§7 · Agent 7 · Adversarial audit · 2026-10-17T03:21:52Z

Audit 7.5 · Audit of the function-level theorem under Proposition III's conditions: (SG) over \(\F_q(\!(t)\!)\) holds without (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies., under D0D0 · Orbit geometry over F̄_q, p ≠ 2 (Chen–Yi, Prop. 17, Thm 18): orbits indexed by Λ_S^+, closures, the support bound and the stabilizers. and D3D3 · Lang's theorem for the arc group L⁺G and its stabilizers over F_q.

proved audit: correct

Second audit: correct. Agree: the audited step is valid under D0D0 · Orbit geometry over F̄_q, p ≠ 2 (Chen–Yi, Prop. 17, Thm 18): orbits indexed by Λ_S^+, closures, the support bound and the stabilizers., D3D3 · Lang's theorem for the arc group L⁺G and its stabilizers over F_q., q odd, T split (re-derived for Lemma 4.8). The remark on field-independence for the covered labels matches Proposition 1.4 step 1 (the double-coset bound φ(λ) − ν ∈ Z≥0 Φ̌+ via the projection ½(1−θ)), given an elementary orbit classification.

An in-run check that the dominant-lift argument gives (SG) over \(\F_q(\!(t)\!)\) for the families satisfying Proposition III, with Lang's theorem in place of Borel–Serre; it notes that for the covered labels the support bound is independent of the field.

Audit 7.5.

The step audited: Corollary C with labels \(\Lambda_S^+\), orbits \(\cO_\mu(\F_q)=Kx_\mu\), a dominant lift \(\lambda\), \(t^\lambda x_0=x_{\varphi(\lambda)}\), and $$Kt^\lambda X(\cO)\subseteq(L^+G\,t^\lambda L^+X)(\F_q)\subseteq\bigsqcup_{\nu\le\varphi(\lambda)}\cO_\nu(\F_q).$$ Outcome: Lemma 4.8 holds under D0D0 · Orbit geometry over F̄_q, p ≠ 2 (Chen–Yi, Prop. 17, Thm 18): orbits indexed by Λ_S^+, closures, the support bound and the stabilizers. (Chen–Yi Prop. 17 and Thm. 18 over \(\overline{\F}_q\), \(p\ne2\); the geometric support bound of Proposition III, step 2, whose proof is the double-coset bound on \(\Gr\) plus Thm. 18), D3D3 · Lang's theorem for the arc group L⁺G and its stabilizers over F_q. (Lang), \(q\) odd and \(T\) split over \(\F_q\). Severity: none; the (A)“Under (A)”: assuming all of Sakellaridis's assumptions (A1)–(A5) for X at the given p-adic place, so that his Theorem 8.0.2 applies.-free function column for the spheres (all \(n\)) and \(\GL_n/(\GL_p\times\GL_q)\) over \(\F_q(\!(t)\!)\) stands.

  1. \(x_\mu\) is rational because \(\tau(x_\mu)=t^\mu\in S(F)\) and \(\tau\) is injective.
  2. Compact-openness of \(\cO_\mu(\F_q)\) (Lemma 4.2, step 4) is what makes \(C_K=\operatorname{span}\{\one_{\cO_\mu(\F_q)}\}\) without Borel–Serre.
  3. Remark. For the covered labels the support bound can be proved independently of the field by the double-coset argument (Theorem 3.9; Proposition 1.4, step 1), which does not need Theorem 18. So for families where \(K\backslash X(F)\simeq\Lambda_S^+\) is known by elementary means (symplectic forms, spheres with \(q\) odd, the group), the conclusion holds over every non-archimedean \(F\).
  4. The orbit classification over \(F\) asked for in item 3 of Summary 1.9 is supplied for \(F=\F_q(\!(t)\!)\) by Lemma 4.2.

Builds on

  • Lemma 4.8 · Under the two conditions of Proposition III, (SG) holds over \(\F_q(\!(t)\!)\) without Sakellaridis's hypotheses

Built on by

  • Summary 7.19 · First audit verdict: conclusions (i)–(v) survive with their hypotheses; the only candidate divergence is \(E_6/F_4\)
Original record, as the agent wrote it at 03:21 UTC

A4-8: Prop III conditions imply (SG) over F_q((t)) without (A), under D0 and D3

Step audited. Corollary C with \(\Lambda_X^+=\Lambda_S^+\), \(O_\mu=\mathcal O_\mu(\mathbb F_q)=Kx_\mu\), dominant lift \(\lambda\), \(t^\lambda x_0=x_{\varphi(\lambda)}\), and \(Kt^\lambda X(\mathcal O)\subseteq(L^+G\,t^\lambda L^+X)(\mathbb F_q)\subseteq\bigsqcup_{\nu\le\varphi(\lambda)}\mathcal O_\nu(\mathbb F_q)\).

Outcome: holds under H = D0 (Chen–Yi Prop. 17/Thm. 18 over \(\overline{\mathbb F}_q\), \(p\ne2\); the geometric support bound of criterion.html step 2, whose proof is the double-coset bound on \(\mathrm{Gr}\) plus Thm. 18), D3 (Lang), \(q\) odd, \(T\) split over \(\mathbb F_q\). \(x_\mu\) is rational since \(\tau(x_\mu)=t^\mu\in S(F)\) and \(\tau\) is injective. Compact-openness of \(\mathcal O_\mu(\mathbb F_q)\) (A4-2 step 4) is what makes \(C_K=\mathrm{span}\{\mathbf 1_{\mathcal O_\mu(\mathbb F_q)}\}\) without Borel–Serre.

Remark. For the covered labels the support bound can even be done field-independently by the double-coset argument (A3-9, A1-4 step 1), which does not need Thm. 18; so for families where \(K\backslash X(F)\simeq\Lambda_S^+\) is known by elementary means (symplectic forms, spheres with \(q\) odd, the group) the conclusion holds over every non-archimedean \(F\). Severity: none; the (A)-free function column for spheres (all \(n\)) and \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) over \(\mathbb F_q(\!(t)\!)\) stands. A1-9's item 3 ('orbit classification over \(F\) not on the pages') is answered for \(F=\mathbb F_q(\!(t)\!)\) by A4-2.

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