The agent was asked to attack the other agents' proofs: check every transfer of hypotheses between \(\mathbb C\), \(\overline{\mathbb F}_q\) and \(p\)-adic fields, look for circularity and wrong normalizations, and recompute the key finite computations by its own methods. It worked in three passes as results arrived.
In the first pass it confirmed the central-inertia proof for \(\mathrm{SL}_2/\mathrm{SO}_2\), recomputed the \(\mathrm{GL}_4/\mathrm{Sp}_4\) Hecke values for \(q=2,3\) (Audit 7.6), showed that three write-ups of the transfer theorem are one proof, and redid the \(E_6\) Weyl-group conjugacy from a re-entered Cartan matrix (Audit 7.9). It flagged a false side remark, stale table cells and missing field caveats. In the second pass it confirmed the lattice-count theorems and noted that, fed into the transfer theorem, they give sheaf-level equality over \(\overline{\mathbb F}_q\) for \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) and \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) (Audit 7.23). In the third pass, on the \(E_6/F_4\) sheaf proof, it found the dimension count for the covering fibre wrong and replaced it by a torus-fixed-point argument showing that this fibre is a single point (Objection 7.28). Its final verdict: \(E_6/F_4\) is the one divergence, with sheaves over \(\overline{\mathbb F}_q\) and functions over \(p\)-adic fields under Sakellaridis's hypotheses (Summary 7.32).
The second audit found errors in this agent's own checks: its repair of a codimension citation fails in many components of \(L(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m})\), a remark on the \(\mathrm{SO}_2\) model of \(\mathrm{SL}_2/\mathrm{SO}_2\) is false, and it passed an input that fails in residue characteristic 2 as well as two tables with wrong entries. Corrections fix each, and no conclusion changed. Its final verdict is the basis of the synthesis.
Result as the agent reported it partial
A8's sheaf-level theorem survives audit: for E_6/F_4, every IC_mu is a retract of a Satake object (IC_{3k omega_1} of V_{((3k-1)/2)omega_1+omega_6} * IC_0; mirror by the 1<->6 rerun; covered labels by Prop. III step 6), hence C^Sat = C over F_q-bar (p != 2) under D0–D2The 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. and Chen–Yi Prop. 64, Betti via 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).. Verified independently: two-element interval and codimension 8, both fibres in the open cell, 8 torus-fixed points, dim Z = 36, stabilizer 29, so dim F_k >= 7; the detection criterion dim F - dim F' >= codim/2 is normalization-free. Lemma 8.5's dimension bookkeeping is wrong as written (group-case counterexample) but F_k' = {t^{lambda_k}} follows from a T'-fixed-point argument. Final: all five team conclusions survive; the sheaf column is decided in every row; E_6/F_4 is the unique row with C^Sat = C while (SG) fails, the latter only for p-adic F under Sakellaridis's (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..
Gap
(1) Function side of E_6/F_4: Sakellaridis (A2)(A2) Integral model: X is a quasi-affine G-scheme over the integers whose open Borel orbit is smooth and surjective over them, with a G-eigenmeasure on X and a B-invariant measure on the open orbit., (A3)(A3) Sakellaridis's three axioms over O, relabelled: (A3a) rank-one models over O (Axiom 2.2.1); (A3b) the Cartan decomposition (Axiom 2.4.1); (A3c) Iwahori orbits (Axiom 2.4.2). For E6/F4 all three hold in both characteristics (VI run). at the place and (A4)(A4) Spherical roots and colours: no spherical root of type N, every Levi L_γ of a simple spherical root classical, and his Statements 6.3.1 and 7.1.5 on the spherical roots and the virtual coloured weights. beyond 'L_gamma classical' unverified; (SG) over F_q((t)) undecided (needs Theorem 8.0.2 in characteristic p, or failure of multiplicity one at q^{rho-omega_4}). (2) 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). (Betti) for all positive sheaf rows: standard by BBD §6 spreading out, not written out. (3) Cosmetic: normalize the diagram automorphism sigma to preserve G^theta (or use the rerun); cite tau^{-1}(L^+G) = L^+X as the closed-immersion property, not Chen–Yi Lemma 40; replace Lemma 8.5 step 2 by the fixed-point argument.
Blocked routes
- Deciding 'C^Sat = C ⟹ (SG)' through the trace dictionary: impossible in general (Proposition 4.7 shape, Sakellaridis Example 8.0.1, and now E_6/F_4 where a q-point Weyl coincidence kills surjectivity of r while multiplicities in q^{1/2} can stay nonzero).
- Proving the SL_2/SO_2 sheaf failure from the Hecke-parity obstruction: the Z/2-grading does not survive cofibres (localisation triangle of IC_1), so only the sign local systems give a sheaf-level obstruction (Theorem 2.4).
- Reading a divergence off Table 1 by pairing a Betti cell with a '(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.' cell: not meaningful without the two bridges of Audit 7.16.
- Deciding E_6/F_4 at function level over F_q((t)) by A5's method without an orbit classification of E_6(O) on the norm-one Albert hypersurface: not available; and Theorem 6.6 predicts the triangular system fails at 3ω_1^∨ rather than closes.
- Applying Proposition B with reverse dominance globally on all symplectic labels: lower sets are infinite; the argument must stay inside the finite sets S_{d,c} (as Theorem 5.7 effectively does).
- Bounding dim F' by K_N-quotient dimensions of L^+G-orbits: the left K_N-quotients do not see the codimensions (in the group case dim K_N\O_lambda = dim X_N for every lambda); use a renormalized dimension theory or torus fixed points.
- Deciding (SG) over F_q((t)) for E_6/F_4 from the sheaf result: impossible by the dictionary alone (Proposition 4.7); C^Sat = C gives only nonzero rows of the multiplicity matrix.
Brief as sent
Audit the claimed proofs in returns/A2.json (central inertia splits \(\mathcal C\); Satake objects even; sign local systems odd; finite birational convolution map; function-level dimension count), returns/A4.json (trace dictionary; Theorem A4-6 (SG) \(\Rightarrow\mathcal C^{\mathrm{Sat}}=\mathcal C\) under D0–D5; A4-7; A4-8; GL\(_4/\)Sp\(_4\) lattice counts; A4-11) and returns/A6.json (E\(_6/F_4\): r not onto via the W-conjugate pair; spheres: r onto). Checklist: hypothesis transfer (\(\mathbb C\) vs \(\overline{\mathbb F}_q\) vs p-adic, char F = 0 in (A1), q odd, placid vs finite type), circularity, special case as general, an analogue certifying an inference at the other level, normalizations (shifts, sign of \(\delta(X)^{1/2}\), left/right, vol K = 1), thick vs linear span, the naturality and bimultiplicativity of the central involution, the direction of 'nonzero trace coefficient implies nonzero multiplicity space', unitriangularity, independent recomputation of the GL\(_4/\)Sp\(_4\) counts and of the E\(_6\) Weyl conjugacy. Return check/objection nodes with severity, then a verdict on which team-level conclusions survive. Further claims (A1, A3, A5) sent later by message.