item 1, \(T'=h^{-1}Th\), so \(hB''h^{-1}=n^{-1}Bn\), \(n\in N_G(T)\); then \(gh^{-1}n^{-1}\in B\) and \(z\in Bnx_0=B\dot vx_0\).
4. **Same orbit iff same \(x\).** If \(v'=vu\), \(u\in C_W(w_L)\), lift \(u\) to \(h\in N_H(T)\) ([[A2-2]], 6): \(\dot v'x_0\in\dot vhTx_0=T\dot vx_0\). Conversely, with \(\sigma_0\) lifting \(w_0\) and \(\tilde\tau(z)=\tau(z)\sigma_L\sigma_0\): \(\theta(B)=\sigma_L\sigma_0B(\sigma_L\sigma_0)^{-1}\), so \(\tilde\tau(bz)=b\tilde\tau(z)\beta(b)^{-1}\) with \(\beta(b)\in B\), and \(\tilde\tau(By_x)\subseteq B\dot w\theta(\dot w)^{-1}\sigma_L\sigma_0B\), the Bruhat cell of \(w(w_Lww_L)^{-1}w_Lw_0=xw_0\). Cells are disjoint.
5. **Item 3.** \(B_x=(B\cap\theta_xB)^{\theta_x}=T^{\theta_x}\ltimes(U_{\Phi^+\cap\theta_x\Phi^+})^{\theta_x}\) (over \(\mathbb Z\): [[A2-4]], step 2), of dimension \(4+12+c_x\) by [[A2-2]], 2–3. The \(B\)-eigencharacters of \(k(By_x)\) are the characters of \(T\) trivial on \(T^{\theta_x}\): rank \(2\). \(\ell(x)\) counts the \(12\) positive roots of \(\Phi_x\) and the \(2c_x\) positive \(\gamma\) with \(-x\gamma>0\).
6. **Item 4.** \(\dot s_\alpha\dot w\) lifts \(s_\alpha w\) and \((s_\alpha w)w_L(s_\alpha w)^{-1}=s_\alpha xs_\alpha\).

Remark: Knop's action is left multiplication on \(W/C_W(w_L)\); the stabilizer of the open orbit is \(C_W(w_L)=W_X\ltimes W_L\), so the count \(45\) of S4-5(3) holds in every characteristic \(\ne2\).
SOURCES: Steinberg, Mem. AMS 80 (1968), Thms. 7.5, 8.1 (cited, not checked against a local source); conrad-reductive-group-schemes.pdf, Thm. 1.4.12; CGP Prop. A.8.10; Edixhoven 1992 Prop. 3.4 (cited); scripts e6_orbits.py
--- A2-4 · Lemma · Smooth stabilizers; orbit immersions \(B/B_x\hookrightarrow X\) over \(\mathbb Z[1/2]\)
STATUS CLAIMED: proved
PARENTS: A2-2, A2-3
STATEMENT:
Let \(n\in N_G(T)(\mathbb Z[1/2])\) lift \(v\), \(y=nx_0\), \(\theta_n=\operatorname{Int}(n)\theta\operatorname{Int}(n)^{-1}\), \(x=vw_Lv^{-1}\), \(B''=\theta_n(B)\). Over \(\mathbb Z[1/2]\), hence over every \(\mathcal O\ni\frac12\):
1. \(B_y=B\cap G^{\theta_n}=(T\ltimes U_{\Phi^+\cap\theta_n\Phi^+})^{\theta_n}\) is smooth with fibres of dimension \(16+c_x\).
2. \(BB''\subseteq G\) is smooth locally closed; with \(\mathcal F\) the fixed locus of \(g\mapsto\theta_n(g)^{-1}\) on it, \(\mu(b)=b\theta_n(b)^{-1}\) is flat onto an open \(\mathcal O_n\subseteq\mathcal F\) and \(B/B_y\simeq\mathcal O_n\).
3. With \(\tau_n(z)=\tau(z)\tau(y)^{-1}\), \(Y_n:=\tau_n^{-1}(\mathcal O_n)\) is locally closed in \(X\) and \(b\mapsto by\) induces \(B/B_y\simeq Y_n\): smooth of relative dimension \(26-c_x\), fibres the orbits \(B_ky\).
