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Definitions

• Groups, levels and the sheaf statement

  \(q\) is a power of an odd prime, \(k=\overline{\mathbb F}_q\) with \(\overline{\mathbb Q}_\ell\)-coefficients. \(G\supseteq B\supseteq T\) is split connected reductive over \(\mathbb F_q\), \(X\) a smooth affine \(G\)-variety over \(\mathbb F_q\), \(K=L^+G\supseteq I\) the Iwahori. \(\widetilde W=X_*(T)\rtimes W\) with \(W\) the finite Weyl group; \(s\) runs over the simple affine reflections, \(P_s=I\cup IsI\) is the parahoric of \(s\), and \(\Omega\) is the set of length-zero elements.

  $$\mathcal C_I=\bigcup_{N,M}\mathcal C_I^{(N,M)},\qquad\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathrm{IC}_w\ast_I\mathrm{IC}_0:w\in\widetilde W\}.$$

  \(\mathcal D\) is the set of pairs (piece of a fine level, irreducible equivariant local system); the simple objects are \(\mathcal S_I=\{\mathrm{IC}_Q:Q\in\mathcal D\}\). \(\mathcal M_I\) collects those that are shifted direct summands of some generator, \(\mathcal M_I^+\supseteq\mathcal M_I\) those that are perverse constituents of one.

• Functions over \(F=\mathbb F_q(\!(t)\!)\): (IG) and (SG)

  \(\mathcal O=\mathbb F_q[\![t]\!]\). \(\mathcal H_I=C_c(I\backslash G(F)/I)\) with \(\operatorname{vol}I=1\) and basis \(T_w=\mathbf 1_{IwI}\), so \((T_s-q)(T_s+1)=0\). The functions and the basic function:

  $$C_I=C_c^\infty(X(F))^I\supseteq C_K=C_c^\infty(X(F))^K,\qquad\varphi_0=\mathbf 1_{X(\mathcal O)} .$$

  Iwahori and spherical generation:

  $$\text{(IG)}\quad\mathcal H_I\ast\varphi_0=C_I,\qquad\qquad\text{(SG)}\quad\mathcal H_K\ast\varphi_0=C_K .$$
• Weil classes and Chen–Yi's module \(\mathbb M\)

  \(A=\overline{\mathbb Q}_\ell^\times/\mu\), \(\mu\) the roots of unity (Chen–Yi §8.2). \(\mathbb M\) is the Grothendieck group of Weil objects of \(\mathcal C_I\) with Frobenius eigenvalues in \(q^{\mathbb Z/2}\subset A\), and \(\mathbb H\) the same for \(\mathcal H_{c,I}\); \(\mathbb H\) is an algebra acting on \(\mathbb M\) by convolution (Chen–Yi (27), Prop. 70). With \(v[\mathcal F]=[\mathcal F[1](\tfrac12)]\), \(\mathbb M\) is free over \(\mathbb Z[v^{\pm1}]\) on the standard classes \(\mathbf 1_Q=[j_{Q!}\mathcal L_Q]\) (Theorem 1, where \(q^{1/2}=-v^{-1}\)).

  $$T_w=[j_{w!}k],\qquad C'_w=[\mathrm{IC}_w],\qquad\underline H_s=C'_s=v(T_s+1),\qquad m_0=[\mathrm{IC}_0],\qquad c_Q=[\mathrm{IC}_Q],$$

  each intersection complex with its Weil structure pure of weight \(0\). Then \(C'_wm_0=[\mathrm{IC}_w\ast_I\mathrm{IC}_0]\).

• IC coordinates, supports and specialization

  The \(c_Q\) form a \(\mathbb Z[v^{\pm1}]\)-basis of \(\mathbb M\), unitriangular against the \(\mathbf 1_Q\) (Chen–Yi, Theorems 3–4). Coordinates and supports:

  $$x=\sum_Q\kappa_Q(x)\,c_Q,\qquad\operatorname{supp}_c(x)=\{Q\in\mathcal D:\kappa_Q(x)\ne0\}.$$

  The specialization at the residue field:

  $$\mathrm{sp}_q:\mathbb Z[v^{\pm1}]\to\overline{\mathbb Q}_\ell,\ \ v\mapsto-q^{-1/2},\qquad\mathbb M_q=\mathbb M\otimes_{\mathrm{sp}_q}\overline{\mathbb Q}_\ell .$$
• Sign type and positive certificates

  A \(\mathbb Z[v^{\pm1}]\)-linear \(\Lambda:\mathbb M\to\mathbb Z[v^{\pm1}]\) is of sign type if, for every simple affine \(s\), every \(\omega\in\Omega\) and every \(x\in\mathbb M\),

  $$\Lambda(\underline H_sx)=0,\qquad\Lambda(T_\omega x)=\pm\Lambda(x);\qquad\text{equivalently}\ \ \Lambda(T_sx)=-\Lambda(x).$$

