{"agent": "S4", "family": "Sakellaridis hypotheses for E6/F4: Cartan decomposition, integral involution", "nodes": [{"id": "S4-10", "t": "2026-10-17T21:53:35Z", "kind": "check", "status": "proved", "label": "Audit", "name": "Part 2 standing assumption added", "title": "Part 2 assumes no type-\\(N\\) reflections; this holds for \\(E_6/F_4\\)", "summary": "Adds to the list of [[S4-1]] the assumption that opens Part 2 of Sakellaridis's paper, under which Theorem 6.3.2 and everything after it are proved: no pair (maximal-rank Borel orbit, simple root) of type \\(N\\), with only unramified parameters considered. It is stronger in form than 'no spherical root of type \\(N\\)' and holds for \\(E_6/F_4\\).", "statement": "**(S0) Opening of Part 2, §6 (p. 41).** \"From now on we assume that \\(X\\) is a spherical variety without type-\\(N\\) reflections, i.e. such that for no orbit \\(Y\\) of maximal rank and simple root \\(\\alpha\\) the pair \\((Y,\\alpha)\\) is of type \\(N\\). Moreover, the character \\(\\delta(X)^{-1/2}\\tilde\\chi\\) of \\(A_X\\) is assumed to be unramified.\"\n\nCorrection to [[S4-1]] and [[S4-2]]: (S0) is a standing hypothesis of Statement 6.3.1, Theorem 6.3.2, Theorem 7.2.1 and §8, alongside (S7) (\"no spherical root of type \\(N\\), \\(L_\\gamma\\) classical\", p. 58). In the pages' list it belongs to the requirement on spherical roots and colours, (A4) of the classical page, which the pages state only as \"no spherical root of type \\(N\\)\". The second sentence restricts the parameters \\(\\tilde\\chi\\) to unramified ones; Theorem 8.0.2 uses only those, so it imposes nothing on \\(X\\). Also, for affine \\(X\\) every colour is relevant (Lemma 7.1.1, p. 58).\n\n**For \\(E_6/F_4\\).** (S0) holds: no pair \\((Y,\\alpha)\\) with \\(Y\\) of maximal rank has type \\(T\\) or \\(N\\) ([[S4-5]], part 3). Both colours are relevant, consistent with Lemma 7.1.1: \\(D_1\\) belongs to \\(\\gamma_1\\) and \\(D_6\\) to \\(\\gamma_6\\) ([[S4-3]]).", "parents": ["S4-1", "S4-2", "S4-3", "S4-5"], "corrects": ["S4-1", "S4-2"], "body": "1. **Text.** The quoted sentences open Part 2 on p. 41, before §6.1; everything in §§6–8 is under them. Statement 6.3.1's types are \\(G\\), \\(T\\) split, \\(T\\) non-split and \\((U,\\psi)\\) only, which reflects this assumption.\n2. **Strength.** \"No spherical root of type \\(N\\)\" concerns only the simple spherical reflections \\(w_\\gamma\\) and their Brion paths. (S0) concerns every maximal-rank orbit and every simple root, so it is stronger in form.\n3. **Check for \\(E_6/F_4\\).** By [[S4-5]](3), for each of the \\(45\\) maximal-rank orbits \\(Y\\) and each simple \\(\\alpha\\) a pair of type \\(T\\) or \\(N\\) would force a root into \\(w(\\mathbb Q\\omega_1\\oplus\\mathbb Q\\omega_6)\\), and \\(E_6\\) has none. This uses Knop's transport of weight lattices along his \\(W\\)-action (cited in [[S4-5]], not checked). Hence (S0) holds.\n4. **Unramified parameters.** In the proof of Theorem 8.0.2 (p. 64), \\(P_{\\check\\lambda}\\) is a polynomial in the unramified \\(\\chi\\) on \\(\\delta(X)^{1/2}A_X^*\\), so only unramified \\(\\delta(X)^{-1/2}\\tilde\\chi\\) occur.\n5. **Lemma 7.1.1.