4. \(Y_n\) depends only on \(x\) (write \(Y_x\ni y_x\)).
PROOF / BODY:
1. **Root groups (any base).** For closed \(\Psi\) with \(\Psi\cap-\Psi=\emptyset\), \(\prod_{\gamma\in\Psi}U_\gamma\simeq U_\Psi\), a smooth closed subgroup (Conrad, Prop. 5.1.16); \(U_{-\Phi'^+}\times T\times U_{\Phi'^+}\to G\) is an open immersion for every positive system (Conrad, Thm. 5.1.13).
2. **\(BB''\).** With \(\Phi''^+=\theta_n\Phi^+\), inside the open cell \(\Omega''=U_{\Phi''^-}\times T\times U_{\Phi''^+}\):
$$B=U_{\Phi^+\cap\Phi''^-}\times T\times U_{\Phi^+\cap\Phi''^+},\quad B''=1\times T\times U_{\Phi''^+},\quad BB''=U_{\Phi^+\cap\Phi''^-}\times T\times U_{\Phi''^+}.$$
So \(BB''\) is closed in \(\Omega''\), smooth affine, \(T_1(BB'')=\mathfrak b+\mathfrak b''\), and \(B\cap B''=T\ltimes U_{\Phi^+\cap\Phi''^+}\).
3. **Item 1.** \(B_y\) is the fixed locus of \(\theta_n\) on the smooth \(B\cap\theta_nB\); \(\mathbb Z/2\simeq\mu_2\) over \(\mathbb Z[1/2]\), so it is smooth with Lie algebra the fixed vectors (CGP A.8.10(2); Edixhoven 3.4): \(4+12+c_x\) by [[A2-2]], 2–3, for every \(p\ne2\).
4. **\(\mathcal F\).** \(\sigma(g)=\theta_n(g)^{-1}\) preserves \(BB''\) (as \(\theta_nB''=B\)); \(\mathcal F\) is smooth with \(T_1\mathcal F=(\mathfrak b+\mathfrak b'')^{-\theta_n}\). \(\mu\) is the orbit map at \(1\) of \(b*g=bg\theta_n(b)^{-1}\) on \(\mathcal F\), with scheme-theoretic stabilizer \(B_y\).
5. **Fibres.** \(d\mu_1=1-\theta_n\) has kernel \(\operatorname{Lie}B_y\) and image \(T_1\mathcal F\) (\(Z=b_1+\theta_nb_2\), \(\theta_nZ=-Z\) gives \(Z=(1-\theta_n)\frac{b_1-b_2}2\)). So \(B_k/B_{y,k}\to\mathcal F_k\) is an étale monomorphism, an open immersion (EGA IV 17.9.1).
6. **Item 2.** \(\mu\) is flat by the fibrewise criterion (EGA IV 11.3.10), hence open; \(B\to\mathcal O_n\) is fppf with \(B\times_{\mathcal O_n}B\simeq B\times B_y\), so \(\mathcal O_n\simeq B/B_y\).
7. **Item 3.** \(\tau_n(by)=bn\theta(n)^{-1}\theta(b)^{-1}\theta(n)n^{-1}=b\theta_n(b)^{-1}\). \(\mathcal O_n\) is locally closed in \(G\), so \(Y_n\) is locally closed; \(\tau_n|_{Y_n}\) is a monomorphism whose composite with \(B\to Y_n\) is the epimorphism \(\mu\), hence an isomorphism.
8. **Item 4.** Lifts of \(vC_W(w_L)\) are \(nht\), \(h\in N_H(T)\) ([[A2-2]], 6), and \(nhtx_0\in T\cdot y\). All steps commute with base change \(\mathbb Z[1/2]\to\mathcal O\).
SOURCES: conrad-reductive-group-schemes.pdf, Thm. 5.1.13, Prop. 5.1.16; CGP Prop. A.8.10; Edixhoven 1992 Prop. 3.4 (cited); EGA IV 11.3.10, 17.9.1 (cited)
--- A2-5 · Proposition · Rank-one quotients: a point (type \(G\)) or \(\mathbb P^1_{\mathcal O}\) (type \(U\))
STATUS CLAIMED: proved
PARENTS: A2-2, A2-3, A2-4
STATEMENT:
\(\mathcal O\) a DVR, \(2\in\mathcal O^\times\); \(x\in\mathcal C\), \(\alpha\in\Delta\), \(y=y_x\), \(\pi:P_\alpha\to P_\alpha/R(P_\alpha)\simeq\mathrm{PGL}_2\) (\(\pi(t)=\operatorname{diag}(\alpha(t),1)\)).
