Re-check V1 · Re-check by G1

Lemma S1.1 (holds with change): I checked the counting identity of step 1 on 198 random W-stable, negation-closed multisets (repeated and half-integral weights, A2/B2/G2/A1^3/A3); it is right as an identity of multisets, since membership in N_Theta depends only on the vector and multiplicities are carried by W- and (-1)-stability. D_gamma=N_Theta(w_gamma) is literally the second clause of 7.1.5, and 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). => (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). is valid; I also checked 7.1.5, m_gamma=1, the (d) decomposition for every (w,I) and the coboundary exactly on E6/F4, T\PGL2 (split and non-split), Example 7.2.4 and my own B2 type-G plus type-T example. Changes, none altering a claim: (b) also uses Statement 6.3.1 (a Brion path for w_gamma) and H3H3 · No spherical root of type N. to put the 6.3.1 colours into D_gamma, which the Setting does not list, and 'so m_gamma>=1' needs the line that the remaining elements sum to (m_gamma-1)gamma-check in T_col, so m_gamma>=1 by strict convexity. (c) holds for 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). with T strictly convex, as S1-1 states it and as A1-2 needs; A5-20's wording 'T cap C^-={0}' alone does not give strict convexity (in A2 the half-plane spanned by R and -gamma-check_1 meets C^- only in 0).

Remark S1.4 (holds): (a) and (b) are right. The 'usual inductive argument' of p. 64 is A1-2's steps 4-5 and needs a_J!=0. Since >2 is reflexive, the p. 62 Remark says nothing about the coefficient of e^lambda, and read strictly (a_J=|W_J|) it is false at walls: I computed a{gamma1}=1+q^-4!=2 for E6/F4 and c=1-q^-1!=2 for T\PGL2. (c) In characteristic 0, Lemma 7.1.4 together with the minimality of D_V and the exponent clause of 6.3.1 gives (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)., which the coboundary step on p. 61 already presupposes; with S1-1 (checked) and S1-3 (graded by G6), the printed proof is completed. In S1-3's positivity and support argument I found no flaw: it uses only the homogeneous integral model, (A3b'), (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x., (7.4) and the adjunction, all available in characteristic 0. (d) E6/F4 is unchanged (a_J in {1, 1+t, c} recomputed).

Theorem S1.5 (holds with change): Each of my five gaps is now a stated hypothesis or proved: Lemma 7.1.4 uniqueness is (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) is S1-1(d); the F-rational adapted data are 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.; the leading coefficients and c!=0 are S1-3; type N is H3H3 · No spherical root of type N.. Step 4 is right: for D_gamma=N_Theta(w_gamma), beta/(beta o w_gamma)=-e^{gamma-check} prod over D_gamma of (1-sigma q^-r e^-theta)/(1-sigma q^-r e^theta), and 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). and H3H3 · No spherical root of type N. this is (6.3)-(6.5), using 6.3.1's exponent clause for a type-T split pair; I checked both exactly on four examples. Change: the statement takes Delta_X 'as in Sakellaridis, read over F-bar' and drops A1-9's clause that the w_gamma generate W_X'. Step 4 (extension of the coboundary from generators) and the chamber C^- need Delta_X to be the normalized simple roots of W_X' for C^-; in characteristic 0 this is Knop's description of W_X, and in characteristic p it must be stated. Likewise the split/non-split subtypes and the signs sigma are F-rational. No other characteristic-0 input found.