** It is stated for affine \\(X\\) and proved by a case analysis (sketch, p. 58). For \\(E_6/F_4\\), [[S4-3]] verifies directly that each of the two colours belongs to a spherical root, so no appeal to the lemma is needed.", "concept": "bridge", "key": false, "sources": ["sakellaridis-v3.pdf, opening of Part 2, §6, p. 41; §7 p. 58; Lemma 7.1.1 p. 58"]}, {"id": "S4-11", "t": "2026-10-17T21:53:56Z", "kind": "lemma", "status": "proved", "label": "Proposition", "name": "An E IV involution defined over Z", "title": "An involution of type E IV of \\(E_6\\) defined over \\(\\mathbb Z\\)", "summary": "Settles the existence of the integral model: the Chevalley involution composed with the Tits lift of the longest element of the black \\(D_4\\) is an involution of the split simply connected \\(E_6\\) over \\(\\mathbb Z\\), acting by \\(-w_L\\) on the torus, trivially on \\(\\mathrm{Spin}_8\\), with 52-dimensional fixed group \\(F_4\\) on every fibre of characteristic \\(\\ne2\\), and with the standard \\(T\\subseteq B\\subseteq P\\) adapted to it.", "statement": "Let \\(G_{\\mathbb Z}\\) be the split simply connected Chevalley group of type \\(E_6\\) with pinning \\((T,B,\\{x_{\\alpha_i}\\})\\) (Bourbaki labels). Let \\(C\\) be the Chevalley involution: \\(C(t)=t^{-1}\\) on \\(T\\) and \\(C(x_{\\pm\\alpha_i}(c))=x_{\\mp\\alpha_i}(-c)\\). Put \\(\\sigma_\\alpha=x_\\alpha(1)x_{-\\alpha}(-1)x_\\alpha(1)\\), let \\(\\sigma_L\\) be the Tits lift of \\(w_L\\) (the longest element of \\(W(D_4)\\), black nodes \\(2,\\dots,5\\)), and set\n$$\\theta_0:=\\operatorname{Int}(\\sigma_L)\\circ C .$$\nThen:\n1. \\(\\theta_0\\) is an automorphism of \\(G_{\\mathbb Z}\\) with \\(\\theta_0^2=\\mathrm{id}\\) and \\(\\theta_0|_T=-w_L\\).\n2. \\(\\theta_0\\) is the identity on the derived group \\(\\mathrm{Spin}_8\\) of the black Levi \\(L\\).\n3. On every geometric fibre of characteristic \\(\\ne2\\), \\(\\theta_0\\) is an outer involution with \\(\\dim\\mathfrak g^{\\theta_0}=52\\); hence it has type E IV and \\(G^{\\theta_0}\\) is connected of type \\(F_4\\).\n4. \\(\\theta_0(U_P)=U_P^-\\) for the standard parabolic \\(P\\supseteq B\\) of \\(\\{2,3,4,5\\}\\), so \\(P\\) is minimal \\(\\theta_0\\)-split with \\(\\theta_0\\)-stable Levi \\(L=Z_G(S)\\), \\(S=(T^{-\\theta_0})^\\circ\\) of rank \\(2\\).\n\nSo the model \\(\\mathcal M_\\theta\\) of [[S4-4]] exists, with \\(\\theta=\\theta_0\\) defined over \\(\\mathbb Z\\); its existence is no longer a cited input.", "parents": ["S4-4"], "corrects": ["S4-4"], "body": "1. **\\(C\\) and the Tits group.** \\(C\\) exists over \\(\\mathbb Z\\) by the isomorphism theorem for pinned split reductive group schemes (SGA 3, Exp. XXIII; cited): it realizes \\(-1\\) on \\(\\mathcal X(T)\\) with the pinning \\(\\{-X_{-\\alpha_i}\\}\\). In \\(\\mathrm{SL}_2\\), \\(\\sigma=x(1)y(-1)x(1)=\\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix}\\) and \\(C(\\sigma)=y(-1)x(1)y(-1)=\\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix}=\\sigma\\). Hence \\(C\\) fixes every \\(\\sigma_{\\alpha_i}\\), hence \\(\\sigma_L\\); and \\(C^2=\\mathrm{id}\\), being trivial on \\(T\\) and on the simple root groups.\n2. **Order.** \\(C\\circ\\operatorname{Int}(\\sigma_L)=\\operatorname{Int}(C(\\sigma_L))\\circ C=\\operatorname{Int}(\\sigma_L)\\circ C\\), so \\(\\theta_0^2=\\operatorname{Int}(\\sigma_L^2)\\). On \\(T\\), \\(\\theta_0=w_L\\circ(-1)=-w_L\\).\n3. **\\(\\theta_0\\) on \\(\\mathrm{Spin}_8\\)** (proves 2). Restricted to \\(\\mathrm{Spin}_8\\), \\(\\theta_0=\\operatorname{Int}(\\sigma_{w_0})\\circ C\\) for the induced pinning of \\(D_4\\). In the split model \\(\\mathfrak{so}_8\\) with \\(f_i=e_i^{\\mathsf T}\\) and \\(C(X)=-X^{\\mathsf T}\\), exact integer computation along the reduced word \\(s_1s_2s_1s_3s_2s_1s_4s_2s_1s_3s_2s_4\\) (\\(D_4\\) labels, centre \\(2\\)) gives \\(\\operatorname{Ad}(\\sigma_{w_0})(C(e_i))=e_i\\) and \\(\\operatorname{Ad}(\\sigma_{w_0})(C(f_i))=f_i\\) for all \\(i\\) (c7_theta0.py). So \\(\\theta_0(x_{\\pm\\alpha}(c))=x_{\\pm\\alpha}(c)\\) for the black simple roots, over \\(\\mathbb Z\\), and \\(\\theta_0\\) is trivial on \\(\\mathrm{Spin}_8\\).