Proposition VIII and proof

Click a dotted symbol or a label such as H5 to recall its statement. The function half now holds over \(\mathbb F_q(\!(t)\!)\), the same field family as the sheaf half’s \(\overline{\mathbb F}_q\): by Proposition VI, Sakellaridis’s Theorem 8.0.2 holds for the model \(\mathcal M_\theta\) over \(\mathbb F_q(\!(t)\!)\) and every \(p\)-adic field, \(p\) odd (audited run sakellaridis-char-p). The sheaf half is \(\ell\)-adic over \(\overline{\mathbb F}_q\). Proposition I and Proposition II are used for the sheaf half; Proposition VI, which proves Theorem 8.0.2 for \(\mathcal M_\theta\), and Proposition VII, which shows that \(C_K\) needs exactly two generators, for the function half.

Proposition VIII as a paper · statement and complete proof (PDF) →

Definitions
  • The symmetric space \(E_6/F_4\)

    \(G\) is the split simply connected group of type \(E_6\) (Bourbaki labels: the chain \(1\)–\(3\)–\(4\)–\(5\)–\(6\), node \(2\) attached to node \(4\)). \(\theta\) is the involution of type E IV, with connected fixed group \(G^\theta=F_4\); \(X=G/G^\theta\), \(x_0=eG^\theta\). The Satake diagram has black nodes \(2,3,4,5\), a \(D_4\), and white nodes \(1,6\).

    $$\Lambda_T=X_*(T)=\textstyle\bigoplus_j\mathbb Z\alpha_j^\vee,\qquad\theta=-w_L,\qquad\varphi=1-\theta,\qquad L=Z_G(S),\qquad Z_{G^\theta}(S)=\mathrm{Spin}_8\ \text{(connected)}.$$

    Write \((a,b):=a\omega_1^\vee+b\omega_6^\vee\). \(\varphi\) kills \(\alpha_2^\vee,\dots,\alpha_5^\vee\), and the images of the white coroots form a basis of the split lattice:

    $$e_1=\varphi(\alpha_1^\vee)=(2,-1),\qquad e_6=\varphi(\alpha_6^\vee)=(-1,2),\qquad\Lambda_S=\mathbb Ze_1\oplus\mathbb Ze_6=\{(a,b):a\equiv b\bmod3\}.$$

    \(\sigma\) is the diagram automorphism \(1\leftrightarrow6\), \(3\leftrightarrow5\). Root data on the family page →

  • Orbit labels, Hecke labels and the uncovered labels

    Orbit labels are the dominant split coweights; Hecke labels the dominant coweights. In fundamental-coweight coordinates \(\lambda=\sum_ic_i\omega_i^\vee\):

    $$\Lambda_S^+=\{(a,b):a,b\ge0,\ a\equiv b\bmod3\},\qquad\varphi(\lambda)=(2c_1+c_2+2c_3+2c_4+c_5,\ c_2+c_3+2c_4+2c_5+2c_6).$$

    A dominant \(\lambda\) with \(\varphi(\lambda)=(a,0)\) has \(c_2=\dots=c_6=0\), so \(a=2c_1\) is even. With \(a\equiv b\bmod 3\) this gives the uncovered labels, where H5 fails:

    $$\Lambda_S^+\setminus\varphi(\Lambda_T^+)=\{(3k,0),\ (0,3k):k\ \text{odd}\}.$$

    For odd \(k\), the uncovered label \(\mu_k\), the label \(\mu_k'\) just above it, and a Hecke label lifting \(\mu_k'\):

    $$\mu_k=(3k,0),\qquad\mu_k'=\mu_k+e_6=(3k-1,2),\qquad\lambda_k=\tfrac{3k-1}2\,\omega_1^\vee+\omega_6^\vee\in\Lambda_T^+.$$

    Example. \(k=1\): \(\mu_1=3\omega_1^\vee\), \(\mu_1'=(2,2)\), \(\lambda_1=\omega_1^\vee+\omega_6^\vee\), the highest weight of the 650-dimensional representation of the dual group.

  • Sheaf level over \(\overline{\mathbb F}_q\)

    \(k=\overline{\mathbb F}_q\), \(p\ne2\), coefficients \(\overline{\mathbb Q}_\ell\). \(LX=X(k(\!(t)\!))\) is the union of the orbits \(\mathcal O_\mu=L^+G\cdot x_\mu\), \(\tau(x_\mu)=t^\mu\), with \(\overline{\mathcal O_\mu}=\bigsqcup_{\nu\le\mu}\mathcal O_\nu\) for the dominance order. \(\delta(\mu)\) is the codimension of \(\mathcal O_0\) in \(\overline{\mathcal O_\mu}\), and \(\mathrm{IC}_\nu\) is normalized by \(\mathrm{IC}_\nu|_{\mathcal O_\nu}=\overline{\mathbb Q}_\ell[\delta(\nu)]\).

    \(\mathcal C_K\) is the category of bounded \(L^+G\)-equivariant constructible complexes on \(LX\) supported on finitely many orbit closures. The convolution map and the Satake objects:

    $$m:\operatorname{Gr}^{\le\lambda}\mathbin{\widetilde\times}L^+X\longrightarrow LX,\quad[g,y]\longmapsto gy,\qquad V_\lambda\ast\mathrm{IC}_0=m_!\big(V_\lambda\mathbin{\widetilde\boxtimes}\mathrm{IC}_0\big)\simeq\bigoplus_{\nu,n}\mathrm{IC}_\nu[-n]\otimes E_{\nu,n}(\lambda),$$ $$\mathcal C_K^{\mathrm{Sat}}=\operatorname{Thick}_{\mathcal C_K}\{V_\lambda\ast\mathrm{IC}_0:\lambda\in\Lambda_T^+\},\qquad F_k=m^{-1}(x_{\mu_k}),\quad F_k'=m^{-1}(x_{\mu_k'})\ \ (\lambda=\lambda_k).$$

    The decomposition is input D2. \(\operatorname{Thick}\) is closed under isomorphisms, shifts, cofibers and retracts.

