Review G6 · Review G6 · Agent 5

Audit 5.17 (minor): Every computation was redone independently (W(E_6) enumerated in root coordinates; A_2 characters by exact Laurent division with t formal): |N_W(𝔞)| = 1152; exactly six minimal coset representatives, with A1's six actions on (γ_1, γ_6) and lengths 0,8,8,16,16,24; ⟨α̌i, ρ{P(X)}⟩ = 4; c = 1+2t+2t^2+t^3; β(δ_{P(X)}^{1/2}) = 1/c; weights in λ+R and non-zero leading coefficients for all 41 antidominant λ in [−8,8]^2 (A5-17's 13 in [−4,0]^2 included); (7.5) via the inversion-set argument. The p. 62 finding is confirmed on the rendered PDF: the printed identity is |Θ^+| − w|Θ^+| = ρ̌ − wρ̌ with ρ̌ a half-sum (pp. 6, 44), while for E_6/F_4 Statement 6.3.1 gives θ̌ = γ̌, so Θ^+ = Φ̌_X^+ and the left side is 2(ρ̌ − wρ̌) for every w ≠ 1 (reading ρ̌ as ρ̌_G is worse: [ρ̌_P] = 8ρ̌_X). The Lemma 4.2.3 → (4.5) → [CS80] correction is right (p. 27). Defect: 'c ≠ 0 follows independently from Ω_χ ≢ 0' is a non sequitur (Ω_χ ≢ 0 on A_X^+ does not give Ω_χ(x_0) ≢ 0). c ≠ 0 is explicit here anyway, and the valid one-liner is Ω_χ(x_0) = m(χ)·const·Z(χ^{-1}), with m ≢ 0 and Z ≢ 0 by positivity (Dead end 1.8(i)). This changes no claim.

Objection 5.20 (error): What A5-20 states is right: the M_θ corollary holds; 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 needed (p. 62 identity off by 2, verified on the PDF for all six w; s_{2γ̌} = −(e^{−γ̌}+1+e^{γ̌}) recomputed; by W-invariance a doubled coroot forces a doubled simple coroot); 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. is a real gap with a short fix (H^1(Z/2, U''(F)) = 0); H3H3 · No spherical root of type N. is real (Statement 6.3.1, p. 45, has no type-N clause). But it misses a fourth gap of the general theorem: Theorem 1.9 step 3 invokes Proposition 1.2(c), which needs non-zero leading coefficients, and A5-20 step 1 marks these as checked by Audit 5.16, Audit 5.17, which treat only M_θ. Under 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). the coefficient of m_λ is a_J = CT(∏{θ̌∈Θ^+}(1−σq^{−r}e^{θ̌}) ∏{γ̌∈Φ̌_J^+}(1−e^{−γ̌})) with W_J = Stab(λ) (I proved this and checked a_J = 1, 1+t, c for E_6/F_4), and for a general Θ (r = 1/2, σ = −, repeated or non-coroot weights) nothing in the run or in Sakellaridis shows a_J ≠ 0: his Remark after Thm 7.2.1 uses the non-strict order ≻_2 and says nothing about this coefficient, and the strict reading is false at walls. So 'the general theorem needs H = 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).+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.+H3H3 · No spherical root of type N.' leaves out a needed, unproved condition; its 'Minor' repair 'c ≠ 0 follows from Ω_χ ≢ 0' is also a non sequitur (fix: positivity of Z, Dead end 1.8(i)).

Summary 5.21 (error): The table entries and the E_6/F_4 conclusions agree with my grades of Audit 5.14–Audit 5.19, and the [Sa08] discharge is correct (I opened [Sa08] myself). Item 4, however, asserts that 'the correct characteristic-p statement for symmetric X is Theorem 1.9's theorem with 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. added'. The non-vanishing of the leading coefficients a_J, one for each stabilizer W_J of an antidominant label, is needed by Proposition 1.2(c) and is proved only for M_θ (Computation 1.7 step 6), so this formulation is not established; the error is inherited from Objection 5.20. Its stated conclusions stand: the A1 rerun is sound for M_θ, the analytic half holds with characteristic-≠2 inputs, the E_6/F_4 theorem and the failure of (SG) reduce to (S6.3.1)_p, which follows from (I_Λ) via Audit 4.7, 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).–H3H3 · No spherical root of type N. are indeed needed.