VI PAGE BRIEF (root of the sakellaridis-char-p run → page writer for Proposition VI)

You rewrite the Proposition VI page of the satake-two-levels workspace so that it states the split proposition below and proves it, using the audited results of the run sakellaridis-char-p. The page is a proof-concept-graph page (panels G, A, B, C, D). Read first:
- the skill: /Users/miltonlin/Projects/Trees/.claude/skills/proof-concept-graph/SKILL.md and its references/ui-design.md; explain-math-html's conventions are inherited (light palette, KaTeX, no inline font sizes);
- the current page /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/satake-two-levels/props/sakellaridis.html (it is the template: keep its shell, panels, recall chips, style);
- its assets: props/assets/sakellaridis-graph.js, sakellaridis-literature.js, sakellaridis-glossary.js, sakellaridis-sources.js (generated), props/sources/sakellaridis-index.json, props/pdf/sakellaridis.tex (+ preamble.tex), props/assets/prepare_sources.py (run with `uv run --with pymupdf python3 props/assets/prepare_sources.py sakellaridis`);
- the run: /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/sakellaridis-char-p/returns/{A1,A2,A3,A4,A5,A6}.json (A2 = (A3a); A3 = hypotheses over F_q((t)); A4 = close-field transfer; A5 = audit of A3, A4 (repairs A5-3, A5-4, A5-10, A5-11); A6 = audit of A2 (repairs A6-3, A6-5, A6-6); A1 = direct rerun, partial, its audit is in progress). Node pages will exist at ../../sakellaridis-char-p/nodes/<id>.html (relative to props/); link run steps there.

FILES YOU OWN (edit only these): props/sakellaridis.html; props/assets/sakellaridis-graph.js, -literature.js, -glossary.js, -sources.js; props/sources/sakellaridis-index.json and the files prepare_sources.py writes for this slug (sources/sakellaridis-*.png|txt, sources/sakellaridis-manifest.json, sources/draft-sakellaridis-*); props/pdf/sakellaridis.tex and sakellaridis.pdf. In props/assets/glossary.js (shared by all proposition pages) you may add or change only the entries for the labels (A1ₚ), (A3a), (A3b), (A3c) and "Axiom 2.2.1 / 2.4.1 / 2.4.2". Do not edit data.json, any other props page, style.css, or any build script; leave the <nav class="site-nav">…</nav> block exactly as it is (a build script rewrites it), and do not add a sub-tab row (the root adds it). A new local source is provided: props/sources/sakellaridis-2008-v2.pdf = Sakellaridis, On the unramified spectrum of spherical varieties over p-adic fields, arXiv:math/0606130v2 (Compositio 2008) — §1.7 on PDF p. 7 and §4.1 on p. 17 say "local non-archimedean field in characteristic zero" with no unramified assumption; Theorem 5.2.1 is on p. 31.

RELABELLING (use everywhere on the page, in the glossary and in the PDF):
- (A1) Field and group: characteristic-0 field (Sakellaridis §1.1), G split with connected fibres and simply connected derived group (§1.7). Unchanged meaning.
- (A1ₚ) the same with F = F_q((t)) (characteristic p); not in the paper.
- (A2) Integral model (§1.7). Unchanged.
- (A3a) Rank-one models over 𝒪 = Sakellaridis's Axiom 2.2.1 (p. 9), with his scheme structure on Y and the complement clause (read inside closures: A6-7 shows the X-reading is unsatisfiable).
- (A3b) Cartan decomposition = Axiom 2.4.1 (p. 10).  (A3c) Iwahori orbits = Axiom 2.4.2 (p. 11).
- (A4), (A5) unchanged.  The old "(I)" and "Axiom 2.2.1 over O" become "(A3a)". Keep anchors: id="hyp-a3" must still exist (put it on the (A3a) item or an empty span before it); new ids hyp-a1p, hyp-a3a, hyp-a3b, hyp-a3c.
- Hypotheses of the characteristic-p transfer, (T1)–(T5) (A4-9, audited A5-7), for a symmetric X = G/G^θ over Λ = W(F_q), p odd, θ stabilizing a split maximal torus T ⊇ S (S maximal θ-split, split over Λ), H = G^θ, X = G/H, x0 = eH, Λ_S^+ = X_*(S) ∩ Λ_T^+, test fields F = F_q((t)) and F'_e = Q_q(p^{1/e}):
  (T1) H has connected fibres and is split over Λ; (T2) X(E) = G(E)x0 for E = F and E = F'_e, e ≫ 0; (T3) X(O_E) = {y ∈ X(E) : τ(y) ∈ G(O_E)}, τ(gx0) = gθ(g)^{-1}; (T4) X(F'_e) = ⊔_{κ∈Λ_S^+} K x_κ for e ≫ 0; (T5) Theorem 8.0.2 holds over F'_e for e ≫ 0 (e.g. (A1)–(A5) over O'_e, with ψ' of conductor O'_e), with Sakellaridis's labelling independent of e.

