§5 · Agent 5 · Audit: transfer, hypotheses · 2026-10-18T04:03:11Z

Audit 5.6 · Audit of A4-8: Theorem 8.0.2 for M_θ over F_q((t)); uniqueness; (SG)

proved audit: correct ★ key step

Second audit: correct. Re-derived Theorem 4.8: bijectivity from the two bases, the equivariance chain through one F'e per pair (lambda, mu), and uniqueness through the non-zero ideal f = {g: g(y1) = g(sigma y1) = 0} contained in R inside the domain S. S4-7 (read in S4.json) is unconditional algebra for every real q > 1, and S4-8 needs only an isomorphism with phi_0 -> 1 plus P{3omega1v} not in R, which holds in either label convention. The convention remark and remark (b) (identifying S, W_X, Lambda_X^+ with characteristic-p invariants needs Proposition 3.6, Proposition 3.7) are accurate. Correct.

The module transfer, the uniqueness argument and the consequences for (SG) are correct. The theorem is conditional on \((I_\Lambda)\), as stated, and also inherits \(H_{\mathrm{Sa08}}\) from Audit 4.7. Its claim to use 'no statement over \(\bar F\)' holds for the isomorphism but not for identifying the target with Sakellaridis's invariants in characteristic \(p\).

Audit 5.6.

Outcome: holds, conditionally on \((I_\Lambda)\) and \(H_{\mathrm{Sa08}}\) (Audit 5.5). Steps 1–5 of Theorem 4.8 are correct. (i) \(\Psi(\mathbf 1_{Kx_\kappa})=P_\kappa\) is a linear bijection \(C_K\to S\). It is \(\mathcal H_K\)-equivariant for every pair \((\lambda,\mu)\), each pair checked against one \(F'_e\) with \(e\ge\max(D(\lambda,\mu),D'(\lambda))\), where \(D'\) is as repaired in Objection 5.4. (ii) It is the unique \(\mathcal H_K\)-isomorphism with \(\varphi_0\mapsto1\): the conductor ideal \(\mathfrak f\subseteq R=\operatorname{im}r\) is non-zero by S4-7, which is unconditional algebra valid for every real \(q>1\). (iii) (SG) fails over \(\mathbb F_q(\!(t)\!)\), and \(C_K\) needs exactly two generators. Remarks. (a) No characteristic-\(p\) conclusion feeds back into a hypothesis (checklist item 10). (b) \(S\), \(W_X=S_3\), \(\Lambda_X^+\) and \(\delta(X)\) enter as the combinatorial data of the \(p\)-adic side. To call the result "Sakellaridis's theorem for \(X_F\)" one needs these to equal the characteristic-\(p\) invariants of \(X_{\bar F}\). That is Proposition 3.6, Proposition 3.7 (with Audit 5.10, Objection 5.11), so the claim "no statement over \(\bar F\) is used" applies to the proof of the isomorphism only.

  1. Bases. By Proposition 4.5, \(\{b_\kappa=\mathbf 1_{Kx_\kappa}\}_{\kappa\in\Lambda_S^+}\) is a basis of \(C_K(F)\). Over one \(F'_e\), Theorem 8.0.2 (available under \((I_\Lambda)\) by Audit 4.7(b)) makes \(\{P_\kappa\}\) a basis of \(S\), and Audit 5.5 step 5 shows it is the same set for every \(e\). So \(\Psi\) is bijective. \(P_0=1\) is stated in the proof on p. 64, and \(X(\mathcal O)=Kx_0\).
  2. Equivariance. For fixed \((\lambda,\mu)\): \(\Psi(T_\lambda*b_\mu)=\sum_{\nu\in N(\lambda,\mu)}c^\nu_{\lambda\mu}(F)P_\nu\) (Proposition 4.6(a)). This equals \(\sum_\nu c^\nu_{\lambda\mu}(F'_e)P_\nu\) (Proposition 4.6(b)), which is \(\Phi_{F'_e}(T'_\lambda*b'_\mu)=r(\mathcal ST'_\lambda)P_\mu\) by Theorem 8.0.2 over \(F'_e\), and \(\mathcal S T'_\lambda=\mathcal ST_\lambda\) (Proposition 4.6(c)). The \(T_\lambda\) span \(\mathcal H_K\) and the \(b_\mu\) span \(C_K\). Only finitely many \(\nu\) occur for each pair, so one \(e\) suffices per pair. The map \(r\) depends only on \(q\): \(\delta(X)^{1/2}\) is a power of \(q\).
  3. Uniqueness. If \(\Psi'\) is another such isomorphism, \(\Delta=\Psi'\Psi^{-1}\) is \(R\)-linear with \(\Delta(1)=1\), so \(\Delta|_R=\mathrm{id}\). S4-7(3) gives \(R=\{g:g(y_1)=g(\sigma y_1)\}\), so \(\mathfrak f=\{g:g(y_1)=g(\sigma y_1)=0\}\) is an ideal of \(S\) contained in \(R\), and it is non-zero because \(\operatorname{Spec}S\) is an irreducible surface. For \(0\ne a\in\mathfrak f\) and any \(s\): \(a\Delta(s)=\Delta(as)=as\). \(S\) is a domain (the coordinate ring of a translate of a torus, taken \(W_X\)-invariants), so \(\Delta(s)=s\).
  4. Consequences. \(\Psi(\mathcal H_K\varphi_0)=R\) and \(\dim S/R=1\) (S4-7), so (SG) fails. S4-8 needs only an isomorphism \(C_K\simeq S\) with \(\varphi_0\mapsto1\) and \(P_{3\omega_1^\vee}\notin R\). The latter is a statement about the explicit polynomial (lowest-weight Remark, p. 62) and holds in either label convention. Here \(3\omega_1^\vee=2e_1+e_6\in\Lambda_S^+\) (Audit 5.2).
  5. Convention (checklist item 8). The identity "\(f\) acts as multiplication by \(r(\mathcal Sf)\)" is transported in whatever convention Theorem 8.0.2 holds over \(F'_e\), because \(c^\nu_{\lambda\mu}\) and \(n_{\lambda\nu}\) are defined by the same formulas over both fields. The paper does not fix the conventions for \(\mathcal S\) or the action. Matching them with the left-action formulas of Proposition 4.6 can change \(r\) only by automorphisms of source and target (S4-8 step 1), which leaves (SG) and the two-generator statement unchanged.
  6. What remains. \((I_\Lambda)\) (Axiom 2.2.1 for \(\mathcal M_\theta\) over \(W(\mathbb F_q)\), for 45 orbits and 6 simple roots; Dead end 4.10) and \(H_{\mathrm{Sa08}}\).

Builds on

  • Theorem 4.8 · Theorem: Theorem 8.0.2 for \(\mathcal M_\theta\) over \(\mathbb F_q((t))\); (SG) fails

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