  A positive certificate for \(Q\in\mathcal D\) is a linear \(\Lambda\) with

  $$\Lambda(C'_wm_0)=0\ \ \forall w,\qquad\Lambda(c_{Q'})\in\mathbb Z_{\ge0}[v^{\pm1}]\ \ \forall Q',\qquad\Lambda(c_Q)\ne0 .$$
• Closed and non-closed \(I\)-orbits of a symmetric \(X\)

  For an \(I\)-orbit \(O\) and a simple affine \(s\), \(P_sO\) is a union of at most three \(I\)-orbits; Chen–Yi, Lemma 28, sorts \(O\) into types I, IIa, IIb, IIIa, IIIb, IVa, IVb for \(s\):

  $$O\ \text{open in}\ P_sO\iff O\ \text{of type I, IIb, IIIb or IVb for}\ s .$$

  Lemma 31: \(O\) is closed iff its twisted involution \(w_O\) lies in \(\Omega\) iff \(O\) is of type IIb, IIIb or IVb for no simple affine \(s\). On a closed orbit \(c_O=v^{\delta(O)}\mathbf 1_O\).

Hypotheses (Lev), I1, I2, (DT), (Dict), (Wt) and (Sym)

(Lev), I1, I2, (DT) and (Dict) are the hypotheses of Propositions I and II, copied here. (Wt) and (Sym) are this proposition's.

• (Lev) · Level axioms.

  In the level presentation: each \(Y_{N,M'}\to Y_{N,M}\) (\(M'\ge M\)) is a Zariski-locally trivial affine-space fibration, equivariant for \(I/I_{N,M'}\to I/I_{N,M}\); each \(Y_N\subseteq Y_{N+1}\) is a closed embedding with \(Y_N=\pi_{N+1,M}^{-1}(Y_{N,M})\) for \(M\gg0\); each \(I/I_{N,M}\) is a connected linear algebraic group and \(I_{N,M}\) is pro-unipotent; and $$\pi_{N,M}^*:\ D^b_c\big(I/I_{N,M}\backslash Y_{N,M}\big)\longrightarrow\mathcal C_I\quad\text{is fully faithful with image }\mathcal C_I^{(N,M)} .$$

  Where it holds

  For the group, \(\mathbb A^n\) and \(\mathbb G_m\) (Proposition III). For the symmetric rows it is the open obligation (L_X) for Chen–Yi's placid presentation (A8-11).

• I1 · Finite levels.

  \(LX\) has a level presentation \(LX=\varinjlim_NY_N\), \(Y_N=\varprojlim_MY_{N,M}\), such that every object of \(\mathcal C_I\) is pulled back from some \(Y_{N,M}\) and each level has finitely many pieces: $$\mathcal C_I=\bigcup_{N,M}\mathcal C_I^{(N,M)},\qquad|\mathcal P_{N,M}|<\infty .$$

  Where it holds

  The finiteness half holds for symmetric \(X\) (Chen–Yi, Corollary 27(i)). The strict finite-quotient form is assumed, not proved, for the symmetric rows and for \(\mathrm{GL}_2/T\) with \(N\ge1\) (A9-5).

• I2 · Connected stabilizers.

  Every stabilizer of \(I/I_{N,M}\) on \(Y_{N,M}\) is connected; so every \(I\)-equivariant local system on a piece is constant.

  Without it

  Simple objects carry local systems; the bridge then needs (IG) over every \(\mathbb F_{q^n}(\!(t)\!)\) (A3-7), and trace tests are expanded over Frobenius orbits of constituents (A8-12).

• (DT) · Decomposition.

  Every generator is, in the equivariant category, a finite direct sum of shifted simple objects: $$\mathrm{IC}_w\ast_I\mathrm{IC}_0\ \simeq\ \bigoplus_{b}S_b[s_b]\quad\text{in }\mathcal C_I,\qquad S_b\in\mathcal S_I .$$ Over \(\overline{\mathbb F}_q\) it follows from a finite-type proper model of the convolution by purity and the stack decomposition theorem (Sun, Thm 1.2); a non-equivariant splitting does not suffice.

  Where it holds

  For symmetric \(X\): Chen–Yi, Proposition 70(ii). Under (DT), \(\mathcal M_I=\mathcal M_I^+\).

• (Dict) · Trace dictionary over \(\mathbb F_q\).

