A5 audited the two returns that carried the characteristic-\(p\) theorem, A4's close-field transfer and A3's re-verification of the hypotheses, and then A1's direct rerun. It re-derived every step against the run's checklist and recomputed the lattice data, the orbit combinatorics and the \(A_2\) Weyl characters in exact arithmetic.
In the first task no claim failed; four needed a line more. The Satake support needed a bound independent of the field, supplied by a primitive-row argument ( (Objection 5.4)). "Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits. holds" needed the valuation cone in characteristic \(p\) ( (Audit 5.3)). The orbit action needed a converse, and the colour weights the diagram automorphism of \(E_6\). One citation it could not open, Sakellaridis's 2008 paper over ramified fields ( (Audit 5.5)); the root checked that the paper covers every field of characteristic 0.
In the second task it confirmed that the analytic half of Sakellaridis's proof reruns over \(\mathbb F_q(\!(t)\!)\) with characteristic-\(\ne2\) inputs ( (Audit 5.14), (Audit 5.15)). It verified the error on p. 62 and the repair that covers \(E_6/F_4\) ( (Audit 5.17)), and showed that the rerun's last open input follows from (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary. ( (Audit 5.19)). It also found that the general statement needs three more hypotheses ( (Objection 5.20)).
A fresh review found the first audit correct throughout. In the second, A5 had missed one gap: the general theorem also needs non-zero leading coefficients ( (Objection 5.20)), which the salvage then proved ( (Lemma S1.3)).
Result as the agent reported it partial
Audit of Lemma 4.2, Computation 4.4, Proposition 4.5, Proposition 4.6, Audit 4.7, Theorem 4.8, Audit 4.3, Proposition 4.9 and Audit 3.1 … Proposition 3.10: no claim fails. Four repairs are supplied: (1) Proposition 4.5's 'Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits. holds' needs the char-p valuation cone Proposition 3.7(3); the parametrization K\X(F) = ⊔_{κ∈Λ_S^+} K x_κ stands as proved. (2) Proposition 4.6(c) needs a uniform bound on ν; supplied: ⟨λ,w_0ω_i⟩ ≤ ⟨ν,w_0ω_i⟩ ≤ ⟨λ,ω_i⟩ by a primitive-row argument. (3) Proposition 3.6(2): the converse Y_w = Y_w' ⇒ w^{-1}w' ∈ W(X) is supplied by conjugacy of maximal tori in the stabilizer. (4) Proposition 3.7 step 3: the colour weights {ω_1, ω_6} do not follow from 'a basis of dominant weights'; they are forced by the diagram automorphism δ, which commutes with θ_0. Consequences: Theorem 8.0.2 for M_θ over F_q((t)), p odd (p = 3 included), and the failure of (SG) there, are proved by the close-fields transfer modulo (I_Λ) (Axiom 2.2.1 over W(F_q)) and H_Sa08 (ramification-independence of [Sa08, Thm 5.2.1], quoted as Theorem 3.7.1). The hypotheses (A1)(A1) Field and group: F a non-archimedean local field of characteristic 0 (Sakellaridis §1.1), G split reductive over O with connected fibres and simply connected derived group (§1.7)., (A2)(A2) Integral model: X quasi-affine over O, its open Borel orbit smooth and surjective over O with B-homogeneous special fibre; a G-eigenmeasure on X and a B-invariant measure on the open orbit (§1.7)., (A3b)(A3b) Cartan decomposition (Sakellaridis's Axiom 2.4.1): the X-antidominant torus points represent every K-orbit on X(F), and points with distinct labels lie in distinct K-orbits., (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x., (A4)(A4) Spherical roots and colours: no pair of maximal rank of type N; classical Levis L_γ; Statements 6.3.1 and 7.1.5.