  • Function level over a \(p\)-adic field

    \(F\) is \(\mathbb F_q(\!(t)\!)\) or a \(p\)-adic field, \(p\) odd, with ring of integers \(\mathcal O\) and residue field \(\mathbb F_q\); \(K=G(\mathcal O)\). Spherical generation:

    $$\text{(SG)}\qquad\mathcal H_K\cdot\varphi_0=C_K,\qquad\mathcal H_K=C_c(K\backslash G(F)/K)\simeq\mathbb C[A^*]^W,\quad C_K=C_c^\infty(X(F))^K,\quad\varphi_0=\mathbf 1_{X(\mathcal O)}.$$

    The dual torus of \(X\), its coordinates, the shift and the little Weyl group:

    $$A_X^*=\operatorname{Hom}(\Lambda_S,\mathbb C^\times)\hookrightarrow A^*=\operatorname{Hom}(\Lambda_T,\mathbb C^\times),\quad u=y(e_1),\ v=y(e_6),\qquad\delta(X)^{1/2}=q^{\rho_L},\qquad W_X=S_3 .$$

    The restriction map and the orbit sums:

    $$r:\mathbb C[A^*]^W\longrightarrow\mathbb C[\Lambda_S]^{W_X},\quad r(f)(y)=f(q^{\rho_L}y),\qquad M_\kappa=\sum_{\kappa'\in W_X\kappa}y^{\kappa'},\qquad y_1=(q^{-5},q),\ y_2=(q^{-5},q^4).$$
  • The multiplicity matrix over \(\mathbb F_q(\!(t)\!)\)

    For a Satake object defined over \(\mathbb F_q\), the Frobenius traces of the multiplicity spaces give a matrix with rows indexed by orbit labels and columns by Hecke labels:

    $$m_{\nu\lambda}=\sum_i(-1)^i\operatorname{tr}\big(\mathrm{Fr}\mid E_{\nu,i}(\lambda)\big),\qquad\nu\in\Lambda_S^+,\ \lambda\in\Lambda_T^+ .$$

    A row \(\nu\) is nonzero when some \(E_{\nu,\bullet}(\lambda)\ne0\); the columns span when every finitely supported vector on \(\Lambda_S^+\) is a combination of columns.

Inputs: Sakellaridis’s hypotheses (A), D0–D3 and D5
  • (A) · Sakellaridis’s hypotheses, proved for \(\mathcal M_\theta\).

    Sakellaridis’s hypotheses for an \(\mathcal O\)-model of \(X\) at \(F\), as stated on Proposition VI: (A1) \(F\) of characteristic \(0\) and \(G\) split with simply connected derived group ((A1ₚ): the same with \(F=\mathbb F_q(\!(t)\!)\)); (A2) an integral model whose open Borel orbit is smooth, surjective and \(B\)-homogeneous over \(\mathcal O\), with eigenmeasures; (A3a) rank-one models over \(\mathcal O\) (Axiom 2.2.1), (A3b) the Cartan decomposition (Axiom 2.4.1) and (A3c) Iwahori orbits (Axiom 2.4.2); (A4) no maximal-rank pair \((Y,\alpha)\) of type \(N\), classical \(L_\gamma\), Statements 6.3.1 and 7.1.5; (A5) \(X\) homogeneous and affine with one open \(B(F)\)-orbit. They are used only in part (a).

    Status for \(E_6/F_4\): proved for \(\mathcal M_\theta\)

    For the model \(\mathcal M_\theta=(G_{\mathbb Z}/G_{\mathbb Z}^{\theta_0})\otimes\mathcal O\), \(\theta_0=\operatorname{Int}(\sigma_L)\circ C\) of type E IV defined over \(\mathbb Z\), of Proposition VII, \(p\) odd:

    The cited inputs not re-proved are listed in Proposition VII, step 11.

  • D0 · Orbit geometry over \(\overline{\mathbb F}_q\).

    Chen–Yi’s results over \(\overline{\mathbb F}_q\), \(p\ne2\): the orbit parametrization (Prop. 17), the closure order (Thm. 18), the support bound and the stabilizer lemma:

    $$\overline{\mathcal O_\mu}=\bigsqcup_{\nu\le\mu}\mathcal O_\nu,\qquad\operatorname{supp}(V_\lambda\ast\mathrm{IC}_0)\subseteq\overline{\mathcal O_{\varphi(\lambda)}} .$$
    Caveat: Prop. 17(i) is proved over \(\mathbb C\)

    Chen–Yi prove Prop. 17(i), the orbit parametrization behind D0, by citing Gaitsgory–Nadler, Thm. 8.2.9, and Nadler 2004, Thm. 4.2, which are over \(\mathbb C\). Two agents of the run sakellaridis-char-p found this independently, and the audit confirmed it (Audit 3.1, Audit 4.3, Audit 5.7). For \(E_6/F_4\) the function-level parametrization \(K\backslash X(\mathbb F_q(\!(t)\!))\simeq\Lambda_S^+\) is now proved directly (Theorem 3.8, Proposition 4.5).

  • D1 · Proper base change.

    Proper base change and Grothendieck–Lefschetz for the convolution map on finite-type models; the Frobenius-compatible perverse t-structure.

  • D2 · Decomposition.

    Each Satake object is a finite direct sum of shifted IC complexes with constant coefficients, on finite-type equivariant models (Sun, Thm. 1.2; Chen–Yi, Prop. 70):

    $$V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus_{\nu,n}\mathrm{IC}_\nu[-n]\otimes E_{\nu,n}(\lambda).$$
  • D3 · Lang’s theorem.

    Lang’s theorem for \(L^+G\) and the stabilizers, so that \(K\backslash X(\mathbb F_q(\!(t)\!))=\Lambda_S^+\). Used only in part (c).

  • D5 · Comparison with \(\mathbb C\).

    For each pair of labels, “\(\mathrm{IC}_\mu\) is a direct summand of \(V_\lambda\ast\mathrm{IC}_0\)” holds over \(\overline{\mathbb F}_p\) for large \(p\) exactly when it holds in the Betti category over \(\mathbb C\) (spreading out, BBD §6.1). This is the open item 1 of the spherical workspace’s Proposition III.

Proposition VIII

Setting. \(X=E_6/F_4\), with the root data, labels and maps of the definitions.