  1. Rationality. \(G\supseteq B\supseteq T\) are split over \(\mathbb F_q\), \(q\) odd; \(X\), the action, the level presentation and its transition maps are defined over \(\mathbb F_q\), so that \(LX(\mathbb F_q)=X(F)\); the ranks \(d(M,M')\) of \(Y_{N,M'}\to Y_{N,M}\) do not depend on \(N\).
  2. Topology. \(Y_N(\mathbb F_q)\) is compact open in \(X(F)\), \(\bigcup_NY_N(\mathbb F_q)=X(F)\), and the fibres of \(\pi_{N,M}\) on \(Y_N(\mathbb F_q)\), \(M\ge0\), form a basis of the topology of \(Y_N(\mathbb F_q)\).
  3. Rational orbits. Every \(I/I_{N,M}\)-orbit on \(Y_{N,M}\) is Frobenius-stable.
  4. Bounded convolution. For \(w\in\widetilde W\) and \(N\) there is \(N'\) with \(\overline{IwI}\cdot Y_N\subseteq Y_{N'}\), and for each \(M\) an \(M''\) such that \((g,x)\mapsto\pi_{N,M}(g^{-1}x)\), on \(\{(g,x)\in\overline{IwI}\times Y_{N'}:g^{-1}x\in Y_N\}\), factors through \(\overline{IwI}\times Y_{N',M''}\).

  What it gives here

  The identification \(\mathbb M_q\simeq C_I\otimes\overline{\mathbb Q}_\ell\) of \(\mathcal H_I\)-modules, \(\mathbf 1_Q\mapsto\mathbf 1_{Q(\mathbb F_q)}\) (A1-7, step 2). With (DT) it gives (Wt).

• (Wt) · Weights.

  \(X\) and the level presentation are defined over \(\mathbb F_q\); every simple object \(\mathrm{IC}_Q\) is Frobenius-stable and carries its weight-\(0\) Weil structure \(j_{!*}\mathcal L_Q[\delta](\tfrac{\delta}{2})\); \(\mathrm{IC}_w\) and \(\mathrm{IC}_0\) carry Weil structures pure of weight \(0\); and every \(\mathrm{IC}_w\ast_I\mathrm{IC}_0\) lies in \(\mathcal C_I\) and is the proper pushforward of a pure complex on a finite-type model.

  Where it holds

  (Dict) and (DT) give it (BBD 5.1.14). For symmetric \(X\): pointwise purity of the \(\mathrm{IC}_{\xi,v}\) is Chen–Yi, Theorem 81; the finite-type models need I1 (A9-8).

• (Sym) · Symmetric setting.

  \(X=G/G^\theta\) for an involution \(\theta\) over \(\mathbb F_q\), \(p\ne2\), \(G^\theta\) connected, \((B_0,T_0)\) a \(\theta\)-stable Borel pair over \(\mathbb F_q\) with \(T_0\) split (Chen–Yi §8.2), and every \(I\)-orbit on \(LX\) Frobenius-stable with connected stabilizer; so \(\mathcal D\) is the set of \(I\)-orbits \(O\), each \(\mathrm{IC}_O\) with constant coefficients.

  Where it holds

  \(G^\theta\) is connected for every symmetric row of Table 1. Connected \(I\)-stabilizers exclude the orbits of type IV with non-constant local systems; for \(\mathrm{GL}_2/T\) they are connected (A6-10, step 4).

Archived supporting argument · omitted from the concise verdict.

Proof

Step 1 names the two questions. Steps 2–3 read both in \(\mathbb M\); steps 4–5 prove (a) and show what it cannot exclude; step 6 places \(E_6/F_4\). Steps 7–9 turn exclusion tests into the positive-certificate lemma (b); steps 10–12 give the closed-orbit test (c). Step 13 concludes, with the open rows of Table 1; two remarks close the page.

1. Purpose Two questions about one module

   Both levels are statements about the classes \(C'_wm_0=[\mathrm{IC}_w\ast_I\mathrm{IC}_0]\) in \(\mathbb M\); a divergence is one of the two mixed cases:

   $$\text{sheaves: }\bigcup_w\operatorname{supp}_c(C'_wm_0)\overset?=\mathcal D,\qquad\text{functions: }\operatorname{span}\{\mathrm{sp}_q(C'_wm_0)\}\overset?=\mathbb M_q .$$
   Plan

   • Four cases. Both hold or both fail: the levels agree. Sheaves generate and (IG) fails, or the reverse: a divergence.
   • Reading. Steps 2–3 prove the two equivalences of (a).
   • Exclusion. Step 4 excludes “(IG) holds, sheaves fail”; step 5 shows that “sheaves generate, (IG) fails” cannot be excluded this way.
   • Tests. Steps 7–12 give sufficient conditions for “sheaves fail”. Run node A1-7, step 3.
2. Method Lemma: multiplicities are nonnegative

