The agent was asked for a split counterexample other than \(E_6/F_4\). It first proved that (SG) makes every Hecke eigenspace of \(C^\infty(X(F))^K\) at most one-dimensional, over any local field (Lemma 3.1), so two independent periods on one unramified representation refute (SG). It then made Sakellaridis's Example 8.0.1 concrete. \(X=\mathrm{SL}_2^3\backslash(\mathrm{Sp}_4\times\mathrm{Sp}_4)\) is the space of triples \((W_1,W_2,\varphi)\): two nondegenerate planes and an isometry between them. The dual group map \(\mathrm{SL}_2\times\mathrm{SL}_2^{\mathrm{diag}}\times\mathrm{SL}_2\to\mathrm{SO}_5\times\mathrm{SO}_5\) fuses \((a,b,a)\) with \((b,a,b)\), and no power of \(q\) enters (Lemma 3.3). A Mackey analysis of a generic unramified principal series of \(\mathrm{Sp}_4\), restricted to \(\mathrm{Sp}_2\times\mathrm{Sp}_2\), gives two non-isomorphic \(\mathrm{SL}_2\)-quotients (Lemma 3.4). Hence \(\pi\boxtimes\pi\) has two independent periods, and (SG) fails over \(\mathbb F_q(\!(t)\!)\) for every \(q\) and every \(n\) (Theorem 3.5).
On the sheaf side it computed the \(L^+G\)-orbits on \(LX\): they carry labels \((d_1,d_2,e)\) and have connected stabilizers. It also computed the convolution fibres and their dimensions. 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 as a summand (Proposition 3.8). Given the sheaf-theoretic inputs D1–D2 for this \(X\), that yields \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) (Corollary 3.9). The one missing input is the placidity of orbit closures in \(LX\) (Dead end 3.10). In Proposition II's symmetric setting it found no counterexample.
Result as the agent reported it partial
For Sakellaridis's split spherical variety X = SL_2^3\(Sp_4 x Sp_4) (Example 8.0.1; X = triples (W_1, W_2, phi) of nondegenerate planes and an isometry), (SG) fails over F_q((t)) for every q, and every (SG)n fails: for generic unramified pi of Sp_4(F), pi (x) pi has two independent H(F)-periods (Mackey theory, characteristic-free), so C_K is not cyclic over H_K. On the sheaf side over F_q-bar (q odd): L^+G-orbits on LX are labelled by (d_1,d_2,e) with e = d_1 - d_2 mod 2, stabilizers are connected, orbit dimensions are 3d_1+3d_2+e, closures contain finitely many orbits, every convolution map Gr{<=lambda} x~ L^+X -> LX is semismall, and the number of top components of the fibre over x_nu equals the branching multiplicity of the SL_2^3-module V_(d_1,e,d_2) in Res^{SO_5 x SO_5} V_lambda; every orbit label is relevant for an explicit lambda. Consequently, if the sheaf-theoretic inputs D1-D2 hold for this X, then V_lambda*IC_0 = sum_nu IC_nu^{c_{nu lambda}}, C_k^Sat = C_k, and m^(n) = D_n c D_n' so that (SG)_n is the surjectivity of R(SO_5^2) -> R(SL_2^3/mu_2), which fails by the q-independent fusion (a,b,a) ~ (b,a,b). Formal mechanism: the trace of a Karoubi-generating functor need not be onto (Res: Rep G^ -> Rep G^_X is always Karoubi-generating; its trace, restriction of class functions, is onto only without fusion). In Proposition II's symmetric setting no counterexample is produced.
Gap
An unconditional C_k^Sat = C_k for this non-symmetric X needs D1-D2 for X: the placidity of L^+G-orbit closures in LX (ind-topology) with finite-type models, i.e. an analogue of Chen-Yi Section 5 and App. A.5; then BBD/Sun Thm 1.2 and the proved semismallness give the rest. In Proposition II's setting (symmetric, (R), (C)) the converse remains open; the obligation is (SG) over F_q((t)) for E_6/F_4 or another (R),(C)-symmetric space with fusion (Sakellaridis's Thm 8.0.2 in characteristic p, or a direct Hecke computation).
Blocked routes
- Unconditional C_k^Sat = C_k for Sakellaridis's X: needs the placidity of L^+G-orbit closures in LX and finite-type models of the target of the convolution (Chen-Yi Section 5 analogue); the compactification LX in L^+IG(2,8) gives the wrong topology and the SL_2-torsor X -> Y_1 x Y_2 is only Zariski-locally trivial.
- Mackey/generic-multiplicity-two argument inside Proposition II's setting: under (R),(C) generic unramified pi are expected to have one G^theta-period (single open orbit of rational points), so fusion can only occur at special q-dependent parameters where principal series are reducible and the Ext-vanishing step fails; would need Sakellaridis's Theorem 8.0.2 over F_q((t)) or explicit Hecke counts.
- Frobenius-cancellation (trace 0 at n = 1) shape in the split example: excluded because every top component of every convolution fibre of X is geometrically irreducible and defined over F_q; would need fibres with Frobenius-permuted components over split base points.
Brief as sent
Target: parts (1) and (3) with G split, not E_6/F_4. Starting points: (1) prove: (SG) ⇒ every Hecke eigenspace of C^∞(X(F))^K has dimension ≤ 1; (2) Sakellaridis's Example 8.0.1, X = SL_2^3\(Sp_4 × Sp_4): an explicit model, (SG) over F_q((t)) decided directly; (3) C_k^Sat = C_k for that X, or what blocks it; (4) smaller split examples of the fusion shape, or an n-dependent shape (Frobenius trace 0 on a multiplicity space at n = 1); (5) for each candidate, (R), (C), D0–D3 and what remains. Shared setting, question and rules: scripts/R/shared-brief.md.