SETTING (Proposition IId of the satake-two-levels pages: the converse of IIc).

Fields. p is an odd prime, q a power of p, k = F̄_q. F = F_q((t)), O = F_q[[t]]; F_n = F_{q^n}((t)) is the unramified extension of degree n.

Spherical varieties. G is a connected reductive group over F_q, H ⊂ G a closed subgroup over F_q, X = G/H homogeneous and spherical (a Borel subgroup of G_k has an open orbit on X_k), x_0 = eH. Symmetric case: H = G^θ for an involution θ over F_q. T is a maximal torus of G_k, Λ_T = X_*(T), Λ_T^+ the dominant coweights.

Sheaves over k. LX, LG, L^+G are the loop and arc spaces over k, and L^+G acts on LX. When every L^+G-orbit closure in LX contains finitely many orbits (true for symmetric spaces: Chen–Yi, arXiv:2310.20006v4, Prop. 17, Thm 18), put
  C_k      = bounded L^+G-equivariant constructible Q̄_ℓ-complexes on LX, each supported on finitely many orbit closures;
  IC_0     = the constant sheaf on L^+X = L^+G·x_0;   V_λ = the Satake sheaf on Gr_{≤λ}, λ ∈ Λ_T^+;
  V_λ*IC_0 = m_!(V_λ ⊠~ IC_0), m : LG ×^{L^+G} L^+X → LX the convolution map;
  C_k^Sat  = Thick_{C_k}{ V_λ*IC_0 : λ ∈ Λ_T^+ }   (Thick = closed under isomorphisms, shifts, cones and direct summands).
The categorical statement is C_k^Sat = C_k. It uses ALL λ ∈ Λ_T^+ and depends only on X_k with its G_k-action, not on the F_q-structure.

Functions over F. K = G(O); H_K = C_c(K\G(F)/K) (Haar measure with vol K = 1) acts on C_K = C_c^∞(X(F))^K by (f*φ)(x) = Σ_{gK ∈ G(F)/K} f(g) φ(g^{-1}x); φ_0 = 1_{X(O)}. Spherical generation:
  (SG) over F :   H_K * φ_0 = C_K .
(SG)_n is the same statement over F_n. Coefficients C or Q̄_ℓ: (SG) does not depend on the characteristic-0 coefficient field.

Trace of Frobenius. A Weil object (M, Φ : Fr^*M ≅ M) of C_k has trace functions tr_n(M) on X(F_n) = LX(F_{q^n}). Over F_q((t)) the pages prove (part IIa): tr_n(V_λ*IC_0) = tr_n(V_λ) * φ_0, and the tr_n(V_λ) span H_{K_n}. By the decomposition theorem (D2), V_λ*IC_0 ≅ ⊕_{ν,i} IC_ν[−i] ⊗ E_{ν,i}(λ); Frobenius induces A_{ν,i}(λ) on E_{ν,i}(λ), and m^{(n)}_{νλ} = Σ_i (−1)^i tr(A_{ν,i}(λ)^n). The categorical statement says that every row of the support pattern {E_{ν,·}(λ) ≠ 0} is nonzero (Proposition I); (SG)_n says that the columns of m^{(n)} span ⊕_ν Q̄_ℓ (part IIb).

Standing hypotheses of Proposition II (symmetric X):
  (R)  G, θ, X are defined over F_q, q odd; T ⊇ S split over F_q (S a maximal θ-split torus); G has a θ-stable Borel pair (B_0, T_0) over F_q with T_0 split.
  (C)  G^θ and Z_{G^θ}(S) are connected.
  D0   Chen–Yi's orbit geometry over k: orbits O_μ ↔ μ ∈ Λ_S^+, closures, the support bound, stabilizers Z_{G^θ}(μ) ⋉ (pro-unipotent).
  D1   proper base change and the Grothendieck–Lefschetz formula on convolution fibres; a Frobenius-compatible perverse t-structure whose simple objects are the IC_ν.
  D2   V_λ*IC_0 is pure and a finite sum of shifted IC_ν (Sun, Thm 1.2; Chen–Yi, Prop. 70).
  D3   Lang's theorem for L^+G and for the stabilizers.
Proposition IIc (proved on the pages): under (R), (C) and D0–D3, if (SG)_n holds for one n, then C_k^Sat = C_k.

TEAM QUESTION (IId). Does C_k^Sat = C_k imply (SG) over F_q((t))?
 (1) Decide it by a counterexample: a spherical variety X = G/H over F_q, q odd (symmetric or not), with C_k^Sat = C_k proved over k = F̄_q and (SG) proved to FAIL over F = F_q((t)) (for some odd q, or for all). E_6/F_4 is excluded as an answer. State which of (R), (C), D0–D3 the example satisfies.
 (2) Identify the formal mechanism by which the trace of Frobenius (Laumon's function–sheaf dictionary; the categorical trace Tr(Fr, −) of Gaitsgory, Arinkin–Gaitsgory–Kazhdan–Raskin–Rozenblyum–Varshavsky, Ben-Zvi–Nadler, Zhu) fails to turn C_k^Sat = C_k into (SG), and formal conditions that force a counterexample.
 (3) In the setting of Proposition II (with (R) and (C)): a counterexample, or the exact remaining obligation.

DOES NOT COUNT. E_6/F_4. A function-level failure proved only over p-adic fields, or only under Sakellaridis's (A1)–(A5) (his Theorem 8.0.2, arXiv:0905.4244v3, assumes characteristic 0), presented as a failure over F_q((t)). A categorical statement deduced from an unproved local geometric Langlands conjecture (Ben-Zvi–Sakellaridis–Venkatesh) presented as proved. A mechanism showing only that "the dictionary does not imply it" presented as a counterexample. Numerical evidence for small q only.