   Under (Lev) and (Wt), with \(\mathcal F=\mathrm{IC}_w\ast_I\mathrm{IC}_0\), \(V^n_Q=\operatorname{Hom}(\mathrm{IC}_Q,{}^pH^n\mathcal F)\) and \(d^n_Q=\dim V^n_Q\):

   $$C'_wm_0=\sum_Qm_{Q,w}\,c_Q,\qquad m_{Q,w}=\sum_nd^n_Q\,v^{-n}\in\mathbb Z_{\ge0}[v^{\pm1}],\qquad m_{Q,w}\ne0\iff\mathrm{IC}_Q\ \text{is a perverse constituent of}\ \mathcal F .$$
   Proof

   • Purity. \(\mathcal F\) is a proper pushforward of a pure complex of weight \(0\), hence pure of weight \(0\) (BBD 5.1.14). So \({}^pH^n\mathcal F\) is pure of weight \(n\) (BBD 5.4.1) and geometrically semisimple (BBD 5.3.8); it is semisimple in the equivariant heart because \(I/I_{N,M}\) is connected.
   • Isotypic parts. \(\operatorname{End}\mathrm{IC}_Q=\overline{\mathbb Q}_\ell\) and \(\mathrm{IC}_Q\) is Frobenius-stable, so \({}^pH^n\mathcal F\simeq\bigoplus_Q\mathrm{IC}_Q\otimes V^n_Q\) as Weil objects, with \(V^n_Q\) pure of weight \(n\).
   • Eigenvalues. They occur on stalks of \({}^pH^n\mathcal F\), hence on stalks of \(\mathcal F\) (the perverse spectral sequence degenerates, BBD 5.4.5), and \([\mathcal F]\in\mathbb M\): they lie in \(q^{n/2}\mu\). So \([V^n_Q]=d^n_Q[q^{n/2}]=d^n_Q(-1)^nv^{-n}\).
   • Output. The truncation triangles give \([\mathcal F]=\sum_n(-1)^n[{}^pH^n\mathcal F]=\sum_Q\big(\sum_nd^n_Qv^{-n}\big)c_Q\); a coefficient vanishes iff every \(d^n_Q=0\). Audit A9-8, step 2.
3. Result Sheaves read the support, functions the span

   The two equivalences of (a):

   $$\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\iff\bigcup_w\operatorname{supp}_c(C'_wm_0)=\mathcal D,\qquad\text{(IG)}\iff\operatorname{span}\{\mathrm{sp}_q(C'_wm_0)\}_w=\mathbb M_q .$$
   Proof

   • Support. Assume (Lev), I1, (Wt), (DT). By step 2, \(Q\in\operatorname{supp}_c(C'_wm_0)\) iff \(\mathrm{IC}_Q\) is a perverse constituent of \(\mathrm{IC}_w\ast_I\mathrm{IC}_0\), iff (by (DT)) a shifted summand. So \(\mathcal M_I=\mathcal M_I^+=\{\mathrm{IC}_Q:Q\in\bigcup_w\operatorname{supp}_c(C'_wm_0)\}\), and the criterion \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\iff\mathcal M_I=\mathcal S_I\) (Proposition I) concludes.
   • Specialization. Under (Dict) the trace of Frobenius identifies \(\mathbb M_q\) with \(C_I\otimes\overline{\mathbb Q}_\ell\) as an \(\mathcal H_I\)-module: \(\mathbf 1_Q\mapsto\mathbf 1_{Q(\mathbb F_q)}\), \(T_w\mapsto T_w\), and \(m_0\) goes to a nonzero multiple of \(\varphi_0\); the coefficients of Chen–Yi's (29) are point counts.
   • Span. \(C'_w=v^{\ell(w)}T_w+\sum_{y\lt w}p_{y,w}T_y\) specializes to an element of \(\mathcal H_I\) unitriangular in the \(T_y\), so \(\operatorname{span}\{\mathrm{sp}_q(C'_wm_0)\}=\mathcal H_I\ast\varphi_0\otimes\overline{\mathbb Q}_\ell\). (IG) over \(\mathbb C\) and over \(\overline{\mathbb Q}_\ell\) agree: a rank condition on integral matrices (A3-4(iii), audited in A7-7).
4. Result Span ⇒ support: functions generate, so sheaves generate

