§2 · Agent 2 · Integral models · 2026-10-18T03:22:27Z

Audit 2.9 · Audit of the 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.: characteristic, \(p=3\), scheme structure, citations

proved audit: correct

Second audit: correct. Its inventory matches the proof: Steinberg Thms 7.5, 8.1 (any algebraically closed field; numbers confirmed via citing papers), the Bruhat decomposition, Conrad 5.1.13, 5.1.14, 5.1.16 (any base, checked locally), CGP A.8.10/Edixhoven 3.4 (2 invertible), EGA IV 11.3.10, 11.10.10, 17.9.1, 18.12.6 (general), plus S4-11 (whose sign convention I re-verified) and S4-13 step 2; no characteristic-0, p-adic or Knop-type input enters, only 2∈O^× is used, and p=3 is covered. Nits that change nothing: S4-4 is cited by Strategy 2.1 and Theorem 2.8 but omitted from the inventory (only for facts the proof does not need); the hedge on divisibility of α|_{T^{θ_x}} is moot (every root restricts to a primitive character of T^{θ_x} for all 45 x, checked); item 4 records the quotients as fppf quotients without reconciling them with Sakellaridis's geometric-quotient convention (the gap graded in Proposition 2.5).

No characteristic-0 theorem is used; every citation holds over any base with \(2\) invertible or over any field; stabilizers are smooth and orbit maps are immersions over \(\mathcal O\), also for \(p=3\).

Audit 2.9.

Besides the proved nodes S4-11 (\(\theta_0\) an involution over \(\mathbb Z\), \(\theta_0|_T=-w_L\)) and S4-13 step 2, the proof uses only: Steinberg, Mem. AMS 80, Thms. 7.5, 8.1 (any algebraically closed field); Bruhat decomposition and conjugacy of tori/Borels (any field); Conrad Thm. 5.1.13, Props. 5.1.14, 5.1.16 (any base); CGP A.8.10 / Edixhoven 3.4 (\(2\) invertible); EGA IV 11.3.10, 11.10.10, 17.9.1, 18.12.6. Not used: Knop, Brion, Luna, BLV, [Sa06], Richardson–Springer, Vust, Helminck–Wang, Theorem 8.0.2.

  1. Transfer. The count 45, maximal rank and types (S4-5(3), via Knop) are re-proved for char \(\ne2\) (Proposition 2.3, Proposition 2.5); torus conjugacy uses only \(\dim\mathfrak g^{\pm\theta}\) and rank-\(\le3\) root systems.
  2. Residue vs. generic characteristic. Only \(2\in\mathcal O^\times\); Proposition 2.3 applies to \(\overline F\) (char \(0\) or \(p\)) and \(\overline\kappa\).
  3. \(\mathcal O\)-schemes. \(Y^{\rm can}\) is identified as a subscheme (Theorem 2.8, 2–4); sets via Lemma 2.7, schemes via reducedness.
  4. Types vs. \(\mathcal O\)-isomorphisms. Quotients are fppf quotients over \(\mathcal O\), equivariant (Proposition 2.5).
  5. Non-reduced stabilizers. \(B_x\), \((P_\alpha)_y\) smooth over \(\mathcal O\); orbit maps immersions.
  6. \(B(F)\)-orbits, Galois cohomology. Not used; all 45 orbits are defined over \(\mathbb Z[1/2]\).

7–10. No normalizations; left/right via \(YP_\alpha\leftrightarrow P_\alpha Y\); no "almost every place"; no circularity.

  1. \(p=3\). \(T^\theta\simeq\mathbb G_m^4\) over \(\mathbb Z\); divisibility of \(\alpha|_{T^{\theta_x}}\) is harmless (only its fppf image is used); all Lie-algebra counts hold for \(p\ne2\).

\(2\in\mathcal O^\times\) is used for smoothness of fixed loci, \(\mathfrak g=\mathfrak g^\theta\oplus\mathfrak g^{-\theta}\), and \(\frac12(1-\theta_n)\). Steinberg's theorem numbers were not checked against a local source; S4-11 step 3 was re-checked (so8_theta.py).

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