§5 · Agent 5 · Audit: transfer, hypotheses · 2026-10-18T04:04:55Z

Audit 5.9 · Audit of A3-4, A3-5, A3-8: θ-split tori, Delorme–Sécherre, (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. in char p

proved audit: correct

Second audit: correct. Re-derived Proposition 3.4: the dimension identity giving I >= 20+2d, the A3 bound for d >= 3, the SL2 trick, the Levi sizes {0,2,...,14,20,24} with the unique 24 at {2,3,4,5} (confirmed by my script), open H-orbits in G/P, and descent through the smooth transporter with its cocycle in Spin8. I read Delorme-Secherre Sec. 2-4: Thm 4.8 uses Prop. 4.4, BT 1.13(2) and Helminck-Helminck Lemma 2.2 only (not Prop. 3.2 or Thm 3.10, which cite Helminck-Wang), and its inputs are characteristic-free given odd residue characteristic; the E6/F4 replacement of HH 2.2 is valid. Theorem 3.8's dictionary (<phi(mu),gamma_i> = 2<mubar,gamma_i>) and the realization of W_X by n_i in H(Z) acting as s_{gamma_i} check. Correct.

The direct proof that the maximal \((\theta,F)\)-split tori form one \(H(F)\)-class is valid in every characteristic \(\ne2\). Delorme–Sécherre's Theorem 4.8 is stated and proved for residue characteristic \(\ne2\). Hence \(G(F)=K\,T(F)\,H(F)\), and (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. holds over \(\mathbb F_q(\!(t)\!)\) independently of the transfer of Proposition 4.5.

Audit 5.9.

Outcome: Proposition 3.4, Audit 3.5 and Theorem 3.8 hold. The building theory inside Delorme–Sécherre is cited and was not opened: Bruhat–Tits I–II and Landvogt, whose standing hypotheses are a complete discretely valued field with perfect residue field. In Theorem 3.8, the identification of \(A_X^+\) with \(\{x_\kappa:\kappa\in\Lambda_S^+\}\) uses Proposition 3.7(3), the characteristic-\(p\) valuation cone. This is exactly the citation missing from Proposition 4.5 (Audit 5.3 (iv)), so Theorem 3.8 is the complete form of (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. over \(\mathbb F_q(\!(t)\!)\).

Proposition 3.4.

  1. Dimension identity. Root spaces are lines, and \(d\theta=\pm1\) on imaginary ones (\(p\ne2\)). This gives \(52=(6-d)+c+\tfrac{72-I}2\le42-d+\tfrac I2\). For \(d\ge3\) the imaginary roots lie in a rank-\(\le3\) subsystem, so \(I\le12\) (\(A_3\) is the largest such), contradicting \(I\ge26\).
  2. \(\mathrm{SL}_2\) trick. \(\operatorname{Aut}(\mathbb G_a)=\mathbb G_m\) in every characteristic (an additive polynomial with an inverse is linear). \(\begin{pmatrix}1&1\\1&-1\end{pmatrix}\) has determinant \(-2\in k^\times\). So \(\varepsilon=-1\) would produce a larger \((\theta,k)\)-split torus. Hence \([M,M]\subseteq H\).
  3. Levi count, recomputed. For subsets of at most four nodes, the sizes are \(\{0,2,4,6,8,10,12,14,20,24\}\), and exactly one subset, \(\{2,3,4,5\}\), gives \(24\). So \(|\Phi(M)|=20+2d\) forces \(d=2\) and \(M\) of type \(D_4\).
  4. Conjugacy over \(\bar k\). \(H\cap Q\subseteq Q\cap\theta Q=M\), so \(\dim H\cdot Q=52-28=24=\dim G/P\). Two open \(H\)-orbits in the irreducible \(G/P\) coincide.
  5. Descent over \(F\). The transporter is a torsor under the smooth \(N_H(S)\) (normalizer of a torus, Conrad 2.1.2/2.3.1), so it has \(F^s\)-points; \(\bar F\)-points alone would not suffice in characteristic \(p\). The cocycle lies in \(Z_H(S)=\mathrm{Spin}_8\), killed by Bruhat–Tits III. Checklist items 5–6 are handled.

Audit 3.5 (Delorme–Sécherre, arXiv v1; text read from the copy downloaded by A3, scratchpad/solve-vi/A3-dl-ds/ds.txt).

  1. Hypothesis. The abstract, §2 and §3 state "non Archimedean locally compact field of residue characteristic different from 2". The characteristic of \(k\) appears only in Lemma 3.6(iii) (finite index of squares), which is not used in §4.
  2. Inputs. Lemma 2.3 uses only that \(U^\pm_\Omega\) are pro-\(p\), \(p\) odd. Lemma 4.1 uses \(w=v^2\) in \(U_{-a}(k)\cong(k,+)\), i.e. \(v=w/2\) with \(|2|=1\). Lemma 4.7 uses \(\operatorname{Aut}(\mathbb G_a)\) = scalars. Theorem 4.8 uses Prop. 4.4, Bruhat–Tits 1.13(2) and Helminck–Helminck Lemma 2.2.
  3. Replacement of Helminck–Helminck. If \(S'^-=hSh^{-1}\) with \(h\in H(F)\), then \(h^{-1}S'h\) is a maximal torus containing \(S\), hence inside \(Z_G(S)=L=S\cdot\mathrm{Spin}_8\), hence equal to \(S\cdot T_8'\). Split maximal tori of \(\mathrm{Spin}_8\) are \(\mathrm{Spin}_8(F)\)-conjugate (Borel–Tits), and \(\mathrm{Spin}_8(F)\subseteq H(F)\). Correct. \(N(F)K=T(F)K\) because the Tits lifts lie in \(G(\mathbb Z)\).

Theorem 3.8.

  1. Covering: \(X(F)=G(F)x_0\) (Lemma 3.3) and \(G(F)=KT(F)H(F)\) (Audit 3.5). \(W_X\) is realized in \(K\) by \(\dot n_i=\sigma_{\alpha_i}\theta(\sigma_{\alpha_i})\in H(\mathbb Z)\cap N_G(T)\), which act on \(\Lambda_S\) by \(s_{\gamma_i}\) (recomputed; see the statement of Audit 5.10).
  2. Separation: \(\tau\) and the Cartan decomposition of \(G(F)\). Same label implies same orbit because \(A_X(\mathcal O)=\operatorname{im}T(\mathcal O)\).
  3. The dictionary \(x_0\cdot e^{\lambda}(\varpi)\leftrightarrow e^{-\lambda}(t)x_0\) (checklist item 8) follows §2.1 of the paper; compare the sign remark in Audit 5.5 step 6.

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  • Summary 5.13 · Verdict of audit A5: transfer route A4-2…A4-9 and char-p hypotheses A3-1…A3-10

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