(a) Functions fail. Let \(p\) be odd, let \(F\) be \(\mathbb F_q(\!(t)\!)\) or a \(p\)-adic field, and let \(X=\mathcal M_\theta\), the model of Proposition VII; or let \(F\) be \(p\)-adic and \(X\) any \(\mathcal O\)-model satisfying (A). Then \(q^{\rho_L}y_1\) and \(q^{\rho_L}y_2\) are \(W(E_6)\)-conjugate, \(y_1\) and \(y_2\) are not \(W_X\)-conjugate, and

$$M_{(3,0)}(y_1)=q^{-9}+q^{3}+q^{6}\ \ne\ q^{-6}+q^{-3}+q^{9}=M_{(3,0)}(y_2).$$

Hence \(\operatorname{im}r\subseteq\{g:g(y_1)=g(y_2)\}\subsetneq\mathbb C[\Lambda_S]^{S_3}\), \(r\) is not onto, and (SG) fails, for every \(q\); more precisely, \(C_K\) needs exactly two generators (Proposition VII). For \(\mathcal M_\theta\) no hypothesis remains open: (A) is proved, and Theorem 8.0.2 holds over both fields (Proposition VI).

(b) Sheaves hold. Over \(\overline{\mathbb F}_q\), \(p\ne2\), under D0–D2, for every odd \(k\):

$$\varphi(\lambda_k)=\mu_k',\qquad\{\nu:\mu_k\le\nu\le\mu_k'\}=\{\mu_k,\mu_k'\},\qquad\operatorname{codim}=8,\qquad F_k,F_k'\subseteq\operatorname{Gr}^{\lambda_k},\qquad F_k'=\{t^{\lambda_k}\},\qquad\dim F_k\ge7 .$$

Consequently \(\mathrm{IC}_{3k\omega_1^\vee}\) is a direct summand of \(V_{\lambda_k}\ast\mathrm{IC}_0\). With the mirror labels \(3k\omega_6^\vee\) and the covered labels, every \(\mathrm{IC}_\mu\) is a summand of a Satake object, and \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\); through D5 the same holds in the Betti category over \(\mathbb C\).

(c) Why the levels split. Over \(\mathbb F_q(\!(t)\!)\), \(q\) odd, under D0–D3, \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) says that every row of the multiplicity matrix is nonzero, while (SG) says that its columns span. For \(E_6/F_4\) every row is nonzero, by (b), and the columns do not span, by (a) over \(\mathbb F_q(\!(t)\!)\): the first does not imply the second. Both halves now concern \(\mathbb F_q(\!(t)\!)\) and its geometric counterpart over \(\overline{\mathbb F}_q\).

$$\boxed{\ \mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\ \text{over }\overline{\mathbb F}_q\ (\text{D0–D2}),\qquad\text{(SG) fails, }C_K\text{ needs two generators, over }\mathbb F_q(\!(t)\!)\text{ and every }p\text{-adic }F\ (p\ \text{odd},\ \mathcal M_\theta)\ }$$
Sources

References

  • Sakellaridis, Spherical functions on spherical varieties, v3: §1.1 (characteristic zero), §1.7 (assumptions, \(\delta(X)\)), §2.1 (\(L(X)\), \(A_X\), \(A_X^*\)), Theorem 6.3.2, Statement 7.1.5 and Theorem 7.2.1, Example 8.0.1, Theorem 8.0.2, Corollary 8.0.4 and the remark after it, Appendix A (steps 1–4, 14–15).
  • Chen–Yi, v4: Proposition 17 (part (i) proved by citing Gaitsgory–Nadler, Thm. 8.2.9, and Nadler 2004, Thm. 4.2, over \(\mathbb C\)) and Theorem 18 (orbits, closures), Corollary 14 (degree bound), Proposition 64 (codimension), Proposition 70 (decomposition), Theorem 87 and Example 88 (connected stabilizers for \(E_6/F_4\)), Lemma 90, §1.2.6 (steps 5–12).
  • Sun, v3, Theorem 1.2 (D2). A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Astérisque 100 (1982), Corollary 5.4.6, Definition 1.4.22 and §6.1 (D5). Bezrukavnikov–Finkelberg, v2, §6.5 Proposition 5 (comparison across fields).
  • Zhu, v2: Proposition 2.1.5 and (2.1.17) (cells, bounded convolution), §5.1 (IC normalization, stabilizers), proof of Lemma 5.3.7 (fixed points of a regular cocharacter), §5.6 (functions–sheaves dictionary).
  • Project: Proposition VI (Sakellaridis’s hypotheses; Theorem 8.0.2 over \(\mathbb F_q(\!(t)\!)\)), Proposition VII (the Hecke module), the audited run sakellaridis-char-p (nodes Theorem 2.8, Theorem 3.8, Summary 3.12, Theorem 4.8; audited in Audit 5.7, Summary 5.13, Summary 6.9), the audit record S4, the \(E_6/F_4\) family page (root data, uncovered labels), Proposition III, Theorem E, and the run’s verdict; these locate the target and are not literature.

References checked 17 October 2026. The reader in D holds located extracts.

Proof

Steps 1–4 prove (a) from Theorem E, which holds for \(\mathcal M_\theta\) over \(\mathbb F_q(\!(t)\!)\) and every \(p\)-adic field. Steps 5–12 prove (b) by computing two convolution fibres; step 13 is a heuristic refinement, not used. Steps 14–16 prove (c) and conclude.

  1. For \(\mathcal M_\theta\) over \(\mathbb F_q(\!(t)\!)\) or a \(p\)-adic field, \(p\) odd, and for any model satisfying (A) at a \(p\)-adic field, Theorem E identifies \(C_K\) with the invariant functions on the shifted torus, with \(\varphi_0\mapsto1\); so \(\mathcal H_K\varphi_0\) corresponds to \(\operatorname{im}r\):

    $$C_K\simeq\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X},\qquad\text{(SG)}\iff r\ \text{onto}.$$
    Proof
    • Theorem E. Sakellaridis, Theorem 8.0.2 gives the isomorphism of \(\mathcal H_K\)-modules, with \(\mathcal H_K\simeq\mathbb C[A^*]^W\) acting through \(r\); Corollary 8.0.4(1) is “onto ⇒ (SG)”. Theorem E on the classical page →
    • Over \(\mathbb F_q(\!(t)\!)\). Sakellaridis assumes characteristic \(0\). For \(\mathcal M_\theta\) the same isomorphism, with \(\varphi_0\mapsto1\), holds over \(\mathbb F_q(\!(t)\!)\) by Proposition VI (run sakellaridis-char-p, Theorem 4.8; audited in Audit 5.6); with it, “onto ⇔ (SG)” holds there too.
    • Invariants. For symmetric \(X\), \(L(X)=Z_G(S)=L\), \(A_X=L/L^\theta\simeq S\), and \(T\twoheadrightarrow A_X\) is \(\varphi\) on cocharacters, so \(\Lambda_X=\varphi(\Lambda_T)=\Lambda_S\). \(\delta(X)\) is the modular character of \(L\cap B\), so \(\delta(X)^{1/2}=q^{\pm\rho_L}\).
    • Sign. \(x\mapsto x^{-1}\) commutes with \(W\) and carries \(q^{\rho_L}A_X^*\) to \(q^{-\rho_L}A_X^*\); it intertwines the two restriction maps, so surjectivity does not depend on the sign.
    • Transport. The minimal representative \(w\) of each class of \(W_X=N_W(\Lambda_S)/W_L\) satisfies \(w\Phi_L^+=\Phi_L^+\), hence \(w\rho_L=\rho_L\); so \(\operatorname{im}r\subseteq\mathbb C[\Lambda_S]^{W_X}\).