1. **(G)** If \(\alpha\in\Phi_x\): \(\varphi_\alpha(\mathrm{SL}_2)\subseteq G^{\theta_x}\), \(P_\alpha Y_x=Y_x\), \(Y_x/R(P_\alpha)=\operatorname{Spec}\mathcal O\).
2. **(U, \(Y_x\) open)** If \(\alpha\notin\Phi_x\), \(x\alpha>0\): with \(x'=s_\alpha xs_\alpha\), \(B'=\dot s_\alpha B\dot s_\alpha^{-1}=\pi^{-1}(B_2^-)\): (a) \((P_\alpha)_y=B'_y\), smooth, \(\pi((P_\alpha)_y)=B_2^-\); (b) \(P_\alpha/(P_\alpha)_y\to X_{\mathcal O}\) is an immersion onto a smooth \(\mathcal P=P_\alpha Y_x=P_\alpha Y_{x'}\), \(|\mathcal P|=|Y_x|\sqcup|Y_{x'}|\), \(\dim Y_{x'}=\dim Y_x-1\); (c) \(py\mapsto\pi(p)[0:1]\) gives \(P_\alpha\)-equivariantly
$$\mathcal P/R(P_\alpha)\simeq\mathrm{PGL}_2/B_2^-\simeq\mathrm{PGL}_2/B_2=SU\backslash\mathrm{PGL}_2\ (S=T_2),$$
with \(Y_x\) over \(\{[z:1]\}\) and \(Y_{x'}\) over \([1:0]\).
3. **(U, \(Y_x\) small)** If \(\alpha\notin\Phi_x\), \(x\alpha<0\): \((s_\alpha xs_\alpha,\alpha)\) is as in 2.
PROOF / BODY:
1. **\(\pi\).** \(Z(L_\alpha)=\ker\alpha\) is a split torus (\(\alpha\) primitive in \(X^*(T)\)), \(L_\alpha=\ker\alpha\cdot\varphi_\alpha(\mathrm{SL}_2)\), intersection \(\alpha^\vee(\mu_2)\); so \(L_\alpha/\ker\alpha\simeq\mathrm{PGL}_2\).
2. **(G).** \(\theta_x\) fixes \(U_{\pm\alpha}\) pointwise ([[A2-2]], 2), so \(\varphi_\alpha\) and \(\theta_x\varphi_\alpha\) agree on \(U^\pm\), on \(\alpha^\vee(s)=w(s)w(1)^{-1}\), on the open cell, hence everywhere (schematic density, EGA IV 11.10.10). As \(P_\alpha=B\cdot\varphi_\alpha(\mathrm{SL}_2)\), \(P_\alpha y=By\); and \(\pi((P_\alpha)_y)=\mathrm{PGL}_2\).
3. **\(\pi((P_\alpha)_y)\supseteq B_2^-\).** \(\pi(T^{\theta_x})=T_2\) since \(\alpha|_{T^{\theta_x}}\ne0\) ([[A2-2]], 5). With \(\delta=-\theta_x\alpha\in\Phi^+\smallsetminus\{\alpha\}\), \(-\alpha+\delta\notin\Phi\cup\{0\}\) ([[A2-2]], 4), so \(U_{-\alpha},U_\delta\) commute (Conrad 5.1.14) and \(u(c)=x_{-\alpha}(c)\theta_x(x_{-\alpha}(c))=x_{-\alpha}(c)x_\delta(\pm c)\in(P_\alpha)_y\), \(\pi(u(c))=\pi(x_{-\alpha}(c))\).
