Re-check V2 · Re-check by G6

Proposition S1.2 (holds): I re-derived the inversion-sum formula: 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)., S1-1(d) gives |N_Θ(w)| = |N_Φ̌(w)|, and for u ∈ W_J the only I ⊆ Θ^+ with |I| = ρ̌ − uρ̌ is N_Θ(u^{-1}) (both leftover multisets lie in a strictly convex cone and sum to 0), so it equals my constant-term formula; the (P) recursion N_Θ(ww_γ) = w_γN_Θ(w) ⊔ D_γ is right. Before reading the scripts I computed, by exact Laurent division and by products of rank-one characters: E_6/F_4 gives 1, 1+t, c and Example 7.2.4 gives 1, 1−q^{-1}, 1−q^{-1}, 1−q^{-2} with Π_0 = 1−q^{-2} constant, matching (b), (d) and the agent's s1_aJ.py. Macdonald's product formula in (c) is cited and checked only numerically; the B_2 sample c = (1+t_1)(1+t_2)(1+t_1t_2) checks out, which is enough for the non-vanishing claim.

Lemma S1.3 (holds): (1) The conventions are consistent: (3.2) is ∫φ·χ̃'^{-1}|ω| with χ̃' = δ^{1/2}χ̃|c|, so Ŝ_φ(χ^{-1}) = ∫φ·χ̃(a(y))·δ_P^{-1/2}|c|^{-1}|ω_X|, the same χ̃ to the power +1 as the w = 1 term of (6.2) (Lemma 3.9.1 evaluates this integral); Sakellaridis's convention that regular B-eigencharacters are dominant (p. 61, f(yb) = η(b)f(y)) gives |f_η(x_0a)| = q^{−⟨ν,η⟩}|f_η(x_0)| for the same label ν, Θ^+ ⊆ T_col by the same pairing (p. 60), and Lemma 4.2.3 pins the orientation of (7.2) (B_{w_γ}(δ_P^{1/2}) = 0 forces e^{γ̌}(δ_P^{1/2}) = q^{−r}). The stray 'x_λ̌ = e^{−λ̌}(ϖ)' after Thm 4.2.2 is a sign slip in the source, which S1-3 sidesteps by defining labels intrinsically. (2) The adjunction with vol(x_λK) and m ≢ 0, (3) |f_η(x_0)| = 1 for primitive f_η (X_{F_q} integral, x̄0B{F_q} dense) together with dual of the weight cone = T_col for homogeneous affine X, and (4) uniqueness of Laurent coefficients on the φ-independent tube region R of Proposition 1.5(a) all check out. Independent test: in the SL_2 group case (q = 3, λ = −α̌, exact enumeration mod t^3), Π_λŜ_0 = cC_λŜ_λ holds on the coefficients of z^{-1}, z^0, z^1 (3/4, 1, 19/12), and (v) gives a_∅ = 1.

Summary S1.6 (holds): Its verdicts are right: A5-20 holds with change (its outcome, the M_θ corollary and the list 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).–H3H3 · No spherical root of type N., is true; its step-1 attribution of non-zero leading coefficients to M_θ-only checks and its 'c ≠ 0 from Ω_χ ≢ 0' are replaced by S1-3, 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). is sharpened 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). by S1-1, which I re-derived), and A5-21 holds as stated (its item 4 is now the proved S1-5). The rewrite of R-4(b) and of its open items is consistent with S1-1, S1-3 and S1-5. The remark that part (a) needs no 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). caveat rests on S1-4 (Lemma 7.1.4 ⇒ (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). in characteristic 0), which I did not grade; note that S1-4's transfer of S1-3 to a general (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). model also needs the integral-special-fibre step of S1-3 step 2.