  2. The shift has a simple form in the coordinates \(u,v\), the target ring has three generators, and only one is in question:

    $$x=q^{\rho_L}y:\quad x^{\alpha_1^\vee}=q^{-3}u,\ \ x^{\alpha_6^\vee}=q^{-3}v,\ \ x^{\alpha_j^\vee}=q\ (2\le j\le5);\qquad r\ \text{onto}\iff M_{(3,0)}\in\operatorname{im}r .$$
    Proof
    • Shift. \(2\rho_L=6\alpha_2+6\alpha_3+10\alpha_4+6\alpha_5\), so \(\langle\rho_L,\alpha_j^\vee\rangle_{j=1..6}=(-3,1,1,1,1,-3)\); and \(x^\lambda=q^{\langle\rho_L,\lambda\rangle}y^{\varphi(\lambda)}\).
    • Little Weyl group. \(w_1=w_0(D_5^{\{1..5\}})w_0(D_4)\) and \(w_6=w_0(D_5^{\{2..6\}})w_0(D_4)\) fix \(\rho_L\), commute with \(\varphi\), and act by \(s_1(a,b)=(-a,a+b)\), \(s_6(a,b)=(a+b,-b)\); on \(A_X^*\), \(s_1(u,v)=(u^{-1},uv)\), \(s_6(u,v)=(uv,v^{-1})\). So \(W_X=S_3\); the restricted root system is \(A_2\) with all multiplicities 8.
    • Generators. \((1,1),(3,0),(0,3)\) generate the monoid \(\Lambda_S^+\), and \(M_\kappa M_{\kappa'}=M_{\kappa+\kappa'}+\) lower terms with finite lower sets, so \(M_{(1,1)},M_{(3,0)},M_{(0,3)}\) generate \(\mathbb C[\Lambda_S]^{S_3}\).
    • Two of three. \(r(m_{\omega_2^\vee})=c_8(q)M_{(1,1)}+\mathrm{const}\), \(c_8=q^{-3}+q^{-2}+q^{-1}+2+q+q^2+q^3\ne0\), so \(M_{(1,1)}\in\operatorname{im}r\). \(\sigma\) fixes \(\rho_L\) and \(r\circ\sigma=\sigma\circ r\) exchanges \(M_{(3,0)}\) and \(M_{(0,3)}\).

  3. The shifted points \(q^{\rho_L}y_1\), \(q^{\rho_L}y_2\) are \(W(E_6)\)-conjugate, so every element of \(\operatorname{im}r\) agrees at \(y_1,y_2\); \(M_{(3,0)}\) does not:

    $$q^{\rho_L}y_1=q^{\rho-9\omega_1-3\omega_6}\ \sim_W\ q^{\rho-\omega_4}\ \sim_W\ q^{\rho-9\omega_1}=q^{\rho_L}y_2,\qquad M_{(3,0)}(y_1)\ne M_{(3,0)}(y_2).$$
    Proof
    • Weights. By step 2, \(q^{\rho_L}y_1\) has \(\alpha_j^\vee\)-exponents \((-8,1,1,1,1,-2)\) and \(q^{\rho_L}y_2\) has \((-8,1,1,1,1,1)\), i.e. the weights \(\rho-9\omega_1-3\omega_6\) and \(\rho-9\omega_1\).
    • Conjugacy. Reflecting at negative coordinates, \(s_1s_3s_4s_2s_5s_4s_3s_1s_6s_5s_4s_2s_3s_1s_4s_3s_5s_6\) and \(s_1s_3s_4s_2s_5s_4s_3s_1s_6s_5s_4s_2s_3\) carry them to the dominant weight \((1,1,1,0,1,1)=\rho-\omega_4\). A \(W\)-invariant function takes equal values at \(W\)-conjugate points.
    • Not \(W_X\)-conjugate. In \((\log_qu,\log_qv)\): \(W_Xy_1=\{(-5,1),(-4,-1),(-1,5),(1,4),(4,-5),(5,-4)\}\) and \(W_Xy_2=\{(-5,4),(-4,5),(-1,-4),(1,-5),(4,1),(5,-1)\}\), disjoint.
    • Separation. \(W_X(3,0)=\{(3,0),(-3,3),(0,-3)\}\) and \(y^{(a,b)}=u^{(2a+b)/3}v^{(a+2b)/3}\), so \(M_{(3,0)}=u^2v+u^{-1}v+u^{-1}v^{-2}\), with values \(q^{-9}+q^6+q^3\) and \(q^{-6}+q^9+q^{-3}\), different for \(q\gt1\).
    • Conclusion. \(M_{(3,0)}\notin\operatorname{im}r\), so \(r\) is not onto and, by step 1, (SG) fails. Only the \(W\)-conjugacy of integral weights is used, so this holds for every \(q\). Also \(\sigma y_1\in W_Xy_2\): \(y_1\) is glued to its \(\sigma\)-image.
    • Sharpened. Proposition VII shows that \(\operatorname{im}r\) is exactly \(\{g:g(y_1)=g(\sigma y_1)\}\), of codimension one; this step stands as stated.

    Within the family of restricted root system \(A_2\) with multiplicity \(d\) (the Hermitian \(3\times3\) matrices over composition algebras of dimension \(d\)), \(r\) is onto for \(d=4\), \(\mathrm{SL}_6/\mathrm{Sp}_6\), and not onto for \(d=8\), \(E_6/F_4\).

  4. For the model \(\mathcal M_\theta\) of Proposition VII, \(p\) odd, every hypothesis of Theorem 8.0.2 holds over every \(p\)-adic \(F\), and Theorem 8.0.2 holds over \(\mathbb F_q(\!(t)\!)\) by transfer; so (a) holds for \(\mathcal M_\theta\) with no open hypothesis, granted cited inputs.

