Summary 6.9 · 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 \(\mathcal M_\theta\) over every DVR with \(2\) invertible

Summary 6.9.

Outcomes, claim by claim:

• Strategy 2.1 (strategy, reading of p. 9): holds; the reading of the complement clause inside closures is forced (Audit 6.7).
• Lemma 2.2 (sign data, \(T^\theta\), \(C_W(w_L)\)): holds, for the sign convention \(C(x_{\pm\alpha_i}(c))=x_{\mp\alpha_i}(-c)\) of S4-11; with the opposite sign the same formula gives type E I (Audit 6.2).
• Proposition 2.3 (orbits over \(\bar k\), \(\operatorname{char}\ne2\)): holds with a repair: two Borel subgroups share a maximal torus, derived from the Bruhat decomposition (Audit 6.3 step 2).
• Lemma 2.4 (smooth stabilizers, orbit immersions over \(\mathbb Z[1/2]\)): holds (Audit 6.4).
• Proposition 2.5 (rank-one quotients): holds with a repair: the special fibre of \(\overline{B'y}\) misses \(\dot s_\alpha Y_x\), by Lemma 2.7 item 3 applied to \(x'\) (Audit 6.5 step 4); the quotient is also categorical.
• Computation 2.6 (the \(45\times6\) table, closure sets, 821 Bruhat relations): holds, recomputed independently (Audit 6.1).
• Lemma 2.7 (special fibres of closures): holds with a repair: \(P_\alpha Z_{x'}\) is closed in \(X_{\mathcal O}\) by properness of \(P_\alpha/B\) (Audit 6.6 step 4).
• Theorem 2.8 (the theorem): holds (Audit 6.7).
• Audit 2.9 (self-audit): holds, with one correction: \(\alpha|_{T^{\theta_x}}\) is always primitive, so the divisibility caveat is moot (Audit 6.2 step 5, Audit 6.8).
• Summary 2.10 (verdict on (Q2)): holds (Audit 6.8).

Established: for \(\mathcal M_\theta\) and every DVR \(\mathcal O\) with \(2\in\mathcal O^\times\), the \(B_F\)-orbits are the 45 orbits \(Y_{x,F}\), all of maximal rank; Sakellaridis's \(\mathcal O\)-scheme of each is the smooth orbit scheme \(Y_x\simeq B/B_x\); \(YP_\alpha/R(P_\alpha)\) is \(\mathrm{PGL}_2\backslash\mathrm{PGL}_2\) for the 90 pairs with \(\alpha\in\Phi_x\) and \(B_2\backslash\mathrm{PGL}_2\simeq\mathbb P^1_{\mathcal O}\) for the other 180, \(P_\alpha\)-equivariantly; the complement clause holds for all 270 pairs. So (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 over \(\mathbb F_q[[t]]\) and over the ring of integers of every \(p\)-adic field, \(p\) odd.