\n4. **\\(\\sigma_L^2=1\\).** By step 3, \\(\\operatorname{Int}(\\sigma_L^2)=\\theta_0^2\\) is trivial on \\(\\mathrm{Spin}_8\\), so \\(\\sigma_L^2\\in Z(\\mathrm{Spin}_8)\\simeq\\mu_2\\times\\mu_2\\). Its image in \\(\\mathrm{SO}_8\\) (the vector representation) is \\(I\\) (same computation). Triality \\(\\tau\\) preserves the \\(D_4\\) pinning and \\(w_0\\), so it fixes the Tits lift \\(\\sigma_{w_0}\\) (Tits lifts do not depend on the reduced word). Hence \\(\\sigma_L^2\\) also acts trivially in the two half-spin representations (vector \\(\\circ\\,\\tau^{\\pm1}\\)). The three kernels meet trivially in \\(Z(\\mathrm{Spin}_8)\\), so \\(\\sigma_L^2=1\\) and \\(\\theta_0^2=\\mathrm{id}\\). (This agrees with Tits's formula \\(\\sigma_{w_0}^2=(2\\rho^\\vee)(-1)\\) and \\(\\rho_L^\\vee=3\\alpha_2^\\vee+3\\alpha_3^\\vee+5\\alpha_4^\\vee+3\\alpha_5^\\vee\\in\\Lambda_T\\).)\n5. **Type** (proves 3). \\(\\theta_0\\) acts on \\(\\mathcal X(T)\\) by \\(-w_L\\notin W(E_6)\\) (as \\(-1\\notin W(E_6)\\)), so it is outer. \\(\\dim\\mathfrak g^{\\theta_0}\\) is the sum of: \\(4\\) from \\(\\mathfrak t\\) (the \\(+1\\)-eigenspace of \\(-w_L\\) is the black span); \\(24\\) from the \\(48\\) roots outside \\(L\\), which \\(-w_L\\) pairs without fixed points (\\(-w_L\\alpha=\\alpha\\) iff \\(\\alpha\\) lies in the black span), giving one fixed line per pair; and \\(24\\) from the root spaces of \\(L\\), fixed by step 3. Total \\(52\\). The outer involutions of \\(E_6\\) have fixed algebras of type \\(C_4\\) (dimension \\(36\\)) or \\(F_4\\) (dimension \\(52\\)), so \\(\\theta_0\\) has type E IV. On a fibre of characteristic \\(p\\ne2\\), \\(G^{\\theta_0}\\) is smooth, connected (Steinberg, Mem. AMS 80, Thm 8.1) and reductive of rank \\(4\\) and dimension \\(52\\), hence of type \\(F_4\\).\n6. **Parabolic** (proves 4). For \\(\\alpha\\in\\Phi^+\\smallsetminus\\Phi_L\\), \\(w_L\\) does not change the \\(\\alpha_1\\)- and \\(\\alpha_6\\)-coefficients, so \\(\\theta_0\\alpha=-w_L\\alpha\\) has them negated, one of them non-zero, and is negative; hence \\(\\theta_0(U_P)=U_P^-\\) and \\(P\\cap\\theta_0(P)=L\\). \\(T^{-\\theta_0}\\) has cocharacters the \\(+1\\)-eigenspace of \\(w_L\\), of rank \\(2\\), the rank of E IV.", "concept": "e6f4-functions", "key": true, "sources": ["SGA 3 Exp. XXIII (isomorphism theorem for pinned groups; cited)", "Steinberg, Mem. AMS 80 (1968), Thm 8.1 (cited)", "families/e6-f4.html §§1–4"]}, {"id": "S4-12", "t": "2026-10-17T21:54:58Z", "kind": "lemma", "status": "proved", "label": "Proposition", "name": "E6/F4: G(F) = K T(F) H(F) for p odd", "title": "\\(E_6/F_4\\), \\(p\\) odd: \\(G(F)=K\\,T(F)\\,H(F)\\) by Delorme–Sécherre", "summary": "States Delorme–Sécherre's Cartan decomposition for split \\(p\\)-adic symmetric spaces exactly, checks each hypothesis for the model with \\(\\theta_0\\), and shows that its index set (classes of maximal split \\(\\theta\\)-anti-invariant tori under \\(H(F)\\)) has one element because the centralizer \\(\\mathrm{Spin}_8\\) has trivial Galois cohomology.", "statement": "**Theorem (Delorme–Sécherre, arXiv:math/0612545v1, Theorem 4.8 = Theorem 0.4; cited, statement read from the arXiv PDF).** Let \\(k\\) be a non-archimedean locally compact field of residue characteristic \\(\\ne2\\), \\(G\\) a connected reductive \\(k\\)-split group, \\(\\sigma\\) an involutive \\(k\\)-automorphism, \\(H\\) an open \\(k\\)-subgroup of \\(G^\\sigma\\). Let \\(A\\) be a maximal \\((\\sigma,k)\\)-split torus (a \\(k\\)-split torus on which \\(\\sigma\\) acts by inversion, maximal for this property), \\(S\\) a \\(\\sigma\\)-stable maximal \\(k\\)-split torus containing \\(A\\), \\(\\{A_j\\}_{j\\in J}\\) representatives of the \\(H_k\\)-conjugacy classes of maximal \\((\\sigma,k)\\)-split tori, \\(y_j\\in G_k\\) with \\(y_jAy_j^{-1}=A_j\\), and \\(K\\) the stabilizer in \\(G_k\\) of a special point \\(x\\) of the building. Then\n$$G_k=\\bigcup_{j\\in J}H_k\\,y_j\\,S_k\\,K .