   Under (Lev), I1, I2, (Dict) and (DT), the lower-left case does not occur:

   $$\text{(IG) over }\mathbb F_q(\!(t)\!)\ \Longrightarrow\ \bigcup_w\operatorname{supp}_c(C'_wm_0)=\mathcal D\ \Longrightarrow\ \mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\ \text{over}\ \overline{\mathbb F}_q .$$
   Proof

   • Basis. The \(\mathrm{sp}_q(c_Q)\) are a basis of \(\mathbb M_q\) (unitriangular), and the \(Q\)-coordinate of \(\mathrm{sp}_q(C'_wm_0)\) is \(\mathrm{sp}_q(m_{Q,w})\).
   • Coordinates. If the \(\mathrm{sp}_q(C'_wm_0)\) span \(\mathbb M_q\) (step 3), the coordinate form \(\kappa_Q\) is nonzero on one of them: \(\mathrm{sp}_q(m_{Q,w})\ne0\), hence \(m_{Q,w}\ne0\) and \(Q\in\operatorname{supp}_c(C'_wm_0)\). Step 3 concludes.
   • Witness. If \(\mathrm{IC}_Q\notin\mathcal M_I\), the coordinate form kills \(\mathcal H_I\ast\varphi_0\) and not \(\operatorname{tr}\mathrm{IC}_Q\): (IG) fails.
   • Source. This is Proposition II: run node A3-6, audited in A7-7. With disconnected stabilizers, (IG) over every \(\mathbb F_{q^n}(\!(t)\!)\) is needed (A3-7). Nothing is claimed over \(\mathbb C\).
5. Ruled out Support ⇒ span

   The upper-right case is not excluded by this argument. Two simple objects and columns \(c_1+c_2\), \(2c_1+2c_2\): every row is nonzero, the columns span a line:

   $$m=\begin{pmatrix}1&2\\1&2\end{pmatrix},\qquad\text{rows}\ne0,\qquad\operatorname{rank}m=1 .$$
   Proof

   • One \(n\). Cancellation across degrees: \(2a-b=0\) with \(|b|=q^{1/2}|a|\) at \(q=4\). Traces over all \(\mathbb F_{q^n}\) determine a class; one \(n\) does not (Laumon, Theorem 1.1.2).
   • What sheaves give. From \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\), only nonzero rows of \(m^{(n)}\), each at some \(n\le\dim\bigoplus_iE_{S,i}(w)\) (A3-10(a)).
   • Spherical instance. For \(E_6/F_4\) sheaves generate at level \(K\) and \(C_K=\mathcal H_K\ast\varphi_0\oplus\mathbb C\mathbf 1_{O_{3\omega_1^\vee}}\) over every \(\mathbb F_{q^n}(\!(t)\!)\) (satake Proposition IX). No Iwahori-level instance is known.
   • Common sufficient condition. Weighted triangularity: a Weil object \(\mathcal G\ast_IY\), \(Y\in\{\mathrm{IC}_0\}\cup\{\Delta_{P'}:P'\lt P\}\), supported on \(\overline P\) with nonzero trace on \(P(\mathbb F_q)\), for every piece \(P\), gives (IG) by induction on the finite poset of pieces (A3-10(c)). Audit A7-10: holds as stated. This completes (a).
6. Result E₆/F₄: (IG) fails

   For \(X=E_6/F_4\), (IG) fails over \(\mathbb F_q(\!(t)\!)\) and over \(p\)-adic \(F\), \(p\) odd; so \(E_6/F_4\) is a divergence iff \(\mathcal M_I=\mathcal S_I\):

   $$\text{(IG)}\ \Longrightarrow\ \text{(SG)},\qquad C_K=\mathcal H_K\ast\varphi_0\oplus\mathbb C\,\mathbf 1_{O_{3\omega_1^\vee}} .$$
   Proof

   • (IG) ⇒ (SG). With \(e_K=[K:I]^{-1}\mathbf 1_K\): \(C_K=e_K\ast C_I=e_K\ast\mathcal H_I\ast\varphi_0=\mathcal H_K\ast\varphi_0\) (classical page, Proposition B; HKP §4.1).
   • (SG) fails. Over every \(\mathbb F_{q^n}(\!(t)\!)\): satake run, Propositions VI, VII and IX(a); over \(p\)-adic \(F\), \(p\) odd, through Sakellaridis's Theorem 8.0.2.
   • Sheaves. Spherical sheaf generation holds (satake Proposition VIII(b)); the Iwahori sheaf level is open. Run node A3-12, item 4.
7. Method Exclusion in trace form

   Under the inputs of (a) and purity, sheaf generation is a statement about functions over all \(F_n=\mathbb F_{q^n}(\!(t)\!)\); an exclusion must control every \(n\):

   $$\mathrm{IC}_Q\in\mathcal M_I\iff\exists n:\ \kappa^{(n)}_Q\big(\mathcal H_{I_n}\ast\varphi_0\big)\ne0 .$$
   Proof