4. **(a).** Geometrically, \(\operatorname{Lie}(P_\alpha)_y=(\mathfrak t\oplus\bigoplus_\Psi\mathfrak g_\gamma)^{\theta_x}\), \(\Psi=\Phi(P_\alpha)\cap\theta_x\Phi(P_\alpha)=(\Phi^+\cap\theta_x\Phi^+)\sqcup\{-\alpha,\delta\}=s_\alpha\Phi^+\cap\theta_xs_\alpha\Phi^+\), dimension \(17+c_x\), equal to \(\operatorname{Lie}B'_y\) with \(B'_y\) smooth ([[A2-4]] for \(\dot s_\alpha^{-1}\dot w\)). So \((P_\alpha)_{y,\bar k}\) is smooth and \(\pi\) of it is \(B_2^-\) or \(\mathrm{PGL}_2\). If \(\mathrm{PGL}_2\), \(P_\alpha y=R(P_\alpha)y\subseteq By\), so \(By_{x'}=By_x\), \(x'=x\), \(x\alpha=\pm\alpha\): excluded. Hence \((P_\alpha)_{y,\bar k}=B'_{y,\bar k}\) on both fibres. Over \(\mathcal O\) the ideal \(I\) of \(B'_y\subseteq(P_\alpha)_y\) has \(I/\varpi I=0\) (\(B'_y\) flat) and \(\varpi^NI=0\) (\(I\otimes F=0\)), so \(I=0\).
5. **(b).** On fibres, \(py\mapsto\pi(p)[0:1]\) has \(R(P_\alpha)\)-orbit fibres, \(R(P_\alpha)\subseteq B\): \(\{[z:1]\}\leftrightarrow By\), \([1:0]\leftrightarrow B\dot s_\alpha y=Y_{x'}\). \(c_{x'}=c_x+1\) (\(s_\alpha\) maps the \(\theta_x\)-pairs in \((\Phi^+\smallsetminus\{\alpha\})\cup\{-\alpha\}\) onto the \(\theta_{x'}\)-pairs in \(\Phi^+\)). \(B'y=\dot s_\alpha Y_{x'}\) is locally closed ([[A2-4]]). \(\mathcal W=\{(gB',z):g^{-1}z\in B'y\}\subseteq\mathbb P^1\times X\) is \(\simeq P_\alpha/(P_\alpha)_y\) and \(\psi=\mathrm{pr}_2|_{\mathcal W}\) is the orbit map, a monomorphism. With \(Z'=\overline{B'y}\) (fibres of dimension \(\le\dim Y_x-1\)) and \(\partial'=Z'\smallsetminus B'y\): \(Z'_k\) misses the larger \(B'\)-orbit \(\dot s_\alpha Y_x\) of \(P_\alpha y_k\), so \(V=X\smallsetminus P_\alpha\partial'\) is an open neighbourhood of \(|\mathcal P|\) over which \(\mathcal W\) is closed in \(\mathbb P^1\times V\). So \(\psi\) is a proper monomorphism into \(V\), a closed immersion (EGA IV 18.12.6).
6. **(c).** \((P_\alpha)_yR(P_\alpha)=B'\), so the fppf quotient is \(P_\alpha/B'\); right translation by \(\bar s\) gives \(\mathrm{PGL}_2/B_2\).
7. **3.** \(x'\alpha=s_\alpha(-x\alpha)>0\) as \(-x\alpha\in\Phi^+\smallsetminus\{\alpha\}\), and \(\alpha\notin s_\alpha\Phi_x=\Phi_{x'}\).
SOURCES: sakellaridis-v3.pdf, §2.2 and Axiom 2.2.1, p. 9; conrad-reductive-group-schemes.pdf, Prop. 5.1.14; EGA IV 11.10.10, 18.12.6 (cited)
--- A2-6 · Computation · The \(45\times6\) pairs: 90 of type \(G\), 180 of type \(U\)
STATUS CLAIMED: proved
PARENTS: A2-2, A2-3
STATEMENT:
1. \(|\mathcal C|=45\), \(w_0\in\mathcal C\). Open orbit \(x=w_L\) (\(c=0\)): type U (\(Y\) open) for \(\alpha_1,\alpha_6\), type \(G\) for \(\alpha_2,\dots,\alpha_5\).
2. \(\dim Y_x=26-c_x=32-\frac12\ell(x)\) takes the values \(14,\dots,26\) with multiplicities \(1,2,3,4,5,5,5,5,5,4,3,2,1\).
