A6 audited A2's proof of (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. and redid the computation before reading A2's scripts. Its own enumeration of the 45 orbits and the 270 pairs reproduced every number, including 821 closure relations that agree with the Bruhat order ( (Audit 6.1)). It re-checked that the E IV involution is trivial on \(\mathrm{Spin}_8\) for the sign convention used, and noted that the opposite sign gives an involution of type E I ( (Audit 6.2)).
Step by step it confirmed the torus-conjugacy argument in every characteristic \(\ne2\), the smooth orbit schemes over \(\mathbb Z[1/2]\) and the rank-one quotients ( (Audit 6.3), (Audit 6.5)). In three places it added a line of proof: that two Borel subgroups share a maximal torus, a fibre-dimension bound, and why the saturations \(P_\alpha Z\) are closed over \(\mathcal O\) ( (Audit 6.6)). It also settled how Sakellaridis's complement clause must be read: inside closures, the only reading a non-open orbit can satisfy ( (Audit 6.7)).
Verdict: (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. holds for \(E_6/F_4\) over every discrete valuation ring in which 2 is invertible, in mixed and equal characteristic, \(p=3\) included ( (Summary 6.9)).
A fresh review confirmed every step and every repair.
Result as the agent reported it proved
All ten claims Strategy 2.1 to Summary 2.10 survive the audit (three need a one-line repair, supplied in Audit 6.3, Audit 6.5, Audit 6.6; one depends on S4-11's sign convention for the Chevalley involution). For \(\mathcal M_\theta\) (\(E_6/F_4\)) and every DVR \(\mathcal O\) with \(2\in\mathcal O^\times\), mixed or equal characteristic, \(p=3\) included, Sakellaridis's Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. holds as a statement about \(\mathcal O\)-schemes: the 45 \(B_F\)-orbits are all of maximal rank, Sakellaridis's \(\mathcal O\)-scheme of each is the smooth orbit scheme \(Y_x\simeq B/B_x\), \(YP_\alpha/R(P_\alpha)\simeq\mathrm{PGL}_2\backslash\mathrm{PGL}_2\) for the 90 pairs with \(\alpha\in\Phi_x\) and \(\simeq B_2\backslash\mathrm{PGL}_2=\mathbb P^1_{\mathcal O}\) for the other 180, \(P_\alpha\)-equivariantly, and the complement clause (read inside closures, the only satisfiable reading) holds for all 270 pairs. The 45 x 6 table, the closure sets (821 Bruhat relations), \(T^\theta\simeq\mathbb G_m^4\), \(|C_W(w_L)|=1152\) and the triviality of \(\theta_0\) on \(\mathrm{Spin}_8\) were recomputed independently.
Gap
No mathematical gap found. Citations not verified against a local source: Steinberg, Mem. AMS 80, Thms. 7.5, 8.1; CGP Prop. A.8.10(2) / Edixhoven Prop. 3.4; EGA IV 11.3.10, 11.10.10, 17.9.1, 18.12.6. The triviality of \(\theta_0\) on \(\mathrm{Spin}_8\) holds for S4-11's convention \(C(x_{\pm\alpha_i}(c))=x_{\mp\alpha_i}(-c)\); with the opposite sign the same formula gives an involution of type E I.
Blocked routes
- Closing Proposition 2.5 step 5 by a fibre-dimension bound for the special fibre of the flat closure of \(B'y\): needs a cited equidimensionality theorem for special fibres of \(\mathcal O\)-flat integral schemes of finite type (not among the local sources); replaced by Lemma 2.7 item 3 applied to \(x'\) (Audit 6.5 step 4).