$$\n(The proof, p. 25 of the arXiv version, uses that \\(x\\) lies in the apartment of \\(S\\), via \\(N_kK=S_kK\\) and Bruhat–Tits property 1.13(2); this is the setting of their §4.1, and it is how the theorem is applied here.)\n\n**Application.** Let \\(F\\) be \\(p\\)-adic, \\(p\\ne2\\), \\(\\theta=\\theta_0\\) as in [[S4-11]], \\(H=G^\\theta=F_4\\), \\(K=G(\\mathcal O)\\). Then \\(|J|=1\\), and\n$$G(F)=K\\,T(F)\\,H(F),\\qquad X(F)=K\\cdot T(F)x_0 .$$", "parents": ["S4-11", "S4-4", "A6-9"], "corrects": [], "body": "**Hypotheses.**\n1. \\(k=F\\) is locally compact non-archimedean with odd residue characteristic. \\(G=E_6\\) is \\(F\\)-split and connected. \\(\\sigma=\\theta_0\\) is an involutive \\(F\\)-automorphism ([[S4-11]]). \\(H=G^{\\theta_0}\\) is all of \\(G^\\sigma\\) and connected (Steinberg), so it is open in it.\n2. \\(A:=S\\), the split torus with cocharacters \\(\\Lambda_S\\). It is \\((\\sigma,F)\\)-split. It is maximal because every \\(\\theta\\)-split torus has dimension at most the rank \\(2\\) of E IV, which \\(S\\) attains.\n3. \\(S_{\\mathrm{DS}}:=T\\), the split maximal torus of the Chevalley model. It is \\(\\theta_0\\)-stable and contains \\(S\\).\n4. \\(x:=\\) the hyperspecial vertex fixed by \\(G(\\mathcal O)\\). It lies in the apartment of \\(T\\) because \\(T\\) is split over \\(\\mathcal O\\). Since \\(G\\) is simply connected, its stabilizer is the parahoric \\(G(\\mathcal O)=K\\) (Bruhat–Tits; cited).\n\n**\\(|J|=1\\).** Let \\(A'\\) be any maximal \\((\\theta,F)\\)-split torus.\n5. By Helminck–Wang, Adv. Math. 99 (1993), Prop. 10.3 (quoted by Delorme–Sécherre §3.4; cited), \\(A'\\) is \\(G(F)\\)-conjugate to \\(S\\), so \\(\\dim A'=2\\) and \\(A'_{\\bar F}\\) is a maximal \\(\\theta\\)-split torus.\n6. Over \\(\\bar F\\) (characteristic \\(0\\)), maximal \\(\\theta\\)-split tori are conjugate under \\((G^\\theta)^\\circ=H\\) (Vust 1974; Richardson, Invent. Math. 66 (1982); cited, numbering not checked). So \\(A'=hSh^{-1}\\) with \\(h\\in H(\\bar F)\\).\n7. For \\(\\tau\\in\\operatorname{Gal}(\\bar F/F)\\), \\(\\operatorname{Int}(\\tau(h))|_S=\\tau\\circ\\operatorname{Int}(h)|_S\\circ\\tau^{-1}\\). Both tori are split, so Galois acts trivially on \\(X_*(S)\\) and \\(X_*(A')\\), and \\(\\operatorname{Int}(\\tau(h))|_S=\\operatorname{Int}(h)|_S\\). Hence \\(c_\\tau:=h^{-1}\\tau(h)\\in Z_H(S)(\\bar F)\\), a continuous \\(1\\)-cocycle.\n8. \\(Z_H(S)=Z_G(S)\\cap H=L\\cap H=L^\\theta=\\mathrm{Spin}_8\\) (family page §4: \\(L^\\theta=S[2]\\cdot\\mathrm{Spin}_8=\\mathrm{Spin}_8\\)), which is simply connected, so \\(H^1(F,\\mathrm{Spin}_8)=1\\) (Kneser; cited). Write \\(c_\\tau=z\\tau(z)^{-1}\\) with \\(z\\in\\mathrm{Spin}_8(\\bar F)\\). Then \\(h':=hz\\) satisfies \\(\\tau(h')=h'\\), so \\(h'\\in H(F)\\), and \\(h'Sh'^{-1}=hSh^{-1}=A'\\). So \\(J=\\{1\\}\\) with \\(y_1=1\\).\n\n**Conclusion.**\n9. The theorem gives \\(G(F)=H(F)\\,T(F)\\,K\\); inverting, \\(G(F)=K\\,T(F)\\,H(F)\\).\n10. \\(X(F)=G(F)x_0\\), because \\(\\ker(H^1(F,F_4)\\to H^1(F,E_6))\\) is trivial (\\(H^1(F,F_4)=1\\), Kneser). Hence \\(X(F)=K\\,T(F)\\,x_0\\).