   • (i) Coordinates. Weights separate the degrees in \(m^{(n)}_{Q,w}=\sum_i(-1)^i\operatorname{tr}A_{Q,i}(w)^n\), and a Vandermonde determinant bounds \(n\) (A3-11(a)).
   • (ii) Sign module. For \(W\ne1\), a form \(\lambda_n\) on \(C_{I_n}\) with \(\lambda_n(T_s\ast f)=-\lambda_n(f)\) kills \(\mathcal H_{I_n}\ast\varphi_0\) (as \(T_s\ast\varphi_0=q^n\varphi_0\) for finite \(s\)) and every \(P_s\)-equivariant trace; if at every \(n\) it vanishes on the other non-equivariant simple objects and not on \(S_0\), then \(S_0\notin\mathcal M_I\) (A3-11(c), audited in A7-10).
   • (iii) Local systems. With I2 dropped: \(\lambda_r(h\ast f)=\varepsilon_r(h)\lambda_r(f)\), \(\lambda_r(\operatorname{tr}_rS)=0\) for every \(F^r\)-stable simple \(S\not\simeq S_0\), \(\lambda_r(\operatorname{tr}_rS_0)=c\beta^r\), \(c\ne0\), \(S_0\) pure: then \(S_0\notin\mathcal M_I^+\). The sign-obstruction lemma A2-10(e), with the repair of A8-12: traces are expanded over Frobenius orbits of constituents, and an orbit of length \(d\) contributes \(0\) to \(\operatorname{tr}_r\) unless \(d\mid r\).
   • Output. Each test needs control at every \(n\). Positivity replaces it by one condition on the classes \(c_Q\) (steps 8–9).
8. Method Lemma: sign maps kill the generators

   If \(\Lambda\) is of sign type, and \(W\ne1\) or \(\Lambda(m_0)=0\), then

   $$\Lambda(m_0)=0\qquad\text{and}\qquad\Lambda(C'_wm_0)=0\quad(w\in\widetilde W).$$
   Proof

   • \(m_0\). For a finite simple \(s\in W\), \(\mathrm{IC}_0\) is \(K\)-equivariant and \(P_s/I\simeq\mathbb P^1\), so \(\mathrm{IC}_s\ast_I\mathrm{IC}_0\simeq\mathrm{IC}_0[1](\tfrac12)\oplus\mathrm{IC}_0[-1](-\tfrac12)\), i.e. \(\underline H_sm_0=(v+v^{-1})m_0\). Then \((v+v^{-1})\Lambda(m_0)=0\) in the domain \(\mathbb Z[v^{\pm1}]\).
   • Generators. The \(\underline H_s\) and \(T_\omega\) generate \(\mathbb H\); write \(C'_w\) in monomials \(\underline H_{s_1}\cdots\underline H_{s_r}T_\omega\). A monomial with \(r\ge1\) is killed by its first factor, and \(\Lambda(T_\omega m_0)=\pm\Lambda(m_0)=0\).
   • Repair. A finite simple reflection is needed (audit A9-8, step 1): for \(\mathrm{GL}_1\) (\(\mathbb A^1\)) \(\Lambda(m_0)=0\) must be assumed. Run node A6-7(a). This is (b)(i).
9. Result Positive certificates exclude, and are sharp

   Under (Lev), I1 and (Wt), (b)(ii)–(iii):

   $$\Lambda(C'_wm_0)=0\ \forall w,\ \ \Lambda(c_{Q'})\in\mathbb Z_{\ge0}[v^{\pm1}]\ \forall Q',\ \ \Lambda(c_Q)\ne0\ \Longrightarrow\ \mathrm{IC}_Q\notin\mathcal M_I^+\ \Longrightarrow\ \mathrm{IC}_Q\notin\mathcal C_I^{\mathrm{Iwa}} .$$
   Proof