3. \(\theta_x\Phi^+=\Phi^+\) only for \(x=w_0\) (dimension 14); every other \(x\) has a simple \(\alpha\notin\Phi_x\) with \(x\alpha>0\).
4. Of \(270\) pairs: \(90\) with \(\alpha\in\Phi_x\) (G), \(90\) with \(x\alpha\in\Phi^+\smallsetminus\Phi_x\) (U, \(Y\) open), \(90\) with \(x\alpha\in\Phi^-\smallsetminus\Phi_x\) (U, \(Y\) small); none with \(\theta_x\alpha=-\alpha\).
5. In the 90 open type-U pairs: \(\Phi(P_\alpha)\cap\theta_x\Phi(P_\alpha)\cap\{\pm\alpha\}=\{-\alpha\}\), \(-\alpha-\theta_x\alpha\notin\Phi\cup\{0\}\), \(c_{s_\alpha xs_\alpha}=c_x+1\), \(\dim(P_\alpha)_y=\dim B_y+1\).
6. \(\Sigma_{w_0}=\{w_0\}\), \(\Sigma_x=\Sigma_{x'}\cup\{s_\alpha vs_\alpha:v\in\Sigma_{x'},\alpha\notin\Phi_v\}\) (\(x'=s_\alpha xs_\alpha\), \(\alpha\) as in 3) is independent of the choices and equals \(\{z\in\mathcal C:x\le z\}\) (Bruhat order; 821 pairs).
PROOF / BODY:
**Computed.** Roots of \(E_6\) from the Cartan matrix; \(W\) as permutations of the \(72\) roots; \(w_L=s_2s_3s_4s_2s_3s_4s_5s_4s_2s_3s_4s_5\), \(\theta=-w_L\) (\(\theta\alpha_1=-(\alpha_1+\alpha_2+2\alpha_3+2\alpha_4+\alpha_5)\)); \(\mathcal C\) by conjugation-closure with representatives \(w\). For each \(x\): \(-x=w\theta w^{-1}\), \(24\) imaginary roots, no real roots, no \(\gamma+\theta_x\gamma\in\Phi\), \(\ell(x)=12+2c_x\), the type of each \(\alpha\) (agreeing with \(w^{-1}\alpha\in\Phi_L\) for type \(G\)), and the partner \(s_\alpha xs_\alpha\) (\(=x\) iff type \(G\); otherwise open/small exchanged, \(c\) changes by \(\pm1\)). Item 5 from the fixed-vector count \(4+\#(\Psi\cap\Phi_x)+\frac12\#(\Psi\smallsetminus\Phi_x)\). Item 6 by recursion for every choice rule, and Deodhar's criterion for the Bruhat order.

**Why it suffices.** Items 1–5 are the finite inputs of [[A2-3]], [[A2-5]], [[A2-7]]; item 6 gives the closure sets of [[A2-7]] (which fixes one choice). All are statements about \(W(E_6)\), valid over every field. Scripts e6_orbits.py, e6_checks.py (under one second).
SOURCES: scripts e6_orbits.py, e6_checks.py; families/e6-f4.html §§1-3
--- A2-7 · Lemma · Orbit closures over \(\mathcal O\) have the expected special fibres
STATUS CLAIMED: proved
PARENTS: A2-3, A2-4, A2-5, A2-6
STATEMENT:
\(Z_x\) := closure of \(Y_{x,F}\) in \(X_{\mathcal O}\) (reduced, flat, \(B\)-stable); \(\Sigma_x\) as in [[A2-6]], 6, for a fixed choice.
1. \(\overline{Y_{x,k}}=\bigcup_{v\in\Sigma_x}Y_{v,k}\) for every algebraically closed \(k\) over \(\mathcal O\).
2. \(Y_{w_0}\) is closed in \(X_{\mathcal O}\).
3. \(Z_x\cap X_\kappa=\overline{Y_{x,\kappa}}\).
PROOF / BODY:
1. \(Z_x\) is \(B\)-stable (\(B\times Z_x\) flat) and contains the section \(y_x\), so \(Y_{x,\kappa}\subseteq Z_x\).
