§4 · Agent 4 · Audit: counterexample and mechanism · 2026-10-17T23:00:25Z

Audit 4.17 · Audit: the close-field transfer of a (SG) witness to highly ramified \(p\)-adic fields

proved

Lemma 1.7 is correct under its explicit hypotheses (H-int) and (H-Cartan). The counts \(N_{\mathcal O}(\lambda,\mu)\) depend only on \((\mathcal O/\varpi^{m_0},\bar\varpi)\). The support of the difference function lies in finitely many orbits, by the \(\tau\)-bound and (H-Cartan). And \(\mathcal O'/\varpi'^m\cong\mathbb F_q[t]/t^m\) once \(e\ge m\). It transfers only failures at unbounded ramification under (H-Cartan), which the agent states it does not have.

Audit 4.17.

Under (H-int) (smooth affine models over \(W(\mathbb F_q)\), \(p\) odd, \(X\subset V\) with \(T\) diagonal, \(x_0\in X(W(\mathbb F_q))\)) and (H-Cartan): (a) \(N_{\mathcal O}(\lambda,\mu)=(T_\lambda\ast\varphi_0)(x_\mu)\) depends only on the \(W(\mathbb F_q)\)-algebra \(\mathcal O/\varpi^{m_0}\) and \(\bar\varpi\), for an \(m_0=m_0(\lambda,\mu)\) independent of \(\mathcal O\); (b) an identity \(\mathbf 1_{Kx_\mu}=\sum_{\lambda\in\Lambda_0}c_\lambda T_\lambda\ast\varphi_0\) over \(\mathbb F_q(\!(t)\!)\) persists over every \(\mathcal O'\) with residue field \(\mathbb F_q\), \(e(\mathcal O')\ge m_1\), satisfying (H-Cartan); (c) contrapositive.

Outcome: holds (conditional, as stated; no unconditional application is claimed).

  1. \(x_\mu\) is rational. \(\mu(-1)\in S[2]\), and \(\theta=\) inversion on \(S\) fixes \(S[2]\), so \(S[2]\subseteq H\). Hence \(\gamma(\mu(\sqrt\varpi))x_0=\mu(\sqrt\varpi)\mu(\pm1)x_0=x_\mu\), and \(\tau(x_\mu)=\mu(\sqrt\varpi)\theta(\mu(\sqrt\varpi))^{-1}=\mu(\varpi)\). Rationality forces \(\langle\chi,\mu\rangle\) even on the support of \(x_0\), so the coordinates are \(\varpi^{\langle\chi,\mu\rangle/2}\cdot\)(constants in \(W(\mathbb F_q)\)), the same for all \(\mathcal O\).
  2. (a). \(K(c)\subseteq K_\lambda=K\cap t^\lambda Kt^{-\lambda}\) for \(c=\max_\alpha|\langle\alpha,\lambda\rangle|\), by the factorization of \(K(c)\) into \(T(1+\varpi^c\mathcal O)\) and root subgroups \(U_\alpha(\varpi^c\mathcal O)\). Membership in \(K_\lambda\) is a divisibility condition on matrix coefficients in a faithful representation (closed embedding \(G\subset\mathrm{GL}(W)\): \(G(\mathcal O)=G(F)\cap\mathrm{GL}(W_{\mathcal O})\)). The incidence condition \(t^{-\lambda}k^{-1}x_\mu\in V(\mathcal O)\) is equivalent to \(g^{-1}x_\mu\in X(\mathcal O)\), because \(X\) is closed in \(V\) and \(\mathcal O\) is a domain. It is right \(K_\lambda\)-invariant and depends on \(k\) modulo \(K(b)\), \(b\) the largest pole order. Smoothness of \(G\) makes \(G(\mathcal O)\to G(\mathcal O/\varpi^m)\) onto. So \(N_{\mathcal O}\) is a count in \(G(\mathcal O/\varpi^{m_0})/(K_\lambda/K(m_0))\) of solutions of conditions defined over \((\mathcal O/\varpi^{m_0},\bar\varpi)\).
  3. Close rings. If \(e(\mathcal O')\ge m\) then \(p\in\varpi'^m\mathcal O'\), so \(\mathcal O'/\varpi'^m\) is a local \(\mathbb F_p\)-algebra with perfect residue field \(\mathbb F_q\). Its Teichmüller representatives form a coefficient field (additive in characteristic \(p\)), equal to the image of \(W(\mathbb F_q)\). Hence \(\mathcal O'/\varpi'^m\cong\mathbb F_q[t]/t^m\), \(\varpi'\mapsto t\), as \(W(\mathbb F_q)\)-algebras.
  4. (b), finiteness of the support. \(\tau:X\to G\) is defined over \(W(\mathbb F_q)\), so \(\tau(X(\mathcal O'))\subseteq K'\) and \(\tau(kt^\lambda x)=kt^\lambda\tau(x)t^{-\theta\lambda}\theta(k)^{-1}\in K't^\lambda K't^{-\theta\lambda}K'\subseteq\bigsqcup_{\nu\le\lambda+(-\theta\lambda)^+}K't^\nu K'\). By (H-Cartan) every \(K'\)-orbit in \(G(F')X(\mathcal O')\) is some \(K'x'_\nu\), with \(\tau(x'_\nu)=\nu(\varpi')\), and distinct dominant \(\nu\) give distinct double cosets, hence distinct orbits. So \(f'\) is supported on the finitely many \(K'x'_\nu\), \(\nu\in N_0\), and \(\mathbf 1_{K'x'_\mu}(x'_\nu)=\delta_{\mu\nu}\). For \(e(\mathcal O')\ge\max m_0\) over \(\Lambda_0\times N_0\), step 3 and (a) give \(f'(x'_\nu)=\sum c_\lambda N_{\mathbb F_q[\![t]\!]}(\lambda,\nu)-\delta_{\mu\nu}=0\). So \(f'=0\). Rational coefficients: IIb step 7.
  5. Scope. The transfer needs, on the \(p\)-adic side, a failure proved at places with residue field \(\mathbb F_q\), unbounded ramification and (H-Cartan). Sakellaridis's Theorem 8.0.2 additionally needs (A1)–(A5), with (A3) known only at almost all places of a global field. The agent says explicitly that no such case is established, so Lemma 1.7 makes no illegitimate \(p\)-adic-to-\(\mathbb F_q(\!(t)\!)\) claim. That is consistent with the brief's 'does not count' rule.

Builds on

  • Lemma 1.7 · Close-field transfer: witnesses of (SG) persist over highly ramified \(p\)-adic fields

Built on by

  • Summary 4.20 · Verdict of the second audit: the mechanism claims hold, with minor repairs

Sources