§6 · Agent 6 · Audit: integral models · 2026-10-18T04:01:44Z

Audit 6.8 · Audit of the citations, the eleven-point checklist and the scope (claims A2-9, A2-10)

proved audit: correct

Second audit: correct. Conrad 1.4.12 (over an algebraically closed field; Audit 6.8 says 'over a field', used over k-bar), 5.1.13, 5.1.14 and 5.1.16 (any non-empty base) were checked in the local text; the non-local citations (Steinberg 7.5, 8.1; CGP A.8.10(2); Edixhoven 3.4; EGA IV 11.3.10, 11.10.10, 17.9.1, 18.12.6) are honestly flagged, their recalled statements are the standard ones and their hypotheses hold. I re-ran the eleven checklist items: at p = 3 the non-reduced centre mu_3 meets B_y trivially (theta_n inverts it), ker alpha is a torus, and the Lie counts hold mod 3 (dim g^{theta_x} = 52 and dim b^{theta_x} = 16+c_x computed over F_3). The correction to Audit 2.9 item 11 and the scope of Summary 2.10 (S4-13 item (I) is exactly (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 O) are accurate.

Every citation in Lemma 2.2–Theorem 2.8 is used within its standing hypotheses; the four Conrad results were checked in the local PDF, the others (Steinberg, CGP, Edixhoven, EGA IV) are standard but not available locally. All eleven checklist items pass, and the scope statement of Summary 2.10 is accurate. Outcome: holds.

Audit 6.8.

  1. Conrad, Reductive group schemes: Thm. 1.4.12 (Bruhat decomposition, over a field), Thm. 5.1.13 (open cell \(U_{-\Phi^+}\times T\times U_{\Phi^+}\), any non-empty base), Prop. 5.1.14 (commutation of \(U_a,U_b\), \(b\ne\pm a\), any base), Prop. 5.1.16 (\(U_\Psi\) for closed \(\Psi\), \(\Psi\cap-\Psi=\emptyset\), any base) say what A2 uses, over the bases where A2 uses them (local PDF checked). Thm. 1.4.12 needs the one-line corollary of Audit 6.3 step 2.
  2. Not checkable locally, statements recalled and consistent with the use: Steinberg, Mem. AMS 80, Thm. 7.5 and Thm. 8.1 (algebraically closed fields of any characteristic); CGP Prop. A.8.10(2) and Edixhoven Prop. 3.4 (smooth fixed loci for \(\mu_2\simeq\mathbb Z/2\) when \(2\) is invertible); EGA IV 11.3.10 (fibrewise flatness), 11.10.10 (fibrewise schematic density), 17.9.1 (flat monomorphism of finite presentation is an open immersion), 18.12.6 (proper monomorphism is a closed immersion).
  3. Checklist items 1–11 pass (below). One correction to Audit 2.9 item 11: no divisibility of \(\alpha|_{T^{\theta_x}}\) occurs at all (Audit 6.2 step 5).
  4. Summary 2.10 claims nothing about (A1)(A1) Field and group: F a non-archimedean local field of characteristic 0 (Sakellaridis §1.1), G split reductive over O with connected fibres and simply connected derived group (§1.7)., (A2)(A2) Integral model: X quasi-affine over O, its open Borel orbit smooth and surjective over O with B-homogeneous special fibre; a G-eigenmeasure on X and a B-invariant measure on the open orbit (§1.7)., (A3b)(A3b) Cartan decomposition (Sakellaridis's Axiom 2.4.1): the X-antidominant torus points represent every K-orbit on X(F), and points with distinct labels lie in distinct K-orbits., (A3c)(A3c) Iwahori orbits (Sakellaridis's Axiom 2.4.2): xJ ⊆ xB(O) for every X-antidominant torus point x., (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., (A5)(A5) X homogeneous and affine, and the open Borel orbit has a single B(F)-orbit of F-points (§8). or Theorem 8.0.2 in characteristic \(p\), which is accurate. Outcome: holds.

Checklist.

  1. Hypothesis transfer. No characteristic-0 theorem is used: the orbit count, the maximal rank and the types are proved from \(\dim\mathfrak g^{\pm\theta}\), Steinberg's theorems and the Bruhat decomposition (Audit 6.3); Knop, Brion, Luna, BLV, Richardson–Springer and Theorem 8.0.2 do not occur in Lemma 2.2–Theorem 2.8.
  2. Residue characteristic versus characteristic of \(F\). Only \(2\in\mathcal O^\times\) is used; every geometric-fibre argument is stated over an arbitrary algebraically closed field of characteristic \(\ne2\), so it applies to \(\bar F\) (characteristic 0 or \(p\)) and \(\bar\kappa\).
  3. Generic fibre, special fibre, \(\mathcal O\)-scheme. Sakellaridis's \(Y\) is identified as a subscheme of \(X_{\mathcal O}\), with its special fibre computed through flat closures (Audit 6.6, Audit 6.7).
  4. Type over \(\bar F\) versus \(\mathcal O\)-isomorphism. The rank-one quotients are \(\mathcal O\)-schemes with \(P_\alpha\)-equivariant isomorphisms to \(\operatorname{Spec}\mathcal O\) and \(\mathbb P^1_{\mathcal O}\), and are categorical quotients (Audit 6.5).
  5. Non-reduced stabilizers, inseparable orbit maps. \(B_y\) and \((P_\alpha)_y\) are smooth over \(\mathcal O\); orbit maps are immersions; \(T^\theta\simeq\mathbb G_m^4\) and root restrictions are primitive, so no \(\mu_p\) appears (Audit 6.2, Audit 6.4).
  6. Finiteness of \(B(F)\)-orbits, Galois cohomology. Not used: all 45 orbits have \(\mathbb Z[1/2]\)-points.
  7. Normalizations. None enter (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..
  8. Left versus right. \(YP_\alpha\) (right action) is \(P_\alpha Y\) under \(x\cdot g=g^{-1}x\); the four models are symmetric under this change.
  9. Almost every place. Not used; the argument is uniform over \(\mathbb Z[1/2]\).
  10. Circularity. Theorem 8.0.2 and its conclusion are not used; Lemma 2.7 uses only the geometric-fibre part of Proposition 2.5, and Proposition 2.5 step 5 can be closed by Lemma 2.7 without a loop (Audit 6.5 step 4).
  11. \(p=3\). \(H\) is smooth and connected (Steinberg 8.1 holds in every characteristic); \(X/(\theta-1)X\) has only 2-torsion in general and none here; every Lie-algebra count uses only \(p\ne2\).

Citations. For each result, the hypotheses are met: Conrad's §5.1 results are for a split reductive group over any non-empty scheme (used over \(\mathbb Z[1/2]\) and \(\mathcal O\)); Thm. 1.4.12 over a field (used over \(\bar k\)); CGP A.8.10(2) needs a group of multiplicative type acting on a smooth separated scheme (\(\mathbb Z/2\simeq\mu_2\) over \(\mathbb Z[1/2]\) acting on \(G\), \(B\cap\theta_nB\), \(BB''\)); Steinberg 7.5 needs a semisimple automorphism (an involution in characteristic \(\ne2\)); Steinberg 8.1 needs \(G\) semisimple simply connected (true); EGA IV 11.3.10 needs the source flat and the target locally of finite presentation over the base (true), 17.9.1 and 18.12.6 need finite presentation (true, everything is of finite type over a Noetherian base). The numbers of the non-local citations were not verified against a source in this audit, as A2 also states for Steinberg.

Outcome: holds.

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