\n\n**Field and model:** \\(F\\) \\(p\\)-adic with \\(p\\) odd; the model \\(\\mathcal M_\\theta\\) with \\(\\theta=\\theta_0\\) (any \\(\\mathcal O\\)-involution of type E IV that preserves \\(K\\) and the split torus \\(T\\) works the same way).", "concept": "e6f4-functions", "key": true, "sources": ["Delorme–Sécherre, arXiv:math/0612545v1, Theorem 0.4 = Theorem 4.8, §3.4, §4.1, Proposition 4.4 (read from the arXiv PDF); published in Pacific J. Math. 251 (2011)", "Helminck–Wang, Adv. Math. 99 (1993), Prop. 10.3 (via Delorme–Sécherre §3.4; not checked directly)", "Kneser (H^1 of simply connected groups over p-adic fields); Vust 1974 / Richardson 1982 (conjugacy of maximal θ-split tori); cited", "families/e6-f4.html §4"]}, {"id": "S4-13", "t": "2026-10-17T21:55:33Z", "kind": "lemma", "status": "proved", "label": "Proposition", "name": "E6/F4: Axiom 2.4.1 holds; only 2.2.1 left", "title": "\\(E_6/F_4\\), \\(p\\) odd: Axiom 2.4.1 holds; only Axiom 2.2.1 over \\(\\mathcal O\\) open", "summary": "Replaces [[S4-6]]: the existence half of Axiom 2.4.1 holds for \\(p\\) odd. Delorme–Sécherre's decomposition puts every \\(K\\)-orbit into \\(T(F)x_0\\), and \\(W_X\\) is realized inside \\(H(\\mathcal O)\\cap N_G(T)\\) by products of two commuting Tits lifts, which moves every label into the dominant chamber. The only hypothesis of Sakellaridis's theorem still open for \\(E_6/F_4\\) is Axiom 2.2.1 as \\(\\mathcal O\\)-schemes.", "statement": "Model \\(\\mathcal M_\\theta\\) with \\(\\theta=\\theta_0\\) ([[S4-11]]), \\(F\\) \\(p\\)-adic, \\(p\\ne2\\).\n1. **\\(W_X\\subseteq K\\).** For \\(i\\in\\{1,6\\}\\) let \\(\\beta_i:=w_L\\alpha_i=-\\theta\\alpha_i\\) (\\(\\beta_1=\\alpha_1+\\alpha_2+2\\alpha_3+2\\alpha_4+\\alpha_5\\)) and\n$$\\dot n_i:=\\sigma_{\\alpha_i}\\,\\theta(\\sigma_{\\alpha_i})\\in G(\\mathbb Z).$$\nThen \\(\\dot n_i\\in H(\\mathcal O)\\cap N_G(T)\\), and on labels \\(\\varphi(\\mu)\\in\\Lambda_S\\) it acts by the simple reflection \\(s_{\\gamma_i}\\) of \\(W_X\\): \\(\\dot n_i\\,t^\\mu x_0=t^{s_{\\alpha_i}s_{\\beta_i}\\mu}x_0\\) with \\(\\varphi(s_{\\alpha_i}s_{\\beta_i}\\mu)=s_{\\gamma_i}\\varphi(\\mu)\\).\n2. **Axiom 2.4.1 holds.** \\(X(F)=\\bigsqcup_{\\kappa\\in\\Lambda_S^+}Kx_\\kappa\\). So \\(A_X^+x_0\\) contains a complete set of representatives of \\(K\\backslash X(F)\\), distinct labels give distinct orbits, and \\(K\\backslash X(F)\\simeq\\Lambda_S^+\\).\n3. **What remains.** With [[S4-3]], [[S4-4]], [[S4-5]], [[S4-10]], [[S4-11]] and [[S4-12]], every hypothesis of Theorem 8.0.2 holds for \\(\\mathcal M_\\theta\\), \\(p\\ne2\\), except possibly\n**(I) Axiom 2.2.1 over \\(\\mathcal O\\) (p. 9):** for each of the \\(45\\) Borel orbits \\(Y_F\\) of maximal rank (with \\(Y\\) its \\(\\mathcal O\\)-scheme \"complement of the closure of the complement of \\(Y_F\\) in its closure\") and each simple root \\(\\alpha\\), \\(YP_\\alpha/R(P_\\alpha)\\simeq SU\\backslash\\mathrm{PGL}_2\\) (\\(S\\subseteq N(U)\\)) when \\((Y_F,\\alpha)\\) has type \\(U\\), and \\(\\simeq\\mathrm{PGL}_2\\backslash\\mathrm{PGL}_2\\) when it has type \\(G\\), as \\(\\mathcal O\\)-schemes; and the complement of \\(YP_\\alpha\\) is the closure of the complement of \\((YP_\\alpha)_F\\). (Types \\(T,N\\) do not occur: [[S4-5]](3).)", "parents": ["S4-12", "S4-11", "S4-5", "S4-4", "A6-4"], "corrects": ["S4-6"], "body": "**Proof of 1.**\n1. \\(\\beta_i\\) is a root, \\((\\alpha_i,\\beta_i)=0\\), and neither \\(\\alpha_i+\\beta_i=\\gamma_i\\) nor \\(\\alpha_i-\\beta_i\\) is a root (c8_wx_in_K.py; for \\(i=1\\), \\(\\alpha_1-\\beta_1=-(\\alpha_2+2\\alpha_3+2\\alpha_4+\\alpha_5)\\) has coefficient \\(2\\) at the outer node \\(3\\) of the black \\(D_4\\)). So the root \\(\\mathrm{SL}_2\\)'s of \\(\\alpha_i\\) and \\(\\beta_i\\) commute.