Re-check V5 · Re-check by V5

Proposition X4.7 (holds): I re-derived every step independently in V5/v5_e6.py, using exact integer Laurent arithmetic in t=q^{1/2} and an E6 built from its Cartan matrix. theta=-w_L, with w_L = -1 on span(D4D4 · Unitriangularity of the Satake traces tr V_λ in the basis T_λ = 1_{Kt^λK} (Zhu (5.6.2), Lemma 5.6.3; Lusztig, Kato).), is an involution that fixes 24 roots, so dim g^{-theta}=26 (type E IV). phi(alpha_1^v)=(2,-1) and phi(alpha_6^v)=(-1,2). <rho_L,alpha_j^v>=(-3,1,1,1,1,-3). w1=w0(D5D5 · Comparison with C: “IC_μ is a summand of V_λ ∗ IC_0” has the same answer over F̄_q and over C (BBD §6.1, for large p).)w0(D4D4 · Unitriangularity of the Satake traces tr V_λ in the basis T_λ = 1_{Kt^λK} (Zhu (5.6.2), Lemma 5.6.3; Lusztig, Kato).) and w6 fix rho_L, commute with phi and induce s1 and s6. sigma commutes with theta, fixes rho_L and acts on (a,b) by swapping. From the W-orbits of sizes 72/270/432/27: r(m_{w2^v})=c8 M11+c0. r(m_{w1^v+w6^v})=M22+c8 P+d1 M11+d0. r(m_{w1^v+w3^v})=c8 M41+c'' M22+d M03+e M11 with no M30 term. r(m_{3w1^v})=M60+c M03, where c=q^{-9}+q^{-6}+q^{-3}+2+q^3+q^6+q^9. M11^2=M22+2P+2M11+6. M30M03=M33+3, M30^2=M60+2M03 and M11M30=M41+M22+M11. Both sigma-odd identities hold exactly: r-sigma*r = D(c8 M11-d) and D(P+2-c), with D=M30-M03. Evaluating at y1 gives d=c8 M11(y1) and c-2=P(y1). D(y1)=-q^{-9}(q^3-1)^3(q^3+1)(q^6+q^3+1), which is nonzero for real q>0 with q!=1. The functional g(y1)-g(sigma y1) kills all four r(m_mu). W_X y1 does not contain sigma y1. lambda1* and its sigma-image both reduce to rho-omega_4. The structural steps also hold. <W_X,sigma>=W(G2) acts on Lambda_S, which is the G2 root lattice and also its weight lattice, with fundamental weights (1,1) and (3,0); Bourbaki VI.3.4 Thm 1 then gives S^sigma=C[M11,P]. S=S^sigma (+) S^sigma D. R is sigma-stable and contains S^sigma. I=n, so R=K=S^sigma (+) nD and S=R (+) C M30. S/R is the character f->f(q^{rho-omega_4}). The units are C^x because a W_X-fixed exponent must be 0. So S is cyclic over C[A^*]^W iff r is onto, which fails. S=R+R M30 gives exactly 2 generators. dim S/m_xi S=2: CRT gives at least 2 and the two generators give at most 2. This is pure algebra about r, valid for every real q>1. No defect found, so the S4 scripts were not needed.

Corollary X4.8 (holds with change): Items 1, 3 and 4 follow from the conclusion of Thm 8.0.2 (C_K~S, H_K acting through r, phi0->1) together with S4-7. The convention changes (sign of the shift, left/right, Satake inversion) act by automorphisms that preserve im r. The character q^{rho-omega_4} is fixed by inversion because -w0=sigma fixes node 4. Criterion (c) is unconditional and correct: a cyclic commutative module H/a has Hom(H/a,chi) of dimension at most 1. Defect: item 2 (that 1_{O_{3w1^v}} is a second generator) does not follow from the bare conclusion the statement assumes. It needs the explicit map 1_{x_lambda K} -> P_lambda and the lowest-weight property, which come from the proof of 8.0.2, i.e. from its hypotheses. The body uses exactly that. Also, the nonzero leading coefficient is asserted in the proof of 8.0.2 on p.64, not in the Remark on p.62, which only gives weights >_2 lambda. I checked it directly from (7.4), with S4-3's Theta+ = positive A2 coroots, sigma=+, r=4 (V5/v5_plambda.py). P_0 is constant. For lambda=(0,-3) or (-3,0), P_lambda is proportional to (1+q^{-4})M_kappa + (1-q^{-8})M11 + (1-q^{-4}-q^{-8}+q^{-12}). Note the coefficient is 1+q^{-4}, not 1. Its sigma-odd part is a nonzero constant times D, so it lies outside R in every convention, and item 2 holds under the hypotheses. Fix: item 2 should assume the hypotheses (S1)-(S10), or the proof's explicit isomorphism; items 1, 3 and 4 need only the conclusion. Not verified by me: the parenthetical 'holds if (E) and (I)', which rests on S4-3 to S4-6 and on cited integral-model facts, and item (b), S4-4/S4-5. Correct: char 0 is a standing assumption (p.2), so the paper gives nothing over F_q((t)).

Summary X4.9 (holds with change): The algebraic row (im r = K, codimension 1, every q) is verified (see S4-7). The row 'fails; not cyclic for any generator; second generator 1_{O_{3w1^v}}; cokernel is the character q^{rho-omega_4}' is correct under Thm 8.0.2 including its proof (see S4-8). It correctly stays p-adic and leaves F_q((t)) undecided. Changes: (1) The cell says 'conditional only on (E) and (I)'. It also rests on inputs that S4's own gap list marks as cited but unchecked: existence of the O-involution (the Albert model), smoothness and connectedness of G_O^theta (Edixhoven, Steinberg), quasi-affineness (Alper, result number unchecked), and for F ramified over Q_p, Sakellaridis's 'trivial modifications' remark. The table mentions these; the cell sentence should too. (2) Verdict on A6-6: its gluing, M30 not in im r, and '(SG) fails under (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.' are all correct as stated; I re-derived the gluing and the disjoint W_X-orbits. S4-7 and S4-8 strengthen that conclusion rather than correct a defect, so 'holds' is the accurate grade, not 'holds with change'. (3) Verdict on A7-32, 'holds with change: cell sharpened, hypotheses reduced to (E) and (I)', is acceptable conditional on S4-3 to S4-6. It does not repeat the earlier audit's point that A7-32's 'equivalently, exhibit a 2-dim eigenspace' is only a sufficient criterion, but S4-8(c) states that correctly. 'Contrary to unverified in A6-9/A7-32' describes an update, not a contradiction. Not verified by me: S4-2 to S4-6, which carry every hypothesis-status row of the table.