    Proof
    • (A1), (A1ₚ), field and group. \(E_6\) is split and simply connected; \(F\) is \(p\)-adic, or \(F=\mathbb F_q(\!(t)\!)\).
    • (A2), the integral model. For \(p\) odd, \(\theta_0=\operatorname{Int}(\sigma_L)\circ C\) is defined over \(\mathbb Z\), and \(G_{\mathcal O}^\theta\) is smooth with connected fibres of type \(F_4\), \(X\) is quasi-affine, and the open Borel orbit is \(U_P\times A_X\), smooth and surjective over \(\mathcal O\) with one \(B_{\mathbb F_q}\)-orbit as special fibre.
    • (A3a), rank-one models over \(\mathcal O\). The 45 Borel orbits \(B\dot wx_0\) are all of maximal rank, and \(YP_\alpha/R(P_\alpha)\) is \(\mathrm{PGL}_2\backslash\mathrm{PGL}_2\) (90 pairs) or \(B_2\backslash\mathrm{PGL}_2=\mathbb P^1_{\mathcal O}\) (180 pairs), over every discrete valuation ring with \(2\) invertible (run sakellaridis-char-p, Theorem 2.8; audited in Summary 6.9).
    • (A3b), (A3c), Cartan decomposition and Iwahori orbits. \(G(F)=K\,T(F)\,G^\theta(F)\) by Delorme–Sécherre with \(H^1(F,\mathrm{Spin}_8)=1\) and \(W_X\) realized in \(H(\mathcal O)\); \(Jx\subseteq B(\mathcal O)x\) at every label. Over \(\mathbb F_q(\!(t)\!)\) both are reproved in characteristic \(p\) (Theorem 3.8, Audit 3.9).
    • (A4), spherical roots and colours. The spherical roots \(\gamma_1=2\omega_1-\omega_6\), \(\gamma_6=-\omega_1+2\omega_6\) have type \(G\), and no maximal-rank pair \((Y,\alpha)\) has type \(N\) (Sakellaridis’s §6 assumption); their supports \(\{1,\dots,5\}\), \(\{2,\dots,6\}\) give \(L_\gamma\) of type \(D_5\); Statement 7.1.5 holds with \(\Theta=\check\Phi_X\times\{(+,4)\}\), and Statement 6.3.1 follows by Sakellaridis’s Theorem 6.3.2. Over \(\mathbb F_q(\!(t)\!)\) all but the analytic part of Statement 6.3.1 is reproved (Summary 3.12); the transfer does not need that part.
    • (A5), one open orbit. For \(g=ul\in B\cap G^\theta\), \(u\in U_P\), \(l\in B_L\): \(\theta(u)=u\,l\,\theta(l)^{-1}\in U_{P^-}\cap P=1\), so \(B\cap G^\theta=B_L\cap L^\theta\), a Borel subgroup of \(L^\theta=\mathrm{Spin}_8\). It is connected, so \(H^1(F,B\cap G^\theta)=1\) and \(\mathring X(F)\) is one \(B(F)\)-orbit.
    • Theorem 8.0.2 over \(\mathbb F_q(\!(t)\!)\). Transfer from the totally ramified fields \(\mathbb Q_q(p^{1/e})\), \(e\gg0\), where \(\mathbb Q_q\) is the unramified extension of \(\mathbb Q_p\) with residue field \(\mathbb F_q\) (Lemma 4.2, Proposition 4.5, Proposition 4.6, Theorem 4.8; audited in Audit 5.1, Audit 5.3, Objection 5.4, Audit 5.6).
    • Cited, not re-proved. Delorme–Sécherre; Bruhat–Tits; Hensel and Lang; Conrad, Edixhoven and Steinberg; EGA IV; SGA 3, Exp. XXIII (Proposition VII, step 11). \(E_6/F_4\) is not in Sakellaridis’s table, which lists only classical \(\mathfrak g\); his axioms are proved at almost every place of a global field, and here at every place with \(p\) odd.

  5. Over \(\overline{\mathbb F}_q\), D0 gives H1–H3 of Proposition I for \(E_6/F_4\), D2 gives the decomposition of each Satake object, and D1 computes its stalks by fibre cohomology.

    Proof
    • H1 from the closure order (Chen–Yi, Thm. 18) and finite-type models (Thm. 36); H2 from the operations on those models; H3 from connected stabilizers (Ex. 88), i.e. \(Z_{G^\theta}(S)=\mathrm{Spin}_8\) connected. Over \(\overline{\mathbb F}_q\) these come from Chen–Yi directly, not from Proposition III, which is proved over \(\mathbb C\).
    • D2 with H3 gives constant coefficients in \(V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus\mathrm{IC}_\nu[-n]\otimes E_{\nu,n}(\lambda)\).

  6. \(\lambda_k\) lifts the label just above \(\mu_k\), nothing lies between the two, and the codimension is 8:

    $$\varphi(\lambda_k)=\mu_k',\qquad[\mu_k,\mu_k']=\{\mu_k,\mu_k'\},\qquad\delta(\mu_k')-\delta(\mu_k)=8,\qquad\dim\operatorname{Gr}^{\lambda_k}=\delta(\mu_k')=8(3k+1).$$
    Proof
    • Cone. \(e_1=2\alpha_1^\vee+\alpha_2^\vee+2\alpha_3^\vee+2\alpha_4^\vee+\alpha_5^\vee\) and its mirror \(e_6\) are positive. Conversely \(p=\tfrac12(1-\theta)\) fixes \(\Lambda_S\) and sends \(\sum_jc_j\alpha_j^\vee\) to \(\tfrac{c_1}2e_1+\tfrac{c_6}2e_6\); with \(\Lambda_S=\mathbb Ze_1\oplus\mathbb Ze_6\): $$\nu\le\mu\iff\mu-\nu\in\mathbb Z_{\ge0}e_1+\mathbb Z_{\ge0}e_6 .$$ \(\mu_k'-\mu_k=e_6\) spans an edge of this cone, so no label lies strictly between.
    • Codimension. Chen–Yi, Prop. 64 (\(LX\) connected since \(G\) is simply connected; its proof is valid for every \(p\ne2\)): \(\operatorname{codim}_{\overline{\mathcal O_\mu}}\mathcal O_\nu=\langle\rho,\mu-\nu\rangle\). The coefficients of \(\alpha_1,\alpha_6\) in \(2\rho\) are 16, so \(\delta(a,b)=8(a+b)\) and \(\langle\rho,e_6\rangle=8\).
    • Hecke label. \(c_k=\tfrac{3k-1}2\in\mathbb Z_{\ge0}\); \(\omega_1^\vee,\omega_6^\vee\) have classes \(1,2\) in \(\mathbb Z/3\), so \(\lambda_k\) has class \(c_k+2=\tfrac{3(k+1)}2\equiv0\) and lies in \(\Lambda_T\). \(\theta\) negates \(\omega_1^\vee,\omega_6^\vee\), so \(\varphi(\lambda_k)=(2c_k,2)=\mu_k'\), and \(\langle2\rho,\lambda_k\rangle=16(c_k+1)=8(3k+1)\).