\n2. \\(\\theta(\\sigma_{\\alpha_i})\\) lies in \\(\\theta(\\mathrm{SL}_{2,\\alpha_i})=\\mathrm{SL}_{2,\\theta\\alpha_i}=\\mathrm{SL}_{2,\\beta_i}\\), normalizes \\(T\\), and lifts \\(s_{\\beta_i}\\). It commutes with \\(\\sigma_{\\alpha_i}\\), so \\(\\theta(\\dot n_i)=\\theta(\\sigma_{\\alpha_i})\\,\\sigma_{\\alpha_i}=\\dot n_i\\) (using \\(\\theta^2=\\mathrm{id}\\)). Hence \\(\\dot n_i\\in H\\); since \\(\\theta=\\theta_0\\) is defined over \\(\\mathbb Z\\), \\(\\dot n_i\\in G(\\mathbb Z)\\subseteq K\\).\n3. \\(\\dot n_i\\) lifts \\(w:=s_{\\alpha_i}s_{\\beta_i}\\). For \\(\\lambda\\in\\Lambda_S\\): \\(\\langle\\beta_i,\\lambda\\rangle=\\langle-\\theta\\alpha_i,\\lambda\\rangle=-\\langle\\alpha_i,\\theta\\lambda\\rangle=\\langle\\alpha_i,\\lambda\\rangle\\) and \\(\\beta_i^\\vee=-\\theta\\alpha_i^\\vee\\), so \\(w\\lambda=\\lambda-\\langle\\alpha_i,\\lambda\\rangle(\\alpha_i^\\vee-\\theta\\alpha_i^\\vee)=\\lambda-\\langle\\alpha_i,\\lambda\\rangle e_i\\): the reflection \\(s_{\\gamma_i}\\), i.e. \\((a,b)\\mapsto(-a,a+b)\\) for \\(i=1\\) and \\((a,b)\\mapsto(a+b,-b)\\) for \\(i=6\\) ([[A6-4]](3); checked on sample vectors in c8_wx_in_K.py). \\(w\\) commutes with \\(\\theta\\) on \\(\\Lambda_T\\) (it is a product of reflections in a \\(\\theta\\)-stable pair), hence with \\(\\varphi=1-\\theta\\).\n4. \\(\\dot n_it^\\mu x_0=(\\dot n_it^\\mu\\dot n_i^{-1})\\,\\dot n_ix_0=t^{w\\mu}x_0\\), since \\(\\dot n_i\\in H\\).\n\n**Proof of 2.**\n5. By [[S4-12]], every \\(x\\in X(F)\\) is \\(K\\)-conjugate to some \\(t\\,t^\\mu x_0\\) with \\(t\\in T(\\mathcal O)\\), \\(\\mu\\in\\Lambda_T\\); \\(T(\\mathcal O)\\subseteq K\\) removes \\(t\\).\n6. \\(W_X=\\langle s_{\\gamma_1},s_{\\gamma_6}\\rangle\\) and every \\(\\kappa\\in\\Lambda_S\\) is \\(W_X\\)-conjugate to a dominant one. Applying the \\(\\dot n_i\\) (step 4) moves \\(t^\\mu x_0\\) inside its \\(K\\)-orbit to a point with dominant label, i.e. into \\(A_X^+x_0\\).\n7. Distinctness and the identification with \\(\\Lambda_S^+\\) are [[S4-5]](1).\n\n**Where [[S4-6]] stood.** It listed (E) as open. (E) is the content of steps 5–6, so [[S4-6]] is replaced by this node; its route (ii), \"need \\(\\mathcal F=\\{1\\}\\)\", is what \\(|J|=1\\) in [[S4-12]] provides. (I) is unchanged: it is a finite check (\\(45\\times6\\) pairs), not done here.\n\n**Cited inputs used:** Delorme–Sécherre Theorem 4.8 and Helminck–Wang Prop. 10.3 (through [[S4-12]]); conjugacy of maximal \\(\\theta\\)-split tori over \\(\\bar F\\) (Vust, Richardson); Kneser's \\(H^1(F,\\mathrm{Spin}_8)=H^1(F,F_4)=1\\); Bruhat–Tits for \\(K=\\operatorname{Stab}(x)\\).", "concept": "e6f4-functions", "key": true, "sources": ["Delorme–Sécherre, arXiv:math/0612545v1, Theorem 4.8 (via S4-12)", "sakellaridis-v3.pdf, Axiom 2.2.1 p. 9; Axiom 2.4.1 p. 10", "families/e6-f4.html §4"]}, {"id": "S4-14", "t": "2026-10-17T21:56:09Z", "kind": "verdict", "status": "proved", "label": "Summary", "name": "E6/F4 function cell, re-derived again", "title": "\\(E_6/F_4\\) function cell: two generators, modulo Axiom 2.2.1 over \\(\\mathcal O\\)", "summary": "Re-states the \\(E_6/F_4\\) function cell after the review. With an explicit integral involution, the Delorme–Sécherre decomposition and the little Weyl group realized in \\(K\\), every hypothesis of Sakellaridis's Theorem 8.0.2 holds for \\(p\\) odd except Axiom 2.2.1 as \\(\\mathcal O\\)-schemes; under it \\(C_K\\) is not cyclic and needs exactly two generators.", "statement": "**\\(E_6/F_4\\), functions.** \\(F\\) a \\(p\\)-adic field (characteristic \\(0\\)), \\(p\\ne2\\); \\(X=G_{\\mathbb Z}/G_{\\mathbb Z}^{\\theta_0}\\) base-changed to \\(\\mathcal O\\), with \\(\\theta_0=\\operatorname{Int}(\\sigma_L)\\circ C\\) ([[S4-11]]); \\(K=G(\\mathcal O)\\); \\(\\varphi_0=\\mathbf 1_{X(\\mathcal O)}=\\mathbf 1_{Kx_0}\\).