Proposition IX · Does sheaf generation imply function generation?

Fresh audit · 21 October 2026

15 errors in 73 reviewed steps; three reviewers and three fresh salvage agents, compared with five agents in the original run. All repairs passed re-check: 18 root inputs held, five were salvaged, none was lost. The general negative verdict is unchanged.

Final derivation → · Reviews and re-checks → · Restructured proof →

The norm-torus proof is unconditional. The split categorical example retains (O1)–(O3); the separate E₆/F₄ implication retains VI–VIII and D0–D2.

1 · The answer

No. Sheaf generation over \(\overline{\mathbb F}_q\) does not imply spherical generation over \(\mathbb F_q(\!(t)\!)\): $$\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\quad\not\Longrightarrow\quad\mathcal H_K\ast\varphi_0=C_K\ \text{ over }F=\mathbb F_q(\!(t)\!).$$

The first family is a complete counterexample in characteristic \(p\), outside (R). The second line of attack gives a split, non-symmetric counterexample whose function half is proved and whose sheaf half rests on one named input. Inside (R) and (C), for symmetric \(X\), the question stays open and reduces to the labels without dominant lifts.

This is Proposition IX of the satake-two-levels workspace, the converse of Proposition IIc and of separate interest: statement and attempt tree · the trace formalism. The agents' steps call it IId, its working name.

2 · The counterexamples

Unitary twists (Theorem 2.6, Counterexample 1.5; audit Audit 4.5, Summary 4.12). Let \(H\ne1\) be split connected reductive over \(\mathbb F_q\), \(q\) odd, with a Chevalley involution \(\vartheta\). Put \(G=\operatorname{Res}_{\mathbb F_{q^2}/\mathbb F_q}H\), let \(\sigma\) be its Galois involution, \(\theta=\vartheta\circ\sigma\) and \(X=G/G^\theta\). Then, with \(E=\mathbb F_{q^2}(\!(t)\!)\), $$X(F)=\{x\in H(E):\vartheta(\sigma x)=x^{-1}\},\qquad g\cdot x=g\,x\,\vartheta(\sigma g)^{-1}.$$ For \(H=\mathrm{GL}_n\) these are the nondegenerate hermitian matrices, acted on by \(x\mapsto gxg^*\).

1. Sheaves. Over \(k\), \(G_k=H\times H\) and \(X_k\cong H\) is the group case. The convolution gives \(V_{(c,0)}\ast\mathrm{IC}_0\cong\mathrm{IC}_c\), and all stabilizers are connected, so \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) (Lemma 2.5).
2. Functions. The rational Hecke labels are \((a,a)\), \(a\in\Lambda_H^+\). The translates satisfy \(\operatorname{supp}(T_a\ast\varphi_0)\subseteq\bigcup_{c\le2a}O_c\) with value \(\ge1\) at \(t^{2a}\). Since \(a\mapsto2a\) is injective, the leading-term lemma gives

3. Over \(F_n\). For \(n\) even, \(G\) splits and \((\mathrm{SG})_n\) holds; for \(n\) odd, the computation over \(F\) repeats.
4. Hypotheses. (C) and D0–D3 hold, and so does every clause of (R) except "\(T\supseteq S\) split" and "\(T_0\) split". The failure is witnessed by a Hecke eigenfunctional vanishing at \(\varphi_0\), since \(C_K\) is a finitely generated Hecke module (Audit 4.16).

Sakellaridis's split variety (Theorem 3.5, Corollary 3.9; audit Audit 5.5, Summary 5.12). \(X=\mathrm{SL}_2^3\backslash(\mathrm{Sp}_4\times\mathrm{Sp}_4)\) is the space of triples \((W_1,W_2,\varphi)\): nondegenerate planes \(W_i\subset V_i\) and an isometry \(\varphi:W_1\to W_2\) (Lemma 3.3).

1. Functions, proved. Restricted to \(\mathrm{Sp}_2\times\mathrm{Sp}_2\), a generic unramified principal series \(\pi=I(\chi)\) of \(\mathrm{Sp}_4\) has two non-isomorphic \(\mathrm{SL}_2\)-quotients, \(\sigma_{\chi_1\chi_2^{-1}}\) and \(\sigma_{\chi_1\chi_2}\) (Mackey theory, valid in every characteristic; Lemma 3.4, Audit 5.3). So \(\pi\boxtimes\pi\) has two independent \(H(F)\)-periods. Since (SG) forces multiplicity at most one (Lemma 3.1), (SG) fails over every non-archimedean \(F\), every \(q\), every \(n\).
2. Sheaves, modulo one input. The \(L^+G\)-orbits on \(LX\) are labelled by \((d_1,d_2,e)\) with \(e\equiv d_1-d_2\pmod 2\). Their stabilizers are connected and their dimensions are \(3d_1+3d_2+e\). Every convolution map is semismall, its multiplicities are the branching numbers for \(\mathrm{SO}_5\times\mathrm{SO}_5\downarrow\mathrm{SL}_2^3\), and every label occurs (Proposition 3.6–Proposition 3.8, confirmed by brute-force counts in Computation 5.7). Given Chen–Yi's \(\ell\)-adic framework for this non-symmetric \(X\), this yields \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\). That framework is the one missing input, split into (O1) Sakellaridis–Wang's degeneration in odd characteristic, (O2) a dimension theory, and (O3) Chen–Yi's Appendix A and Prop. 70 verbatim (Objection 5.10).
3. Hypotheses. \(G\) and \(H\) are split, Frobenius fixes every label, and the stabilizers are connected: the analogues of (R), (C), D0 and D3 hold. \(X\) is not symmetric.

