The agent was asked for the Iwahori analogue of "functions generate ⇒ sheaves generate" over \(\mathbb F_q(\!(t)\!)\), with every input of the sheaf–function dictionary named. It worked one finite level at a time. It glued the perverse hearts with Weil structures ( (Setup 3.1)) and identified the Iwahori-invariant functions at a level with combinations of indicators of pieces, by Lang's theorem ( (Lemma 3.2)). It proved that the trace of a convolution is the convolution of the traces, with \(\operatorname{vol}I=1\).
The traces of the simple objects form a basis of \(C_I\), unitriangular in the piece indicators, and each generator's trace expands in them with coefficients that vanish off its summands ( (Lemma 3.4)). So if \(\mathcal H_I\ast\varphi_0=C_I\), every simple object has a nonzero coefficient somewhere, hence is a summand, and the criterion gives \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\) ( (Theorem 3.6)). With disconnected stabilizers one needs (IG) over every \(\mathbb F_{q^n}(\!(t)\!)\).
The converse fails in general: sheaves give nonzero rows of the coefficient matrix, while functions need spanning columns ( (Proposition 3.10)); the spherical \(E_6/F_4\) case is the model. The agent also checked the bridge's inputs for \(\mathbb A^n\) and \(\mathbb G_m\), and gave a sign-and-weights proof for the group. The audit confirmed the bridge over \(\overline{\mathbb F}_q\) and recorded that nothing is proved over \(\mathbb C\).
Result as the agent reported it proved
Over Fbar_q (l-adic sheaves) and F_q((t)), p odd, under (L1) level presentation over F_q, (L2) finitely many Fr-stable orbits per level with connected stabilizers, (D1) bounded convolution, (D2) Lang and Lefschetz, (D3) decomposition of IC_w *I IC_0: (IG) implies that every simple object is a shifted summand of some IC_w *_I IC_0, hence C_I^Iwa = C_I (Theorem 3.6, with pointwise form and witness kappa_S); with disconnected stabilizers, (IG) over all F{q^n}((t)) suffices (Proposition 3.7). Reverse: sheaves give only non-zero rows of m^(n); (WT) gives (IG) (Proposition 3.10); with purity, C_I^Iwa = C_I iff every IC-coordinate is non-zero on H_{I_n}*phi_0 for some n (Theorem 3.11). Second proofs with all inputs checked: A^n, G_m sheaves from functions (Corollary 3.8); group: no Delta_omega is a summand, Delta_e not in C_I^Iwa, (IG) fails, by sign characters and weights (Proposition 3.9). E6/F4: (IG)_n fails for all n (Summary 3.12).
Gap
Nothing over C (transfer of the l-adic summand statements needs spreading out uniform in w). For the rows other than group, A^n, G_m: the inputs (L1), (L2)/(Proposition 3.7), (D1), (D3) are unchecked, and C_I^Iwa = C_I is decided only by the summand condition M_I = Irr (equivalently the test of Theorem 3.11); no Iwahori-level example of 'sheaves hold, (IG) fails' is established (E6/F4 is the candidate). The E6/F4 function failure is conditional on the satake run's Propositions VI-VII.
Blocked routes
- Sheaves => functions by traces alone: needs spanning columns of m^(1), i.e. a triangular, cancellation-free system (WT); non-zero rows (from C_I^Iwa = C_I with purity) do not suffice, as E6/F4 shows at spherical level (satake run, Prop. IX(a)).
- Betti version over C of (II_I): needs, for each simple S, a w realising S over Fbar_p with p outside the BBD 6.1 bad set for that w (spreading out uniform in w); not available.
- (II_I) from (IG) over F_q((t)) alone with disconnected stabilizers: non-Fr-stable simple objects have zero trace; needs (IG) over F_{q^n}((t)) for n making them Fr^n-stable (Proposition 3.7).
Brief as sent
Target: State and prove the functions ⇒ sheaves bridge (II_I): over \overline{F}_q, with G, X, the level presentation and the finite-level orbits defined over F_q, if (IG) holds over F = F_q((t)) then every simple object of C_I lies in M_I, hence (with the summand criterion) C_I^{Iwa} = C_I. Name every dictionary input. State also what the trace map gives in the reverse direction: why sheaves ⇒ functions fails in general, and which extra hypothesis would make it hold.
Approach: Frobenius traces at a finite level and unitriangularity. Starting points: (1) at level (N,M) identify the functions in C_I supported in Y_N(F) and constant on fibres of π_{N,M} with the I(F_q[[t]])-invariant functions on Y_{N,M}(F_q), and show they have basis 1_{P(F_q)} over the Frobenius-stable pieces P (Lang's theorem for the connected group I/I_{N,M} and connected stabilizers); (2) show the traces of the IC_w span H_I and that tr(G ∗I M) = tr(G) ∗ tr(M) with vol(I) = 1 (proper base change, Grothendieck–Lefschetz); (3) Frobenius on the decomposition of IC_w ∗_I IC_0: isotypic parts, non-F-stable constituents contribute zero trace, multiplicity spaces carry Weil structures; (4) linear independence of the traces of the F-stable simple objects at a common level (unitriangular in the 1{P(F_q)}); (5) conclude and list which of the satake run's inputs D0–D3 are replaced by level-wise inputs; (6) state what fails for the converse (summands invisible to traces), with the spherical counterexample E_6/F_4 of the satake run as the model, and whether the converse holds when the decomposition has no multiplicity cancellation.