Review G8 · Review G8 · root

Remark 7.2 (minor): The chain to be audited (Theorem 2.8 → (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary. over W(F_q) and O'_e; Audit 4.7 → Theorem 8.0.2 over Q_q(p^{1/e}); Theorem 4.8 → F_q((t)); Proposition VII → (SG) fails) and the launch of A6 with an independent recomputation are accurate, and the consequence is correctly stated as conditional. Defect: the summary says A2 claims (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary. 'over every ring with 2 invertible' and the title 'over Z[1/2]', whereas Theorem 2.8 and Summary 2.10 claim it for every DVR with 2 invertible: the orbit schemes and stabilizers are built over Z[1/2] (Lemma 2.4), but Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O., with its complement clause read against the generic fibre, is a statement over a DVR. 'By base change' is also unnecessary, since A2-8 covers every O'_e directly. Nothing downstream uses the overstatement.

Audit 7.3 (minor): I opened [Sa08] (arXiv:math/0606130v2, local PDF and text), read §1.7 and §§3–6 and searched the whole text: there is no 'unramified over Q_p' assumption anywhere, the only field hypothesis is characteristic 0 (Hironaka/Igusa/Denef in §4.5, trivial H^1 of unipotent groups in Lemma 3.4.1, Borel 15.11 in Thm 3.1.1). Every result Sakellaridis 2009 takes from it (Prop. 3.2.1 in his §2.1, the measure modification §3.8 in his §1.7, the morphisms S^Y_χ̃ of §4.4–4.5 in his §3.2, Thm 5.3.2 = his Thm 3.2.2, Thm 5.2.1 in his Lemma 3.3.5 and Thm 3.7.1, §4.6 in his §5, Thm 6.2.1 in his §8) is stated for every locally compact non-archimedean field of characteristic 0; so H_Sa08 of Audit 5.5 holds for every Q_q(p^{1/e}) and, with Theorem 2.8 and Audit 4.7, Theorem 4.8 holds without conditions. Quotes 2–4 are verbatim and correctly located (PDF pp. 7, 17, 31); quote 1 is verbatim but on PDF p. 2, not p. 1. Also slightly loose: Sa09's Thm 3.7.1 contains a (U,ψ) bullet that is [Sa08, Thm 5.4.3] (proof omitted there), not Thm 5.2.1, and Sa09 §2.1 applies 'the argument of [Sa08, Prop. 3.2.1]' over the residue field F_q; both are irrelevant for M_θ (no (U,ψ) pairs; x0 = eH is an O-point).

Theorem 7.4 (minor): Re-derived from the parents: over every p-adic F, Theorem 8.0.2 for M_θ follows from (A1)–(A5)Sakellaridis's standing assumptions for Theorem 8.0.2: (A1) characteristic-0 field and split group; (A2) integral model; (A3a) rank-one models over O, (A3b) Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) spherical roots and colours; (A5) homogeneous, affine, one open Borel orbit. over O_F (S4/S4b with Summary 3.12's characteristic-free proofs, (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary. by Theorem 2.8/Summary 6.9), the normalizations of Audit 4.7/Audit 5.5, Audit 7.3, and (7.5) by the inversion-set argument of Computation 1.7 (factor-2 error on p. 62 and the repair re-derived by hand); over F_q((t)) it follows from the transfer Theorem 4.8/Audit 5.6 with (I_Λ) = Theorem 2.8 at W(F_q); my own scripts confirm that the W-orbit of ρ−ω4 meets ρ_L + a_X^* in two W_X-orbits (so r is not onto; a scan of half-integral points finds no other collision) and that P for the labels of 3ω1^∨ separates them (q = 3, 5, 7). Defects, none changing an E6/F4 claim: (1) in (c), 'the other hypotheses hold over F_q((t)) (Summary 3.12, Summary 5.13)' and the summary's 'every hypothesis in both characteristics' need Audit 5.19 for (A4)(A4) Spherical roots and colours: no pair of maximal rank of type N; classical Levis L_γ; Statements 6.3.1 and 7.1.5.(iii)_p, which those steps exclude, and 'So (T1)–(T5)Hypotheses of the characteristic-p transfer for symmetric X over W(F_q): (T1) H split with connected fibres; (T2) X(E) = G(E)x0; (T3) τ detects integral points; (T4) the Cartan decomposition over the ramified fields F'_e; (T5) Sakellaridis's theorem over F'_e = Q_q(p^{1/e}), e ≫ 0. hold' skips the list closing Proposition 4.9 ((T2) from Proposition 4.5/A3-3, (T3) from A4-3, (T5) from Audit 4.7 + Audit 7.3 + Theorem 2.8). (2) (a) cites Theorem 8.0.2 under (A1)–(A5)Sakellaridis's standing assumptions for Theorem 8.0.2: (A1) characteristic-0 field and split group; (A2) integral model; (A3a) rank-one models over O, (A3b) Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) spherical roots and colours; (A5) homogeneous, affine, one open Borel orbit. for all X with no caveat, although the root's own finding with Objection 5.20 is that the printed proof of (7.5) fails and is repaired only for multiplicity-free Θ^+ ⊆ Φ̌_X^+, so for general X (a) (and the 'e.g.' route to (T5)) needs 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).. (3) In (d), 'the direct route reaches the same conclusion, with Statement 6.3.1 in characteristic p supplied by (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary.': for E6/F4 that input is (A3a)(A3a) Rank-one models over O (Sakellaridis's Axiom 2.2.1): for every Borel orbit Y of maximal rank and every simple root α, the O-scheme YP_α/R(P_α) is one of the four standard PGL2-spaces over O, with his scheme structure on Y, and YP_α is the closure of its generic fibre minus the closure of the generic boundary. over W(F_q) pushed through Theorem 8.0.2 at F'_e and the close-field point counts (Audit 5.19: route B coincides with A4's transfer), so it is not independent confirmation. (4) (b) omits A4-9's standing data (θ-stable split T containing a maximal θ-split S, split over Λ) and its target ring is built from the p-adic invariants, identified with the characteristic-p ones only for M_θ (Audit 5.6 remark (b)); the Chen–Yi finding is cited to Audit 3.1, which does not contain it (it is in A3's blocked routes and in A4-3).