\n\n| claim | status | where |\n|---|---|---|\n| \\(\\operatorname{im}r=\\{g:g(y_1)=g(\\sigma y_1)\\}\\), codimension one | proved, pure algebra, every \\(q\\) | [[S4-7]] |\n| \\(C_K\\) not cyclic; \\(C_K=\\mathcal H_K\\varphi_0\\oplus\\mathbb C\\mathbf 1_{O_{3\\omega_1^\\vee}}\\); cokernel the character \\(q^{\\rho-\\omega_4}\\) | proved under Theorem 8.0.2 | [[S4-8]] |\n| integral model with an E IV involution over \\(\\mathbb Z\\), \\(T\\subseteq B\\subseteq P\\) adapted | proved | [[S4-11]] |\n| §1.1/§1.7: \\(F\\), \\(G\\), \\(X\\) over \\(\\mathcal O\\), \\(\\mathring X\\), eigenmeasures | proved for \\(p\\ne2\\), modulo cited integral facts | [[S4-4]] |\n| Part 2 opening: no type-\\(N\\) reflections | proved | [[S4-10]] |\n| §7: no type-\\(N\\) roots, \\(L_\\gamma=D_5\\) classical, Statements 6.3.1, 7.1.5 | proved | [[S4-3]] |\n| §8: homogeneous, affine, one open \\(B(F)\\)-orbit | proved | [[S4-4]] |\n| Axiom 2.4.1: \\(K\\backslash X(F)\\simeq\\Lambda_S^+\\) via \\(A_X^+\\) | proved for \\(p\\ne2\\), via Delorme–Sécherre | [[S4-12]], [[S4-13]], [[S4-5]] |\n| Axiom 2.4.2 | proved for \\(p\\ne2\\) | [[S4-5]] |\n| Axiom 2.2.1: types (only \\(G\\), \\(U\\)) | proved over \\(\\bar F\\) (Knop's lattice transport cited) | [[S4-5]] |\n| Axiom 2.2.1 as \\(\\mathcal O\\)-schemes, 45 orbits × 6 simple roots | **open** | [[S4-13]] (I) |\n\n**Cell.** \"Fails, and \\(C_K\\) needs exactly two generators (\\(\\varphi_0\\) and \\(\\mathbf 1_{O_{3\\omega_1^\\vee}}\\)), with cokernel the character \\(q^{\\rho-\\omega_4}\\); for \\(p\\)-adic \\(F\\) with \\(p\\) odd and the model \\(G/G^{\\theta_0}\\); conditional only on Axiom 2.2.1 over \\(\\mathcal O\\) (and the cited inputs listed in the gap).\" Over \\(\\mathbb F_q(\\!(t)\\!)\\) the cell is undecided: Sakellaridis assumes characteristic \\(0\\).", "parents": ["S4-7", "S4-8", "S4-10", "S4-11", "S4-12", "S4-13", "S4-3", "S4-4", "S4-5", "A6-6", "A7-32"], "corrects": ["S4-9"], "body": "1. **Change from [[S4-9]].** That cell was conditional on two statements: (E), the existence half of Axiom 2.4.1, and (I), Axiom 2.2.1 over \\(\\mathcal O\\). (E) is now proved for \\(p\\) odd ([[S4-12]], [[S4-13]]); (I) remains. The model's involution is now explicit and defined over \\(\\mathbb Z\\) ([[S4-11]]), so its existence is no longer a cited input. The Part 2 standing assumption missing from [[S4-1]] is added and verified ([[S4-10]]).\n2. **Unchanged.** The algebra of \\(r\\) ([[S4-7]]) and its transport under Theorem 8.0.2 ([[S4-8]]) do not depend on the review.\n3. **Fields.** The function side remains a \\(p\\)-adic statement, so the divergence with the sheaf side over \\(\\overline{\\mathbb F}_q\\) is still a comparison across fields, as [[A7-32]] records.\n4. **Inputs.** [[A6-6]]: the gluing \\(y_1\\leftrightarrow\\sigma y_1\\) is algebra about \\(r\\) and holds as stated. [[A7-32]]: its \\(E_6/F_4\\) function cell holds with the change recorded in the inputs field.", "concept": "synthesis", "key": true, "sources": ["data.json: A6-6, A7-32", "sakellaridis-v3.pdf, Theorem 8.0.2 p. 64; Part 2 opening p. 41; Axiom 2.2.1 p. 9", "Delorme–Sécherre, arXiv:math/0612545v1, Theorem 4.8"], "inputs": {"A6-6": "holds as stated (the gluing of y_1 and σy_1, and M_(3,0) ∉ im r, are algebra about r); its conclusion '(SG) fails under (A1)–(A5)' (Sakellaridis's field/group, O-model, axioms, spherical-root/colour and affine-one-orbit hypotheses) holds with change: under Theorem 8.0.2 C_K is not cyclic, needs exactly two generators, and im r is the codimension-one ring K ([[S4-7]], [[S4-8]]); for the model G/G^{θ_0} with p odd, the theorem's hypotheses all