THE PROPOSITION (state it in this form; p odd throughout; F a non-archimedean local field with residue field F_q):
 (a) VIa · characteristic 0 (Sakellaridis, Theorem 8.0.2, Corollary 8.0.4). Under (A1)–(A5): C_K ≅ S := C[δ(X)^{1/2}A_X^*]^{W_X} as H_K-modules (f acts by r(Sf)), 1_{X(O)} ↦ 1; hence (SG) ⟺ r onto. Status: proved, cited. For F ramified over Q_p: the paper's "unramified" simplification only fixes (ψ, dx), which §1.7 already fixes with conductor O (A4-7, audited A5-5), and the inputs it takes from [Sa08] are proved there for every local field of characteristic 0 (Sa08 §1.7, §4.1; checked by the root).
 (b) VIb · characteristic p, the form the verdict needs. If F = F_q((t)) and X = G/G^θ is symmetric over W(F_q) with (T1)–(T5), then K\X(F) ≅ Λ_S^+ and the same conclusion holds: C_K ≅ S with 1_{X(O)} ↦ 1, the images P_κ of the orbit indicators being those over F'_e (they depend only on q). Status: proved by the run (A4-9 with A4-5, A4-6, A4-8; audited A5-3, A5-4, A5-6, A5-7; repairs included).
     Proof idea, in four steps: F and F'_e are e-close (O/ϖ^m ≅ O'/ϖ'^m for m ≤ e); membership in Cartan and Iwasawa cells is read from valuations of matrix coefficients in the fundamental Weyl modules, so every count used is decided modulo ϖ^D with D independent of the field (A4-2); counting K-orbits in K t^λ X(O) transfers the Cartan decomposition to F (A4-5); every structure constant (T_λ * 1_{Kx_μ})(x_ν) and every Satake coefficient of T_λ agrees with its value over F'_e for e ≥ D(λ, μ) (A4-6, with the ν-bound of A5-4); so 1_{Kx_κ} ↦ P_κ is H_K-equivariant over F (A4-8).
     One further sentence: a second route reruns Sakellaridis's proof directly over F_q((t)) (A1); it needs (A2), (A5), (A3b), (A3c) and Statements 6.3.1, 7.1.5 read in characteristic p, and not (A3a); for E6/F4 its one open input, Statement 6.3.1 in characteristic p, follows from Statement 6.3.1 over ramified p-adic fields by A1's level lemma (A1-8) now that (A3a) holds there. Mark this sentence "partial · audit in progress" (the root will tell you if the status changes).
 (c) VIc · E6/F4 satisfies the hypotheses. For the model M_θ and every discrete valuation ring O with 2 ∈ O^× (every complete DVR of odd residue characteristic, mixed or equal characteristic, p = 3 included): (A3a) holds (A2-8, audited A6-9: the 45 Borel orbits B ẇ x0 are all the orbits and all of maximal rank in every characteristic ≠ 2; Sakellaridis's O-scheme of each is the smooth orbit scheme B/B_x with B_x = (B ∩ θ_x B)^{θ_x}; YP_α/R(P_α) ≅ PGL2\PGL2 for the 90 pairs with α ∈ w(Φ_L) and ≅ B2\PGL2 = P^1_O for the other 180; the complement clause for all 270 pairs); (A1)–(A5) over p-adic O (S4, S4b) and (A1ₚ), (A2), (A3b), (A3c), (A4)(i),(ii),(iv), (A5) over F_q[[t]] (A3, audited A5-8…A5-12); (T1)–(T5) hold for M_θ (A4-9's reading: T1 by S4-4/A4-4, T2 by A4-5 (E3), T3 by A4-3, T4 by S4-12/S4-13, T5 by A4-7 with (A3a)).
 (d) VId · the criterion for E6/F4. For M_θ, p odd, over F_q((t)) and over every p-adic F (any ramification): C_K ≅ S with φ0 ↦ 1, so (SG) ⟺ r onto. Status: proved (from (a)–(c)). With Proposition VII (r is not onto) this gives that (SG) fails and C_K = H_Kφ0 ⊕ C·1_{Kx_{3ω1∨}}; say it in one sentence and link e6-f4-module.html.

