§7 · Root · root: registry, redirects, synthesis · 2026-10-18T03:35:10Z

Remark 7.2 · Registry after the integral-model return: (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. claimed over \(\mathbb Z[1/2]\); launch its audit

open audit: minor defect

Minor defect (second audit). 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 Theorem 2.8 covers every O'_e directly. Nothing downstream uses the overstatement.

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. for \(E_6/F_4\) over every ring with 2 invertible. Together with the transfer, this would give Sakellaridis's theorem over \(\mathbb F_q(\!(t)\!)\) and over every \(p\)-adic field. A fresh auditor is assigned.

Chain to be audited: Theorem 2.8 gives (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 \(W(\mathbb F_q)\) and every \(\mathcal O'_e\) by base change; Audit 4.7 then gives Theorem 8.0.2 over \(\mathbb Q_q(p^{1/e})\); Theorem 4.8 carries it to \(\mathbb F_q(\!(t)\!)\); Proposition VII gives that (SG) fails there. The same (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. closes the open item of the \(p\)-adic theorem. Launch agent A6 on A2-1 … A2-10, with an independent recomputation of the table.

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