   • No cancellation. By step 2, \(0=\Lambda(C'_wm_0)=\sum_Qm_{Q,w}\Lambda(c_Q)\), each term in \(\mathbb Z_{\ge0}[v^{\pm1}]\). A sum of Laurent polynomials with nonnegative coefficients vanishes only termwise, and \(\mathbb Z[v^{\pm1}]\) is a domain; so \(\Lambda(c_Q)\ne0\) forces \(m_{Q,w}=0\) for every \(w\): \(\mathrm{IC}_Q\notin\mathcal M_I^+\).
   • Membership. The objects whose perverse cohomology has all constituents in \(\mathcal M_I^+\) form a thick subcategory \(\mathcal E\) (Stacks, Lemma 13.6.3) containing every generator, hence \(\mathcal C_I^{\mathrm{Iwa}}\), and not \(\mathrm{IC}_Q\).
   • (⇐) is the above. (⇒) Take \(\Lambda=\kappa_Q\): \(\kappa_Q(C'_wm_0)=m_{Q,w}=0\) by step 2, \(\kappa_Q(c_{Q'})\in\{0,1\}\), \(\kappa_Q(c_Q)=1\).
   • Scope. Positivity is needed on every simple object; \(\kappa_Q\) is not of sign type, so the sharp form gives no test computable from (29) alone. Run A6-7(b),(c); audit A9-8.
10. Method Lemma: non-closed orbits are invisible

    Under (Sym), a sign-type \(\psi\) vanishes on every orbit that is not closed, (c)(i):

    $$O\ \text{not closed}\ \Longrightarrow\ \overline O=\overline{P_sO}\ \text{for some}\ s\ \Longrightarrow\ \underline H_sc_O=(v+v^{-1})c_O\ \Longrightarrow\ \psi(c_O)=0 .$$
    Proof

    • Lemma 31. \(O\) is closed iff it is of type IIb, IIIb or IVb for no simple affine \(s\).
    • Lemma 28. An orbit of type IIb, IIIb or IVb for \(s\) is open in \(P_sO\), so \(\overline O\) is \(P_s\)-stable.
    • Equivariance. With constant coefficients (Sym), \(\mathrm{IC}_O\) is \(P_s\)-equivariant: \(P_s\) acts at a finite level through a connected group. Step 8's computation applies to \(c_O\).
    • Source. Run A6-10, step 1; audit A9-10, step 1, with both lemmas checked in Chen–Yi.
11. Result Closed-orbit test: the sheaf half

    Under (Sym), \(W\ne1\) and the hypotheses of (b), (c)(ii):

    $$\psi(c_O)\in\mathbb Z_{\ge0}[v^{\pm1}]\ \ \text{for all closed}\ O,\quad\psi(c_{O_1})\ne0\ \Longrightarrow\ \mathrm{IC}_{O_1}\notin\mathcal C_I^{\mathrm{Iwa}} .$$
    Proof

    • Positivity everywhere. \(\psi(c_O)=0\) on non-closed orbits (step 10) and \(\ge0\) on closed ones; under (Sym) the \(c_O\) are all the simple classes.
    • Certificate. \(\psi\) kills every \(C'_wm_0\) (step 8, \(W\ne1\)), so it is a positive certificate for \(O_1\); step 9 concludes.
    • Precision. A unit multiple \(\pm v^j\psi\) may be tested instead. A row of this kind can be a divergence “sheaves generate” only if no nonzero sign-type \(\psi\) has a unit multiple that is nonnegative on all closed-orbit classes (A9-10).
12. Result Closed-orbit test: the function half

    Under (Sym), \(W\ne1\) and (Dict), (c)(iii): a nonzero specialization makes \(\psi_q=\mathrm{sp}_q\circ\psi\) a Hecke-module witness,

    $$\psi_q:\ C_I\otimes\overline{\mathbb Q}_\ell\longrightarrow\overline{\mathbb Q}_\ell,\qquad\psi_q(T_s\ast f)=-\psi_q(f),\quad\psi_q(\varphi_0)=0,\quad\psi_q\ne0 .$$
    Proof

    • Character. \(\underline H_s=v(T_s+1)\) with \(v\) a unit gives \(\psi_q(T_s\ast f)=-\psi_q(f)\); \(\psi_q(T_\omega\ast f)=\pm\psi_q(f)\). The character \(T_s\mapsto-1\) respects \((T_s-q)(T_s+1)=0\).
    • Kernel. \(\psi_q(\varphi_0)=0\) since \(\psi(m_0)=0\) (step 8); if \(\mathrm{sp}_q\psi(c_O)\ne0\) for a closed \(O\), then \(\mathcal H_I\ast\varphi_0\subseteq\ker\psi_q\subsetneq C_I\otimes\overline{\mathbb Q}_\ell\) (classical page, Lemma C(1)).
    • The hypothesis is needed. On a closed orbit \(c_O=v^{\delta(O)}\mathbf 1_O\), and the values of \(\psi\) on closed orbits are constrained only by the module relations. A positive value \(v+2v^2\) specializes at \(q=4\) to \(-\tfrac12+2\cdot\tfrac14=0\).
    • Source. Objection A9-10 to A6-10(c) and step 5, with its repair: \(\psi\) is \(\mathbb Z[v^{\pm1}]\)-valued on Chen–Yi's \(\mathbb M\), where evaluation at \(q^{1/2}\) is defined.
13. Conclusion Where a divergence can occur

    Steps 3–5 prove (a), steps 8–9 prove (b), steps 10–12 prove (c). Over \(\mathbb F_q(\!(t)\!)\), under the inputs of Proposition II, a divergence can only be “sheaves generate, (IG) fails”. The open rows of Table 1:

    Table 2. Table 1's open rows. The sheaf side is open in each; every row needs (Lev) for Chen–Yi's presentation.

