The agent was asked to turn the trace of Frobenius into exact implications between spherical generation over \(\mathbb F_q(\!(t)\!)\) and the categorical statement over \(\overline{\mathbb F}_q\). It fixed the setting and listed six inputs D0–D5The standing inputs of the sheaf–function dictionary over F̄_q, p ≠ 2: D0 Chen–Yi's orbit geometry; D1 traces of Frobenius and the perverse t-structure; D2 decomposition of Satake objects into shifted IC sheaves; D3 Lang's theorem; D4 unitriangularity of Satake traces; D5 comparison of F̄_q with C.: Chen–Yi's orbit geometry over \(\overline{\mathbb F}_q\), proper base change and the perverse t-structure, the decomposition of Satake objects, Lang's theorem, unitriangularity of Satake traces, and the comparison between \(\overline{\mathbb F}_q\) and \(\mathbb C\) (Setup 4.1).
With these, the trace of a Satake object is a Hecke translate of the basic function, and its coefficients in the basis of IC traces are Frobenius traces on multiplicity spaces. The main theorem: if \(G^\theta\) and \(Z_{G^\theta}(S)\) are connected, spherical generation over \(\mathbb F_q(\!(t)\!)\) for one \(q\) makes every IC sheaf a summand of a Satake object, hence \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\) (Theorem 4.6). The converse does not follow: sheaf generation says every row of the multiplicity matrix is nonzero, while spherical generation needs its columns to span (Proposition 4.7). Applications: two lattice counts settle \(\mathrm{GL}_4/\mathrm{Sp}_4\) at both levels (Theorem 4.9); Proposition III's conditions give generation over \(\mathbb F_q(\!(t)\!)\) without Sakellaridis's hypotheses; and for \(\mathrm{SL}_2/\mathrm{SO}_2\) a square-class symmetry shows that the sign-twisted IC sheaves are not generated (Theorem 4.11). An attempt at \(\mathrm{GL}_6/\mathrm{Sp}_6\) stopped.
The audit found one step of the \(\mathrm{SL}_2/\mathrm{SO}_2\) proof circular as written; the theorem is true, and a correction supplies the step. The stopped \(\mathrm{GL}_6/\mathrm{Sp}_6\) table had four wrong entries, without changing its conclusion. The transfer theorem was judged correct. It is Theorem 3 of the verdict: it rules out the reverse divergence and turns later function-level results into sheaf-level ones.
Result as the agent reported it partial
For symmetric X=G/G^theta with G^theta and Z_{G^theta}(S) connected, defined over F_q (q odd, T split), modulo the dictionary obligations D0-D5The standing inputs of the sheaf–function dictionary over F̄_q, p ≠ 2: D0 Chen–Yi's orbit geometry; D1 traces of Frobenius and the perverse t-structure; D2 decomposition of Satake objects into shifted IC sheaves; D3 Lang's theorem; D4 unitriangularity of Satake traces; D5 comparison of F̄_q with C. (Chen-Yi geometry over F_q-bar, proper base change and the perverse t-structure on the placid quotient, purity/decomposition of V_lambda*IC_0, Lang for L^+G, Lusztig-Kato unitriangularity, comparison F_q-bar/C): (1) (SG) over F_{q^n}((t)) for one n implies C^Sat = C (proof via tr(V_lambda*IC_0) = tr(V_lambda)*phi_0, the IC-basis expansion whose coefficients are traces of Frobenius on multiplicity spaces, and the Serre-subcategory thickness argument); (2) C^Sat = C implies only that every IC_mu is detected by some Hecke translate over some F_{q^n}((t)), not (SG) (retract-closure vs K_0; Example 8.0.1 shape), so no converse is proved; (3) under Prop. III's two conditions (SG) holds over F_q((t)) with no use of (A1)-(A5)Sakellaridis's standing assumptions for his Theorem 8.0.2 (the pages' Theorem E): (A1) a characteristic-0 field and a split group with simply connected derived group; (A2) an integral model of X with a smooth open Borel orbit and invariant measures; (A3a) rank-one models over O, (A3b) the Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) conditions on the spherical roots and colours; (A5) X homogeneous and affine with one open Borel orbit. (Lang + geometric support bound + Cor. C); (4) for GL_4/Sp_4 the uncovered labels (k,k,k,k) are explicit Hecke translates (two lattice counts: rows (1,(q+1)(q^2+1)) and (q^2+q,0), machine-checked for q=3,5), so (SG) holds over F_q((t)) and hence C^Sat = C, deciding that 'unknown' row; (5) for SL_2/SO_2 the sign-twisted IC_{sgn,mu} are not in C^Sat (square-class automorphism over every F_{q^n} + Vandermonde on Frobenius eigenvalues + Serre subcategory), so C^Sat != C, deciding the 'undecided' row. Among Table 1 rows no case with (SG) over F_q((t)) and C^Sat != C can exist; no row exhibits C^Sat = C with (SG) failing; GL_{2m}/Sp_{2m} (m>=3), SL_{2n}/Sp_{2n} (n>=3), E_6/F_4 remain open, reduced to a finite lattice computation over F_q((t)).
