A1 was asked to rerun Sakellaridis's own proof over \(\mathbb F_q(\!(t)\!)\) and to replace every input that needs characteristic 0. Its first attempt ended with nothing saved, because one answer exceeded the output limit; relaunched with instructions to save each step as it went, it finished in under an hour.
It began with an inventory of the 27 external inputs of the proof, sorted into those valid over any local field, those that depend only on the residue field, and those that need replacing ( (Strategy 1.1)). For symmetric varieties with \(p\) odd it described the Levi of the open orbit and the smooth Borel stabilizers without Brion–Luna–Vust, and showed that the \(\mathbb F_q(\!(t)\!)\)-points fall into finitely many Borel orbits. It then built the open-orbit functionals \(S_\chi\) without Sakellaridis's 2008 paper: convergence, uniqueness for generic \(\chi\), rational continuation ( (Proposition 1.5)). From these came the functional equations and the sum formula, with the little Weyl group defined as a stabilizer instead of through Knop's action ( (Proposition 1.6)). For \(E_6/F_4\) it computed that group, the \(L\)-factor \(\beta\) and the constant \(c\), and noticed that the identity Sakellaridis uses on p. 62 is off by a factor 2; it proved the needed inequality another way ( (Computation 1.7)).
One input stayed open, the cocycle value of the two spherical reflections in characteristic \(p\) (Statement 6.3.1). A level lemma reduced it to ramified \(p\)-adic fields ( (Dead end 1.8)), where (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., proved by A2, supplies it. The audit A5 confirmed every step for \(E_6/F_4\); for general symmetric varieties it added three hypotheses that the stated theorem had left implicit ( (Theorem 1.9)).
A fresh review then found one more gap in the general theorem: non-zero leading coefficients, used without proof. A salvage agent proved them by positivity, which completes the list ( (Lemma S1.3), (Theorem S1.5)).
Result as the agent reported it partial
Over F = F_q((t)), p odd, for symmetric X = G/G^θ (θ an involution over O): Theorem 8.0.2 holds, i.e. there is a canonical H_K-isomorphism C_c^∞(X(F))^K ≅ C[δ(X)^{1/2}A_X^*]^{W_X′}, f acting as multiplication by r(Sf) and 1_{x_0K} ↦ 1, hence (SG) ⟺ r onto, under (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)., (A5)(A5) X homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points (§8)., (A3b′) (Axiom 2.4.1 with antidominant labels for W_X′ := N_W(a)∩[W/W_L]), (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x., Statements 6.3.1 and 7.1.5 read in characteristic p, and cone(Θ^+)∩C^- = {0}. Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. ((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.) and (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) are not used. Every analytic step (§§3–5 for the open orbit and §8) is proved over F_q((t)) without characteristic-0 citations: the integrals (3.2) converge, are unique up to scalar for generic χ (finitely many B(F)-orbits, smooth stabilizers, p odd) and continue rationally (Bernstein's principle); the functional equations and the sum formula (6.2) hold with W_X′; the final step is formal. The external inputs are characteristic-≠2 results: Springer 1985, Conrad–Gabber–Prasad A.8.10 and 2.1.10, Bernstein–Zelevinsky, Bernstein's principle. For E6/F4: W_X′ ≅ W(A_2) (computed in W(E_6)), the colour hypotheses are unnecessary, β = ∏(1−e^{γ̌})/(1−q^{-4}e^{γ̌}), and c = (1+q^{-4})(1+q^{-4}+q^{-8}). The theorem holds modulo (S6.3.1)_p, (A3b′) and (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x., and then Proposition VII gives that (SG) fails over F_q((t)).
Gap
(1) (S6.3.1)p for M_θ: the characteristic-p cocycle value B{w_γ} = −e^{γ̌}(1−q^{-4}e^{−γ̌})/(1−q^{-4}e^{γ̌}), γ = γ_1, γ_6. Route A needs Lemmas M1 (rationality and rank-one functional equations for the integrals over the Brion-path orbits of Spin_9\Spin_10 over F_q((t))) and M2 (that orbit structure over F̄ of characteristic p). Route B needs Lemma M4 (Statement 6.3.1 / (1.4) for M_θ over every totally ramified p-adic E with ψ of conductor O_E, which needs (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. over O_E). (2) (A3b′) and (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x. for M_θ over F_q((t)), which are (Q3). (3) For general symmetric X: Theorem 6.3.2 and the colour lemmas 7.1.1–7.1.4 in characteristic p, i.e. the [Was96]/[Ak83] rank-≤2 classification in characteristic p.
Blocked routes
- Rerun §6.19 (Spin_9\Spin_10 Brion path, n = 5) in characteristic p: needs M1, rationality of S^Y and the Fubini identity T_αS^Z = S^Y on each path orbit (generic uniqueness for the families D_Y), and M2, the B-orbit structure of the split D_5 affine quadric in characteristic ≠ 2.
- Close-fields transfer of Z_F(χ) = S_χ(1_{X(O)})(1) from totally ramified p-adic fields (the level lemma is proved): needs Statement 6.3.1 for M_θ over ramified O_E, which rests on Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. over O_E, open even in characteristic 0.
- Theorem 6.3.2 ((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) ⇒ Statement 6.3.1) for general symmetric X in characteristic p: needs Lemma 6.2.2 and Proposition 6.2.1, i.e. the rank ≤ 2 classification [Was96], [Ak83] in characteristic p.
- Using Knop's W-action and 'V is a fundamental domain for W_X' [Kn94–Kn96] in characteristic p: not established here (Knop's 2014 characteristic-≠2 result recalled, not opened); avoided by defining W_X′ = Stab_W(δ(X)^{1/2}A_X^*) and labelling Cartan by its antidominant chamber.
Brief as sent
Target (Q1): Sakellaridis's Theorem 8.0.2 over \(\mathbb F_q(\!(t)\!)\), \(p\) odd, for symmetric \(X\), at least for \(E_6/F_4\). Approach: rerun his own argument and replace every input that uses \(\operatorname{char}F=0\) (the external inputs of §§2–8 and their standing hypotheses; characteristic-\(\ne2\) replacements for symmetric varieties; the analytic steps of §3, §3.8, §4, §5.4, §6.19 over \(\mathbb F_q(\!(t)\!)\)). A first attempt, launched at 02:06:51, ended without output when one response exceeded the output limit; the agent was relaunched at 03:28:40 with instructions to save each step as it went.