3 · Why the trace of Frobenius cannot give the converse

What the trace of Frobenius gives (Theorem T, Theorem 1.3; audit Audit 4.15). Under (R), (C), D0–D3, Frobenius-twisted Hochschild homology computes both sides, concentrated in degree 0: $$\operatorname{Tr}(\mathrm{Fr},\mathcal H)\cong\mathcal H_K,\qquad\operatorname{Tr}(\mathrm{Fr},\mathcal C_k)\cong C_K,\qquad\operatorname{Tr}(\mathrm{Fr},\,F\mapsto F\ast\mathrm{IC}_0)=(f\mapsto f\ast\varphi_0).$$ So (SG) says that the third map is onto. AGKRRV's theorem, \(\operatorname{Tr}(\mathrm{Frob},\mathrm{Shv}_{\mathrm{Nilp}}(\mathrm{Bun}_G))\cong\mathrm{Funct}_c(\mathrm{Bun}_G(\mathbb F_q))\), is of the second kind: it computes the trace of a whole category. The categorical statement yields only \(\operatorname{Tr}(\mathrm{Fr},\mathcal C^{\mathrm{Sat}}_k)=\operatorname{Tr}(\mathrm{Fr},\mathcal C_k)\), which is a tautology once the two categories coincide. Functoriality produces the map \(\operatorname{Tr}(\mathcal H)\to\operatorname{Tr}(\mathcal C_k)\) but never makes it onto.

Where the image falls short. By Morita invariance, \(\operatorname{Tr}(\mathrm{Fr},\mathcal C_k)\) is spanned by classes \([N,\beta]\) of retracts \(N\) of Satake objects with arbitrary \(\beta:\mathrm{Fr}^*N\to N\). The Hecke category contributes only whole Satake objects with their canonical Frobenius: $$[V_\lambda\ast\mathrm{IC}_0,\Phi]=\sum_{\nu,i}(-1)^i\operatorname{tr}A_{\nu,i}(\lambda)\,[\mathrm{IC}_\nu].$$ Thick closure adds retracts; the trace records classes. In terms of the multiplicity matrix \(M\) over \(R=\mathbb Z[\overline{\mathbb Q}_\ell^\times]\) (Theorem A, Theorem 1.2; audit Audit 4.14): $$\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\iff\text{no row of }M\text{ is }0,\qquad(\mathrm{SG})_n\iff\operatorname{coker}(M)\otimes_{R,\mathrm{ev}_n}\overline{\mathbb Q}_\ell=0,\qquad\mathrm{Tri}=\mathcal C_k\iff\operatorname{coker}(M)\otimes_{R,\varepsilon}\mathbb Z=0 .$$ A single \(n\) is a single point of \(\operatorname{Spec}R\); Laumon's Theorem 1.1.2 recovers a class only from all \(n\) together.

The two ways the image misses, both realized.

1. Satake objects without Weil structure. If Frobenius moves \(\lambda\), then \(V_\lambda\) is not Weil, while its summands \(\mathrm{IC}_c\) are Weil objects of \(\mathcal C_k\) reached by no Weil Satake object (Lemma 2.8). This is the unitary twist, where failures occur only at odd \(n\).
2. Fusion. The columns of \(M\) are restricted characters \(\chi\mapsto\chi|_{\check G_X}\). Every irreducible representation of \(\check G_X\) occurs in some restriction, so every row is nonzero. But when two \(\check G_X\)-classes fuse in \(\check G\), the restriction of class functions is not onto. This is Sakellaridis's variety, where \((a,b,a)\) and \((b,a,b)\) fuse and (SG) fails at every \(n\).

A third shape, cancellation of Frobenius traces at \(n=1\) inside a multiplicity space, is possible in principle but did not occur (Proposition 1.8, Remark 3.11).

A self-contained account, from the definition of the categorical trace in Gaitsgory–Kazhdan–Rozenblyum–Varshavsky and Arinkin–Gaitsgory–Kazhdan–Raskin–Rozenblyum–Varshavsky to the toy models of each mechanism: Proposition IX · The trace formalism.