    \(X\)	(IG)	what decides the row
    \(E_6/F_4\)	fails over \(\mathbb F_q(\!(t)\!)\) and \(p\)-adic \(F\), \(p\) odd	\(\mathcal M_I=\mathcal S_I\)? A divergence iff yes.
    \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), \(n\ge2\)	fails, \(\operatorname{char}F=0\)	(IG) over \(\mathbb F_q(\!(t)\!)\); then \(\mathcal M_I=\mathcal S_I\)?
    \(\mathrm{GL}_{2p}/\mathrm{GL}_p^2\), \(\mathrm{GL}_3/\mathrm{GL}_1\times\mathrm{GL}_2\)	fails, \(\operatorname{char}F=0\)	(IG) over \(\mathbb F_q(\!(t)\!)\); then \(\mathcal M_I=\mathcal S_I\)?
    \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(q\gt p\), \(n\ge4\); \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\), \(n\ge4\)	open	\(\mathcal M_I=\mathcal S_I\)? If not, (IG) fails too, given the inputs of (a).

    In every row, a nonzero sign-type \(\psi\) that is nonnegative on the closed orbits decides the sheaf side as “sheaves fail” (step 11).

    Proof

    • Function half through (a). Where the inputs of (a) hold as well, (a) turns the sheaf half into a failure of (IG) over \(F\) without the specialization, through the coordinate form of step 4; the specialization makes \(\psi_q\) itself the witness.
    • Rows. Function-level entries as on the verdict; the characteristic-\(0\) failures do not reach (a), which needs \(\mathbb F_q(\!(t)\!)\). Run nodes A1-7 (Tables A–B) and A3-12 (obligations O1–O3).

Remarks

1. Remark A functional of mixed sign is not enough

   For \(X=\mathrm{GL}_2/T\) (closed orbits \(A_\pm\)), the Steinberg functional \(\lambda\) and its twist \(\psi\) both kill the generators; only \(2\psi\) is positive:

   $$\lambda(c_{A_\pm})=\pm1,\qquad2\psi(c_{A_\pm})=1,\qquad\lambda=\psi=0\ \text{on the other simple classes}.$$
   Proof

   • Values. \(\lambda(\mathbf 1_{A_\pm})=\pm1\), \(\lambda(\mathbf 1_{O_{i,m}})=0\); \(2\psi(\mathbf 1_{A_\pm})=1\), \(2\psi(\mathbf 1_{O_{i,m}})=2(-1)^{m+1}\). Both satisfy \(\Lambda\circ T_s=-\Lambda\); \(\lambda\circ T_\omega=-\lambda\), \(\psi\circ T_\omega=\psi\).
   • \(\lambda\). \(0=\lambda(C'_wm_0)=m_{A_+,w}-m_{A_-,w}\): a summand \(\mathrm{IC}_{A_+}\oplus\mathrm{IC}_{A_-}\) would be invisible to it.
   • \(2\psi\). A positive certificate: \(\mathrm{IC}_{A_\pm}\notin\mathcal C_I^{\mathrm{Iwa}}\), given I1 (the sheaf half of Proposition IV again). \(\mathrm{sp}_q(2\psi)(c_{A_+})=1\ne0\), so (IG) fails too. \(\lambda+2\psi\) and \(2\psi-\lambda\) certify one closed orbit each.
   • Functions. Either functional witnesses the failure of (IG) (classical page, Proposition F, uses \(\lambda\)). Run A6-8; audit A9-9.
2. Remark The spherical model

   At level \(K\) the same two statements are rows and columns of one matrix. Functions imply sheaves over \(\mathbb F_q(\!(t)\!)\), and the converse fails for \(E_6/F_4\):

   $$\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K,\qquad C_K=\mathcal H_K\ast\varphi_0\oplus\mathbb C\,\mathbf 1_{O_{3\omega_1^\vee}} .$$
   Proof

   • Sources. Satake run, Proposition II (functions ⇒ sheaves) and Proposition IX (sheaves ⇏ functions).
   • Iwahori form. Part (a) is the Iwahori form of II. The Iwahori form of IX is open, with \(E_6/F_4\) the candidate (step 6).