(i),(ii),(iv), (A5)(A5) X homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points (§8). for M_θ hold over F_q((t)) for every odd p, with char-free inputs. All numerical root-system claims were recomputed exactly and agree. Second task: Audit of Strategy 1.1 … Theorem 1.9: no claim fails for M_θ. Over F_q((t)), p odd, the open-orbit analytic half of Sakellaridis's proof holds with characteristic-≠2 inputs only. That half consists of convergence, generic uniqueness and rationality of S_χ (Proposition 1.5); the functional equations and the sum formula over W_X′ = N_W(a) ∩ [W/W_L], with neither Knop's action nor (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary. (Proposition 1.6); the formal last step (Proposition 1.2); and the M_θ algebra W_X′ ≅ S_3, B_w = β/(β∘w), c = (1+q^{-4})(1+q^{-4}+q^{-8}), triangularity (Computation 1.7), all recomputed exactly. Repairs and corrections: Proposition 1.5 step 7 refined for central tori; Lemma 1.3(e)'s Springer lemma replaced by Steinberg 7.5; Strategy 1.1 row 25 and Computation 1.7/A1-9 misdescribe Lemma 4.2.3 (its proof uses (4.5), i.e. [CS80]), which is harmless because c ≠ 0 follows from Ω ≢ 0. Theorem 1.9's general symmetric theorem needs three tacit hypotheses. H1H1 · Sakellaridis's triangularity (7.5) for the actual Θ of X. Equivalent to (U): the virtual colours belonging to each simple spherical root sum to its coroot (S1-1).: the inequality (7.5) for general Θ; the printed identity on p. 62 is off by a factor 2, and the inversion-set fix covers only multiplicity-free Θ+ ⊆ Φ̌_X^+. H2H2 · F-rational adapted data: a θ-stable split maximal torus T ⊆ B over F containing a maximal θ-split torus, with B adapted and x0 in the open Borel orbit.: F-rational adapted data (T, S, B). H3H3 · No spherical root of type N.: no type N. For M_θ the corollary holds: Theorem 8.0.2 over F_q((t)) and the failure of (SG) hold modulo (S6.3.1)_p, i.e. the identity Z_F = R_q, and (S6.3.1)_p follows from (I_Λ) via Dead end 1.8 route B and Theorem 8.0.2 at totally ramified F′_e. With the coordinator's report on [Sa08], H_Sa08 of the first audit is discharged.
Gap
(I_Λ): Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. for the 45 maximal-rank orbits and 6 simple roots of M_θ over W(F_q) as Λ-schemes (Dead end 4.10). H_Sa08: [Sa08, Thm 5.2.1] and the morphisms of [Sa08, §3] over ramified p-adic fields (not opened). Cited but not opened: Bruhat–Tits III Thm 4.7; Bruhat–Tits I–II and Landvogt inside Delorme–Sécherre; Richardson 1977; SGA 3 for G_Λ ↪ GL(W). Statement 6.3.1 in analytic form over F_q((t)) (Remark 3.11) is not needed by the transfer route. Second task: (I_Λ) (Axiom 2.2.1 for M_θ over W(F_q)), or a direct proof of Z_F = R_q over F_q((t)). For general symmetric X: H1H1 · Sakellaridis's triangularity (7.5) for the actual Θ of X. Equivalent to (U): the virtual colours belonging to each simple spherical root sum to its coroot (S1-1). (7.5), H2H2 · F-rational adapted data: a θ-stable split maximal torus T ⊆ B over F containing a maximal θ-split torus, with B adapted and x0 in the open Borel orbit., H3H3 · No spherical root of type N.. Cited but not opened: Springer 1985 (finiteness of B-orbits, char ≠ 2), CGP 2.1.10 and A.8.10, Bernstein–Zelevinsky, Banks 1998 (Bernstein's principle), [Sa08] (content as reported by the coordinator).
Blocked routes
- Unconditional char-p Theorem 8.0.2 for M_θ via the transfer: blocked on (I_Λ), Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. for M_θ over W(F_q) as Λ-schemes (Dead end 4.10); needs smooth B-stabilizers B_y over Z[1/2] for the 45 orbits and the purity identification of Sakellaridis's O-scheme Y.
- Removing H_Sa08: needs [Sa08, Thm 5.2.1] (functional equations up to rational factors) proved for every finite extension of Q_p; [Sa08] was not available locally.
- General symmetric X via Sakellaridis's p. 62 argument for (7.5): blocked; the identity |Θ+| − w|Θ+| = ρ̌ − wρ̌ is false (factor 2 when Θ+ = Φ̌X^+), and with a repeated coroot in Θ+ the conclusion fails (rank-one example s{2γ̌} = −(e^{−γ̌}+1+e^{γ̌})); needs a proof of (7.5) for the actual Θ of X.
- Unconditional (S6.3.1)_p for M_θ: blocked on (I_Λ) (route B) or on M1–M2 (route A: §6.19 in characteristic p); a direct evaluation of Z_F by point counts over F_q[t]/t^N (Dead end 1.8(ii)) is an untried alternative.