Under Sakellaridis's hypotheses (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., over a \(p\)-adic field, spherical generation is equivalent to surjectivity of a restriction map \(r\) from \(\mathbb C[A^*]^W\) to the \(W_X\)-invariant functions on the shifted torus \(\delta(X)^{1/2}A_X^*\). The pages had not computed \(r\) for \(E_6/F_4\) or the spheres; the agent was asked to.
It fixed the formula \(r(f)(y)=f(q^{\rho_L}y)\), checked it against the pages' Proposition F, and showed that neither the sign of the shift nor the left/right convention matters (Setup 6.1). For the spheres the orbit sum of the roots restricts to a positive multiple of \(u+u^{-1}\) plus a constant, so \(r\) is onto (Proposition 6.3). For \(E_6/F_4\) it found \(W_X=S_3\) and reduced the question to one orbit sum \(M_{(3,0)}\), at the label \(3\omega_1^\vee\), where the sheaf-side hypothesis H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ. also fails (Computation 6.4). It then exhibited two points \(y_1=(q^{-5},q)\) and \(y_2=(q^{-5},q^4)\) that are not \(S_3\)-conjugate but whose shifts are \(W(E_6)\)-conjugate; since \(M_{(3,0)}\) takes different values at them, \(r\) is not onto, for every \(q\) (Theorem 6.6). Of the hypotheses, (A1)(A1) Field and group: F is a local field of characteristic 0 (p-adic), and G is split over the integers, with connected fibres and simply connected derived group. Its characteristic-p counterpart is (A1ₚ). and (A5)(A5) X = H\G is homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points. were verified; (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 given place and most of (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. were not (Lemma 6.9).
The audit recomputed the \(E_6\) data independently and found it correct. Minor defects: one gloss conflated the missing orbit sum with a missing orbit indicator (the stronger reading also holds 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.), and the agent's summary dropped the qualifier '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.' in its answers. This is the function half of the \(E_6/F_4\) divergence (Theorem 6 of the verdict), and it prompted the direct attack on the sheaf side.
Result as the agent reported it partial
Under Sakellaridis's hypotheses (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. (Theorem E): (i) for the spheres Spin_{2n}/Spin_{2n-1}, n ≥ 2, the restriction map r is onto — r(m_{ε1+ε2}) = c_n(q)(u+u^{-1}) + const with c_n(q) = Σ_{j=2}^n (q^{n-j}+q^{j-n}) > 0 and the target is C[u+u^{-1}] — so (SG) holds; (ii) for E6/F4, r is NOT onto: the points y1 = (q^{-5}, q) and σ(y1) = (q, q^{-5}) of A_X^* (coordinates u = y(φα1ᵛ), v = y(φα6ᵛ)) are not W_X-conjugate, yet q^{ρ_L}y1 = q^{ρ-9ω1-3ω6} and q^{ρ_L}σ(y1) ~{W_X} q^{ρ-9ω1} are W(E6)-conjugate (both to q^{ρ-ω4}); hence the W_X-orbit sum M{3ω1ᵛ} (with M_{3ω1ᵛ}(y1) = q^6+q^3+q^{-9} ≠ q^9+q^{-3}+q^{-6}) is not in im r and (SG) fails for every q. The sign of the shift δ(X)^{1/2}, left/right conventions and q do not affect either conclusion. (A5)(A5) X = H\G is homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points. is verified for both varieties (generic B-stabilizer is a Borel of Spin_8, resp. Spin_{2n-2}). Consequently: no divergence at the sphere rows; at E6/F4 'SG holds but C^Sat ≠ C' is excluded, and 'C^Sat = C but SG fails' is open exactly to the extent that the sheaf statement is open (H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ. fails at 3ω1ᵛ, the same label as the function-level obstruction).
Gap
(1) Hypotheses (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). (Axioms 2.4.1–2.4.2 at the given place) and the parts of (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. other than 'L_γ classical' (no type N, Statements 6.3.1 and 7.1.5) are not verified for E6/F4 or the spheres; Theorem E is used as cited. (2) The sharper statement im r = {g : g(y1) = g(σy1)} (cokernel one-dimensional) rests on window computations up to norm 24, an exhaustive rank-2 point search and a generic-point test of the rank-1 curve cases; the unramifiedness (tangent) condition is not computed. (3) The sheaf-level status of E6/F4 is untouched.
Blocked routes
- Unconditional (SG) failure for E6/F4 without Theorem E: would need an independent description of K\X(F) (strong Cartan decomposition, Axiom 2.4.1Sakellaridis's Axiom 2.4.1, relabelled (A3b): the Cartan decomposition of X(F) into K-orbits. for E6/F4) and of the Hecke action on orbit indicators, i.e. the content of Sakellaridis Theorem 8.0.2 for this X.
- Exact image of r (im r = K): would need the closed-immersion criterion in full — injectivity on Zariski tangent spaces of δ(X)^{1/2}A_X^/W_X → A^/W at every point (an orbifold tangent computation at the singular points of both quotients) plus a Zariski-density argument for the 126 rank-1 curve components tested only at generic points.
- Using the function-level failure to decide the sheaf level: no lemma transports a Weyl-conjugacy coincidence of q-points into a statement about Thick{V_λ ∗ IC_0}; the dictionary on these pages proves no implication between levels.
Brief as sent
Compute \(r:\mathbb C[A^*]^W\to\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\) for \(E_6/F_4\) and for \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\), where the pages say r is not computed. (1) Fix the shift \(\delta(X)^{1/2}\) as a point of \(A^*\), calibrated on Proposition F's two GL cases; (2) identify \(\mathbb C[A^*]^W\) for the dual of simply connected \(E_6\); (3) compute r on generators in Python (weights of the 27, 78, …, restricted to \(A_X^*\) with the shift); (4) decide surjectivity by an exact argument (explicit generation or invariant theory: closed immersion of \(\delta^{1/2}A_X^*/W_X\to A^*/W\)); (5) state what of (A1)–(A5) remains to be verified. Shared setting and rules: the problem card of the hub.