2. **Item 2.** \(\tilde\tau\) maps \(Y_{v,k}\) into the cell \(B\dot{(vw_0)}B\) ([[A2-3]], step 4), which is \(B\) iff \(v=w_0\); so the closed \(\tilde\tau^{-1}(B)\) has underlying set \(|Y_{w_0}|\).
3. **Item 1** (induction on \(\dim Y_x\)). Base \(x=w_0\): step 2. Else take \(\alpha\) as in [[A2-6]], 3, \(x'=s_\alpha xs_\alpha\): \(P_\alpha\overline{Y_{x',k}}\) is closed (proper map from \(P_\alpha\times^B\overline{Y_{x',k}}\)), irreducible, of dimension \(\dim Y_x\), contains \(Y_{x,k}\), so it is \(\overline{Y_{x,k}}\); by [[A2-5]], \(P_\alpha Y_v=Y_v\) (\(\alpha\in\Phi_v\)) or \(Y_v\sqcup Y_{s_\alpha vs_\alpha}\), giving \(\Sigma_x\).
4. **Item 3** (same induction). Base: \(Z_{w_0}\subseteq|Y_{w_0}|\) by step 2. Step: \(P_\alpha Z_{x'}\) is closed and contains \(Y_{x,F}\), hence \(Z_x\); its special fibre is \(P_\alpha\overline{Y_{x',\kappa}}=\overline{Y_{x,\kappa}}\). Equality by step 1.
5. \(\Sigma_v\subseteq\Sigma_x\) for \(v\in\Sigma_x\); item 1 holds with the same \(\Sigma_x\) over \(\overline F\) and \(\overline\kappa\), whatever their characteristics (\(\ne2\)).
SOURCES: sakellaridis-v3.pdf, remark after Axiom 2.2.1, p. 9; conrad-reductive-group-schemes.pdf, Thm. 1.4.12
--- A2-8 · Theorem · Theorem: Axiom 2.2.1 holds for \(\mathcal M_\theta\) over every DVR with \(2\) invertible
STATUS CLAIMED: proved
PARENTS: A2-3, A2-4, A2-5, A2-6, A2-7
STATEMENT:
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), \(X_{\mathcal O}=(G_{\mathbb Z}/G_{\mathbb Z}^{\theta_0})\otimes\mathcal O\).
1. 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 \(\mathbb Z[1/2]\).
2. Sakellaridis's \(Y^{\rm can}\) of \(Y_{x,F}\) is \(Y_x\simeq B/B_x\): smooth, \(B\)-homogeneous fibres, \(\mathcal O\)-point \(y_x\).
3. 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 \(\mathcal O\).
PROOF / BODY:
1. **Item 1:** [[A2-3]] over \(\overline F\) and [[A2-4]].
2. **\(Y_x\subseteq Y^{\rm can}\).** \(Y_x\) 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 \(\overline{\partial_F}\cap Y_x=\emptyset\).
3. **Same points.** \(\partial_F=\bigcup_{v\in\Sigma_x\smallsetminus\{x\}}Y_{v,F}\), \(\overline{\partial_F}=\bigcup Z_v\), and by [[A2-7]] \(Z_v\cap X_\kappa=\bigcup_{\Sigma_v}Y_{u,\kappa}\) with \(\Sigma_v\subseteq\Sigma_x\smallsetminus\{x\}\). So \(Y^{\rm can}\cap X_\kappa=\overline{Y_{x,\kappa}}\smallsetminus\bigcup_{\Sigma_x\smallsetminus\{x\}}Y_{v,\kappa}=Y_{x,\kappa}\).
4. **Schemes.** \(Y_x\) (smooth) and \(Y^{\rm can}\) (open in reduced \(Z_x\)) are reduced locally closed with the same points, hence equal.
5. **\(\alpha\in\Phi_x\):** \(P_\alpha Y_x=Y_x\), quotient \(\operatorname{Spec}\mathcal O\) ([[A2-5]], 1).
6. **\(\alpha\notin\Phi_x\):** with \(u\in\{x,s_\alpha xs_\alpha\}\) the open one, \(u'=s_\alpha us_\alpha\), \(\mathcal P=P_\alpha Y_x\) ([[A2-5]]) 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: [[A2-5]], 2(c).