  7. The fibres over \(x_{\mu_k}\) and \(x_{\mu_k'}\) avoid the singular part of \(\operatorname{Gr}^{\le\lambda_k}\), so the stalks of the Satake object there are fibre cohomology:

    $$F_k,\ F_k'\subseteq\operatorname{Gr}^{\lambda_k},\qquad(V_{\lambda_k}\ast\mathrm{IC}_0)_x\simeq R\Gamma(F_x,\overline{\mathbb Q}_\ell)[\delta(\mu_k')].$$
    Proof
    • Bound. \(F_x\simeq\{gL^+G\in\operatorname{Gr}^{\le\lambda_k}:g^{-1}x\in L^+X\}\) is closed in a projective variety. If such \(g\) lies in \(\operatorname{Gr}^{\lambda''}\), then \(x\in L^+Gt^{\lambda''}L^+X\), and the support bound (D0) with \(\varphi(\alpha_j^\vee)\in\{0,e_1,e_6\}\) gives \(\mu_k\le\varphi(\lambda'')\le\mu_k'\).
    • Lower label. \(\varphi(\lambda'')=\mu_k\) is impossible: \(\mu_k\) has no dominant lift.
    • Upper label. If \(\varphi(\lambda'')=\mu_k'\) and \(\lambda''\lt\lambda_k\), then \(\kappa=\lambda_k-\lambda''=\sum_{j=2}^5c_j\alpha_j^\vee\ne0\), \(c_j\ge0\). Since \(\langle\alpha_j,\lambda_k\rangle=0\) for black \(j\) and \(\lambda''\) is dominant, \(\langle\alpha_j,\kappa\rangle\le0\) for all black \(j\), contradicting \((\kappa,\kappa)=\sum_jc_j(\alpha_j^\vee,\kappa)\gt0\).
    • Stalks. On the smooth cell \(V_{\lambda_k}=\overline{\mathbb Q}_\ell[\delta(\mu_k')]\) and the twisted product restricts to the same; proper base change (D1) gives the formula. No stalk of \(\mathrm{IC}(\operatorname{Gr}^{\le\lambda_k})\) enters.

  8. A torus fixing \(x_{\mu_k'}\) has a single fixed point on \(F_k'\), so \(F_k'\) is that point, and \(\mathrm{IC}_{\mu_k'}\) occurs once, unshifted:

    $$F_k'=\{t^{\lambda_k}\},\qquad E_{\mu_k',n}(\lambda_k)=\overline{\mathbb Q}_\ell\ (n=0),\quad0\ (n\ne0).$$
    Proof
    • Replacement. The run bounded \(\dim F_k'\) by a dimension count in congruence quotients, which is wrong as written: every \(\mathcal O_\nu/K_N\) has dimension \(N\dim X\), and the convolution does not descend (A8-5, repaired in A7-28). The argument below uses no dimension theory.
    • Torus. \(T'=(T^\theta)^\circ\), the torus of the black coroots, fixes every \(x_\mu\): \(\tau(sx_\mu)=st^\mu\theta(s)^{-1}=st^\mu s^{-1}=t^\mu\). So \(F_k'\) is \(T'\)-stable.
    • Regularity. A root vanishing on \(T'\) would be \(a\omega_1+b\omega_6\) with \(2(a^2+ab+b^2)=3\), impossible. So a generic cocharacter \(\eta\) of \(T'\) has \(Z_G(\eta)=T\), and its fixed points on \(\operatorname{Gr}\) are the \(t^\nu\).
    • One fixed point. A fixed point of \(F_k'\subseteq\operatorname{Gr}^{\lambda_k}\) is \(t^\nu\) with \(\nu\in W\lambda_k\) and \(\varphi(\nu)=\mu_k'\). Then \(\nu-\lambda_k\in\ker\varphi\), which is orthogonal to \(\lambda_k\), and \(|\nu|=|\lambda_k|\) force \(\nu=\lambda_k\).
    • Finiteness. In a \(T\)-equivariant projective embedding with linear action (the lattice model of \(\operatorname{Gr}^{\le\lambda_k}\) and Plücker coordinates), a non-fixed point has two distinct fixed limits under \(\eta\), in its own component. So \(F_k'=\{t^{\lambda_k}\}\); by step 7 its stalk is \(\overline{\mathbb Q}_\ell\) in degree \(-\delta(\mu_k')\), and restricting the decomposition to \(\mathcal O_{\mu_k'}\) gives \(E_{\mu_k',n}\).

  9. The group \(Z_k=Z_{G^\theta}(\mu_k)^\circ\simeq\mathrm{Spin}_9\) acts on \(F_k\) with seven-dimensional orbits through its eight torus-fixed points:

    $$\dim Z_k=36,\qquad\dim\big(\operatorname{Lie}Z_k\cap\operatorname{Lie}P(-\nu)\big)=29,\qquad\dim F_k\ge36-29=7 .$$
    Proof
    • Action. For \(h\in Z_k\), \(\tau(hx_{\mu_k})=ht^{\mu_k}h^{-1}=t^{\mu_k}\), so \(Z_k\) fixes \(x_{\mu_k}\) and acts on \(F_k\).
    • Lie algebra. \(Z_G(\mu_k)\) is the Levi \(D_5\) on nodes \(2,\dots,6\); \(\theta\) fixes the 24 black roots and pairs the other 16 into 8 orbits: $$\operatorname{Lie}Z_k=\mathfrak t^\theta\oplus\bigoplus_{\alpha\in\Phi_L}\mathfrak g_\alpha\oplus\bigoplus_{\{\alpha,\theta\alpha\}}k\,(X_\alpha+\theta X_\alpha),\qquad4+24+8=36\ \ (B_4).$$
    • Fixed points. The points \(t^\nu L^+G\) of \(F_k\) have \(\nu\in W\lambda_k\), \(\varphi(\nu)=\mu_k\). Writing \(\nu=\tfrac{3k}2\omega_1^\vee+m\), norms force \(|m|^2=1\), so \(m=\pm\varepsilon_j'\) (\(j=2,\dots,5\)): eight points, with \(\langle\rho_L,\nu\rangle\in\{-3,-2,-1,0,0,1,2,3\}\).
    • Orbit dimension. The stabilizer of \(t^\nu L^+G\) in \(G\) is the parabolic \(P(-\nu)\) generated by \(T\) and the \(U_\gamma\) with \(\langle\gamma,\nu\rangle\le0\). Counting \(T'\)-weight spaces gives \(4+18+7=29\), and \(\dim\operatorname{Stab}\le\dim\operatorname{Lie}\operatorname{Stab}\) in every characteristic.

  10. Let \(\mathcal P\simeq\bigoplus_{\nu,n}\mathrm{IC}_\nu[-n]\otimes E_{\nu,n}\) be finite with constant coefficients, and \(\mu\lt\mu'\) with (1) \(E_{\nu,n}=0\) unless \(\nu\le\mu'\), (2) \([\mu,\mu']=\{\mu,\mu'\}\), (3) \(E_{\mu',n}=0\) for \(n\ne0\). Then high-degree stalk classes at \(x_\mu\) are multiplicities of \(\mathrm{IC}_\mu\):

    $$H^j(\mathcal P_{x_\mu})\simeq E_{\mu,\,j+\delta(\mu)}\qquad(j\ge-\delta(\mu)).$$
    Proof
    • By the closure order and (1), (2), only \(\nu\in\{\mu,\mu'\}\) have stalks at \(x_\mu\).
    • \(\mathrm{IC}_\mu[-n]\otimes E_{\mu,n}\) contributes \(E_{\mu,n}\) in degree \(n-\delta(\mu)\).
    • Chen–Yi, Cor. 14(ii): \(\deg P_{\mu;\mu'}\le\tfrac12(\delta(\mu')-\delta(\mu)-1)\), i.e. \(H^j(\mathrm{IC}_{\mu'})_{x_\mu}=0\) for \(j\ge-\delta(\mu)\). By (3) no shifted copy of \(\mathrm{IC}_{\mu'}\) reaches these degrees.

  11. The top class of \(F_k\) lies in a degree only \(\mathrm{IC}_{\mu_k}\) reaches, so \(\mathrm{IC}_{3k\omega_1^\vee}\) is a direct summand of \(V_{\lambda_k}\ast\mathrm{IC}_0\):

    $$E_{\mu_k,n}(\lambda_k)\simeq H^{n+8}(F_k)\ \ (n\ge0),\qquad E_{\mu_k,2\dim F_k-8}\ne0,\qquad2\dim F_k-8\ge6 .$$
    Proof
    • Apply step 10 to \(\mathcal P=V_{\lambda_k}\ast\mathrm{IC}_0\), \(\mu=\mu_k\), \(\mu'=\mu_k'\): (1) is the support bound (D0), (2) is step 6, (3) is step 8.
    • By step 7, \(H^j(\mathcal P_{x_{\mu_k}})=H^{j+\delta(\mu_k')}(F_k)\); with \(n=j+\delta(\mu_k)\) and \(\delta(\mu_k')-\delta(\mu_k)=8\) this is \(H^{n+8}(F_k)\).
    • \(F_k\) is projective of dimension \(d\ge7\) (steps 7, 9), so \(H^{2d}(F_k)\ne0\); also \(H^8(F_k)\ne0\), the fourth power of an ample class. So \(\mathrm{IC}_{\mu_k}[-(2d-8)]\) and \(\mathrm{IC}_{\mu_k}\) are summands, hence retracts, and \(\mathrm{IC}_{\mu_k}\in\mathcal C_K^{\mathrm{Sat}}\).

  12. Every \(\mathrm{IC}_\mu\) is a retract of a Satake object, so Proposition I gives generation over \(\overline{\mathbb F}_q\), and D5 carries it to \(\mathbb C\):

    $$\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K .$$
    Proof
    • Mirror labels \((0,3k)\). Rerun steps 6–11 with nodes 1 and 6 exchanged: \(\lambda=\omega_1^\vee+c_k\omega_6^\vee\), Levi \(D_5\) on nodes \(1,\dots,5\); eight fixed points over \((0,3k)\), one over \((2,3k-1)\), orbit bound 7. Transport by \(\sigma\) would first require normalizing \(\sigma\) to preserve \(G^\theta\), since \(\sigma\theta\sigma^{-1}=\operatorname{Int}(t)\circ\theta\) in general.
    • Covered labels. For \(\mu=\varphi(\lambda)\), the restriction of \(V_\lambda\ast\mathrm{IC}_0\) to \(\mathcal O_\mu\) is a nonzero sum of shifted constant sheaves and its support lies in \(\overline{\mathcal O_\mu}\) (Proposition III, step 6, under H3); only \(\mathrm{IC}_\mu\) meets \(\mathcal O_\mu\), so some \(E_{\mu,n}(\lambda)\ne0\).
    • Generation. The uncovered labels are exactly \((3k,0)\) and \((0,3k)\), \(k\) odd. Proposition I: under H1–H3 and D2, \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) iff every \(\mathrm{IC}_\mu\) is a retract of a Satake object.
    • Betti. Each summand statement concerns finite-type data over \(\mathbb Z[1/2]\); D5 transfers it, label by label, to the Betti category, where Proposition I again gives \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\).

