The fresh review found that the direct rerun's general theorem, Theorem 8.0.2 over \(\mathbb F_q(\!(t)\!)\) for an arbitrary symmetric variety, used one fact it never proved: that the lowest coefficient \(a_J\) of each image \(P_\lambda\) is non-zero. The in-run audit had missed it. This agent was asked to prove it, or to name the hypothesis it needs.
It first showed that the triangularity (7.5) holds for every set \(\Theta\) of virtual colours as soon as the colours of each simple spherical root sum to its coroot, and that this condition is equivalent to (7.5) ( (Lemma S1.1)). It then wrote \(a_J\) as a signed sum over the stabilizer of \(\lambda\); for \(E_6/F_4\) the values are \(1\), \(1+q^{-4}\) and \(c=(1+q^{-4})(1+q^{-4}+q^{-8})\) ( (Proposition S1.2)). The decisive step is positivity ( (Lemma S1.3)). The open-orbit integrals are series with non-negative coefficients. Comparing the coefficient of \(e^\lambda\) on both sides of the identity that (7.4) gives shows \(c\neq0\) and \(a_J/c>0\) over every local field, with no condition on \(\Theta\).
So no fourth hypothesis is needed: the corrected theorem is the rerun's list plus the three conditions the in-run audit had added ( (Theorem S1.5)). The same two arguments close the two gaps in Sakellaridis's own characteristic-0 proof ( (Remark S1.4)). The reviewers who had found the errors re-checked each step.
Result as the agent reported it proved
The direct characteristic-p form of Theorem 8.0.2 for symmetric X (F = F_q((t)), p odd) holds under the hypotheses of Theorem 1.9 plus H1-H3 of Objection 5.20 (S1-5). 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). is equivalent to (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1).: the virtual colours of each simple spherical root sum to its coroot (one copy of the coroot for type G, one pair for type T). Under (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1)., (7.5) holds for every Theta, with the cone cone(Theta+) (S1-1). The leading coefficient of P_lambda is a_J = sum over u in W_J of (-1)^l(u) prod over N_Theta(u) of (-sigma q^-r) (S1-2); for E6/F4 the values are 1, 1+t, c with t = q^-4. Non-vanishing needs no extra hypothesis H4: positivity of the open-orbit integrals gives c != 0 and a_J/c > 0 over every local field (S1-3). The same argument closes the two gaps in Sakellaridis's own characteristic-0 proof, the p. 62 factor 2 and the unproved non-vanishing, so part (a) of Proposition VI stands for general X (S1-4). Objection 5.20 holds with change: 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). is characterized as (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1)., and its argument for c != 0 is replaced. Summary 5.21 holds as stated. The E6/F4 conclusions and the verdict Theorem 7.4 are unchanged, apart from the wording of Theorem 7.4(b) and its open items (S1-6).
Gap
None for the target. 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. (adapted data over F) stays a hypothesis: review G6's proposed derivation from (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)., via H^1(Z/2, U''(F)) = 0, was not checked. For a general symmetric X, (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1)., (E) and Statements 6.3.1 and 7.1.5 in characteristic p remain hypotheses, as before.
Blocked routes