7. Counts from [[A2-6]], 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.
SOURCES: sakellaridis-v3.pdf, Axiom 2.2.1 and remark, p. 9; S4.json: S4-4 step 4 (open orbit, x = w_L)
--- A2-9 · Audit · Audit of the proof of (A3a): characteristic, \(p=3\), scheme structure, citations
STATUS CLAIMED: proved
PARENTS: A2-8
STATEMENT:
Besides the proved nodes S4-11 (\(\theta_0\) an involution over \(\mathbb Z\), \(\theta_0|_T=-w_L\)) and S4-13 step 2, the proof uses only: Steinberg, Mem. AMS 80, Thms. 7.5, 8.1 (any algebraically closed field); Bruhat decomposition and conjugacy of tori/Borels (any field); Conrad Thm. 5.1.13, Props. 5.1.14, 5.1.16 (any base); CGP A.8.10 / Edixhoven 3.4 (\(2\) invertible); EGA IV 11.3.10, 11.10.10, 17.9.1, 18.12.6. Not used: Knop, Brion, Luna, BLV, [Sa06], Richardson–Springer, Vust, Helminck–Wang, Theorem 8.0.2.
PROOF / BODY:
1. **Transfer.** The count 45, maximal rank and types (S4-5(3), via Knop) are re-proved for char \(\ne2\) ([[A2-3]], [[A2-5]]); torus conjugacy uses only \(\dim\mathfrak g^{\pm\theta}\) and rank-\(\le3\) root systems.
2. **Residue vs. generic characteristic.** Only \(2\in\mathcal O^\times\); [[A2-3]] applies to \(\overline F\) (char \(0\) or \(p\)) and \(\overline\kappa\).
3. **\(\mathcal O\)-schemes.** \(Y^{\rm can}\) is identified as a subscheme ([[A2-8]], 2–4); sets via [[A2-7]], schemes via reducedness.
4. **Types vs. \(\mathcal O\)-isomorphisms.** Quotients are fppf quotients over \(\mathcal O\), equivariant ([[A2-5]]).
5. **Non-reduced stabilizers.** \(B_x\), \((P_\alpha)_y\) smooth over \(\mathcal O\); orbit maps immersions.
6. **\(B(F)\)-orbits, Galois cohomology.** Not used; all 45 orbits are defined over \(\mathbb Z[1/2]\).
7–10. No normalizations; left/right via \(YP_\alpha\leftrightarrow P_\alpha Y\); no "almost every place"; no circularity.
11. **\(p=3\).** \(T^\theta\simeq\mathbb G_m^4\) over \(\mathbb Z\); divisibility of \(\alpha|_{T^{\theta_x}}\) is harmless (only its fppf image is used); all Lie-algebra counts hold for \(p\ne2\).

\(2\in\mathcal O^\times\) is used for smoothness of fixed loci, \(\mathfrak g=\mathfrak g^\theta\oplus\mathfrak g^{-\theta}\), and \(\frac12(1-\theta_n)\). Steinberg's theorem numbers were not checked against a local source; S4-11 step 3 was re-checked (so8_theta.py).
SOURCES: solve-vi brief: checklist items 1-11; conrad-reductive-group-schemes.pdf
--- A2-10 · Summary · Verdict on (Q2): (A3a) holds for \(\mathcal M_\theta\) over every such \(\mathcal O\)
STATUS CLAIMED: proved
PARENTS: A2-8, A2-9
STATEMENT:
For \(\mathcal M_\theta\) and every DVR \(\mathcal O\ni\frac12\): the \(B_F\)-orbits are the \(45\) orbits \(B\dot wx_0\) (\(ww_Lw^{-1}\in\mathcal C\)), all of maximal rank; Sakellaridis's \(Y\) of each is the smooth \(B/B_x\); \(YP_\alpha/R(P_\alpha)\simeq\mathrm{PGL}_2\backslash\mathrm{PGL}_2\) for the \(90\) pairs with \(\alpha\in w(\Phi_L)\) and \(\simeq B_2\backslash\mathrm{PGL}_2=\mathbb P^1_{\mathcal O}\) for the other \(180\); the complement clause holds for all \(270\) pairs.