Gap
D0D0 · Orbit geometry over F̄_q, p ≠ 2 (Chen–Yi, Prop. 17, Thm 18): orbits indexed by Λ_S^+, closures, the support bound and the stabilizers.: Chen-Yi Prop. 17/Thm 18/support bound/stabilizer lemma over F_q-bar (char != 2); D1D1 · Traces and t-structure: proper base change and the Grothendieck–Lefschetz trace formula on convolution fibres, with a Frobenius-compatible perverse t-structure whose simple objects are the IC sheaves.: proper base change + Grothendieck-Lefschetz on the fibre and the Frobenius-compatible perverse t-structure with simples IC_{chi,mu} on Shv_cons(L^+G\LX); D2D2 · Decomposition: each Satake object V_λ ∗ IC_0 is pure and splits into shifted IC sheaves with finite-dimensional multiplicity spaces (Sun, Thm 1.2; Chen–Yi, Prop. 70).: purity and decomposition of V_lambda*IC_0 and BBD 5.4.1 for ^pH^i in the placid setting; D3D3 · Lang's theorem for the arc group L⁺G and its stabilizers over F_q.: Lang for the pro-algebraic L^+G and stabilizers; D4D4 · Unitriangularity of the Satake traces tr V_λ in the basis T_λ = 1_{Kt^λK} (Zhu (5.6.2), Lemma 5.6.3; Lusztig, Kato).: Lusztig-Kato unitriangularity of tr V_lambda; D5D5 · Comparison with C: “IC_μ is a summand of V_λ ∗ IC_0” has the same answer over F̄_q and over C (BBD §6.1, for large p).: comparison F_q-bar <-> C of 'IC_mu is a summand of V_lambda*IC_0'. Plus: (SG) over F_q((t)) at the uncovered labels of GL_{2m}/Sp_{2m} (m>=3), SL_{2n}/Sp_{2n} (n>=3), E_6/F_4 (finite lattice counts beyond the box of Dead end 4.10, or Theorem E over F_q((t))).
Blocked routes
- C^Sat = C => (SG): would need a positivity/triangularity lemma 'for every uncovered mu some V_lambda*IC_0 has IC_mu as a summand AND a Hecke combination supported in the lower set of mu' (a K_0-level statement); the dictionary cannot supply it (Proposition 4.7).
- Transporting '(SG) under (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies.' (p-adic, Theorem E) to F_q((t)): needs Sakellaridis 8.0.2 in characteristic p for GL_{2m}/Sp_{2m}, or a Q_p <-> F_p((t)) comparison of the spherical Hecke module C_K, neither available (Objection 4.12.2).
- Deciding GL_6/Sp_6 at (1,1,1,1,0,0) by lattice counts: the in-box 3x4 matrix is insufficient; needs the box t^2 O^6 <= M <= t^{-2} O^6 (~q^18 Hermite forms) or a structural count (Dead end 4.10).
- Single-q detection (n=1 in Proposition 4.7, Theorem 4.11): needs Chen-Yi's Tate-type/even-shift assertion in the proof of Thm 93 (Frobenius on multiplicity spaces acts by powers of q^{1/2}), cited but not re-derived (Lemma 4.4 Remark).
Brief as sent
Formulate and prove, with exact hypotheses, implications between (SG) over \(\mathbb F_q(\!(t)\!)\) and \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) over \(\overline{\mathbb F}_q\), for symmetric \(X\) with \(Z_{G^\theta}(S)\) connected: (a) the trace function of \(V_\lambda\ast\mathrm{IC}_0\); (b) whether every Satake object is a direct sum of shifted \(\mathrm{IC}_\nu\) with constant coefficients and by which theorem; (c) when \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) is equivalent to every \(\mathrm{IC}_\mu\) being a summand of some Satake object (Serre subcategory of the heart); (d) (SG) \(\Rightarrow\) \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) or its failure, and the converse or its failure (Sakellaridis's Example 8.0.1 shape; Corollary C under coverage); (e) obstacles: \(\mathbb C\) versus \(\overline{\mathbb F}_q\), \(\operatorname{char}F=0\) in (A1), Weil structures, Lang's theorem, independence of the trace functions. Apply to Table 1. Shared setting and rules: the problem card of the hub.