4 · Conditions that force a counterexample, and when the converse holds

Conditions that force (SG) to fail, over every non-archimedean \(F\) (Proposition 1.6, Lemma 2.2, Lemma 3.1; audits Audit 4.4, Audit 4.16, Audit 5.1):

1. a \(K\)-orbit of \(X(F)\) outside \(G(F)\cdot X(\mathcal O)\), for instance when \(\ker\big(H^1(F,H)\to H^1(F,G)\big)\ne1\);
2. translates \(T_\lambda\ast\varphi_0\), \(\lambda\) rational, with distinct leading orbits that miss a rational orbit label; for tori this is \(\ker\big(X_*(H)_\sigma\to X_*(T)_\sigma\big)\ne0\);
3. two independent \(H(F)\)-periods on one irreducible unramified \(\pi\);
4. a Hecke eigenfunctional on \(C_K\) vanishing at \(\varphi_0\) (equivalent to failure when \(C_K\) is finitely generated over \(\mathcal H_K\));
5. an \(\mathcal H_K\)-linear endomorphism of \(C_K\) that is not a Hecke operator;
6. a nonzero \(c\) with \(\sum_\nu c_\nu m^{(1)}_{\nu\lambda}=0\) for all \(\lambda\).

A failure over \(p\)-adic fields with residue field \(\mathbb F_q\) and unbounded ramification transfers to \(\mathbb F_q(\!(t)\!)\), given integral models and the Cartan decomposition at those fields (Lemma 1.7, Audit 4.17).

When the trace argument does prove the converse (Proposition 1.8). If every label has a dominant lift, \((\mathrm{SG})_n\) holds for every \(n\) (Corollary C of the classical page). More generally it suffices to have a triangular system of summands \(\mathrm{IC}_\mu\subseteq V_{\lambda_\mu}\ast\mathrm{IC}_0\) whose multiplicity spaces are split Tate of one parity.

5 · What remains open

Inside (R) and (C), symmetric \(X\): open. All three searchers stopped at the same obstruction, which the audits confirm (Proposition 1.8, Remark 2.12; Audit 4.18, Audit 4.19):

• Frobenius fixes every label;
• \(X(F_n)=G(F_n)x_0\);
• automorphisms of \((G,\theta)\) act trivially on \(C_K\);
• every top entry \(m^{(n)}_{\varphi(\lambda),\lambda}\) is nonzero;
• the injective-leading-term criterion never applies.

What remains is a spanning condition on the set \(U\) of labels without dominant lifts: $$(\mathrm{SG})\iff\{p_U(T_\lambda\ast\varphi_0):\lambda\in\Lambda_T^+\}\ \text{spans}\ \mathbb C^{(U)} .$$ A counterexample there needs a linear relation among the columns of \(m^{(1)}\) through an uncovered label. Such a relation comes either from gluing of spectral parameters (as \(E_6/F_4\) does \(p\)-adically under Sakellaridis's (A1)–(A5)) or from a cancellation of Frobenius traces at \(n=1\).

For the split variety. (O1)–(O3) of Objection 5.10: Sakellaridis–Wang's affine degeneration and their Lemma 2.3.10 in odd characteristic; a dimension theory compatible with Chen–Yi (8); then Chen–Yi's Appendix A and Prop. 70, which would give D1–D2.

6 · Repairs from the audits

• Dead end 3.10 mislocated the remaining sheaf-theoretic gap (placidity needs no symmetric embedding: Chen–Yi, Remark 37); Objection 5.10 replaces it by (O1)–(O3). Summary 3.13 inherits the old wording.
• Lemma 3.3(d): non-conjugacy of \((a,b,a)\) and \((b,a,b)\) needs \(a\ne\pm b^{\pm1}\), since \(\ker\iota=\{\pm1\}\).
• Lemma 3.1, Lemma 3.4, Proposition 3.8, Corollary 3.9, Remark 3.11: one-line repairs (integral change of variables; the full reducibility locus; \(H^{2f}_c\) and Lang–Weil; drop a remark on even \(n\)).
• Setup 2.4, Lemma 2.5: an \(\mathbb F_q\)-rational \(\theta\)-stable Borel pair exists (Lang–Steinberg, Audit 4.1); purity only for Frobenius-stable \(\lambda\).
• Computation 2.9 step 3 needs \(\operatorname{tr}\mathrm{IC}_{\alpha^\vee}=\mathbf 1_{O_{\alpha^\vee}}+\varphi_0\); Counterexample 1.5 step 6 needs \(\phi_1\phi_2=1\); Dead end 2.11 has an unjustified side remark.
• Theorem 1.3: formality from free even cohomology with scalar Frobenius (not Chen–Yi Thm 93); twisted additivity along the recollement and the definition of LT by functoriality to be written out (Audit 4.15). Proposition 1.8(c) covers only automorphism-induced autoequivalences; Remark 2.12 item 4 sketches one reduction.
• No conclusion changed.

7 · Team and timeline

Team. Three searchers with distinct approaches — the trace formalism (A1), non-split forms (A2) and split varieties with fusion (A3) — and two auditors. A4 checked the non-split counterexample and then the mechanism; A5 checked the split candidate. Five agents in all, against eight in the previous run. No redirect was needed: inside (R) and (C) all three searchers reached the same theorem-strength obstruction, and no new mechanism appeared.

Timeline (UTC, 17 October 2026). 20:24 statement fixed; 20:27 three searchers launched; 21:24 and 21:37 the trace and non-split agents return (the norm torus found twice); 21:40 first audit launched; 22:23 it confirms the counterexample; 22:42 the split agent returns; 22:44 two audits launched; 23:11 the mechanism audit returns; 23:37 the split audit returns; 23:42 verdict.