PROOF / BODY:
1. Proof: [[A2-8]] from [[A2-2]]–[[A2-7]]; audit [[A2-9]].
2. This settles item (I) of S4-13 for every finite \(F/\mathbb Q_p\), \(p\) odd, and supplies (A3a) over \(\mathbb F_q((t))\) for (Q1), (Q3).
3. Not claimed: (A1), (A2), (A3b), (A3c), (A4), (A5) or Theorem 8.0.2 in characteristic \(p\).
SOURCES: sakellaridis-v3.pdf, Axiom 2.2.1, p. 9; S4b.json: S4-13 item (I)

YOUR TASK (adversarial audit).
The claims above assert (Q2): hypothesis (A3a) (Sakellaridis's Axiom 2.2.1, with his scheme structure on Y and the complement clause) holds for the E6/F4 model M_θ over every discrete valuation ring O with 2 ∈ O^×, mixed and equal characteristic, p = 3 included. Re-derive every step and test it against the CHECKLIST, in particular:
 (1) A2-2: the sign data of θ_x on root groups (θ0 trivial on Spin8; no real roots; γ + θ_xγ never a root); T^θ ≅ G_m^4 over Z; C_W(w_L) of order 1152 lifting to N_H(T).
 (2) A2-3: the torus-conjugacy argument (the eigenspace count with a = dim T'_+ and the bound on imaginary roots), that every B-orbit meets N_G(T)x0, and the separation of the 45 orbits through Bruhat cells of τ̃ — in every characteristic ≠ 2, including p = 3, without Knop.
 (3) A2-4: smoothness of B_y over Z[1/2] (fixed points of θ_n on B ∩ θ_nB), the locally closed BB'', the fixed locus 𝓕, flatness of μ and the identification B/B_y ≅ Y_n as schemes over O.
 (4) A2-5: the type-G case (φ_α(SL2) ⊂ G^{θ_x} over Z[1/2]); the type-U case: (P_α)_y = B'_y as O-schemes (the ideal argument), the closed-immersion argument (proper monomorphism into the open V), the quotient 𝓟/R(P_α) ≅ P^1_O as fppf quotient, P_α-equivariantly.
 (5) A2-6: recompute the 45 × 6 table independently (your own script: E6 roots from the Cartan matrix, θ0 = −w_L; types, partners, c_x, dimensions 14–26, the closure sets Σ_x and Bruhat order).
 (6) A2-7, A2-8: the identification of Sakellaridis's O-scheme Y (complement of the closure of the complement of Y_F in its closure) with the orbit scheme Y_x — flatness and reducedness of closures, the special fibre of the closure Z_x, whether P_α Z_{x'} is closed over O, and the complement clause for every pair; whether "complement" is to be read in the closure of YP_α (as the claims do) or in X.
 (7) Every citation (Steinberg Mem. AMS 80 Thms 7.5, 8.1; CGP A.8.10; Edixhoven 3.4; Conrad Thms 1.4.12, 5.1.13, Props 5.1.14, 5.1.16; EGA IV 11.3.10, 11.10.10, 17.9.1, 18.12.6): standing hypotheses, and whether it applies over the base used. The local PDF of Conrad is /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/satake-two-levels/props/sources/conrad-reductive-group-schemes.pdf.
Return nodes of kind check or objection only, one per audited claim or step group, each naming the claim and step audited and the outcome: holds · holds with a repair (give it) · needs hypothesis H · fails at step k (with counterexample or missing lemma). Use ids A6-1, A6-2, …; parents = the audited node ids. End with one verdict node (kind verdict) listing every audited claim with its outcome. Same RULES, timestamps, node format and return schema as above; your files go in scratchpad/solve-vi/A6/.

WORKING DISCIPLINE.
- Never produce one response longer than about 12,000 words, thinking included. Work in short steps with frequent tool calls.
- As soon as a node is finished, write it to its own file /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/75c2c3d9-ef06-4644-b59b-d12583740b0c/scratchpad/solve-vi/A6/nodes/A6-<n>.json (one node per tool call, valid JSON). Keep notes in files.
- At the end, assemble return.json from the node files with a short script, check that it parses, and end with the JSON inside a ```json fence.