PROOF STRUCTURE on the page (keep the existing step style: role label, short statement, a "How" disclosure with the argument, run nodes linked, audits linked):
 - Part (a): the existing steps 1–7 (Sakellaridis's proof), unchanged except the relabelling, and one line in step 1 or 2 recording that [Sa08] covers every characteristic-0 local field.
 - Part (b), new steps: truncated rings and finite determinacy (A4-2, A5-1); the Cartan decomposition over F_q((t)) by counting (A4-5, A5-3; also proved directly from Delorme–Sécherre in characteristic p, A3-4, A3-5, A3-8, A5-9); structure constants and Satake coefficients agree over close fields (A4-6, A5-4); Theorem 8.0.2 over ramified F'_e (A4-7, A5-5, the [Sa08] check); the characteristic-p theorem (A4-8, A4-9, A5-6, A5-7).
 - Part (c): the existing E6/F4 steps (spherical roots and colours; the model and A1, A2, A5; Iwahori orbits, distinct labels, types; Cartan decomposition), each now covering both characteristics (cite A3's nodes for F_q((t)) next to S4's), and the old "one hypothesis still open" step replaced by a Result step "(A3a) holds over every O with 2 invertible" with A2's proof in five lines (θ-stable tori: every θ-stable maximal torus is H-conjugate to T by the eigenspace count 52 = a + (r+c)/2 + i_c, 26 = (6−a) + (r+c)/2 + i_n; the 45 orbits B ẇ x0 and their smooth stabilizers over Z[1/2]; the orbit immersions through τ; the rank-one quotients; closures over O from the closed orbit x = w0).
 - Part (d): the application to E6/F4, now unconditional in both characteristics.
 Keep the number of steps readable; renumber freely (no other page links to #step-N).

GRAPH (panel A, sakellaridis-graph.js): add one node per new step with role, status (proved), statement, argument and joint premises; turn the old open node into a proved Result node; edges MOTIVATES / LEADS_TO / CHALLENGES with justifications; no open node should remain for E6/F4. LITERATURE (panel D): up to three located extracts per new node from local sources (sakellaridis-v3.pdf pp. 2, 7, 9, 23, 32, 64; sakellaridis-2008-v2.pdf pp. 7, 17, 31; conrad-reductive-group-schemes.pdf for Thm 1.4.12, Thm 5.1.13, Props 5.1.14, 5.1.16), or record a shortfall; project drafts and run nodes locate the target and do not count as literature. PDF (panel C and the "as a paper" link): update pdf/sakellaridis.tex to the new statement and proof outline, compile with tectonic, delete the cached draft-sakellaridis-*.png|txt, re-run prepare_sources.py.

TEXT STYLE: plain, short sentences; every symbol defined or recallable; hypothesis labels with their recall chips; statuses stated as "proved · cited", "proved · this run, audited". No "Axiom 2.2.1 over O" left without its new label (A3a).

VERIFY: KaTeX renders without errors (headless Chrome screenshot of G at desktop and phone width; search the rendered DOM for .katex-error), the graph and the literature reader work, every link resolves (../../sakellaridis-char-p/nodes/… pages exist), the PDF compiles. Report back in under 300 words: what changed, the step list, any shortfall, and anything you could not do.