TERMINOLOGY TRAPS. Thick closure contains direct summands (retracts); the stable span does not. C_k^Sat uses all geometric coweights λ ∈ Λ_T^+ of G_k, whereas H_K over F only sees the F-rational structure (Frobenius-fixed coweights for unramified G). One L^+G-orbit can contain several K-orbits of F_q-points when stabilizers are disconnected; K-orbits, G(F)-orbits and L^+G-orbits are different things. "(SG) over F_q((t))" means n = 1 unless (SG)_n is written. The Hecke action is the left convolution above.

FILES (read HTML with  sed -e 's/<[^>]*>/ /g' FILE | tr -s ' \n'  and PDFs with  pdftotext -layout FILE - ).
 Proposition II fragments: /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/satake-two-levels/props/assets/trace-pages/{trace,trace-a,trace-b,trace-c,trace-dictionary,_definitions,_hypotheses}.html
 Previous run (do not re-derive it; E_6/F_4 is excluded): /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/satake-two-levels/verdict.html
 Spherical-generation pages: /Users/miltonlin/Projects/Trees/subpages/tools/proof_graphs/spherical-generation/{symmetric,criterion,generation,classical,summary}.html and families/{group,spin,gl-pq,symplectic,split-forms}.html
 Sources: /Users/miltonlin/Projects/Trees/subpages/tools/solve-maths/satake-two-levels/props/sources/{chen-yi-v4,sakellaridis-v3,zhu-satake-v2,laumon-1987,bf-satake-v2,sun-decomposition,milne-iag-v2,conrad-reductive-group-schemes,bezrukavnikov-yun-v2}.pdf ; /Users/miltonlin/Projects/Trees/subpages/tools/proof_graphs/two-definitions-iwahori/sources/{ponto-shulman-v2,drinfeld-gaitsgory-v3,campbell-dhillon-v1,haines-kottwitz-prasad-v3}.pdf ; Sakellaridis as text: /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/e8c0970d-1ea8-4e12-8549-b0c385d9db05/scratchpad/sak.txt (§8 starts near line 3830).

RULES.
- Return concrete lemmas, constructions, computations or counterexamples, each with a proof or an explicit gap. No status reports, no optimism. Label every claim: proved (here, or cited with exact result number) · heuristic · gap. An analogue at one level does not certify an inference at the other.
- Check the exact meaning of every term you use (see the traps above).
- Do not search the web for a solution of this question; ordinary mathematical background (statements of standard theorems) may be looked up. Prefer the local sources.
- You may run short Python checks with `python3 -I` for finite computations: single process, a few seconds each, no parallelism. Keep scripts in /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/e8c0970d-1ea8-4e12-8549-b0c385d9db05/scratchpad/solve/{ID}/. Do not modify anything under /Users/miltonlin/Projects/Trees.
- Timestamps: at the start of every node run `date -u +%Y-%m-%dT%H:%M:%SZ` and record it in the node. Work in nodes: an idea, a lemma with proof, a computation, a check, an objection, a dead end (status blocked, with the exact missing lemma), or a verdict. Produce 5–14 nodes.
- Write every node as a stage of a mathematics paper: a plain-words name, a one-sentence summary, the precise statement, and the argument as numbered steps, readable by a mathematician outside the subfield. Cite other nodes as [[{ID}-3]], never by bare code in prose; state computations as mathematics (what was computed, the result, why it suffices), not as code or program output.
- Concept keys for nodes: setup · trace · criteria · nonsplit · split · synthesis.
- FINAL ANSWER: only one JSON object inside a ```json fence, with the schema below; also write the identical JSON to /private/tmp/claude-501/-Users-miltonlin-Projects-Trees/e8c0970d-1ea8-4e12-8549-b0c385d9db05/scratchpad/solve/{ID}/return.json. Escape JSON strings properly (double every backslash, escape quotes and newlines). Nothing outside the fence.

RETURN SCHEMA
{"agent":"{ID}",
 "family":"approach family, at most twelve words",
 "nodes":[
  {"id":"{ID}-1","t":"2026-10-17T20:30:12Z",
   "kind":"idea|lemma|computation|check|objection|blocked|verdict",
   "status":"proved|refuted|blocked|open|heuristic|superseded",
   "label":"Setup|Definition|Strategy|Lemma|Proposition|Theorem|Corollary|Computation|Example|Counterexample|Remark|Objection|Audit|Conjecture|Dead end|Summary",
   "name":"tree label: at most 44 characters, plain text (Unicode, no TeX), says what the step establishes",
   "title":"page title, at most 90 characters, inline TeX allowed",
   "summary":"one or two sentences: what this step establishes and why it matters",
   "statement":"the precise claim (Markdown + TeX); empty for Strategy and Remark",
   "parents":["{ID}-0"],
   "body":"the argument: Markdown; math as \\( \\) inline and $$ $$ display; proofs as numbered steps; other nodes as [[{ID}-1]]; results cited with exact numbers and the file they come from",
   "concept":"one of: setup, trace, criteria, nonsplit, split, synthesis",
   "key":false,
   "sources":["chen-yi-v4.pdf, Prop. 17"]}],
 "result":{"status":"proved|partial|refuted|blocked",
           "statement":"precise statement of what was established",
           "gap":"exact remaining obligation, or none"},
 "blocked_routes":["route and the theorem-strength lemma it would need"],
 "scripts":["paths of scripts run"]}