  13. Heuristic. If \(F_k\cong Q^7\), the quadric of isotropic lines in the nine-dimensional representation of \(\mathrm{Spin}_9\), then the multiplicity space is seven-dimensional, and its graded trace is the function-level coefficient \(c_8(q)-1\):

    $$E_{\mu_k,\bullet}(\lambda_k)\simeq\overline{\mathbb Q}_\ell^{\,7}\ \text{in degrees }-6,-4,\dots,6,\qquad P_{\mu_k;\mu_k'}=1,\qquad q^{-3}+\dots+q^{3}=c_8(q)-1 .$$
    Why it is plausible, and why it is not used
    • \(\chi(Q^7)=8\) matches the eight fixed points of step 9, and the \(\mathrm{Spin}_9\)-orbit has dimension \(7=\dim Q^7\).
    • \(c_8(q)-1\) is the coefficient of \(M_{(3,0)}\) in \(r(m_{\omega_1^\vee+\omega_6^\vee})\): the graded multiplicity of the \(\check G_X=\mathrm{PGL}_3\)-representation of highest weight \((3,0)\) in the 650-dimensional representation.
    • \(F_k\cong Q^7\) is not proved, and identifying \(V_\lambda\ast\mathrm{IC}_0\) with Nadler’s specialization needs the real–symmetric equivalence, which is not among the sources. Steps 11–12 need only \(\dim F_k\ge7\).

  14. Inside the trace dictionary of Proposition II, the two statements concern different directions of one matrix:

    $$\text{(SG) over }\mathbb F_q(\!(t)\!)\iff\text{the columns of }(m_{\nu\lambda})\text{ span},\qquad\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\iff\text{every row of }(E_{\nu,\bullet}(\lambda))\text{ is nonzero}.$$
    Proof
    • Columns. Over \(\mathbb F_q(\!(t)\!)\), \(q\) odd, under D0–D3: \(K\backslash X(F)=\Lambda_S^+\) (D3), the \(\operatorname{tr}\mathrm{IC}_\nu\) are unitriangular in the orbit indicators, \(\operatorname{tr}(V_\lambda\ast\mathrm{IC}_0)=\operatorname{tr}(V_\lambda)\ast\varphi_0\), and the \(\operatorname{tr}V_\lambda\) span \(\mathcal H_K\otimes\overline{\mathbb Q}_\ell\). So the column span is \(\mathcal H_K\varphi_0\otimes\overline{\mathbb Q}_\ell\).
    • Rows. By Proposition I, \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) iff each \(\mathrm{IC}_\nu\) is a summand of some Satake object. Proposition II proves that (SG) forces this; the converse does not follow.
    • At \(\mu_k\). Every column with \(E_{\mu_k,\bullet}(\lambda)\ne0\) has \(\varphi(\lambda)\gt\mu_k\), so the summand of step 11 lies on a strictly larger support: it detects \(\mathrm{IC}_{\mu_k}\) but gives no column supported in \(\{\nu\le\mu_k\}\) and nonzero at \(\mu_k\). For \(E_6/F_4\) the columns do not span: (SG) fails over \(\mathbb F_q(\!(t)\!)\) by part (a), and the indicator of the orbit of \(\mu_1=3\omega_1^\vee\) is not in the column span (Proposition VII (b)(4)).
    • The mechanism. Step 3 kills the span through a Weyl coincidence of shifted points, without removing any orbit label; the glued points move with \(q\). In Sakellaridis’s Example 8.0.1, where \(\delta(X)\) is trivial, they do not.

  15. The chain of fields is closed: close local fields carry Theorem 8.0.2 from \(\mathbb Q_q(p^{1/e})\) to \(\mathbb F_q(\!(t)\!)\), so the function half holds over \(\mathbb F_q(\!(t)\!)\), whose geometric counterpart \(\overline{\mathbb F}_q\) carries the sheaf half:

    $$C_K\big(\mathbb F_q((t))\big)\simeq S,\qquad\mathcal H_K*\varphi_0\longleftrightarrow\operatorname{im}r\subsetneq S.$$

    The equal-characteristic identification is the conclusion of VI; its close-field argument compares each fixed finite collection of structure constants. The image of the cyclic submodule is a proper subspace of the full module.

    Proof
    • The bridge. For \(e\gg0\) the rings of integers of \(\mathbb F_q(\!(t)\!)\) and \(\mathbb Q_q(p^{1/e})\) agree modulo a high power of the uniformizer, and membership in Cartan and Iwasawa cells is decided modulo a fixed power. So the Cartan decomposition and every structure constant of the Hecke action on \(C_K\) agree over the two fields, and Theorem 8.0.2 over \(\mathbb Q_q(p^{1/e})\) gives the \(\mathcal H_K\)-isomorphism over \(\mathbb F_q(\!(t)\!)\), with \(\varphi_0\mapsto1\) (run sakellaridis-char-p: Lemma 4.2, Proposition 4.5, Proposition 4.6, Theorem 4.8; audited in Audit 5.1, Audit 5.3, Objection 5.4, Audit 5.6).
    • The former gap. Theorem 8.0.2 assumes characteristic \(0\), and no comparison of \(C_K\) as an \(\mathcal H_K\)-module between \(p\)-adic and Laurent-series fields was available. The bridge is that comparison, for \(\mathcal M_\theta\).
    • Consequence. The computation of \(r\) in steps 2–3 does not depend on the field, so (SG) fails over \(\mathbb F_q(\!(t)\!)\), and \(C_K\) needs exactly two generators there (Proposition VII). Two independent eigenfunctionals with the character of \(q^{\rho-\omega_4}\), which no cyclic module has, exist there too (Proposition VII (b)(3)).
    • Not needed. A Hecke count on the \(E_6(\mathcal O)\)-orbits of the norm-one Albert hypersurface \(\{N=1\}\subset H_3(\mathbb O)\) would be a second, independent route.

  16. By (a) and (b), exactly one of the two statements holds for \(E_6/F_4\), and both concern \(\mathbb F_q(\!(t)\!)\) and its geometric counterpart:

    $$\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\ \ (\overline{\mathbb F}_q,\ p\ne2,\ \text{D0–D2};\ \text{Betti via D5}),\qquad\text{(SG) fails}\ \ (\mathcal M_\theta\ \text{over }\mathbb F_q(\!(t)\!)\ \text{and every }p\text{-adic }F,\ p\ \text{odd}).$$

    On the function side \(C_K\) needs exactly two generators (Proposition VII), with no open hypothesis: Theorem 8.0.2 holds for \(\mathcal M_\theta\) over both fields (Proposition VI), granted the cited inputs of Proposition VII, step 11.

    By Proposition II, no reverse divergence (functions hold, sheaves fail) occurs over \(\mathbb F_q(\!(t)\!)\). The run’s first verdict stated this divergence with the sheaf inputs only; the audit added the function-side hypotheses and the field caveat (A8-10, A7-31, C-1, R-5; verdict §2, §4, §6). The run sakellaridis-char-p proved those hypotheses and removed the field caveat.