§6 · Root · root: registry, redirects, synthesis · 2026-10-17T23:42:00Z

Theorem 6.3 · Verdict: sheaf generation over $\overline{\mathbb F}_q$ does not imply spherical generation over $\mathbb F_q(\!(t)\!)$

proved ★ key step

The converse of IIc fails. Unitary twists of the group case are complete counterexamples in characteristic p outside (R); Sakellaridis's split variety fails at function level for every q and n and generates at sheaf level modulo Chen–Yi's framework for it. Inside (R) and (C), symmetric, the question reduces to the labels without dominant lifts.

Theorem 6.3.

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

Table 2. $X$ and $G$ split? and sheaves over $\overline{\mathbb F}_q$
$X$	$G$ split?	sheaves over $\overline{\mathbb F}_q$	(SG) over $\mathbb F_q(\!(t)\!)$	$(\mathrm{SG})_n$	status
unitary twist of $(H\times H)/H$, any split $H\ne1$	no	generate	fails	holds iff $n$ even	proved, audited twice
· norm torus $\operatorname{Res}\mathbb G_m\to\mathbb G_m$	no	generate	fails (two $G(F)$-orbits)	$n$ even	proved, audited
· hermitian matrices, $\operatorname{Res}\mathrm{GL}_n$	no	generate	fails, for $n\ge2$ already on the base $G(F)$-orbit	$n$ even	proved, audited
· hermitian $\mathrm{SL}_2$, $\det=1$	no	generate	fails with one $G(F)$-orbit and every row of $m^{(1)}$ nonzero	$n$ even	proved, audited
Galois twist $\mathrm{GL}_n(E)/\mathrm{GL}_n(F)$	no	generate	holds	all $n$	proved, audited
$\mathrm{SL}_2^3\backslash(\mathrm{Sp}_4\times\mathrm{Sp}_4)$ (Sakellaridis, Ex. 8.0.1)	yes	generate, given Chen–Yi's framework for this $X$	fails	fails for every $n$	functions proved; sheaves modulo (O1)–(O3)

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.

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 $x\mapsto gxg^*$.

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).
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

$$\mathbf 1_{O_c}\notin\mathcal H_K\ast\varphi_0\qquad\text{for every }c\in\Lambda_H^+\smallsetminus2\Lambda_H .$$

Over $F_n$. For $n$ even, $G$ splits and $(\mathrm{SG})_n$ holds; for $n$ odd, the computation over $F$ repeats.
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).

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 $n$.
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).
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.

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.

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 $n$.
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).

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):

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

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.

Builds on

Theorem 2.6 · Theorem: the unitary twist of $(H\times H)/H$ fails (SG) over $\mathbb F_q(\!(t)\!)$
Computation 2.9 · Computation: unitary $\mathrm{SL}_2$: every row of $m^{(1)}$ nonzero, no spanning
Proposition 2.10 · Proposition: the Galois twist $\mathrm{GL}_n(E)/\mathrm{GL}_n(F)$ satisfies (SG)
Counterexample 1.5 · Counterexample outside (R): the norm torus quotient, and the torus criterion
Summary 4.12 · Verdict of the audit: the unitary twist is a valid counterexample; the repairs are minor
Theorem 3.5 · (SG) fails for Sakellaridis's $X$ over $\mathbb F_q(\!(t)\!)$, every $q$
Proposition 3.8 · Each $m_\lambda$ is semismall; multiplicities are dual branching numbers
Corollary 3.9 · Given D1–D2 for $X$: $\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k$, $m^{(n)}=D_ncD_n'$
Audit 5.5 · Audit of A3-5: two periods on $\pi\boxtimes\pi$, so $\mathcal H_K\ast\varphi_0\ne C_K$
Objection 5.10 · Objection to A3-10: placidity needs no τ (Chen–Yi Remark 37); the gap is elsewhere
Summary 5.12 · Verdict: (SG) fails, proved; $\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k$ modulo (O1)–(O3)
Theorem 1.2 · Theorem A: decategorification is base change of one $R$-matrix
Theorem 1.3 · Theorem T: trace of Frobenius on $\mathcal C_k$ is $C_K$; the missing retract classes
Proposition 1.6 · Forcing criteria: vanishing eigenfunctions, multiplicity two, extra endomorphisms
Lemma 1.7 · Close-field transfer: witnesses of (SG) persist over highly ramified $p$-adic fields
Proposition 1.8 · Under (R), (C), D0–D3: three mechanisms vanish; the obligation sits on uncovered labels
Remark 2.12 · Remark: why the trace misses thick generation, and what (R) removes
Audit 4.16 · Audit: vanishing eigenfunctionals, multiplicity one, extra endomorphisms, row relations
Audit 4.19 · Audit: under (R), (C), injective leading terms force dominant lifts
Summary 4.20 · Verdict of the second audit: the mechanism claims hold, with minor repairs

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Original record, as the agent wrote it at 23:42 UTC

Verdict: sheaf generation over $\overline{\mathbb F}_q$ does not imply spherical generation over $\mathbb F_q(\!(t)\!)$

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).
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