The agent was asked for a counterexample whose \(\mathbb F_q\)-structure is not split. Its strategy (Strategy 2.1) rests on one remark. The categorical statement depends only on the pair over \(k\), so changing the \(\mathbb F_q\)-structure of a pair for which it is known isolates the function side. It proved two elementary obstructions over any local field: Hecke translates of \(\varphi_0\) stay in the \(G(F)\)-orbit of \(X(\mathcal O)\), and when the translates have distinct leading orbits, only those orbits are reached (Lemma 2.2).
The main case is the unitary twist of the group case: \(G=\operatorname{Res}_{\mathbb F_{q^2}/\mathbb F_q}H\) with \(\theta\) a Chevalley involution composed with the Galois action. For \(H=\mathrm{GL}_n\) this is the space of hermitian matrices. Over \(k\) it is the group case, so every IC is the image of a Satake sheaf and \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) (Lemma 2.5). Over \(\mathbb F_q(\!(t)\!)\) the rational Hecke labels \((a,a)\) have leading orbit \(2a\), so no orbit of type \(c\notin2\Lambda_H\) is generated. Hence (SG) fails for every split \(H\ne1\), and it holds over \(\mathbb F_{q^n}(\!(t)\!)\) exactly when \(n\) is even (Theorem 2.6). The smallest instances are the norm torus (Counterexample 2.3), hermitian \(\mathrm{GL}_n\), and hermitian \(\mathrm{SL}_2\) of determinant 1. In the last, \(X(F)\) is a single \(G(F)\)-orbit and every row of \(m^{(1)}\) is nonzero, yet the columns do not span (Computation 2.9). The Galois twist \(\mathrm{GL}_n(E)/\mathrm{GL}_n(F)\) satisfies (SG) (Proposition 2.10), so (SG) is not a property of the pair over \(k\). Under (R) and (C) all of these effects vanish.
Result as the agent reported it proved
IId is false once T is allowed to be non-split. For every nontrivial split connected reductive H over F_q (q odd, or any q), the unitary twist X_u = G/G^θ, G = Res_{F_{q^2}/F_q} H, θ = ϑ∘σ (ϑ a Chevalley involution), is a symmetric space over F_q with connected G^θ and Z_{G^θ}(S). Over k it is the group case (H×H)/H, so \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) (V_{(c,0)}*IC_0 = IC_c, connected stabilizers, Proposition I). Over F_q((t)), Hecke translates of φ_0 have injective leading orbits c = 2a, so 1_{O_c} ∈ H_K*φ_0 forces c ∈ 2Λ_H^+, and (SG) fails; (SG)n fails for odd n and holds for even n. The example satisfies (C) and D0–D3, and (R) except the splitness of T and T_0. Smallest instances: Res G_m acting on G_m through the norm (two G(F)-orbits); Res GL_n acting on hermitian matrices (for n ≥ 2 the failure persists on the base G(F)-orbit, with codimension p{≤n}(2d) − p_{≤n}(d) on each slice); SL_2(E) acting on hermitian matrices of determinant 1 (one G(F)-orbit, every row of m^{(1)} nonzero, columns not spanning). The Galois twist GL_n(E)/GL_n(F) satisfies (SG), so (SG) over F_q((t)) is not determined by (G_k, X_k). Mechanism: H_K sees only σ-fixed coweights; Weil objects induced from non-fixed σ-orbits have trace 0 at n = 1; the retracts used by thick generation carry Weil structures in \(\mathcal C_k\) that come from no Satake object.
Gap
None for the target (part 1 with non-split F_q-structure). For part (3), inside Proposition II's setting ((R) and (C)): Frobenius twisting is void (σ = 1 on Λ_T, all orbits rational), and the forcing criterion never produces a failure: φ restricted to Λ_T^+ is either non-injective modulo trivially acting cocharacters, or injective, in which case (C) forces every label to lift and (SG) holds. By Lemma 4.8 of the run, a counterexample there must have an uncovered label μ ∉ φ(Λ_T^+) with IC_μ a summand of some V_λ*IC_0 and 1_{O_μ(F_q)} ∉ span{T_λ*φ_0}, i.e. a linear dependence of the columns of m^{(1)} that removes μ. This is not decided here.
Blocked routes
- Frobenius twisting under (R): σ = 1 on Λ_T and every L^+G-orbit has the rational point x_μ, so twisting contributes nothing. A counterexample under (R)+(C) needs a theorem-strength computation at an uncovered label: an explicit linear dependence among the columns of m^{(1)} over F_q((t)) that removes μ (for E_6/F_4, e.g. Sakellaridis' Theorem 8.0.2 or a direct Hecke count on the norm-one Albert hypersurface over F_q((t))).
- Leading-term obstruction under (R)+(C): φ is never injective on Λ_T^+ in rank ≥ 1, since injectivity forces T = S and Z_{G^θ}(S) = T[2], which is disconnected. Injectivity modulo trivially acting cocharacters forces (T^θ)° central, and then (C) makes φ: Λ_T → Λ_S onto with dominant lifts, so (SG) holds. The route would need a cancellation analysis among the T_λ sharing a leading orbit.
- G = GL_2 split with H a non-split torus: H = T_E satisfies (SG) (Bruhat–Tits tree recursion) and the sheaf statement; H = T_E^1 has k-form GL_2/SO_2, where the central −I lies in a non-identity stabilizer component, so the sheaf statement fails, and (SG) fails too (two G(F)-orbits). Neither is a counterexample.
- Galois twists of the group case: GL_n(E)/GL_n(F) satisfies (SG) for all n (Proposition B with positive-half lifts), so it cannot give a counterexample.
Brief as sent
Target: part (1) by a counterexample whose F_q-structure is not split, both levels proved. Starting points: (1) F_q-forms of pairs (G_k, X_k) with C_k^Sat = C_k known (group case, tori); (2) the norm torus Res_{F_{q^2}/F_q}G_m acting on G_m, both levels; (3) a non-abelian form of the group case: Res_{E/F}GL_n on hermitian matrices, or on GL_n(E)/GL_n(F); identify X_k with the group case, prove C_k^Sat = C_k, decide (SG); (4) which V_λ are Weil, which orbits are Frobenius-stable, which of (R), (C), D0–D3 hold; (5) G split with H non-split. Shared setting, question and rules: scripts/R/shared-brief.md.