hold except Axiom 2.2.1 over O ([[S4-13]])", "A7-32": "holds with change: the E6/F4 function cell becomes 'fails; C_K needs exactly two generators, φ_0 and 1_{O_{3ω_1^∨}}; p-adic F, p odd, model G/G^{θ_0}'. Its list of unverified hypotheses, '(A2) the O-model, (A3) the axioms at the place, (A4) the spherical-root and colour hypotheses beyond L_γ classical', shrinks to one statement: Axiom 2.2.1 as O-schemes for the 45 maximal-rank Borel orbits ([[S4-13]]). (A2) the O-model holds for p odd ([[S4-4]], [[S4-11]]); in (A3) the axioms, 2.4.1 and 2.4.2 hold ([[S4-12]], [[S4-13]], [[S4-5]]); (A4) the spherical-root and colour hypotheses hold in full ([[S4-3]], [[S4-10]])"}}], "result": {"status": "partial", "statement": "For \\(X=E_6/F_4\\) over a \\(p\\)-adic \\(F\\) with \\(p\\) odd and the integral model \\(G_{\\mathbb Z}/G_{\\mathbb Z}^{\\theta_0}\\), \\(\\theta_0=\\operatorname{Int}(\\sigma_L)\\circ C\\) an involution of type E IV defined over \\(\\mathbb Z\\): (i) unconditionally, \\(\\operatorname{im}r\\) is the codimension-one subring \\(\\{g:g(y_1)=g(\\sigma y_1)\\}\\) of \\(S=\\mathbb C[\\delta(X)^{1/2}A_X^*]^{W_X}\\), and \\(S\\) has only constant units; (ii) under Sakellaridis's Theorem 8.0.2, \\(C_K\\) is not cyclic, \\(C_K=\\mathcal H_K\\varphi_0\\oplus\\mathbb C\\,\\mathbf 1_{O_{3\\omega_1^\\vee}}\\) needs exactly two generators, and \\(C_K/\\mathcal H_K\\varphi_0\\) is the character \\(q^{\\rho-\\omega_4}\\); (iii) every hypothesis of Theorem 8.0.2 holds for this model except Axiom 2.2.1 as \\(\\mathcal O\\)-schemes. These hypotheses are: the conditions of §1.1/§1.7 on \\(F\\), \\(G\\), \\(X\\) and \\(\\mathring X\\); the Part 2 assumption of no type-\\(N\\) reflections; the §7 conditions and Statements 6.3.1 and 7.1.5; the §8 conditions; Axiom 2.4.1 (via Delorme–Sécherre's split Cartan decomposition, whose index set has one element because \\(H^1(F,\\mathrm{Spin}_8)=1\\), and \\(W_X\\) realized in \\(H(\\mathcal O)\\)); Axiom 2.4.2; and the types in Axiom 2.2.1.", "gap": "(I) Axiom 2.2.1 as \\(\\mathcal O\\)-schemes, for the 45 maximal-rank Borel orbits and the 6 simple roots (types \\(U\\) and \\(G\\) only). Cited, not re-checked here: Delorme–Sécherre Theorem 4.8 (statement read from arXiv:math/0612545v1) and Helminck–Wang Prop. 10.3 inside it; conjugacy of maximal \\(\\theta\\)-split tori over \\(\\bar F\\) (Vust, Richardson); Kneser's \\(H^1(F,\\mathrm{Spin}_8)=H^1(F,F_4)=1\\); Bruhat–Tits for \\(K=\\operatorname{Stab}(x)\\); smoothness and connectedness of \\(G^{\\theta_0}_{\\mathcal O}\\) for \\(p\\ne2\\) (Edixhoven, Steinberg); quasi-affineness of \\(G_{\\mathcal O}/G_{\\mathcal O}^{\\theta_0}\\) (Alper); SGA 3 XXIII for the Chevalley involution over \\(\\mathbb Z\\); Knop's transport of weight lattices (types in Axiom 2.2.1, no type-\\(N\\) reflections); the paper's claim that removing '\\(F/\\mathbb Q_p\\) unramified' is trivial. Over \\(\\mathbb F_q(\\!(t)\\!)\\) nothing is decided."}, "blocked_routes": ["Proving Axiom 2.4.1's existence half by removing \\(U_P(F)\\) from \\(X(F)=K\\,U_P(F)\\,T(F)x_0\\) directly: not needed; Delorme–Sécherre's building argument (\\(\\sigma\\)-stable apartments through special points) does it, and the remaining input is \\(|J|=1\\) ([[S4-12]])."], "scripts": ["/private/tmp/claude-501/-Users-miltonlin-Projects-Trees/f0f18b89-aa0d-4ce6-93d2-78c269eb7d7d/scratchpad/salvage/S4/c7_theta0.py", "/private/tmp/claude-501/-Users-miltonlin-Projects-Trees/f0f18b89-aa0d-4ce6-93d2-78c269eb7d7d/scratchpad/salvage/S4/c8_wx_in_K.py", "/private/tmp/claude-501/-Users-miltonlin-Projects-Trees/f0f18b89-aa0d-4ce6-93d2-78c269eb7d7d/scratchpad/salvage/S4/e6core.py"]}